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Metamath Proof Explorer |
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| Ref | Description |
| idi 1 | (_Note_: This inference r... |
| a1ii 2 | (_Note_: This inference r... |
| mp2 9 | A double modus ponens infe... |
| mp2b 10 | A double modus ponens infe... |
| a1i 11 | Inference introducing an a... |
| 2a1i 12 | Inference introducing two ... |
| mp1i 13 | Inference detaching an ant... |
| a2i 14 | Inference distributing an ... |
| mpd 15 | A modus ponens deduction. ... |
| imim2i 16 | Inference adding common an... |
| syl 17 | An inference version of th... |
| 3syl 18 | Inference chaining two syl... |
| 4syl 19 | Inference chaining three s... |
| mpi 20 | A nested modus ponens infe... |
| mpisyl 21 | A syllogism combined with ... |
| id 22 | Principle of identity. Th... |
| idALT 23 | Alternate proof of ~ id . ... |
| idd 24 | Principle of identity ~ id... |
| a1d 25 | Deduction introducing an e... |
| 2a1d 26 | Deduction introducing two ... |
| a1i13 27 | Add two antecedents to a w... |
| 2a1 28 | A double form of ~ ax-1 . ... |
| a2d 29 | Deduction distributing an ... |
| sylcom 30 | Syllogism inference with c... |
| syl5com 31 | Syllogism inference with c... |
| com12 32 | Inference that swaps (comm... |
| syl11 33 | A syllogism inference. Co... |
| syl5 34 | A syllogism rule of infere... |
| syl6 35 | A syllogism rule of infere... |
| syl56 36 | Combine ~ syl5 and ~ syl6 ... |
| syl6com 37 | Syllogism inference with c... |
| mpcom 38 | Modus ponens inference wit... |
| syli 39 | Syllogism inference with c... |
| syl2im 40 | Replace two antecedents. ... |
| syl2imc 41 | A commuted version of ~ sy... |
| pm2.27 42 | This theorem, sometimes ca... |
| mpdd 43 | A nested modus ponens dedu... |
| mpid 44 | A nested modus ponens dedu... |
| mpdi 45 | A nested modus ponens dedu... |
| mpii 46 | A doubly nested modus pone... |
| syld 47 | Syllogism deduction. Dedu... |
| syldc 48 | Syllogism deduction. Comm... |
| mp2d 49 | A double modus ponens dedu... |
| a1dd 50 | Double deduction introduci... |
| 2a1dd 51 | Double deduction introduci... |
| pm2.43i 52 | Inference absorbing redund... |
| pm2.43d 53 | Deduction absorbing redund... |
| pm2.43a 54 | Inference absorbing redund... |
| pm2.43b 55 | Inference absorbing redund... |
| pm2.43 56 | Absorption of redundant an... |
| imim2d 57 | Deduction adding nested an... |
| imim2 58 | A closed form of syllogism... |
| embantd 59 | Deduction embedding an ant... |
| 3syld 60 | Triple syllogism deduction... |
| sylsyld 61 | A double syllogism inferen... |
| imim12i 62 | Inference joining two impl... |
| imim1i 63 | Inference adding common co... |
| imim3i 64 | Inference adding three nes... |
| sylc 65 | A syllogism inference comb... |
| syl3c 66 | A syllogism inference comb... |
| syl6mpi 67 | A syllogism inference. (C... |
| mpsyl 68 | Modus ponens combined with... |
| mpsylsyld 69 | Modus ponens combined with... |
| syl6c 70 | Inference combining ~ syl6... |
| syl6ci 71 | A syllogism inference comb... |
| syldd 72 | Nested syllogism deduction... |
| syl5d 73 | A nested syllogism deducti... |
| syl7 74 | A syllogism rule of infere... |
| syl6d 75 | A nested syllogism deducti... |
| syl8 76 | A syllogism rule of infere... |
| syl9 77 | A nested syllogism inferen... |
| syl9r 78 | A nested syllogism inferen... |
| syl10 79 | A nested syllogism inferen... |
| a1ddd 80 | Triple deduction introduci... |
| imim12d 81 | Deduction combining antece... |
| imim1d 82 | Deduction adding nested co... |
| imim1 83 | A closed form of syllogism... |
| pm2.83 84 | Theorem *2.83 of [Whitehea... |
| peirceroll 85 | Over minimal implicational... |
| com23 86 | Commutation of antecedents... |
| com3r 87 | Commutation of antecedents... |
| com13 88 | Commutation of antecedents... |
| com3l 89 | Commutation of antecedents... |
| pm2.04 90 | Swap antecedents. Theorem... |
| com34 91 | Commutation of antecedents... |
| com4l 92 | Commutation of antecedents... |
| com4t 93 | Commutation of antecedents... |
| com4r 94 | Commutation of antecedents... |
| com24 95 | Commutation of antecedents... |
| com14 96 | Commutation of antecedents... |
| com45 97 | Commutation of antecedents... |
| com35 98 | Commutation of antecedents... |
| com25 99 | Commutation of antecedents... |
| com5l 100 | Commutation of antecedents... |
| com15 101 | Commutation of antecedents... |
| com52l 102 | Commutation of antecedents... |
| com52r 103 | Commutation of antecedents... |
| com5r 104 | Commutation of antecedents... |
| imim12 105 | Closed form of ~ imim12i a... |
| jarr 106 | Elimination of a nested an... |
| jarri 107 | Inference associated with ... |
| pm2.86d 108 | Deduction associated with ... |
| pm2.86 109 | Converse of Axiom ~ ax-2 .... |
| pm2.86i 110 | Inference associated with ... |
| loolin 111 | The Linearity Axiom of the... |
| loowoz 112 | An alternate for the Linea... |
| con4 113 | Alias for ~ ax-3 to be use... |
| con4i 114 | Inference associated with ... |
| con4d 115 | Deduction associated with ... |
| mt4 116 | The rule of modus tollens.... |
| mt4d 117 | Modus tollens deduction. ... |
| mt4i 118 | Modus tollens inference. ... |
| pm2.21i 119 | A contradiction implies an... |
| pm2.24ii 120 | A contradiction implies an... |
| pm2.21d 121 | A contradiction implies an... |
| pm2.21ddALT 122 | Alternate proof of ~ pm2.2... |
| pm2.21 123 | From a wff and its negatio... |
| pm2.24 124 | Theorem *2.24 of [Whitehea... |
| jarl 125 | Elimination of a nested an... |
| jarli 126 | Inference associated with ... |
| pm2.18d 127 | Deduction form of the Clav... |
| pm2.18 128 | Clavius law, or "consequen... |
| pm2.18i 129 | Inference associated with ... |
| notnotr 130 | Double negation eliminatio... |
| notnotri 131 | Inference associated with ... |
| notnotriALT 132 | Alternate proof of ~ notno... |
| notnotrd 133 | Deduction associated with ... |
| con2d 134 | A contraposition deduction... |
| con2 135 | Contraposition. Theorem *... |
| mt2d 136 | Modus tollens deduction. ... |
| mt2i 137 | Modus tollens inference. ... |
| nsyl3 138 | A negated syllogism infere... |
| con2i 139 | A contraposition inference... |
| nsyl 140 | A negated syllogism infere... |
| nsyl2 141 | A negated syllogism infere... |
| notnot 142 | Double negation introducti... |
| notnoti 143 | Inference associated with ... |
| notnotd 144 | Deduction associated with ... |
| con1d 145 | A contraposition deduction... |
| con1 146 | Contraposition. Theorem *... |
| con1i 147 | A contraposition inference... |
| mt3d 148 | Modus tollens deduction. ... |
| mt3i 149 | Modus tollens inference. ... |
| pm2.24i 150 | Inference associated with ... |
| pm2.24d 151 | Deduction form of ~ pm2.24... |
| con3d 152 | A contraposition deduction... |
| con3 153 | Contraposition. Theorem *... |
| con3i 154 | A contraposition inference... |
| con3rr3 155 | Rotate through consequent ... |
| nsyld 156 | A negated syllogism deduct... |
| nsyli 157 | A negated syllogism infere... |
| nsyl4 158 | A negated syllogism infere... |
| nsyl5 159 | A negated syllogism infere... |
| pm3.2im 160 | Theorem *3.2 of [Whitehead... |
| jc 161 | Deduction joining the cons... |
| jcn 162 | Theorem joining the conseq... |
| jcnd 163 | Deduction joining the cons... |
| impi 164 | An importation inference. ... |
| expi 165 | An exportation inference. ... |
| simprim 166 | Simplification. Similar t... |
| simplim 167 | Simplification. Similar t... |
| pm2.5g 168 | General instance of Theore... |
| pm2.5 169 | Theorem *2.5 of [Whitehead... |
| conax1 170 | Contrapositive of ~ ax-1 .... |
| conax1k 171 | Weakening of ~ conax1 . G... |
| pm2.51 172 | Theorem *2.51 of [Whitehea... |
| pm2.52 173 | Theorem *2.52 of [Whitehea... |
| pm2.521g 174 | A general instance of Theo... |
| pm2.521g2 175 | A general instance of Theo... |
| pm2.521 176 | Theorem *2.521 of [Whitehe... |
| expt 177 | Exportation theorem ~ pm3.... |
| impt 178 | Importation theorem ~ pm3.... |
| pm2.61d 179 | Deduction eliminating an a... |
| pm2.61d1 180 | Inference eliminating an a... |
| pm2.61d2 181 | Inference eliminating an a... |
| pm2.61i 182 | Inference eliminating an a... |
| pm2.61ii 183 | Inference eliminating two ... |
| pm2.61nii 184 | Inference eliminating two ... |
| pm2.61iii 185 | Inference eliminating thre... |
| ja 186 | Inference joining the ante... |
| jad 187 | Deduction form of ~ ja . ... |
| pm2.01 188 | Weak Clavius law. If a fo... |
| pm2.01i 189 | Inference associated with ... |
| pm2.01d 190 | Deduction based on reducti... |
| pm2.6 191 | Theorem *2.6 of [Whitehead... |
| pm2.61 192 | Theorem *2.61 of [Whitehea... |
| pm2.65 193 | Theorem *2.65 of [Whitehea... |
| pm2.65i 194 | Inference for proof by con... |
| pm2.21dd 195 | A contradiction implies an... |
| pm2.65d 196 | Deduction for proof by con... |
| mto 197 | The rule of modus tollens.... |
| mtod 198 | Modus tollens deduction. ... |
| mtoi 199 | Modus tollens inference. ... |
| mt2 200 | A rule similar to modus to... |
| mt3 201 | A rule similar to modus to... |
| peirce 202 | Peirce's axiom. A non-int... |
| looinv 203 | The Inversion Axiom of the... |
| bijust0 204 | A self-implication (see ~ ... |
| bijust 205 | Theorem used to justify th... |
| impbi 208 | Property of the biconditio... |
| impbii 209 | Infer an equivalence from ... |
| impbidd 210 | Deduce an equivalence from... |
| impbid21d 211 | Deduce an equivalence from... |
| impbid 212 | Deduce an equivalence from... |
| dfbi1 213 | Relate the biconditional c... |
| dfbi1ALT 214 | Alternate proof of ~ dfbi1... |
| biimp 215 | Property of the biconditio... |
| biimpi 216 | Infer an implication from ... |
| sylbi 217 | A mixed syllogism inferenc... |
| sylib 218 | A mixed syllogism inferenc... |
| sylbb 219 | A mixed syllogism inferenc... |
| biimpr 220 | Property of the biconditio... |
| bicom1 221 | Commutative law for the bi... |
| bicom 222 | Commutative law for the bi... |
| bicomd 223 | Commute two sides of a bic... |
| bicomi 224 | Inference from commutative... |
| impbid1 225 | Infer an equivalence from ... |
| impbid2 226 | Infer an equivalence from ... |
| impcon4bid 227 | A variation on ~ impbid wi... |
| biimpri 228 | Infer a converse implicati... |
| biimpd 229 | Deduce an implication from... |
| mpbi 230 | An inference from a bicond... |
| mpbir 231 | An inference from a bicond... |
| mpbid 232 | A deduction from a bicondi... |
| mpbii 233 | An inference from a nested... |
| sylibr 234 | A mixed syllogism inferenc... |
| sylbir 235 | A mixed syllogism inferenc... |
| sylbbr 236 | A mixed syllogism inferenc... |
| sylbb1 237 | A mixed syllogism inferenc... |
| sylbb2 238 | A mixed syllogism inferenc... |
| sylibd 239 | A syllogism deduction. (C... |
| sylbid 240 | A syllogism deduction. (C... |
| mpbidi 241 | A deduction from a bicondi... |
| biimtrid 242 | A mixed syllogism inferenc... |
| biimtrrid 243 | A mixed syllogism inferenc... |
| imbitrid 244 | A mixed syllogism inferenc... |
| syl5ibcom 245 | A mixed syllogism inferenc... |
| imbitrrid 246 | A mixed syllogism inferenc... |
| syl5ibrcom 247 | A mixed syllogism inferenc... |
| biimprd 248 | Deduce a converse implicat... |
| biimpcd 249 | Deduce a commuted implicat... |
| biimprcd 250 | Deduce a converse commuted... |
| imbitrdi 251 | A mixed syllogism inferenc... |
| imbitrrdi 252 | A mixed syllogism inferenc... |
| biimtrdi 253 | A mixed syllogism inferenc... |
| biimtrrdi 254 | A mixed syllogism inferenc... |
| syl7bi 255 | A mixed syllogism inferenc... |
| syl8ib 256 | A syllogism rule of infere... |
| mpbird 257 | A deduction from a bicondi... |
| mpbiri 258 | An inference from a nested... |
| sylibrd 259 | A syllogism deduction. (C... |
| sylbird 260 | A syllogism deduction. (C... |
| biid 261 | Principle of identity for ... |
| biidd 262 | Principle of identity with... |
| pm5.1im 263 | Two propositions are equiv... |
| 2th 264 | Two truths are equivalent.... |
| 2thd 265 | Two truths are equivalent.... |
| monothetic 266 | Two self-implications (see... |
| ibi 267 | Inference that converts a ... |
| ibir 268 | Inference that converts a ... |
| ibd 269 | Deduction that converts a ... |
| pm5.74 270 | Distribution of implicatio... |
| pm5.74i 271 | Distribution of implicatio... |
| pm5.74ri 272 | Distribution of implicatio... |
| pm5.74d 273 | Distribution of implicatio... |
| pm5.74rd 274 | Distribution of implicatio... |
| bitri 275 | An inference from transiti... |
| bitr2i 276 | An inference from transiti... |
| bitr3i 277 | An inference from transiti... |
| bitr4i 278 | An inference from transiti... |
| bitrd 279 | Deduction form of ~ bitri ... |
| bitr2d 280 | Deduction form of ~ bitr2i... |
| bitr3d 281 | Deduction form of ~ bitr3i... |
| bitr4d 282 | Deduction form of ~ bitr4i... |
| bitrid 283 | A syllogism inference from... |
| bitr2id 284 | A syllogism inference from... |
| bitr3id 285 | A syllogism inference from... |
| bitr3di 286 | A syllogism inference from... |
| bitrdi 287 | A syllogism inference from... |
| bitr2di 288 | A syllogism inference from... |
| bitr4di 289 | A syllogism inference from... |
| bitr4id 290 | A syllogism inference from... |
| 3imtr3i 291 | A mixed syllogism inferenc... |
| 3imtr4i 292 | A mixed syllogism inferenc... |
| 3imtr3d 293 | More general version of ~ ... |
| 3imtr4d 294 | More general version of ~ ... |
| 3imtr3g 295 | More general version of ~ ... |
| 3imtr4g 296 | More general version of ~ ... |
| 3bitri 297 | A chained inference from t... |
| 3bitrri 298 | A chained inference from t... |
| 3bitr2i 299 | A chained inference from t... |
| 3bitr2ri 300 | A chained inference from t... |
| 3bitr3i 301 | A chained inference from t... |
| 3bitr3ri 302 | A chained inference from t... |
| 3bitr4i 303 | A chained inference from t... |
| 3bitr4ri 304 | A chained inference from t... |
| 3bitrd 305 | Deduction from transitivit... |
| 3bitrrd 306 | Deduction from transitivit... |
| 3bitr2d 307 | Deduction from transitivit... |
| 3bitr2rd 308 | Deduction from transitivit... |
| 3bitr3d 309 | Deduction from transitivit... |
| 3bitr3rd 310 | Deduction from transitivit... |
| 3bitr4d 311 | Deduction from transitivit... |
| 3bitr4rd 312 | Deduction from transitivit... |
| 3bitr3g 313 | More general version of ~ ... |
| 3bitr4g 314 | More general version of ~ ... |
| notnotb 315 | Double negation. Theorem ... |
| con34b 316 | A biconditional form of co... |
| con4bid 317 | A contraposition deduction... |
| notbid 318 | Deduction negating both si... |
| notbi 319 | Contraposition. Theorem *... |
| notbii 320 | Negate both sides of a log... |
| con4bii 321 | A contraposition inference... |
| mtbi 322 | An inference from a bicond... |
| mtbir 323 | An inference from a bicond... |
| mtbid 324 | A deduction from a bicondi... |
| mtbird 325 | A deduction from a bicondi... |
| mtbii 326 | An inference from a bicond... |
| mtbiri 327 | An inference from a bicond... |
| sylnib 328 | A mixed syllogism inferenc... |
| sylnibr 329 | A mixed syllogism inferenc... |
| sylnbi 330 | A mixed syllogism inferenc... |
| sylnbir 331 | A mixed syllogism inferenc... |
| xchnxbi 332 | Replacement of a subexpres... |
| xchnxbir 333 | Replacement of a subexpres... |
| xchbinx 334 | Replacement of a subexpres... |
| xchbinxr 335 | Replacement of a subexpres... |
| imbi2i 336 | Introduce an antecedent to... |
| bibi2i 337 | Inference adding a bicondi... |
| bibi1i 338 | Inference adding a bicondi... |
| bibi12i 339 | The equivalence of two equ... |
| imbi2d 340 | Deduction adding an antece... |
| imbi1d 341 | Deduction adding a consequ... |
| bibi2d 342 | Deduction adding a bicondi... |
| bibi1d 343 | Deduction adding a bicondi... |
| imbi12d 344 | Deduction joining two equi... |
| bibi12d 345 | Deduction joining two equi... |
| imbi12 346 | Closed form of ~ imbi12i .... |
| imbi1 347 | Theorem *4.84 of [Whitehea... |
| imbi2 348 | Theorem *4.85 of [Whitehea... |
| imbi1i 349 | Introduce a consequent to ... |
| imbi12i 350 | Join two logical equivalen... |
| bibi1 351 | Theorem *4.86 of [Whitehea... |
| bitr3 352 | Closed nested implication ... |
| con2bi 353 | Contraposition. Theorem *... |
| con2bid 354 | A contraposition deduction... |
| con1bid 355 | A contraposition deduction... |
| con1bii 356 | A contraposition inference... |
| con2bii 357 | A contraposition inference... |
| con1b 358 | Contraposition. Bidirecti... |
| con2b 359 | Contraposition. Bidirecti... |
| biimt 360 | A wff is equivalent to its... |
| pm5.5 361 | Theorem *5.5 of [Whitehead... |
| a1bi 362 | Inference introducing a th... |
| mt2bi 363 | A false consequent falsifi... |
| mtt 364 | Modus-tollens-like theorem... |
| imnot 365 | If a proposition is false,... |
| pm5.501 366 | Theorem *5.501 of [Whitehe... |
| ibib 367 | Implication in terms of im... |
| ibibr 368 | Implication in terms of im... |
| tbt 369 | A wff is equivalent to its... |
| nbn2 370 | The negation of a wff is e... |
| bibif 371 | Transfer negation via an e... |
| nbn 372 | The negation of a wff is e... |
| nbn3 373 | Transfer falsehood via equ... |
| pm5.21im 374 | Two propositions are equiv... |
| 2false 375 | Two falsehoods are equival... |
| 2falsed 376 | Two falsehoods are equival... |
| pm5.21ni 377 | Two propositions implying ... |
| pm5.21nii 378 | Eliminate an antecedent im... |
| pm5.21ndd 379 | Eliminate an antecedent im... |
| bija 380 | Combine antecedents into a... |
| pm5.18 381 | Theorem *5.18 of [Whitehea... |
| xor3 382 | Two ways to express "exclu... |
| nbbn 383 | Move negation outside of b... |
| biass 384 | Associative law for the bi... |
| biluk 385 | Lukasiewicz's shortest axi... |
| pm5.19 386 | Theorem *5.19 of [Whitehea... |
| bi2.04 387 | Logical equivalence of com... |
| pm5.4 388 | Antecedent absorption impl... |
| imdi 389 | Distributive law for impli... |
| pm5.41 390 | Theorem *5.41 of [Whitehea... |
| imbibi 391 | The antecedent of one side... |
| pm4.8 392 | Theorem *4.8 of [Whitehead... |
| pm4.81 393 | A formula is equivalent to... |
| imim21b 394 | Simplify an implication be... |
| pm4.63 397 | Theorem *4.63 of [Whitehea... |
| pm4.67 398 | Theorem *4.67 of [Whitehea... |
| imnan 399 | Express an implication in ... |
| imnani 400 | Infer an implication from ... |
| iman 401 | Implication in terms of co... |
| pm3.24 402 | Law of noncontradiction. ... |
| annim 403 | Express a conjunction in t... |
| pm4.61 404 | Theorem *4.61 of [Whitehea... |
| pm4.65 405 | Theorem *4.65 of [Whitehea... |
| imp 406 | Importation inference. (C... |
| impcom 407 | Importation inference with... |
| con3dimp 408 | Variant of ~ con3d with im... |
| mpnanrd 409 | Eliminate the right side o... |
| impd 410 | Importation deduction. (C... |
| impcomd 411 | Importation deduction with... |
| ex 412 | Exportation inference. (T... |
| expcom 413 | Exportation inference with... |
| expdcom 414 | Commuted form of ~ expd . ... |
| expd 415 | Exportation deduction. (C... |
| expcomd 416 | Deduction form of ~ expcom... |
| imp31 417 | An importation inference. ... |
| imp32 418 | An importation inference. ... |
| exp31 419 | An exportation inference. ... |
| exp32 420 | An exportation inference. ... |
| imp4b 421 | An importation inference. ... |
| imp4a 422 | An importation inference. ... |
| imp4c 423 | An importation inference. ... |
| imp4d 424 | An importation inference. ... |
| imp41 425 | An importation inference. ... |
| imp42 426 | An importation inference. ... |
| imp43 427 | An importation inference. ... |
| imp44 428 | An importation inference. ... |
| imp45 429 | An importation inference. ... |
| exp4b 430 | An exportation inference. ... |
| exp4a 431 | An exportation inference. ... |
| exp4c 432 | An exportation inference. ... |
| exp4d 433 | An exportation inference. ... |
| exp41 434 | An exportation inference. ... |
| exp42 435 | An exportation inference. ... |
| exp43 436 | An exportation inference. ... |
| exp44 437 | An exportation inference. ... |
| exp45 438 | An exportation inference. ... |
| imp5d 439 | An importation inference. ... |
| imp5a 440 | An importation inference. ... |
| imp5g 441 | An importation inference. ... |
| imp55 442 | An importation inference. ... |
| imp511 443 | An importation inference. ... |
| exp5c 444 | An exportation inference. ... |
| exp5j 445 | An exportation inference. ... |
| exp5l 446 | An exportation inference. ... |
| exp53 447 | An exportation inference. ... |
| pm3.3 448 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 449 | Theorem *3.31 (Imp) of [Wh... |
| impexp 450 | Import-export theorem. Pa... |
| impancom 451 | Mixed importation/commutat... |
| expdimp 452 | A deduction version of exp... |
| expimpd 453 | Exportation followed by a ... |
| impr 454 | Import a wff into a right ... |
| impl 455 | Export a wff from a left c... |
| expr 456 | Export a wff from a right ... |
| expl 457 | Export a wff from a left c... |
| ancoms 458 | Inference commuting conjun... |
| pm3.22 459 | Theorem *3.22 of [Whitehea... |
| ancom 460 | Commutative law for conjun... |
| ancomd 461 | Commutation of conjuncts i... |
| biancomi 462 | Commuting conjunction in a... |
| biancomd 463 | Commuting conjunction in a... |
| ancomst 464 | Closed form of ~ ancoms . ... |
| ancomsd 465 | Deduction commuting conjun... |
| anasss 466 | Associative law for conjun... |
| anassrs 467 | Associative law for conjun... |
| anass 468 | Associative law for conjun... |
| pm3.2 469 | Join antecedents with conj... |
| pm3.2i 470 | Infer conjunction of premi... |
| pm3.21 471 | Join antecedents with conj... |
| pm3.43i 472 | Nested conjunction of ante... |
| pm3.43 473 | Theorem *3.43 (Comp) of [W... |
| dfbi2 474 | A theorem similar to the s... |
| dfbi 475 | Definition ~ df-bi rewritt... |
| biimpa 476 | Importation inference from... |
| biimpar 477 | Importation inference from... |
| biimpac 478 | Importation inference from... |
| biimparc 479 | Importation inference from... |
| adantr 480 | Inference adding a conjunc... |
| adantl 481 | Inference adding a conjunc... |
| simpl 482 | Elimination of a conjunct.... |
| simpli 483 | Inference eliminating a co... |
| simpr 484 | Elimination of a conjunct.... |
| simpri 485 | Inference eliminating a co... |
| intnan 486 | Introduction of conjunct i... |
| intnanr 487 | Introduction of conjunct i... |
| intnand 488 | Introduction of conjunct i... |
| intnanrd 489 | Introduction of conjunct i... |
| adantld 490 | Deduction adding a conjunc... |
| adantrd 491 | Deduction adding a conjunc... |
| pm3.41 492 | Theorem *3.41 of [Whitehea... |
| pm3.42 493 | Theorem *3.42 of [Whitehea... |
| simpld 494 | Deduction eliminating a co... |
| simprd 495 | Deduction eliminating a co... |
| simprbi 496 | Deduction eliminating a co... |
| simplbi 497 | Deduction eliminating a co... |
| simprbda 498 | Deduction eliminating a co... |
| simplbda 499 | Deduction eliminating a co... |
| simplbi2 500 | Deduction eliminating a co... |
| simplbi2comt 501 | Closed form of ~ simplbi2c... |
| simplbi2com 502 | A deduction eliminating a ... |
| simpl2im 503 | Implication from an elimin... |
| simplbiim 504 | Implication from an elimin... |
| impel 505 | An inference for implicati... |
| mpan9 506 | Modus ponens conjoining di... |
| sylan9 507 | Nested syllogism inference... |
| sylan9r 508 | Nested syllogism inference... |
| sylan9bb 509 | Nested syllogism inference... |
| sylan9bbr 510 | Nested syllogism inference... |
| jca 511 | Deduce conjunction of the ... |
| jcad 512 | Deduction conjoining the c... |
| jca2 513 | Inference conjoining the c... |
| jca31 514 | Join three consequents. (... |
| jca32 515 | Join three consequents. (... |
| jcai 516 | Deduction replacing implic... |
| jcab 517 | Distributive law for impli... |
| pm4.76 518 | Theorem *4.76 of [Whitehea... |
| jctil 519 | Inference conjoining a the... |
| jctir 520 | Inference conjoining a the... |
| jccir 521 | Inference conjoining a con... |
| jccil 522 | Inference conjoining a con... |
| jctl 523 | Inference conjoining a the... |
| jctr 524 | Inference conjoining a the... |
| jctild 525 | Deduction conjoining a the... |
| jctird 526 | Deduction conjoining a the... |
| iba 527 | Introduction of antecedent... |
| ibar 528 | Introduction of antecedent... |
| biantru 529 | A wff is equivalent to its... |
| biantrur 530 | A wff is equivalent to its... |
| biantrud 531 | A wff is equivalent to its... |
| biantrurd 532 | A wff is equivalent to its... |
| bianfi 533 | A wff conjoined with false... |
| bianfd 534 | A wff conjoined with false... |
| baib 535 | Move conjunction outside o... |
| baibr 536 | Move conjunction outside o... |
| rbaibr 537 | Move conjunction outside o... |
| rbaib 538 | Move conjunction outside o... |
| baibd 539 | Move conjunction outside o... |
| rbaibd 540 | Move conjunction outside o... |
| bianabs 541 | Absorb a hypothesis into t... |
| pm5.44 542 | Theorem *5.44 of [Whitehea... |
| pm5.42 543 | Theorem *5.42 of [Whitehea... |
| ancl 544 | Conjoin antecedent to left... |
| anclb 545 | Conjoin antecedent to left... |
| ancr 546 | Conjoin antecedent to righ... |
| ancrb 547 | Conjoin antecedent to righ... |
| ancli 548 | Deduction conjoining antec... |
| ancri 549 | Deduction conjoining antec... |
| ancld 550 | Deduction conjoining antec... |
| ancrd 551 | Deduction conjoining antec... |
| impac 552 | Importation with conjuncti... |
| anc2l 553 | Conjoin antecedent to left... |
| anc2r 554 | Conjoin antecedent to righ... |
| anc2li 555 | Deduction conjoining antec... |
| anc2ri 556 | Deduction conjoining antec... |
| pm4.71 557 | Implication in terms of bi... |
| pm4.71r 558 | Implication in terms of bi... |
| pm4.71i 559 | Inference converting an im... |
| pm4.71ri 560 | Inference converting an im... |
| pm4.71d 561 | Deduction converting an im... |
| pm4.71rd 562 | Deduction converting an im... |
| pm4.24 563 | Theorem *4.24 of [Whitehea... |
| anidm 564 | Idempotent law for conjunc... |
| anidmdbi 565 | Conjunction idempotence wi... |
| anidms 566 | Inference from idempotent ... |
| imdistan 567 | Distribution of implicatio... |
| imdistani 568 | Distribution of implicatio... |
| imdistanri 569 | Distribution of implicatio... |
| imdistand 570 | Distribution of implicatio... |
| imdistanda 571 | Distribution of implicatio... |
| pm5.3 572 | Theorem *5.3 of [Whitehead... |
| pm5.32 573 | Distribution of implicatio... |
| pm5.32i 574 | Distribution of implicatio... |
| pm5.32ri 575 | Distribution of implicatio... |
| bianim 576 | Exchanging conjunction in ... |
| pm5.32d 577 | Distribution of implicatio... |
| pm5.32rd 578 | Distribution of implicatio... |
| pm5.32da 579 | Distribution of implicatio... |
| sylan 580 | A syllogism inference. (C... |
| sylanb 581 | A syllogism inference. (C... |
| sylanbr 582 | A syllogism inference. (C... |
| sylanbrc 583 | Syllogism inference. (Con... |
| syl2anc 584 | Syllogism inference combin... |
| syl2anc2 585 | Double syllogism inference... |
| sylancl 586 | Syllogism inference combin... |
| sylancr 587 | Syllogism inference combin... |
| sylancom 588 | Syllogism inference with c... |
| sylanblc 589 | Syllogism inference combin... |
| sylanblrc 590 | Syllogism inference combin... |
| syldan 591 | A syllogism deduction with... |
| sylbida 592 | A syllogism deduction. (C... |
| sylan2 593 | A syllogism inference. (C... |
| sylan2b 594 | A syllogism inference. (C... |
| sylan2br 595 | A syllogism inference. (C... |
| syl2an 596 | A double syllogism inferen... |
| syl2anr 597 | A double syllogism inferen... |
| syl2anb 598 | A double syllogism inferen... |
| syl2anbr 599 | A double syllogism inferen... |
| sylancb 600 | A syllogism inference comb... |
| sylancbr 601 | A syllogism inference comb... |
| syldanl 602 | A syllogism deduction with... |
| syland 603 | A syllogism deduction. (C... |
| sylani 604 | A syllogism inference. (C... |
| sylan2d 605 | A syllogism deduction. (C... |
| sylan2i 606 | A syllogism inference. (C... |
| syl2ani 607 | A syllogism inference. (C... |
| syl2and 608 | A syllogism deduction. (C... |
| anim12d 609 | Conjoin antecedents and co... |
| anim12d1 610 | Variant of ~ anim12d where... |
| anim1d 611 | Add a conjunct to right of... |
| anim2d 612 | Add a conjunct to left of ... |
| anim12i 613 | Conjoin antecedents and co... |
| anim12ci 614 | Variant of ~ anim12i with ... |
| anim1i 615 | Introduce conjunct to both... |
| anim1ci 616 | Introduce conjunct to both... |
| anim2i 617 | Introduce conjunct to both... |
| anim12ii 618 | Conjoin antecedents and co... |
| anim12dan 619 | Conjoin antecedents and co... |
| im2anan9 620 | Deduction joining nested i... |
| im2anan9r 621 | Deduction joining nested i... |
| pm3.45 622 | Theorem *3.45 (Fact) of [W... |
| anbi2i 623 | Introduce a left conjunct ... |
| anbi1i 624 | Introduce a right conjunct... |
| anbi2ci 625 | Variant of ~ anbi2i with c... |
| anbi1ci 626 | Variant of ~ anbi1i with c... |
| bianbi 627 | Exchanging conjunction in ... |
| anbi12i 628 | Conjoin both sides of two ... |
| anbi12ci 629 | Variant of ~ anbi12i with ... |
| anbi2d 630 | Deduction adding a left co... |
| anbi1d 631 | Deduction adding a right c... |
| anbi12d 632 | Deduction joining two equi... |
| anbi1 633 | Introduce a right conjunct... |
| anbi2 634 | Introduce a left conjunct ... |
| anbi1cd 635 | Introduce a proposition as... |
| an2anr 636 | Double commutation in conj... |
| pm4.38 637 | Theorem *4.38 of [Whitehea... |
| bi2anan9 638 | Deduction joining two equi... |
| bi2anan9r 639 | Deduction joining two equi... |
| bi2bian9 640 | Deduction joining two bico... |
| anbiim 641 | Adding biconditional when ... |
| bianass 642 | An inference to merge two ... |
| bianassc 643 | An inference to merge two ... |
| an21 644 | Swap two conjuncts. (Cont... |
| an12 645 | Swap two conjuncts. Note ... |
| an32 646 | A rearrangement of conjunc... |
| an13 647 | A rearrangement of conjunc... |
| an31 648 | A rearrangement of conjunc... |
| an12s 649 | Swap two conjuncts in ante... |
| ancom2s 650 | Inference commuting a nest... |
| an13s 651 | Swap two conjuncts in ante... |
| an32s 652 | Swap two conjuncts in ante... |
| ancom1s 653 | Inference commuting a nest... |
| an31s 654 | Swap two conjuncts in ante... |
| anass1rs 655 | Commutative-associative la... |
| an4 656 | Rearrangement of 4 conjunc... |
| an42 657 | Rearrangement of 4 conjunc... |
| an43 658 | Rearrangement of 4 conjunc... |
| an3 659 | A rearrangement of conjunc... |
| an4s 660 | Inference rearranging 4 co... |
| an42s 661 | Inference rearranging 4 co... |
| anabs1 662 | Absorption into embedded c... |
| anabs5 663 | Absorption into embedded c... |
| anabs7 664 | Absorption into embedded c... |
| anabsan 665 | Absorption of antecedent w... |
| anabss1 666 | Absorption of antecedent i... |
| anabss4 667 | Absorption of antecedent i... |
| anabss5 668 | Absorption of antecedent i... |
| anabsi5 669 | Absorption of antecedent i... |
| anabsi6 670 | Absorption of antecedent i... |
| anabsi7 671 | Absorption of antecedent i... |
| anabsi8 672 | Absorption of antecedent i... |
| anabss7 673 | Absorption of antecedent i... |
| anabsan2 674 | Absorption of antecedent w... |
| anabss3 675 | Absorption of antecedent i... |
| anandi 676 | Distribution of conjunctio... |
| anandir 677 | Distribution of conjunctio... |
| anandis 678 | Inference that undistribut... |
| anandirs 679 | Inference that undistribut... |
| sylanl1 680 | A syllogism inference. (C... |
| sylanl2 681 | A syllogism inference. (C... |
| sylanr1 682 | A syllogism inference. (C... |
| sylanr2 683 | A syllogism inference. (C... |
| syl6an 684 | A syllogism deduction comb... |
| syl2an2r 685 | ~ syl2anr with antecedents... |
| syl2an2 686 | ~ syl2an with antecedents ... |
| mpdan 687 | An inference based on modu... |
| mpancom 688 | An inference based on modu... |
| mpidan 689 | A deduction which "stacks"... |
| mpan 690 | An inference based on modu... |
| mpan2 691 | An inference based on modu... |
| mp2an 692 | An inference based on modu... |
| mp4an 693 | An inference based on modu... |
| mpan2d 694 | A deduction based on modus... |
| mpand 695 | A deduction based on modus... |
| mpani 696 | An inference based on modu... |
| mpan2i 697 | An inference based on modu... |
| mp2ani 698 | An inference based on modu... |
| mp2and 699 | A deduction based on modus... |
| mpanl1 700 | An inference based on modu... |
| mpanl2 701 | An inference based on modu... |
| mpanl12 702 | An inference based on modu... |
| mpanr1 703 | An inference based on modu... |
| mpanr2 704 | An inference based on modu... |
| mpanr12 705 | An inference based on modu... |
| mpanlr1 706 | An inference based on modu... |
| mpbirand 707 | Detach truth from conjunct... |
| mpbiran2d 708 | Detach truth from conjunct... |
| mpbiran 709 | Detach truth from conjunct... |
| mpbiran2 710 | Detach truth from conjunct... |
| mpbir2an 711 | Detach a conjunction of tr... |
| mpbi2and 712 | Detach a conjunction of tr... |
| mpbir2and 713 | Detach a conjunction of tr... |
| adantll 714 | Deduction adding a conjunc... |
| adantlr 715 | Deduction adding a conjunc... |
| adantrl 716 | Deduction adding a conjunc... |
| adantrr 717 | Deduction adding a conjunc... |
| adantlll 718 | Deduction adding a conjunc... |
| adantllr 719 | Deduction adding a conjunc... |
| adantlrl 720 | Deduction adding a conjunc... |
| adantlrr 721 | Deduction adding a conjunc... |
| adantrll 722 | Deduction adding a conjunc... |
| adantrlr 723 | Deduction adding a conjunc... |
| adantrrl 724 | Deduction adding a conjunc... |
| adantrrr 725 | Deduction adding a conjunc... |
| ad2antrr 726 | Deduction adding two conju... |
| ad2antlr 727 | Deduction adding two conju... |
| ad2antrl 728 | Deduction adding two conju... |
| ad2antll 729 | Deduction adding conjuncts... |
| ad3antrrr 730 | Deduction adding three con... |
| ad3antlr 731 | Deduction adding three con... |
| ad4antr 732 | Deduction adding 4 conjunc... |
| ad4antlr 733 | Deduction adding 4 conjunc... |
| ad5antr 734 | Deduction adding 5 conjunc... |
| ad5antlr 735 | Deduction adding 5 conjunc... |
| ad6antr 736 | Deduction adding 6 conjunc... |
| ad6antlr 737 | Deduction adding 6 conjunc... |
| ad7antr 738 | Deduction adding 7 conjunc... |
| ad7antlr 739 | Deduction adding 7 conjunc... |
| ad8antr 740 | Deduction adding 8 conjunc... |
| ad8antlr 741 | Deduction adding 8 conjunc... |
| ad9antr 742 | Deduction adding 9 conjunc... |
| ad9antlr 743 | Deduction adding 9 conjunc... |
| ad10antr 744 | Deduction adding 10 conjun... |
| ad10antlr 745 | Deduction adding 10 conjun... |
| ad2ant2l 746 | Deduction adding two conju... |
| ad2ant2r 747 | Deduction adding two conju... |
| ad2ant2lr 748 | Deduction adding two conju... |
| ad2ant2rl 749 | Deduction adding two conju... |
| adantl3r 750 | Deduction adding 1 conjunc... |
| ad4ant13 751 | Deduction adding conjuncts... |
| ad4ant14 752 | Deduction adding conjuncts... |
| ad4ant23 753 | Deduction adding conjuncts... |
| ad4ant24 754 | Deduction adding conjuncts... |
| adantl4r 755 | Deduction adding 1 conjunc... |
| ad5ant13 756 | Deduction adding conjuncts... |
| ad5ant14 757 | Deduction adding conjuncts... |
| ad5ant15 758 | Deduction adding conjuncts... |
| ad5ant23 759 | Deduction adding conjuncts... |
| ad5ant24 760 | Deduction adding conjuncts... |
| ad5ant25 761 | Deduction adding conjuncts... |
| adantl5r 762 | Deduction adding 1 conjunc... |
| adantl6r 763 | Deduction adding 1 conjunc... |
| pm3.33 764 | Theorem *3.33 (Syll) of [W... |
| pm3.34 765 | Theorem *3.34 (Syll) of [W... |
| simpll 766 | Simplification of a conjun... |
| simplld 767 | Deduction form of ~ simpll... |
| simplr 768 | Simplification of a conjun... |
| simplrd 769 | Deduction eliminating a do... |
| simprl 770 | Simplification of a conjun... |
| simprld 771 | Deduction eliminating a do... |
| simprr 772 | Simplification of a conjun... |
| simprrd 773 | Deduction form of ~ simprr... |
| simplll 774 | Simplification of a conjun... |
| simpllr 775 | Simplification of a conjun... |
| simplrl 776 | Simplification of a conjun... |
| simplrr 777 | Simplification of a conjun... |
| simprll 778 | Simplification of a conjun... |
| simprlr 779 | Simplification of a conjun... |
| simprrl 780 | Simplification of a conjun... |
| simprrr 781 | Simplification of a conjun... |
| simp-4l 782 | Simplification of a conjun... |
| simp-4r 783 | Simplification of a conjun... |
| simp-5l 784 | Simplification of a conjun... |
| simp-5r 785 | Simplification of a conjun... |
| simp-6l 786 | Simplification of a conjun... |
| simp-6r 787 | Simplification of a conjun... |
| simp-7l 788 | Simplification of a conjun... |
| simp-7r 789 | Simplification of a conjun... |
| simp-8l 790 | Simplification of a conjun... |
| simp-8r 791 | Simplification of a conjun... |
| simp-9l 792 | Simplification of a conjun... |
| simp-9r 793 | Simplification of a conjun... |
| simp-10l 794 | Simplification of a conjun... |
| simp-10r 795 | Simplification of a conjun... |
| simp-11l 796 | Simplification of a conjun... |
| simp-11r 797 | Simplification of a conjun... |
| pm2.01da 798 | Deduction based on reducti... |
| pm2.18da 799 | Deduction based on reducti... |
| impbida 800 | Deduce an equivalence from... |
| pm5.21nd 801 | Eliminate an antecedent im... |
| pm3.35 802 | Conjunctive detachment. T... |
| pm5.74da 803 | Distribution of implicatio... |
| bitr 804 | Theorem *4.22 of [Whitehea... |
| biantr 805 | A transitive law of equiva... |
| pm4.14 806 | Theorem *4.14 of [Whitehea... |
| pm3.37 807 | Theorem *3.37 (Transp) of ... |
| anim12 808 | Conjoin antecedents and co... |
| pm3.4 809 | Conjunction implies implic... |
| exbiri 810 | Inference form of ~ exbir ... |
| pm2.61ian 811 | Elimination of an antecede... |
| pm2.61dan 812 | Elimination of an antecede... |
| pm2.61ddan 813 | Elimination of two anteced... |
| pm2.61dda 814 | Elimination of two anteced... |
| mtand 815 | A modus tollens deduction.... |
| pm2.65da 816 | Deduction for proof by con... |
| condan 817 | Proof by contradiction. (... |
| biadan 818 | An implication is equivale... |
| biadani 819 | Inference associated with ... |
| biadaniALT 820 | Alternate proof of ~ biada... |
| biadanii 821 | Inference associated with ... |
| biadanid 822 | Deduction associated with ... |
| pm5.1 823 | Two propositions are equiv... |
| pm5.21 824 | Two propositions are equiv... |
| pm5.35 825 | Theorem *5.35 of [Whitehea... |
| abai 826 | Introduce one conjunct as ... |
| pm4.45im 827 | Conjunction with implicati... |
| impimprbi 828 | An implication and its rev... |
| nan 829 | Theorem to move a conjunct... |
| pm5.31 830 | Theorem *5.31 of [Whitehea... |
| pm5.31r 831 | Variant of ~ pm5.31 . (Co... |
| pm4.15 832 | Theorem *4.15 of [Whitehea... |
| pm5.36 833 | Theorem *5.36 of [Whitehea... |
| annotanannot 834 | A conjunction with a negat... |
| pm5.33 835 | Theorem *5.33 of [Whitehea... |
| syl12anc 836 | Syllogism combined with co... |
| syl21anc 837 | Syllogism combined with co... |
| syl22anc 838 | Syllogism combined with co... |
| bibiad 839 | Eliminate an hypothesis ` ... |
| syl1111anc 840 | Four-hypothesis eliminatio... |
| syldbl2 841 | Stacked hypotheseis implie... |
| mpsyl4anc 842 | An elimination deduction. ... |
| pm4.87 843 | Theorem *4.87 of [Whitehea... |
| bimsc1 844 | Removal of conjunct from o... |
| a2and 845 | Deduction distributing a c... |
| animpimp2impd 846 | Deduction deriving nested ... |
| pm4.64 849 | Theorem *4.64 of [Whitehea... |
| pm4.66 850 | Theorem *4.66 of [Whitehea... |
| pm2.53 851 | Theorem *2.53 of [Whitehea... |
| pm2.54 852 | Theorem *2.54 of [Whitehea... |
| imor 853 | Implication in terms of di... |
| imori 854 | Infer disjunction from imp... |
| imorri 855 | Infer implication from dis... |
| pm4.62 856 | Theorem *4.62 of [Whitehea... |
| jaoi 857 | Inference disjoining the a... |
| jao1i 858 | Add a disjunct in the ante... |
| jaod 859 | Deduction disjoining the a... |
| mpjaod 860 | Eliminate a disjunction in... |
| ori 861 | Infer implication from dis... |
| orri 862 | Infer disjunction from imp... |
| orrd 863 | Deduce disjunction from im... |
| ord 864 | Deduce implication from di... |
| orci 865 | Deduction introducing a di... |
| olci 866 | Deduction introducing a di... |
| orc 867 | Introduction of a disjunct... |
| olc 868 | Introduction of a disjunct... |
| pm1.4 869 | Axiom *1.4 of [WhiteheadRu... |
| orcom 870 | Commutative law for disjun... |
| orcomd 871 | Commutation of disjuncts i... |
| orcoms 872 | Commutation of disjuncts i... |
| orcd 873 | Deduction introducing a di... |
| olcd 874 | Deduction introducing a di... |
| orcs 875 | Deduction eliminating disj... |
| olcs 876 | Deduction eliminating disj... |
| olcnd 877 | A lemma for Conjunctive No... |
| orcnd 878 | A lemma for Conjunctive No... |
| mtord 879 | A modus tollens deduction ... |
| pm3.2ni 880 | Infer negated disjunction ... |
| pm2.45 881 | Theorem *2.45 of [Whitehea... |
| pm2.46 882 | Theorem *2.46 of [Whitehea... |
| pm2.47 883 | Theorem *2.47 of [Whitehea... |
| pm2.48 884 | Theorem *2.48 of [Whitehea... |
| pm2.49 885 | Theorem *2.49 of [Whitehea... |
| norbi 886 | If neither of two proposit... |
| nbior 887 | If two propositions are no... |
| orel1 888 | Elimination of disjunction... |
| pm2.25 889 | Theorem *2.25 of [Whitehea... |
| orel2 890 | Elimination of disjunction... |
| pm2.67-2 891 | Slight generalization of T... |
| pm2.67 892 | Theorem *2.67 of [Whitehea... |
| curryax 893 | A non-intuitionistic posit... |
| exmid 894 | Law of excluded middle, al... |
| exmidd 895 | Law of excluded middle in ... |
| pm2.1 896 | Theorem *2.1 of [Whitehead... |
| pm2.13 897 | Theorem *2.13 of [Whitehea... |
| pm2.621 898 | Theorem *2.621 of [Whitehe... |
| pm2.62 899 | Theorem *2.62 of [Whitehea... |
| pm2.68 900 | Theorem *2.68 of [Whitehea... |
| dfor2 901 | Logical 'or' expressed in ... |
| pm2.07 902 | Theorem *2.07 of [Whitehea... |
| pm1.2 903 | Axiom *1.2 of [WhiteheadRu... |
| oridm 904 | Idempotent law for disjunc... |
| pm4.25 905 | Theorem *4.25 of [Whitehea... |
| pm2.4 906 | Theorem *2.4 of [Whitehead... |
| pm2.41 907 | Theorem *2.41 of [Whitehea... |
| orim12i 908 | Disjoin antecedents and co... |
| orim1i 909 | Introduce disjunct to both... |
| orim2i 910 | Introduce disjunct to both... |
| orim12dALT 911 | Alternate proof of ~ orim1... |
| orbi2i 912 | Inference adding a left di... |
| orbi1i 913 | Inference adding a right d... |
| orbi12i 914 | Infer the disjunction of t... |
| orbi2d 915 | Deduction adding a left di... |
| orbi1d 916 | Deduction adding a right d... |
| orbi1 917 | Theorem *4.37 of [Whitehea... |
| orbi12d 918 | Deduction joining two equi... |
| pm1.5 919 | Axiom *1.5 (Assoc) of [Whi... |
| or12 920 | Swap two disjuncts. (Cont... |
| orass 921 | Associative law for disjun... |
| pm2.31 922 | Theorem *2.31 of [Whitehea... |
| pm2.32 923 | Theorem *2.32 of [Whitehea... |
| pm2.3 924 | Theorem *2.3 of [Whitehead... |
| or32 925 | A rearrangement of disjunc... |
| or4 926 | Rearrangement of 4 disjunc... |
| or42 927 | Rearrangement of 4 disjunc... |
| orordi 928 | Distribution of disjunctio... |
| orordir 929 | Distribution of disjunctio... |
| orimdi 930 | Disjunction distributes ov... |
| pm2.76 931 | Theorem *2.76 of [Whitehea... |
| pm2.85 932 | Theorem *2.85 of [Whitehea... |
| pm2.75 933 | Theorem *2.75 of [Whitehea... |
| pm4.78 934 | Implication distributes ov... |
| biort 935 | A disjunction with a true ... |
| biorf 936 | A wff is equivalent to its... |
| biortn 937 | A wff is equivalent to its... |
| biorfi 938 | The dual of ~ biorf is not... |
| biorfri 939 | A wff is equivalent to its... |
| biorfriOLD 940 | Obsolete version of ~ bior... |
| pm2.26 941 | Theorem *2.26 of [Whitehea... |
| pm2.63 942 | Theorem *2.63 of [Whitehea... |
| pm2.64 943 | Theorem *2.64 of [Whitehea... |
| pm2.42 944 | Theorem *2.42 of [Whitehea... |
| pm5.11g 945 | A general instance of Theo... |
| pm5.11 946 | Theorem *5.11 of [Whitehea... |
| pm5.12 947 | Theorem *5.12 of [Whitehea... |
| pm5.14 948 | Theorem *5.14 of [Whitehea... |
| pm5.13 949 | Theorem *5.13 of [Whitehea... |
| pm5.55 950 | Theorem *5.55 of [Whitehea... |
| pm4.72 951 | Implication in terms of bi... |
| imimorb 952 | Simplify an implication be... |
| oibabs 953 | Absorption of disjunction ... |
| orbidi 954 | Disjunction distributes ov... |
| pm5.7 955 | Disjunction distributes ov... |
| jaao 956 | Inference conjoining and d... |
| jaoa 957 | Inference disjoining and c... |
| jaoian 958 | Inference disjoining the a... |
| jaodan 959 | Deduction disjoining the a... |
| mpjaodan 960 | Eliminate a disjunction in... |
| pm3.44 961 | Theorem *3.44 of [Whitehea... |
| jao 962 | Disjunction of antecedents... |
| jaob 963 | Disjunction of antecedents... |
| pm4.77 964 | Theorem *4.77 of [Whitehea... |
| pm3.48 965 | Theorem *3.48 of [Whitehea... |
| orim12d 966 | Disjoin antecedents and co... |
| orim1d 967 | Disjoin antecedents and co... |
| orim2d 968 | Disjoin antecedents and co... |
| orim2 969 | Axiom *1.6 (Sum) of [White... |
| pm2.38 970 | Theorem *2.38 of [Whitehea... |
| pm2.36 971 | Theorem *2.36 of [Whitehea... |
| pm2.37 972 | Theorem *2.37 of [Whitehea... |
| pm2.81 973 | Theorem *2.81 of [Whitehea... |
| pm2.8 974 | Theorem *2.8 of [Whitehead... |
| pm2.73 975 | Theorem *2.73 of [Whitehea... |
| pm2.74 976 | Theorem *2.74 of [Whitehea... |
| pm2.82 977 | Theorem *2.82 of [Whitehea... |
| pm4.39 978 | Theorem *4.39 of [Whitehea... |
| animorl 979 | Conjunction implies disjun... |
| animorr 980 | Conjunction implies disjun... |
| animorlr 981 | Conjunction implies disjun... |
| animorrl 982 | Conjunction implies disjun... |
| ianor 983 | Negated conjunction in ter... |
| anor 984 | Conjunction in terms of di... |
| ioran 985 | Negated disjunction in ter... |
| pm4.52 986 | Theorem *4.52 of [Whitehea... |
| pm4.53 987 | Theorem *4.53 of [Whitehea... |
| pm4.54 988 | Theorem *4.54 of [Whitehea... |
| pm4.55 989 | Theorem *4.55 of [Whitehea... |
| pm4.56 990 | Theorem *4.56 of [Whitehea... |
| oran 991 | Disjunction in terms of co... |
| pm4.57 992 | Theorem *4.57 of [Whitehea... |
| pm3.1 993 | Theorem *3.1 of [Whitehead... |
| pm3.11 994 | Theorem *3.11 of [Whitehea... |
| pm3.12 995 | Theorem *3.12 of [Whitehea... |
| pm3.13 996 | Theorem *3.13 of [Whitehea... |
| pm3.14 997 | Theorem *3.14 of [Whitehea... |
| pm4.44 998 | Theorem *4.44 of [Whitehea... |
| pm4.45 999 | Theorem *4.45 of [Whitehea... |
| orabs 1000 | Absorption of redundant in... |
| oranabs 1001 | Absorb a disjunct into a c... |
| pm5.61 1002 | Theorem *5.61 of [Whitehea... |
| pm5.6 1003 | Conjunction in antecedent ... |
| orcanai 1004 | Change disjunction in cons... |
| pm4.79 1005 | Theorem *4.79 of [Whitehea... |
| pm5.53 1006 | Theorem *5.53 of [Whitehea... |
| ordi 1007 | Distributive law for disju... |
| ordir 1008 | Distributive law for disju... |
| andi 1009 | Distributive law for conju... |
| andir 1010 | Distributive law for conju... |
| orddi 1011 | Double distributive law fo... |
| anddi 1012 | Double distributive law fo... |
| pm5.17 1013 | Theorem *5.17 of [Whitehea... |
| pm5.15 1014 | Theorem *5.15 of [Whitehea... |
| pm5.16 1015 | Theorem *5.16 of [Whitehea... |
| xor 1016 | Two ways to express exclus... |
| nbi2 1017 | Two ways to express "exclu... |
| xordi 1018 | Conjunction distributes ov... |
| pm5.54 1019 | Theorem *5.54 of [Whitehea... |
| pm5.62 1020 | Theorem *5.62 of [Whitehea... |
| pm5.63 1021 | Theorem *5.63 of [Whitehea... |
| niabn 1022 | Miscellaneous inference re... |
| ninba 1023 | Miscellaneous inference re... |
| pm4.43 1024 | Theorem *4.43 of [Whitehea... |
| pm4.82 1025 | Theorem *4.82 of [Whitehea... |
| pm4.83 1026 | Theorem *4.83 of [Whitehea... |
| pclem6 1027 | Negation inferred from emb... |
| bigolden 1028 | Dijkstra-Scholten's Golden... |
| pm5.71 1029 | Theorem *5.71 of [Whitehea... |
| pm5.75 1030 | Theorem *5.75 of [Whitehea... |
| ecase2d 1031 | Deduction for elimination ... |
| ecase3 1032 | Inference for elimination ... |
| ecase 1033 | Inference for elimination ... |
| ecase3d 1034 | Deduction for elimination ... |
| ecased 1035 | Deduction for elimination ... |
| ecase3ad 1036 | Deduction for elimination ... |
| ccase 1037 | Inference for combining ca... |
| ccased 1038 | Deduction for combining ca... |
| ccase2 1039 | Inference for combining ca... |
| 4cases 1040 | Inference eliminating two ... |
| 4casesdan 1041 | Deduction eliminating two ... |
| cases 1042 | Case disjunction according... |
| dedlem0a 1043 | Lemma for an alternate ver... |
| dedlem0b 1044 | Lemma for an alternate ver... |
| dedlema 1045 | Lemma for weak deduction t... |
| dedlemb 1046 | Lemma for weak deduction t... |
| cases2 1047 | Case disjunction according... |
| cases2ALT 1048 | Alternate proof of ~ cases... |
| dfbi3 1049 | An alternate definition of... |
| pm5.24 1050 | Theorem *5.24 of [Whitehea... |
| 4exmid 1051 | The disjunction of the fou... |
| consensus 1052 | The consensus theorem. Th... |
| pm4.42 1053 | Theorem *4.42 of [Whitehea... |
| prlem1 1054 | A specialized lemma for se... |
| prlem2 1055 | A specialized lemma for se... |
| oplem1 1056 | A specialized lemma for se... |
| dn1 1057 | A single axiom for Boolean... |
| bianir 1058 | A closed form of ~ mpbir ,... |
| jaoi2 1059 | Inference removing a negat... |
| jaoi3 1060 | Inference separating a dis... |
| ornld 1061 | Selecting one statement fr... |
| dfifp2 1064 | Alternate definition of th... |
| dfifp3 1065 | Alternate definition of th... |
| dfifp4 1066 | Alternate definition of th... |
| dfifp5 1067 | Alternate definition of th... |
| dfifp6 1068 | Alternate definition of th... |
| dfifp7 1069 | Alternate definition of th... |
| ifpdfbi 1070 | Define the biconditional a... |
| anifp 1071 | The conditional operator i... |
| ifpor 1072 | The conditional operator i... |
| ifpn 1073 | Conditional operator for t... |
| ifptru 1074 | Value of the conditional o... |
| ifpfal 1075 | Value of the conditional o... |
| ifpid 1076 | Value of the conditional o... |
| casesifp 1077 | Version of ~ cases express... |
| ifpbi123d 1078 | Equivalence deduction for ... |
| ifpbi23d 1079 | Equivalence deduction for ... |
| ifpimpda 1080 | Separation of the values o... |
| 1fpid3 1081 | The value of the condition... |
| elimh 1082 | Hypothesis builder for the... |
| dedt 1083 | The weak deduction theorem... |
| con3ALT 1084 | Proof of ~ con3 from its a... |
| 3orass 1089 | Associative law for triple... |
| 3orel1 1090 | Partial elimination of a t... |
| 3orrot 1091 | Rotation law for triple di... |
| 3orcoma 1092 | Commutation law for triple... |
| 3orcomb 1093 | Commutation law for triple... |
| 3anass 1094 | Associative law for triple... |
| 3anan12 1095 | Convert triple conjunction... |
| 3anan32 1096 | Convert triple conjunction... |
| 3ancoma 1097 | Commutation law for triple... |
| 3ancomb 1098 | Commutation law for triple... |
| 3anrot 1099 | Rotation law for triple co... |
| 3anrev 1100 | Reversal law for triple co... |
| anandi3 1101 | Distribution of triple con... |
| anandi3r 1102 | Distribution of triple con... |
| 3anidm 1103 | Idempotent law for conjunc... |
| 3an4anass 1104 | Associative law for four c... |
| 3ioran 1105 | Negated triple disjunction... |
| 3ianor 1106 | Negated triple conjunction... |
| 3anor 1107 | Triple conjunction express... |
| 3oran 1108 | Triple disjunction in term... |
| 3impa 1109 | Importation from double to... |
| 3imp 1110 | Importation inference. (C... |
| 3imp31 1111 | The importation inference ... |
| 3imp231 1112 | Importation inference. (C... |
| 3imp21 1113 | The importation inference ... |
| 3impb 1114 | Importation from double to... |
| bi23imp13 1115 | ~ 3imp with middle implica... |
| 3impib 1116 | Importation to triple conj... |
| 3impia 1117 | Importation to triple conj... |
| 3expa 1118 | Exportation from triple to... |
| 3exp 1119 | Exportation inference. (C... |
| 3expb 1120 | Exportation from triple to... |
| 3expia 1121 | Exportation from triple co... |
| 3expib 1122 | Exportation from triple co... |
| 3com12 1123 | Commutation in antecedent.... |
| 3com13 1124 | Commutation in antecedent.... |
| 3comr 1125 | Commutation in antecedent.... |
| 3com23 1126 | Commutation in antecedent.... |
| 3coml 1127 | Commutation in antecedent.... |
| 3jca 1128 | Join consequents with conj... |
| 3jcad 1129 | Deduction conjoining the c... |
| 3adant1 1130 | Deduction adding a conjunc... |
| 3adant2 1131 | Deduction adding a conjunc... |
| 3adant3 1132 | Deduction adding a conjunc... |
| 3ad2ant1 1133 | Deduction adding conjuncts... |
| 3ad2ant2 1134 | Deduction adding conjuncts... |
| 3ad2ant3 1135 | Deduction adding conjuncts... |
| simp1 1136 | Simplification of triple c... |
| simp2 1137 | Simplification of triple c... |
| simp3 1138 | Simplification of triple c... |
| simp1i 1139 | Infer a conjunct from a tr... |
| simp2i 1140 | Infer a conjunct from a tr... |
| simp3i 1141 | Infer a conjunct from a tr... |
| simp1d 1142 | Deduce a conjunct from a t... |
| simp2d 1143 | Deduce a conjunct from a t... |
| simp3d 1144 | Deduce a conjunct from a t... |
| simp1bi 1145 | Deduce a conjunct from a t... |
| simp2bi 1146 | Deduce a conjunct from a t... |
| simp3bi 1147 | Deduce a conjunct from a t... |
| 3simpa 1148 | Simplification of triple c... |
| 3simpb 1149 | Simplification of triple c... |
| 3simpc 1150 | Simplification of triple c... |
| 3anim123i 1151 | Join antecedents and conse... |
| 3anim1i 1152 | Add two conjuncts to antec... |
| 3anim2i 1153 | Add two conjuncts to antec... |
| 3anim3i 1154 | Add two conjuncts to antec... |
| 3anbi123i 1155 | Join 3 biconditionals with... |
| 3orbi123i 1156 | Join 3 biconditionals with... |
| 3anbi1i 1157 | Inference adding two conju... |
| 3anbi2i 1158 | Inference adding two conju... |
| 3anbi3i 1159 | Inference adding two conju... |
| syl3an 1160 | A triple syllogism inferen... |
| syl3anb 1161 | A triple syllogism inferen... |
| syl3anbr 1162 | A triple syllogism inferen... |
| syl3an1 1163 | A syllogism inference. (C... |
| syl3an2 1164 | A syllogism inference. (C... |
| syl3an3 1165 | A syllogism inference. (C... |
| syl3an132 1166 | ~ syl2an with antecedents ... |
| 3adantl1 1167 | Deduction adding a conjunc... |
| 3adantl2 1168 | Deduction adding a conjunc... |
| 3adantl3 1169 | Deduction adding a conjunc... |
| 3adantr1 1170 | Deduction adding a conjunc... |
| 3adantr2 1171 | Deduction adding a conjunc... |
| 3adantr3 1172 | Deduction adding a conjunc... |
| ad4ant123 1173 | Deduction adding conjuncts... |
| ad4ant124 1174 | Deduction adding conjuncts... |
| ad4ant134 1175 | Deduction adding conjuncts... |
| ad4ant234 1176 | Deduction adding conjuncts... |
| 3adant1l 1177 | Deduction adding a conjunc... |
| 3adant1r 1178 | Deduction adding a conjunc... |
| 3adant2l 1179 | Deduction adding a conjunc... |
| 3adant2r 1180 | Deduction adding a conjunc... |
| 3adant3l 1181 | Deduction adding a conjunc... |
| 3adant3r 1182 | Deduction adding a conjunc... |
| 3adant3r1 1183 | Deduction adding a conjunc... |
| 3adant3r2 1184 | Deduction adding a conjunc... |
| 3adant3r3 1185 | Deduction adding a conjunc... |
| 3ad2antl1 1186 | Deduction adding conjuncts... |
| 3ad2antl2 1187 | Deduction adding conjuncts... |
| 3ad2antl3 1188 | Deduction adding conjuncts... |
| 3ad2antr1 1189 | Deduction adding conjuncts... |
| 3ad2antr2 1190 | Deduction adding conjuncts... |
| 3ad2antr3 1191 | Deduction adding conjuncts... |
| simpl1 1192 | Simplification of conjunct... |
| simpl2 1193 | Simplification of conjunct... |
| simpl3 1194 | Simplification of conjunct... |
| simpr1 1195 | Simplification of conjunct... |
| simpr2 1196 | Simplification of conjunct... |
| simpr3 1197 | Simplification of conjunct... |
| simp1l 1198 | Simplification of triple c... |
| simp1r 1199 | Simplification of triple c... |
| simp2l 1200 | Simplification of triple c... |
| simp2r 1201 | Simplification of triple c... |
| simp3l 1202 | Simplification of triple c... |
| simp3r 1203 | Simplification of triple c... |
| simp11 1204 | Simplification of doubly t... |
| simp12 1205 | Simplification of doubly t... |
| simp13 1206 | Simplification of doubly t... |
| simp21 1207 | Simplification of doubly t... |
| simp22 1208 | Simplification of doubly t... |
| simp23 1209 | Simplification of doubly t... |
| simp31 1210 | Simplification of doubly t... |
| simp32 1211 | Simplification of doubly t... |
| simp33 1212 | Simplification of doubly t... |
| simpll1 1213 | Simplification of conjunct... |
| simpll2 1214 | Simplification of conjunct... |
| simpll3 1215 | Simplification of conjunct... |
| simplr1 1216 | Simplification of conjunct... |
| simplr2 1217 | Simplification of conjunct... |
| simplr3 1218 | Simplification of conjunct... |
| simprl1 1219 | Simplification of conjunct... |
| simprl2 1220 | Simplification of conjunct... |
| simprl3 1221 | Simplification of conjunct... |
| simprr1 1222 | Simplification of conjunct... |
| simprr2 1223 | Simplification of conjunct... |
| simprr3 1224 | Simplification of conjunct... |
| simpl1l 1225 | Simplification of conjunct... |
| simpl1r 1226 | Simplification of conjunct... |
| simpl2l 1227 | Simplification of conjunct... |
| simpl2r 1228 | Simplification of conjunct... |
| simpl3l 1229 | Simplification of conjunct... |
| simpl3r 1230 | Simplification of conjunct... |
| simpr1l 1231 | Simplification of conjunct... |
| simpr1r 1232 | Simplification of conjunct... |
| simpr2l 1233 | Simplification of conjunct... |
| simpr2r 1234 | Simplification of conjunct... |
| simpr3l 1235 | Simplification of conjunct... |
| simpr3r 1236 | Simplification of conjunct... |
| simp1ll 1237 | Simplification of conjunct... |
| simp1lr 1238 | Simplification of conjunct... |
| simp1rl 1239 | Simplification of conjunct... |
| simp1rr 1240 | Simplification of conjunct... |
| simp2ll 1241 | Simplification of conjunct... |
| simp2lr 1242 | Simplification of conjunct... |
| simp2rl 1243 | Simplification of conjunct... |
| simp2rr 1244 | Simplification of conjunct... |
| simp3ll 1245 | Simplification of conjunct... |
| simp3lr 1246 | Simplification of conjunct... |
| simp3rl 1247 | Simplification of conjunct... |
| simp3rr 1248 | Simplification of conjunct... |
| simpl11 1249 | Simplification of conjunct... |
| simpl12 1250 | Simplification of conjunct... |
| simpl13 1251 | Simplification of conjunct... |
| simpl21 1252 | Simplification of conjunct... |
| simpl22 1253 | Simplification of conjunct... |
| simpl23 1254 | Simplification of conjunct... |
| simpl31 1255 | Simplification of conjunct... |
| simpl32 1256 | Simplification of conjunct... |
| simpl33 1257 | Simplification of conjunct... |
| simpr11 1258 | Simplification of conjunct... |
| simpr12 1259 | Simplification of conjunct... |
| simpr13 1260 | Simplification of conjunct... |
| simpr21 1261 | Simplification of conjunct... |
| simpr22 1262 | Simplification of conjunct... |
| simpr23 1263 | Simplification of conjunct... |
| simpr31 1264 | Simplification of conjunct... |
| simpr32 1265 | Simplification of conjunct... |
| simpr33 1266 | Simplification of conjunct... |
| simp1l1 1267 | Simplification of conjunct... |
| simp1l2 1268 | Simplification of conjunct... |
| simp1l3 1269 | Simplification of conjunct... |
| simp1r1 1270 | Simplification of conjunct... |
| simp1r2 1271 | Simplification of conjunct... |
| simp1r3 1272 | Simplification of conjunct... |
| simp2l1 1273 | Simplification of conjunct... |
| simp2l2 1274 | Simplification of conjunct... |
| simp2l3 1275 | Simplification of conjunct... |
| simp2r1 1276 | Simplification of conjunct... |
| simp2r2 1277 | Simplification of conjunct... |
| simp2r3 1278 | Simplification of conjunct... |
| simp3l1 1279 | Simplification of conjunct... |
| simp3l2 1280 | Simplification of conjunct... |
| simp3l3 1281 | Simplification of conjunct... |
| simp3r1 1282 | Simplification of conjunct... |
| simp3r2 1283 | Simplification of conjunct... |
| simp3r3 1284 | Simplification of conjunct... |
| simp11l 1285 | Simplification of conjunct... |
| simp11r 1286 | Simplification of conjunct... |
| simp12l 1287 | Simplification of conjunct... |
| simp12r 1288 | Simplification of conjunct... |
| simp13l 1289 | Simplification of conjunct... |
| simp13r 1290 | Simplification of conjunct... |
| simp21l 1291 | Simplification of conjunct... |
| simp21r 1292 | Simplification of conjunct... |
| simp22l 1293 | Simplification of conjunct... |
| simp22r 1294 | Simplification of conjunct... |
| simp23l 1295 | Simplification of conjunct... |
| simp23r 1296 | Simplification of conjunct... |
| simp31l 1297 | Simplification of conjunct... |
| simp31r 1298 | Simplification of conjunct... |
| simp32l 1299 | Simplification of conjunct... |
| simp32r 1300 | Simplification of conjunct... |
| simp33l 1301 | Simplification of conjunct... |
| simp33r 1302 | Simplification of conjunct... |
| simp111 1303 | Simplification of conjunct... |
| simp112 1304 | Simplification of conjunct... |
| simp113 1305 | Simplification of conjunct... |
| simp121 1306 | Simplification of conjunct... |
| simp122 1307 | Simplification of conjunct... |
| simp123 1308 | Simplification of conjunct... |
| simp131 1309 | Simplification of conjunct... |
| simp132 1310 | Simplification of conjunct... |
| simp133 1311 | Simplification of conjunct... |
| simp211 1312 | Simplification of conjunct... |
| simp212 1313 | Simplification of conjunct... |
| simp213 1314 | Simplification of conjunct... |
| simp221 1315 | Simplification of conjunct... |
| simp222 1316 | Simplification of conjunct... |
| simp223 1317 | Simplification of conjunct... |
| simp231 1318 | Simplification of conjunct... |
| simp232 1319 | Simplification of conjunct... |
| simp233 1320 | Simplification of conjunct... |
| simp311 1321 | Simplification of conjunct... |
| simp312 1322 | Simplification of conjunct... |
| simp313 1323 | Simplification of conjunct... |
| simp321 1324 | Simplification of conjunct... |
| simp322 1325 | Simplification of conjunct... |
| simp323 1326 | Simplification of conjunct... |
| simp331 1327 | Simplification of conjunct... |
| simp332 1328 | Simplification of conjunct... |
| simp333 1329 | Simplification of conjunct... |
| 3anibar 1330 | Remove a hypothesis from t... |
| 3mix1 1331 | Introduction in triple dis... |
| 3mix2 1332 | Introduction in triple dis... |
| 3mix3 1333 | Introduction in triple dis... |
| 3mix1i 1334 | Introduction in triple dis... |
| 3mix2i 1335 | Introduction in triple dis... |
| 3mix3i 1336 | Introduction in triple dis... |
| 3mix1d 1337 | Deduction introducing trip... |
| 3mix2d 1338 | Deduction introducing trip... |
| 3mix3d 1339 | Deduction introducing trip... |
| 3pm3.2i 1340 | Infer conjunction of premi... |
| pm3.2an3 1341 | Version of ~ pm3.2 for a t... |
| mpbir3an 1342 | Detach a conjunction of tr... |
| mpbir3and 1343 | Detach a conjunction of tr... |
| syl3anbrc 1344 | Syllogism inference. (Con... |
| syl21anbrc 1345 | Syllogism inference. (Con... |
| 3imp3i2an 1346 | An elimination deduction. ... |
| ex3 1347 | Apply ~ ex to a hypothesis... |
| 3imp1 1348 | Importation to left triple... |
| 3impd 1349 | Importation deduction for ... |
| 3imp2 1350 | Importation to right tripl... |
| 3impdi 1351 | Importation inference (und... |
| 3impdir 1352 | Importation inference (und... |
| 3exp1 1353 | Exportation from left trip... |
| 3expd 1354 | Exportation deduction for ... |
| 3exp2 1355 | Exportation from right tri... |
| exp5o 1356 | A triple exportation infer... |
| exp516 1357 | A triple exportation infer... |
| exp520 1358 | A triple exportation infer... |
| 3impexp 1359 | Version of ~ impexp for a ... |
| 3an1rs 1360 | Swap conjuncts. (Contribu... |
| 3anassrs 1361 | Associative law for conjun... |
| 4anpull2 1362 | An equivalence of two four... |
| ad5ant245 1363 | Deduction adding conjuncts... |
| ad5ant234 1364 | Deduction adding conjuncts... |
| ad5ant235 1365 | Deduction adding conjuncts... |
| ad5ant123 1366 | Deduction adding conjuncts... |
| ad5ant124 1367 | Deduction adding conjuncts... |
| ad5ant125 1368 | Deduction adding conjuncts... |
| ad5ant134 1369 | Deduction adding conjuncts... |
| ad5ant135 1370 | Deduction adding conjuncts... |
| ad5ant145 1371 | Deduction adding conjuncts... |
| ad5ant2345 1372 | Deduction adding conjuncts... |
| syl3anc 1373 | Syllogism combined with co... |
| syl13anc 1374 | Syllogism combined with co... |
| syl31anc 1375 | Syllogism combined with co... |
| syl112anc 1376 | Syllogism combined with co... |
| syl121anc 1377 | Syllogism combined with co... |
| syl211anc 1378 | Syllogism combined with co... |
| syl23anc 1379 | Syllogism combined with co... |
| syl32anc 1380 | Syllogism combined with co... |
| syl122anc 1381 | Syllogism combined with co... |
| syl212anc 1382 | Syllogism combined with co... |
| syl221anc 1383 | Syllogism combined with co... |
| syl113anc 1384 | Syllogism combined with co... |
| syl131anc 1385 | Syllogism combined with co... |
| syl311anc 1386 | Syllogism combined with co... |
| syl33anc 1387 | Syllogism combined with co... |
| syl222anc 1388 | Syllogism combined with co... |
| syl123anc 1389 | Syllogism combined with co... |
| syl132anc 1390 | Syllogism combined with co... |
| syl213anc 1391 | Syllogism combined with co... |
| syl231anc 1392 | Syllogism combined with co... |
| syl312anc 1393 | Syllogism combined with co... |
| syl321anc 1394 | Syllogism combined with co... |
| syl133anc 1395 | Syllogism combined with co... |
| syl313anc 1396 | Syllogism combined with co... |
| syl331anc 1397 | Syllogism combined with co... |
| syl223anc 1398 | Syllogism combined with co... |
| syl232anc 1399 | Syllogism combined with co... |
| syl322anc 1400 | Syllogism combined with co... |
| syl233anc 1401 | Syllogism combined with co... |
| syl323anc 1402 | Syllogism combined with co... |
| syl332anc 1403 | Syllogism combined with co... |
| syl333anc 1404 | A syllogism inference comb... |
| syl3an1b 1405 | A syllogism inference. (C... |
| syl3an2b 1406 | A syllogism inference. (C... |
| syl3an3b 1407 | A syllogism inference. (C... |
| syl3an1br 1408 | A syllogism inference. (C... |
| syl3an2br 1409 | A syllogism inference. (C... |
| syl3an3br 1410 | A syllogism inference. (C... |
| syld3an3 1411 | A syllogism inference. (C... |
| syld3an1 1412 | A syllogism inference. (C... |
| syld3an2 1413 | A syllogism inference. (C... |
| syl3anl1 1414 | A syllogism inference. (C... |
| syl3anl2 1415 | A syllogism inference. (C... |
| syl3anl3 1416 | A syllogism inference. (C... |
| syl3anl 1417 | A triple syllogism inferen... |
| syl3anr1 1418 | A syllogism inference. (C... |
| syl3anr2 1419 | A syllogism inference. (C... |
| syl3anr3 1420 | A syllogism inference. (C... |
| 3anidm12 1421 | Inference from idempotent ... |
| 3anidm13 1422 | Inference from idempotent ... |
| 3anidm23 1423 | Inference from idempotent ... |
| syl2an3an 1424 | ~ syl3an with antecedents ... |
| syl2an23an 1425 | Deduction related to ~ syl... |
| 3ori 1426 | Infer implication from tri... |
| 3jao 1427 | Disjunction of three antec... |
| 3jaob 1428 | Disjunction of three antec... |
| 3jaobOLD 1429 | Obsolete version of ~ 3jao... |
| 3jaoi 1430 | Disjunction of three antec... |
| 3jaod 1431 | Disjunction of three antec... |
| 3jaoian 1432 | Disjunction of three antec... |
| 3jaodan 1433 | Disjunction of three antec... |
| mpjao3dan 1434 | Eliminate a three-way disj... |
| 3jaao 1435 | Inference conjoining and d... |
| syl3an9b 1436 | Nested syllogism inference... |
| 3orbi123d 1437 | Deduction joining 3 equiva... |
| 3anbi123d 1438 | Deduction joining 3 equiva... |
| 3anbi12d 1439 | Deduction conjoining and a... |
| 3anbi13d 1440 | Deduction conjoining and a... |
| 3anbi23d 1441 | Deduction conjoining and a... |
| 3anbi1d 1442 | Deduction adding conjuncts... |
| 3anbi2d 1443 | Deduction adding conjuncts... |
| 3anbi3d 1444 | Deduction adding conjuncts... |
| 3anim123d 1445 | Deduction joining 3 implic... |
| 3orim123d 1446 | Deduction joining 3 implic... |
| an6 1447 | Rearrangement of 6 conjunc... |
| 3an6 1448 | Analogue of ~ an4 for trip... |
| 3or6 1449 | Analogue of ~ or4 for trip... |
| mp3an1 1450 | An inference based on modu... |
| mp3an2 1451 | An inference based on modu... |
| mp3an3 1452 | An inference based on modu... |
| mp3an12 1453 | An inference based on modu... |
| mp3an13 1454 | An inference based on modu... |
| mp3an23 1455 | An inference based on modu... |
| mp3an1i 1456 | An inference based on modu... |
| mp3anl1 1457 | An inference based on modu... |
| mp3anl2 1458 | An inference based on modu... |
| mp3anl3 1459 | An inference based on modu... |
| mp3anr1 1460 | An inference based on modu... |
| mp3anr2 1461 | An inference based on modu... |
| mp3anr3 1462 | An inference based on modu... |
| mp3an 1463 | An inference based on modu... |
| mpd3an3 1464 | An inference based on modu... |
| mpd3an23 1465 | An inference based on modu... |
| mp3and 1466 | A deduction based on modus... |
| mp3an12i 1467 | ~ mp3an with antecedents i... |
| mp3an2i 1468 | ~ mp3an with antecedents i... |
| mp3an3an 1469 | ~ mp3an with antecedents i... |
| mp3an2ani 1470 | An elimination deduction. ... |
| biimp3a 1471 | Infer implication from a l... |
| biimp3ar 1472 | Infer implication from a l... |
| 3anandis 1473 | Inference that undistribut... |
| 3anandirs 1474 | Inference that undistribut... |
| ecase23d 1475 | Deduction for elimination ... |
| 3ecase 1476 | Inference for elimination ... |
| 3bior1fd 1477 | A disjunction is equivalen... |
| 3bior1fand 1478 | A disjunction is equivalen... |
| 3bior2fd 1479 | A wff is equivalent to its... |
| 3biant1d 1480 | A conjunction is equivalen... |
| intn3an1d 1481 | Introduction of a triple c... |
| intn3an2d 1482 | Introduction of a triple c... |
| intn3an3d 1483 | Introduction of a triple c... |
| an3andi 1484 | Distribution of conjunctio... |
| an33rean 1485 | Rearrange a 9-fold conjunc... |
| 3orel2 1486 | Partial elimination of a t... |
| 3orel2OLD 1487 | Obsolete version of ~ 3ore... |
| 3orel3 1488 | Partial elimination of a t... |
| 3orel13 1489 | Elimination of two disjunc... |
| 3pm3.2ni 1490 | Triple negated disjunction... |
| an42ds 1491 | Inference exchanging the l... |
| nanan 1494 | Conjunction in terms of al... |
| dfnan2 1495 | Alternative denial in term... |
| nanor 1496 | Alternative denial in term... |
| nancom 1497 | Alternative denial is comm... |
| nannan 1498 | Nested alternative denials... |
| nanim 1499 | Implication in terms of al... |
| nannot 1500 | Negation in terms of alter... |
| nanbi 1501 | Biconditional in terms of ... |
| nanbi1 1502 | Introduce a right anti-con... |
| nanbi2 1503 | Introduce a left anti-conj... |
| nanbi12 1504 | Join two logical equivalen... |
| nanbi1i 1505 | Introduce a right anti-con... |
| nanbi2i 1506 | Introduce a left anti-conj... |
| nanbi12i 1507 | Join two logical equivalen... |
| nanbi1d 1508 | Introduce a right anti-con... |
| nanbi2d 1509 | Introduce a left anti-conj... |
| nanbi12d 1510 | Join two logical equivalen... |
| nanass 1511 | A characterization of when... |
| xnor 1514 | Two ways to write XNOR (ex... |
| xorcom 1515 | The connector ` \/_ ` is c... |
| xorass 1516 | The connector ` \/_ ` is a... |
| excxor 1517 | This tautology shows that ... |
| xor2 1518 | Two ways to express "exclu... |
| xoror 1519 | Exclusive disjunction impl... |
| xornan 1520 | Exclusive disjunction impl... |
| xornan2 1521 | XOR implies NAND (written ... |
| xorneg2 1522 | The connector ` \/_ ` is n... |
| xorneg1 1523 | The connector ` \/_ ` is n... |
| xorneg 1524 | The connector ` \/_ ` is u... |
| xorbi12i 1525 | Equality property for excl... |
| xorbi12d 1526 | Equality property for excl... |
| anxordi 1527 | Conjunction distributes ov... |
| xorexmid 1528 | Exclusive-or variant of th... |
| norcom 1531 | The connector ` -\/ ` is c... |
| nornot 1532 | ` -. ` is expressible via ... |
| noran 1533 | ` /\ ` is expressible via ... |
| noror 1534 | ` \/ ` is expressible via ... |
| norasslem1 1535 | This lemma shows the equiv... |
| norasslem2 1536 | This lemma specializes ~ b... |
| norasslem3 1537 | This lemma specializes ~ b... |
| norass 1538 | A characterization of when... |
| trujust 1543 | Soundness justification th... |
| tru 1545 | The truth value ` T. ` is ... |
| dftru2 1546 | An alternate definition of... |
| trut 1547 | A proposition is equivalen... |
| mptru 1548 | Eliminate ` T. ` as an ant... |
| tbtru 1549 | A proposition is equivalen... |
| bitru 1550 | A theorem is equivalent to... |
| trud 1551 | Anything implies ` T. ` . ... |
| truan 1552 | True can be removed from a... |
| fal 1555 | The truth value ` F. ` is ... |
| nbfal 1556 | The negation of a proposit... |
| bifal 1557 | A contradiction is equival... |
| falim 1558 | The truth value ` F. ` imp... |
| falimd 1559 | The truth value ` F. ` imp... |
| dfnot 1560 | Given falsum ` F. ` , we c... |
| inegd 1561 | Negation introduction rule... |
| efald 1562 | Deduction based on reducti... |
| pm2.21fal 1563 | If a wff and its negation ... |
| truimtru 1564 | A ` -> ` identity. (Contr... |
| truimfal 1565 | A ` -> ` identity. (Contr... |
| falimtru 1566 | A ` -> ` identity. (Contr... |
| falimfal 1567 | A ` -> ` identity. (Contr... |
| nottru 1568 | A ` -. ` identity. (Contr... |
| notfal 1569 | A ` -. ` identity. (Contr... |
| trubitru 1570 | A ` <-> ` identity. (Cont... |
| falbitru 1571 | A ` <-> ` identity. (Cont... |
| trubifal 1572 | A ` <-> ` identity. (Cont... |
| falbifal 1573 | A ` <-> ` identity. (Cont... |
| truantru 1574 | A ` /\ ` identity. (Contr... |
| truanfal 1575 | A ` /\ ` identity. (Contr... |
| falantru 1576 | A ` /\ ` identity. (Contr... |
| falanfal 1577 | A ` /\ ` identity. (Contr... |
| truortru 1578 | A ` \/ ` identity. (Contr... |
| truorfal 1579 | A ` \/ ` identity. (Contr... |
| falortru 1580 | A ` \/ ` identity. (Contr... |
| falorfal 1581 | A ` \/ ` identity. (Contr... |
| trunantru 1582 | A ` -/\ ` identity. (Cont... |
| trunanfal 1583 | A ` -/\ ` identity. (Cont... |
| falnantru 1584 | A ` -/\ ` identity. (Cont... |
| falnanfal 1585 | A ` -/\ ` identity. (Cont... |
| truxortru 1586 | A ` \/_ ` identity. (Cont... |
| truxorfal 1587 | A ` \/_ ` identity. (Cont... |
| falxortru 1588 | A ` \/_ ` identity. (Cont... |
| falxorfal 1589 | A ` \/_ ` identity. (Cont... |
| trunortru 1590 | A ` -\/ ` identity. (Cont... |
| trunorfal 1591 | A ` -\/ ` identity. (Cont... |
| falnortru 1592 | A ` -\/ ` identity. (Cont... |
| falnorfal 1593 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1596 | Equality theorem for the a... |
| hadbi123i 1597 | Equality theorem for the a... |
| hadass 1598 | Associative law for the ad... |
| hadbi 1599 | The adder sum is the same ... |
| hadcoma 1600 | Commutative law for the ad... |
| hadcomb 1601 | Commutative law for the ad... |
| hadrot 1602 | Rotation law for the adder... |
| hadnot 1603 | The adder sum distributes ... |
| had1 1604 | If the first input is true... |
| had0 1605 | If the first input is fals... |
| hadifp 1606 | The value of the adder sum... |
| cador 1609 | The adder carry in disjunc... |
| cadan 1610 | The adder carry in conjunc... |
| cadbi123d 1611 | Equality theorem for the a... |
| cadbi123i 1612 | Equality theorem for the a... |
| cadcoma 1613 | Commutative law for the ad... |
| cadcomb 1614 | Commutative law for the ad... |
| cadrot 1615 | Rotation law for the adder... |
| cadnot 1616 | The adder carry distribute... |
| cad11 1617 | If (at least) two inputs a... |
| cad1 1618 | If one input is true, then... |
| cad0 1619 | If one input is false, the... |
| cadifp 1620 | The value of the carry is,... |
| cadtru 1621 | The adder carry is true as... |
| minimp 1622 | A single axiom for minimal... |
| minimp-syllsimp 1623 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1624 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1625 | Derivation of a commuted f... |
| minimp-ax2 1626 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1627 | Derivation of ~ pm2.43 (al... |
| impsingle 1628 | The shortest single axiom ... |
| impsingle-step4 1629 | Derivation of impsingle-st... |
| impsingle-step8 1630 | Derivation of impsingle-st... |
| impsingle-ax1 1631 | Derivation of impsingle-ax... |
| impsingle-step15 1632 | Derivation of impsingle-st... |
| impsingle-step18 1633 | Derivation of impsingle-st... |
| impsingle-step19 1634 | Derivation of impsingle-st... |
| impsingle-step20 1635 | Derivation of impsingle-st... |
| impsingle-step21 1636 | Derivation of impsingle-st... |
| impsingle-step22 1637 | Derivation of impsingle-st... |
| impsingle-step25 1638 | Derivation of impsingle-st... |
| impsingle-imim1 1639 | Derivation of impsingle-im... |
| impsingle-peirce 1640 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1641 | Derivation of ~ ax-2 from ... |
| meredith 1642 | Carew Meredith's sole axio... |
| merlem1 1643 | Step 3 of Meredith's proof... |
| merlem2 1644 | Step 4 of Meredith's proof... |
| merlem3 1645 | Step 7 of Meredith's proof... |
| merlem4 1646 | Step 8 of Meredith's proof... |
| merlem5 1647 | Step 11 of Meredith's proo... |
| merlem6 1648 | Step 12 of Meredith's proo... |
| merlem7 1649 | Between steps 14 and 15 of... |
| merlem8 1650 | Step 15 of Meredith's proo... |
| merlem9 1651 | Step 18 of Meredith's proo... |
| merlem10 1652 | Step 19 of Meredith's proo... |
| merlem11 1653 | Step 20 of Meredith's proo... |
| merlem12 1654 | Step 28 of Meredith's proo... |
| merlem13 1655 | Step 35 of Meredith's proo... |
| luk-1 1656 | 1 of 3 axioms for proposit... |
| luk-2 1657 | 2 of 3 axioms for proposit... |
| luk-3 1658 | 3 of 3 axioms for proposit... |
| luklem1 1659 | Used to rederive standard ... |
| luklem2 1660 | Used to rederive standard ... |
| luklem3 1661 | Used to rederive standard ... |
| luklem4 1662 | Used to rederive standard ... |
| luklem5 1663 | Used to rederive standard ... |
| luklem6 1664 | Used to rederive standard ... |
| luklem7 1665 | Used to rederive standard ... |
| luklem8 1666 | Used to rederive standard ... |
| ax1 1667 | Standard propositional axi... |
| ax2 1668 | Standard propositional axi... |
| ax3 1669 | Standard propositional axi... |
| nic-dfim 1670 | This theorem "defines" imp... |
| nic-dfneg 1671 | This theorem "defines" neg... |
| nic-mp 1672 | Derive Nicod's rule of mod... |
| nic-mpALT 1673 | A direct proof of ~ nic-mp... |
| nic-ax 1674 | Nicod's axiom derived from... |
| nic-axALT 1675 | A direct proof of ~ nic-ax... |
| nic-imp 1676 | Inference for ~ nic-mp usi... |
| nic-idlem1 1677 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1678 | Lemma for ~ nic-id . Infe... |
| nic-id 1679 | Theorem ~ id expressed wit... |
| nic-swap 1680 | The connector ` -/\ ` is s... |
| nic-isw1 1681 | Inference version of ~ nic... |
| nic-isw2 1682 | Inference for swapping nes... |
| nic-iimp1 1683 | Inference version of ~ nic... |
| nic-iimp2 1684 | Inference version of ~ nic... |
| nic-idel 1685 | Inference to remove the tr... |
| nic-ich 1686 | Chained inference. (Contr... |
| nic-idbl 1687 | Double the terms. Since d... |
| nic-bijust 1688 | Biconditional justificatio... |
| nic-bi1 1689 | Inference to extract one s... |
| nic-bi2 1690 | Inference to extract the o... |
| nic-stdmp 1691 | Derive the standard modus ... |
| nic-luk1 1692 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1693 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1694 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1695 | This alternative axiom for... |
| lukshefth1 1696 | Lemma for ~ renicax . (Co... |
| lukshefth2 1697 | Lemma for ~ renicax . (Co... |
| renicax 1698 | A rederivation of ~ nic-ax... |
| tbw-bijust 1699 | Justification for ~ tbw-ne... |
| tbw-negdf 1700 | The definition of negation... |
| tbw-ax1 1701 | The first of four axioms i... |
| tbw-ax2 1702 | The second of four axioms ... |
| tbw-ax3 1703 | The third of four axioms i... |
| tbw-ax4 1704 | The fourth of four axioms ... |
| tbwsyl 1705 | Used to rederive the Lukas... |
| tbwlem1 1706 | Used to rederive the Lukas... |
| tbwlem2 1707 | Used to rederive the Lukas... |
| tbwlem3 1708 | Used to rederive the Lukas... |
| tbwlem4 1709 | Used to rederive the Lukas... |
| tbwlem5 1710 | Used to rederive the Lukas... |
| re1luk1 1711 | ~ luk-1 derived from the T... |
| re1luk2 1712 | ~ luk-2 derived from the T... |
| re1luk3 1713 | ~ luk-3 derived from the T... |
| merco1 1714 | A single axiom for proposi... |
| merco1lem1 1715 | Used to rederive the Tarsk... |
| retbwax4 1716 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1717 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1718 | Used to rederive the Tarsk... |
| merco1lem3 1719 | Used to rederive the Tarsk... |
| merco1lem4 1720 | Used to rederive the Tarsk... |
| merco1lem5 1721 | Used to rederive the Tarsk... |
| merco1lem6 1722 | Used to rederive the Tarsk... |
| merco1lem7 1723 | Used to rederive the Tarsk... |
| retbwax3 1724 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1725 | Used to rederive the Tarsk... |
| merco1lem9 1726 | Used to rederive the Tarsk... |
| merco1lem10 1727 | Used to rederive the Tarsk... |
| merco1lem11 1728 | Used to rederive the Tarsk... |
| merco1lem12 1729 | Used to rederive the Tarsk... |
| merco1lem13 1730 | Used to rederive the Tarsk... |
| merco1lem14 1731 | Used to rederive the Tarsk... |
| merco1lem15 1732 | Used to rederive the Tarsk... |
| merco1lem16 1733 | Used to rederive the Tarsk... |
| merco1lem17 1734 | Used to rederive the Tarsk... |
| merco1lem18 1735 | Used to rederive the Tarsk... |
| retbwax1 1736 | ~ tbw-ax1 rederived from ~... |
| merco2 1737 | A single axiom for proposi... |
| mercolem1 1738 | Used to rederive the Tarsk... |
| mercolem2 1739 | Used to rederive the Tarsk... |
| mercolem3 1740 | Used to rederive the Tarsk... |
| mercolem4 1741 | Used to rederive the Tarsk... |
| mercolem5 1742 | Used to rederive the Tarsk... |
| mercolem6 1743 | Used to rederive the Tarsk... |
| mercolem7 1744 | Used to rederive the Tarsk... |
| mercolem8 1745 | Used to rederive the Tarsk... |
| re1tbw1 1746 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1747 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1748 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1749 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1750 | Justification for ~ rb-imd... |
| rb-imdf 1751 | The definition of implicat... |
| anmp 1752 | Modus ponens for ` { \/ , ... |
| rb-ax1 1753 | The first of four axioms i... |
| rb-ax2 1754 | The second of four axioms ... |
| rb-ax3 1755 | The third of four axioms i... |
| rb-ax4 1756 | The fourth of four axioms ... |
| rbsyl 1757 | Used to rederive the Lukas... |
| rblem1 1758 | Used to rederive the Lukas... |
| rblem2 1759 | Used to rederive the Lukas... |
| rblem3 1760 | Used to rederive the Lukas... |
| rblem4 1761 | Used to rederive the Lukas... |
| rblem5 1762 | Used to rederive the Lukas... |
| rblem6 1763 | Used to rederive the Lukas... |
| rblem7 1764 | Used to rederive the Lukas... |
| re1axmp 1765 | ~ ax-mp derived from Russe... |
| re2luk1 1766 | ~ luk-1 derived from Russe... |
| re2luk2 1767 | ~ luk-2 derived from Russe... |
| re2luk3 1768 | ~ luk-3 derived from Russe... |
| mptnan 1769 | Modus ponendo tollens 1, o... |
| mptxor 1770 | Modus ponendo tollens 2, o... |
| mtpor 1771 | Modus tollendo ponens (inc... |
| mtpxor 1772 | Modus tollendo ponens (ori... |
| stoic1a 1773 | Stoic logic Thema 1 (part ... |
| stoic1b 1774 | Stoic logic Thema 1 (part ... |
| stoic2a 1775 | Stoic logic Thema 2 versio... |
| stoic2b 1776 | Stoic logic Thema 2 versio... |
| stoic3 1777 | Stoic logic Thema 3. Stat... |
| stoic4a 1778 | Stoic logic Thema 4 versio... |
| stoic4b 1779 | Stoic logic Thema 4 versio... |
| alnex 1782 | Universal quantification o... |
| eximal 1783 | An equivalence between an ... |
| nf2 1786 | Alternate definition of no... |
| nf3 1787 | Alternate definition of no... |
| nf4 1788 | Alternate definition of no... |
| nfi 1789 | Deduce that ` x ` is not f... |
| nfri 1790 | Consequence of the definit... |
| nfd 1791 | Deduce that ` x ` is not f... |
| nfrd 1792 | Consequence of the definit... |
| nftht 1793 | Closed form of ~ nfth . (... |
| nfntht 1794 | Closed form of ~ nfnth . ... |
| nfntht2 1795 | Closed form of ~ nfnth . ... |
| gen2 1797 | Generalization applied twi... |
| mpg 1798 | Modus ponens combined with... |
| mpgbi 1799 | Modus ponens on biconditio... |
| mpgbir 1800 | Modus ponens on biconditio... |
| nex 1801 | Generalization rule for ne... |
| nfth 1802 | No variable is (effectivel... |
| nfnth 1803 | No variable is (effectivel... |
| hbth 1804 | No variable is (effectivel... |
| nftru 1805 | The true constant has no f... |
| nffal 1806 | The false constant has no ... |
| sptruw 1807 | Version of ~ sp when ` ph ... |
| altru 1808 | For all sets, ` T. ` is tr... |
| alfal 1809 | For all sets, ` -. F. ` is... |
| alim 1811 | Restatement of Axiom ~ ax-... |
| alimi 1812 | Inference quantifying both... |
| 2alimi 1813 | Inference doubly quantifyi... |
| ala1 1814 | Add an antecedent in a uni... |
| al2im 1815 | Closed form of ~ al2imi . ... |
| al2imi 1816 | Inference quantifying ante... |
| alanimi 1817 | Variant of ~ al2imi with c... |
| alimdh 1818 | Deduction form of Theorem ... |
| albi 1819 | Theorem 19.15 of [Margaris... |
| albii 1820 | Inference adding universal... |
| 2albii 1821 | Inference adding two unive... |
| 3albii 1822 | Inference adding three uni... |
| sylgt 1823 | Closed form of ~ sylg . (... |
| sylg 1824 | A syllogism combined with ... |
| alrimih 1825 | Inference form of Theorem ... |
| hbxfrbi 1826 | A utility lemma to transfe... |
| alex 1827 | Universal quantifier in te... |
| exnal 1828 | Existential quantification... |
| 2nalexn 1829 | Part of theorem *11.5 in [... |
| 2exnaln 1830 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1831 | Theorem *11.25 in [Whitehe... |
| alimex 1832 | An equivalence between an ... |
| aleximi 1833 | A variant of ~ al2imi : in... |
| alexbii 1834 | Biconditional form of ~ al... |
| exim 1835 | Theorem 19.22 of [Margaris... |
| eximi 1836 | Inference adding existenti... |
| 2eximi 1837 | Inference adding two exist... |
| eximii 1838 | Inference associated with ... |
| exa1 1839 | Add an antecedent in an ex... |
| 19.38 1840 | Theorem 19.38 of [Margaris... |
| 19.38a 1841 | Under a nonfreeness hypoth... |
| 19.38b 1842 | Under a nonfreeness hypoth... |
| imnang 1843 | Quantified implication in ... |
| alinexa 1844 | A transformation of quanti... |
| exnalimn 1845 | Existential quantification... |
| alexn 1846 | A relationship between two... |
| 2exnexn 1847 | Theorem *11.51 in [Whitehe... |
| exbi 1848 | Theorem 19.18 of [Margaris... |
| exbii 1849 | Inference adding existenti... |
| 2exbii 1850 | Inference adding two exist... |
| 3exbii 1851 | Inference adding three exi... |
| nfbiit 1852 | Equivalence theorem for th... |
| nfbii 1853 | Equality theorem for the n... |
| nfxfr 1854 | A utility lemma to transfe... |
| nfxfrd 1855 | A utility lemma to transfe... |
| nfnbi 1856 | A variable is nonfree in a... |
| nfnt 1857 | If a variable is nonfree i... |
| nfn 1858 | Inference associated with ... |
| nfnd 1859 | Deduction associated with ... |
| exanali 1860 | A transformation of quanti... |
| 2exanali 1861 | Theorem *11.521 in [Whiteh... |
| exancom 1862 | Commutation of conjunction... |
| exan 1863 | Place a conjunct in the sc... |
| alrimdh 1864 | Deduction form of Theorem ... |
| eximdh 1865 | Deduction from Theorem 19.... |
| nexdh 1866 | Deduction for generalizati... |
| albidh 1867 | Formula-building rule for ... |
| exbidh 1868 | Formula-building rule for ... |
| exsimpl 1869 | Simplification of an exist... |
| exsimpr 1870 | Simplification of an exist... |
| 19.26 1871 | Theorem 19.26 of [Margaris... |
| 19.26-2 1872 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1873 | Theorem ~ 19.26 with tripl... |
| 19.29 1874 | Theorem 19.29 of [Margaris... |
| 19.29r 1875 | Variation of ~ 19.29 . (C... |
| 19.29r2 1876 | Variation of ~ 19.29r with... |
| 19.29x 1877 | Variation of ~ 19.29 with ... |
| 19.35 1878 | Theorem 19.35 of [Margaris... |
| 19.35i 1879 | Inference associated with ... |
| 19.35ri 1880 | Inference associated with ... |
| 19.25 1881 | Theorem 19.25 of [Margaris... |
| 19.30 1882 | Theorem 19.30 of [Margaris... |
| 19.43 1883 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1884 | Obsolete proof of ~ 19.43 ... |
| 19.33 1885 | Theorem 19.33 of [Margaris... |
| 19.33b 1886 | The antecedent provides a ... |
| 19.40 1887 | Theorem 19.40 of [Margaris... |
| 19.40-2 1888 | Theorem *11.42 in [Whitehe... |
| 19.40b 1889 | The antecedent provides a ... |
| albiim 1890 | Split a biconditional and ... |
| 2albiim 1891 | Split a biconditional and ... |
| exintrbi 1892 | Add/remove a conjunct in t... |
| exintr 1893 | Introduce a conjunct in th... |
| alsyl 1894 | Universally quantified and... |
| nfimd 1895 | If in a context ` x ` is n... |
| nfimt 1896 | Closed form of ~ nfim and ... |
| nfim 1897 | If ` x ` is not free in ` ... |
| nfand 1898 | If in a context ` x ` is n... |
| nf3and 1899 | Deduction form of bound-va... |
| nfan 1900 | If ` x ` is not free in ` ... |
| nfnan 1901 | If ` x ` is not free in ` ... |
| nf3an 1902 | If ` x ` is not free in ` ... |
| nfbid 1903 | If in a context ` x ` is n... |
| nfbi 1904 | If ` x ` is not free in ` ... |
| nfor 1905 | If ` x ` is not free in ` ... |
| nf3or 1906 | If ` x ` is not free in ` ... |
| empty 1907 | Two characterizations of t... |
| emptyex 1908 | On the empty domain, any e... |
| emptyal 1909 | On the empty domain, any u... |
| emptynf 1910 | On the empty domain, any v... |
| ax5d 1912 | Version of ~ ax-5 with ant... |
| ax5e 1913 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1914 | If a formula holds for som... |
| nfv 1915 | If ` x ` is not present in... |
| nfvd 1916 | ~ nfv with antecedent. Us... |
| alimdv 1917 | Deduction form of Theorem ... |
| eximdv 1918 | Deduction form of Theorem ... |
| 2alimdv 1919 | Deduction form of Theorem ... |
| 2eximdv 1920 | Deduction form of Theorem ... |
| albidv 1921 | Formula-building rule for ... |
| exbidv 1922 | Formula-building rule for ... |
| nfbidv 1923 | An equality theorem for no... |
| 2albidv 1924 | Formula-building rule for ... |
| 2exbidv 1925 | Formula-building rule for ... |
| 3exbidv 1926 | Formula-building rule for ... |
| 4exbidv 1927 | Formula-building rule for ... |
| alrimiv 1928 | Inference form of Theorem ... |
| alrimivv 1929 | Inference form of Theorem ... |
| alrimdv 1930 | Deduction form of Theorem ... |
| exlimiv 1931 | Inference form of Theorem ... |
| exlimiiv 1932 | Inference (Rule C) associa... |
| exlimivv 1933 | Inference form of Theorem ... |
| exlimdv 1934 | Deduction form of Theorem ... |
| exlimdvv 1935 | Deduction form of Theorem ... |
| exlimddv 1936 | Existential elimination ru... |
| nexdv 1937 | Deduction for generalizati... |
| 2ax5 1938 | Quantification of two vari... |
| stdpc5v 1939 | Version of ~ stdpc5 with a... |
| 19.21v 1940 | Version of ~ 19.21 with a ... |
| 19.32v 1941 | Version of ~ 19.32 with a ... |
| 19.31v 1942 | Version of ~ 19.31 with a ... |
| 19.23v 1943 | Version of ~ 19.23 with a ... |
| 19.23vv 1944 | Theorem ~ 19.23v extended ... |
| pm11.53v 1945 | Version of ~ pm11.53 with ... |
| 19.36imv 1946 | One direction of ~ 19.36v ... |
| 19.36iv 1947 | Inference associated with ... |
| 19.37imv 1948 | One direction of ~ 19.37v ... |
| 19.37iv 1949 | Inference associated with ... |
| 19.41v 1950 | Version of ~ 19.41 with a ... |
| 19.41vv 1951 | Version of ~ 19.41 with tw... |
| 19.41vvv 1952 | Version of ~ 19.41 with th... |
| 19.41vvvv 1953 | Version of ~ 19.41 with fo... |
| 19.42v 1954 | Version of ~ 19.42 with a ... |
| exdistr 1955 | Distribution of existentia... |
| exdistrv 1956 | Distribute a pair of exist... |
| 4exdistrv 1957 | Distribute two pairs of ex... |
| 19.42vv 1958 | Version of ~ 19.42 with tw... |
| exdistr2 1959 | Distribution of existentia... |
| 19.42vvv 1960 | Version of ~ 19.42 with th... |
| 3exdistr 1961 | Distribution of existentia... |
| 4exdistr 1962 | Distribution of existentia... |
| weq 1963 | Extend wff definition to i... |
| speimfw 1964 | Specialization, with addit... |
| speimfwALT 1965 | Alternate proof of ~ speim... |
| spimfw 1966 | Specialization, with addit... |
| ax12i 1967 | Inference that has ~ ax-12... |
| ax6v 1969 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1970 | At least one individual ex... |
| spimw 1971 | Specialization. Lemma 8 o... |
| spimew 1972 | Existential introduction, ... |
| speiv 1973 | Inference from existential... |
| speivw 1974 | Version of ~ spei with a d... |
| exgen 1975 | Rule of existential genera... |
| extru 1976 | There exists a variable su... |
| 19.2 1977 | Theorem 19.2 of [Margaris]... |
| 19.2d 1978 | Deduction associated with ... |
| 19.8w 1979 | Weak version of ~ 19.8a an... |
| spnfw 1980 | Weak version of ~ sp . Us... |
| spfalw 1981 | Version of ~ sp when ` ph ... |
| spvw 1982 | Version of ~ sp when ` x `... |
| 19.3v 1983 | Version of ~ 19.3 with a d... |
| 19.8v 1984 | Version of ~ 19.8a with a ... |
| 19.9v 1985 | Version of ~ 19.9 with a d... |
| spimevw 1986 | Existential introduction, ... |
| spimvw 1987 | A weak form of specializat... |
| spsv 1988 | Generalization of antecede... |
| spvv 1989 | Specialization, using impl... |
| chvarvv 1990 | Implicit substitution of `... |
| 19.39 1991 | Theorem 19.39 of [Margaris... |
| 19.24 1992 | Theorem 19.24 of [Margaris... |
| 19.34 1993 | Theorem 19.34 of [Margaris... |
| 19.36v 1994 | Version of ~ 19.36 with a ... |
| 19.12vvv 1995 | Version of ~ 19.12vv with ... |
| 19.27v 1996 | Version of ~ 19.27 with a ... |
| 19.28v 1997 | Version of ~ 19.28 with a ... |
| 19.37v 1998 | Version of ~ 19.37 with a ... |
| 19.44v 1999 | Version of ~ 19.44 with a ... |
| 19.45v 2000 | Version of ~ 19.45 with a ... |
| equs4v 2001 | Version of ~ equs4 with a ... |
| alequexv 2002 | Version of ~ equs4v with i... |
| exsbim 2003 | One direction of the equiv... |
| equsv 2004 | If a formula does not cont... |
| equsalvw 2005 | Version of ~ equsalv with ... |
| equsexvw 2006 | Version of ~ equsexv with ... |
| cbvaliw 2007 | Change bound variable. Us... |
| cbvalivw 2008 | Change bound variable. Us... |
| ax7v 2010 | Weakened version of ~ ax-7... |
| ax7v1 2011 | First of two weakened vers... |
| ax7v2 2012 | Second of two weakened ver... |
| equid 2013 | Identity law for equality.... |
| nfequid 2014 | Bound-variable hypothesis ... |
| equcomiv 2015 | Weaker form of ~ equcomi w... |
| ax6evr 2016 | A commuted form of ~ ax6ev... |
| ax7 2017 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2018 | Commutative law for equali... |
| equcom 2019 | Commutative law for equali... |
| equcomd 2020 | Deduction form of ~ equcom... |
| equcoms 2021 | An inference commuting equ... |
| equtr 2022 | A transitive law for equal... |
| equtrr 2023 | A transitive law for equal... |
| equeuclr 2024 | Commuted version of ~ eque... |
| equeucl 2025 | Equality is a left-Euclide... |
| equequ1 2026 | An equivalence law for equ... |
| equequ2 2027 | An equivalence law for equ... |
| equtr2 2028 | Equality is a left-Euclide... |
| stdpc6 2029 | One of the two equality ax... |
| equvinv 2030 | A variable introduction la... |
| equvinva 2031 | A modified version of the ... |
| equvelv 2032 | A biconditional form of ~ ... |
| ax13b 2033 | An equivalence between two... |
| spfw 2034 | Weak version of ~ sp . Us... |
| spw 2035 | Weak version of the specia... |
| cbvalw 2036 | Change bound variable. Us... |
| cbvalvw 2037 | Change bound variable. Us... |
| cbvexvw 2038 | Change bound variable. Us... |
| cbvaldvaw 2039 | Rule used to change the bo... |
| cbvexdvaw 2040 | Rule used to change the bo... |
| cbval2vw 2041 | Rule used to change bound ... |
| cbvex2vw 2042 | Rule used to change bound ... |
| cbvex4vw 2043 | Rule used to change bound ... |
| alcomimw 2044 | Weak version of ~ ax-11 . ... |
| excomimw 2045 | Weak version of ~ excomim ... |
| alcomw 2046 | Weak version of ~ alcom an... |
| excomw 2047 | Weak version of ~ excom an... |
| hbn1fw 2048 | Weak version of ~ ax-10 fr... |
| hbn1w 2049 | Weak version of ~ hbn1 . ... |
| hba1w 2050 | Weak version of ~ hba1 . ... |
| hbe1w 2051 | Weak version of ~ hbe1 . ... |
| hbalw 2052 | Weak version of ~ hbal . ... |
| 19.8aw 2053 | If a formula is true, then... |
| exexw 2054 | Existential quantification... |
| spaev 2055 | A special instance of ~ sp... |
| cbvaev 2056 | Change bound variable in a... |
| aevlem0 2057 | Lemma for ~ aevlem . Inst... |
| aevlem 2058 | Lemma for ~ aev and ~ axc1... |
| aeveq 2059 | The antecedent ` A. x x = ... |
| aev 2060 | A "distinctor elimination"... |
| aev2 2061 | A version of ~ aev with tw... |
| hbaev 2062 | All variables are effectiv... |
| naev 2063 | If some set variables can ... |
| naev2 2064 | Generalization of ~ hbnaev... |
| hbnaev 2065 | Any variable is free in ` ... |
| sbjust 2066 | Justification theorem for ... |
| dfsb 2069 | Simplify definition ~ df-s... |
| sbtlem 2070 | In the case of ~ sbt , the... |
| sbt 2071 | A substitution into a theo... |
| sbtru 2072 | The result of substituting... |
| stdpc4 2073 | The specialization axiom o... |
| sbtALT 2074 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2075 | A double specialization us... |
| sbi1 2076 | Distribute substitution ov... |
| spsbim 2077 | Distribute substitution ov... |
| spsbbi 2078 | Biconditional property for... |
| sbimi 2079 | Distribute substitution ov... |
| sb2imi 2080 | Distribute substitution ov... |
| sbbii 2081 | Infer substitution into bo... |
| 2sbbii 2082 | Infer double substitution ... |
| sbimdv 2083 | Deduction substituting bot... |
| sbbidv 2084 | Deduction substituting bot... |
| sban 2085 | Conjunction inside and out... |
| sb3an 2086 | Threefold conjunction insi... |
| spsbe 2087 | Existential generalization... |
| sbequ 2088 | Equality property for subs... |
| sbequi 2089 | An equality theorem for su... |
| sb6 2090 | Alternate definition of su... |
| 2sb6 2091 | Equivalence for double sub... |
| sb1v 2092 | One direction of ~ sb5 , p... |
| sbv 2093 | Substitution for a variabl... |
| sbcom4 2094 | Commutativity law for subs... |
| pm11.07 2095 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2096 | Substitution in an implica... |
| sbbiiev 2097 | An equivalence of substitu... |
| sbievw 2098 | Conversion of implicit sub... |
| sbievwOLD 2099 | Obsolete version of ~ sbie... |
| sbiedvw 2100 | Conversion of implicit sub... |
| 2sbievw 2101 | Conversion of double impli... |
| sbcom3vv 2102 | Substituting ` y ` for ` x... |
| sbievw2 2103 | ~ sbievw applied twice, av... |
| sbco2vv 2104 | A composition law for subs... |
| cbvsbv 2105 | Change the bound variable ... |
| sbco4lem 2106 | Lemma for ~ sbco4 . It re... |
| sbco4 2107 | Two ways of exchanging two... |
| equsb3 2108 | Substitution in an equalit... |
| equsb3r 2109 | Substitution applied to th... |
| equsb1v 2110 | Substitution applied to an... |
| nsb 2111 | Any substitution in an alw... |
| sbn1 2112 | One direction of ~ sbn , u... |
| wel 2114 | Extend wff definition to i... |
| ax8v 2116 | Weakened version of ~ ax-8... |
| ax8v1 2117 | First of two weakened vers... |
| ax8v2 2118 | Second of two weakened ver... |
| ax8 2119 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2120 | An identity law for the no... |
| elsb1 2121 | Substitution for the first... |
| cleljust 2122 | When the class variables i... |
| ax9v 2124 | Weakened version of ~ ax-9... |
| ax9v1 2125 | First of two weakened vers... |
| ax9v2 2126 | Second of two weakened ver... |
| ax9 2127 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2128 | An identity law for the no... |
| elequ2g 2129 | A form of ~ elequ2 with a ... |
| elsb2 2130 | Substitution for the secon... |
| elequ12 2131 | An identity law for the no... |
| ru0 2132 | The FOL statement used in ... |
| ax6dgen 2133 | Tarski's system uses the w... |
| ax10w 2134 | Weak version of ~ ax-10 fr... |
| ax11w 2135 | Weak version of ~ ax-11 fr... |
| ax11dgen 2136 | Degenerate instance of ~ a... |
| ax12wlem 2137 | Lemma for weak version of ... |
| ax12w 2138 | Weak version of ~ ax-12 fr... |
| ax12dgen 2139 | Degenerate instance of ~ a... |
| ax12wdemo 2140 | Example of an application ... |
| ax13w 2141 | Weak version (principal in... |
| ax13dgen1 2142 | Degenerate instance of ~ a... |
| ax13dgen2 2143 | Degenerate instance of ~ a... |
| ax13dgen3 2144 | Degenerate instance of ~ a... |
| ax13dgen4 2145 | Degenerate instance of ~ a... |
| hbn1 2147 | Alias for ~ ax-10 to be us... |
| hbe1 2148 | The setvar ` x ` is not fr... |
| hbe1a 2149 | Dual statement of ~ hbe1 .... |
| nf5-1 2150 | One direction of ~ nf5 can... |
| nf5i 2151 | Deduce that ` x ` is not f... |
| nf5dh 2152 | Deduce that ` x ` is not f... |
| nf5dv 2153 | Apply the definition of no... |
| nfnaew 2154 | All variables are effectiv... |
| nfe1 2155 | The setvar ` x ` is not fr... |
| nfa1 2156 | The setvar ` x ` is not fr... |
| nfna1 2157 | A convenience theorem part... |
| nfia1 2158 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2159 | The setvar ` x ` is not fr... |
| modal5 2160 | The analogue in our predic... |
| nfs1v 2161 | The setvar ` x ` is not fr... |
| alcoms 2163 | Swap quantifiers in an ant... |
| alcom 2164 | Theorem 19.5 of [Margaris]... |
| alrot3 2165 | Theorem *11.21 in [Whitehe... |
| alrot4 2166 | Rotate four universal quan... |
| excom 2167 | Theorem 19.11 of [Margaris... |
| excomim 2168 | One direction of Theorem 1... |
| excom13 2169 | Swap 1st and 3rd existenti... |
| exrot3 2170 | Rotate existential quantif... |
| exrot4 2171 | Rotate existential quantif... |
| hbal 2172 | If ` x ` is not free in ` ... |
| hbald 2173 | Deduction form of bound-va... |
| sbal 2174 | Move universal quantifier ... |
| sbalv 2175 | Quantify with new variable... |
| hbsbw 2176 | If ` z ` is not free in ` ... |
| hbsbwOLD 2177 | Obsolete version of ~ hbsb... |
| sbcom2 2178 | Commutativity law for subs... |
| sbco4lemOLD 2179 | Obsolete version of ~ sbco... |
| sbco4OLD 2180 | Obsolete version of ~ sbco... |
| nfa2 2181 | Lemma 24 of [Monk2] p. 114... |
| nfexhe 2182 | Version of ~ nfex with the... |
| nfexa2 2183 | An inner universal quantif... |
| ax12v 2185 | This is essentially Axiom ... |
| ax12v2 2186 | It is possible to remove a... |
| ax12ev2 2187 | Version of ~ ax12v2 rewrit... |
| 19.8a 2188 | If a wff is true, it is tr... |
| 19.8ad 2189 | If a wff is true, it is tr... |
| sp 2190 | Specialization. A univers... |
| spi 2191 | Inference rule of universa... |
| sps 2192 | Generalization of antecede... |
| 2sp 2193 | A double specialization (s... |
| spsd 2194 | Deduction generalizing ant... |
| 19.2g 2195 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2196 | Inference form of ~ 19.21 ... |
| 19.21bbi 2197 | Inference removing two uni... |
| 19.23bi 2198 | Inference form of Theorem ... |
| nexr 2199 | Inference associated with ... |
| qexmid 2200 | Quantified excluded middle... |
| nf5r 2201 | Consequence of the definit... |
| nf5ri 2202 | Consequence of the definit... |
| nf5rd 2203 | Consequence of the definit... |
| spimedv 2204 | Deduction version of ~ spi... |
| spimefv 2205 | Version of ~ spime with a ... |
| nfim1 2206 | A closed form of ~ nfim . ... |
| nfan1 2207 | A closed form of ~ nfan . ... |
| 19.3t 2208 | Closed form of ~ 19.3 and ... |
| 19.3 2209 | A wff may be quantified wi... |
| 19.9d 2210 | A deduction version of one... |
| 19.9t 2211 | Closed form of ~ 19.9 and ... |
| 19.9 2212 | A wff may be existentially... |
| 19.21t 2213 | Closed form of Theorem 19.... |
| 19.21 2214 | Theorem 19.21 of [Margaris... |
| stdpc5 2215 | An axiom scheme of standar... |
| 19.21-2 2216 | Version of ~ 19.21 with tw... |
| 19.23t 2217 | Closed form of Theorem 19.... |
| 19.23 2218 | Theorem 19.23 of [Margaris... |
| alimd 2219 | Deduction form of Theorem ... |
| alrimi 2220 | Inference form of Theorem ... |
| alrimdd 2221 | Deduction form of Theorem ... |
| alrimd 2222 | Deduction form of Theorem ... |
| eximd 2223 | Deduction form of Theorem ... |
| exlimi 2224 | Inference associated with ... |
| exlimd 2225 | Deduction form of Theorem ... |
| exlimimdd 2226 | Existential elimination ru... |
| exlimdd 2227 | Existential elimination ru... |
| nexd 2228 | Deduction for generalizati... |
| albid 2229 | Formula-building rule for ... |
| exbid 2230 | Formula-building rule for ... |
| nfbidf 2231 | An equality theorem for ef... |
| 19.16 2232 | Theorem 19.16 of [Margaris... |
| 19.17 2233 | Theorem 19.17 of [Margaris... |
| 19.27 2234 | Theorem 19.27 of [Margaris... |
| 19.28 2235 | Theorem 19.28 of [Margaris... |
| 19.19 2236 | Theorem 19.19 of [Margaris... |
| 19.36 2237 | Theorem 19.36 of [Margaris... |
| 19.36i 2238 | Inference associated with ... |
| 19.37 2239 | Theorem 19.37 of [Margaris... |
| 19.32 2240 | Theorem 19.32 of [Margaris... |
| 19.31 2241 | Theorem 19.31 of [Margaris... |
| 19.41 2242 | Theorem 19.41 of [Margaris... |
| 19.42 2243 | Theorem 19.42 of [Margaris... |
| 19.44 2244 | Theorem 19.44 of [Margaris... |
| 19.45 2245 | Theorem 19.45 of [Margaris... |
| spimfv 2246 | Specialization, using impl... |
| chvarfv 2247 | Implicit substitution of `... |
| cbv3v2 2248 | Version of ~ cbv3 with two... |
| sbalex 2249 | Equivalence of two ways to... |
| sbalexOLD 2250 | Obsolete version of ~ sbal... |
| sb4av 2251 | Version of ~ sb4a with a d... |
| sbimd 2252 | Deduction substituting bot... |
| sbbid 2253 | Deduction substituting bot... |
| 2sbbid 2254 | Deduction doubly substitut... |
| sbequ1 2255 | An equality theorem for su... |
| sbequ2 2256 | An equality theorem for su... |
| stdpc7 2257 | One of the two equality ax... |
| sbequ12 2258 | An equality theorem for su... |
| sbequ12r 2259 | An equality theorem for su... |
| sbelx 2260 | Elimination of substitutio... |
| sbequ12a 2261 | An equality theorem for su... |
| sbid 2262 | An identity theorem for su... |
| sbcov 2263 | A composition law for subs... |
| sbcovOLD 2264 | Obsolete version of ~ sbco... |
| sb6a 2265 | Equivalence for substituti... |
| sbid2vw 2266 | Reverting substitution yie... |
| axc16g 2267 | Generalization of ~ axc16 ... |
| axc16 2268 | Proof of older axiom ~ ax-... |
| axc16gb 2269 | Biconditional strengthenin... |
| axc16nf 2270 | If ~ dtru is false, then t... |
| axc11v 2271 | Version of ~ axc11 with a ... |
| axc11rv 2272 | Version of ~ axc11r with a... |
| drsb2 2273 | Formula-building lemma for... |
| equsalv 2274 | An equivalence related to ... |
| equsexv 2275 | An equivalence related to ... |
| sbft 2276 | Substitution has no effect... |
| sbf 2277 | Substitution for a variabl... |
| sbf2 2278 | Substitution has no effect... |
| sbh 2279 | Substitution for a variabl... |
| hbs1 2280 | The setvar ` x ` is not fr... |
| nfs1f 2281 | If ` x ` is not free in ` ... |
| sb5 2282 | Alternate definition of su... |
| equs5av 2283 | A property related to subs... |
| 2sb5 2284 | Equivalence for double sub... |
| dfsb7 2285 | An alternate definition of... |
| sbn 2286 | Negation inside and outsid... |
| sbex 2287 | Move existential quantifie... |
| nf5 2288 | Alternate definition of ~ ... |
| nf6 2289 | An alternate definition of... |
| nf5d 2290 | Deduce that ` x ` is not f... |
| nf5di 2291 | Since the converse holds b... |
| 19.9h 2292 | A wff may be existentially... |
| 19.21h 2293 | Theorem 19.21 of [Margaris... |
| 19.23h 2294 | Theorem 19.23 of [Margaris... |
| exlimih 2295 | Inference associated with ... |
| exlimdh 2296 | Deduction form of Theorem ... |
| equsalhw 2297 | Version of ~ equsalh with ... |
| equsexhv 2298 | An equivalence related to ... |
| hba1 2299 | The setvar ` x ` is not fr... |
| hbnt 2300 | Closed theorem version of ... |
| hbn 2301 | If ` x ` is not free in ` ... |
| hbnd 2302 | Deduction form of bound-va... |
| hbim1 2303 | A closed form of ~ hbim . ... |
| hbimd 2304 | Deduction form of bound-va... |
| hbim 2305 | If ` x ` is not free in ` ... |
| hban 2306 | If ` x ` is not free in ` ... |
| hb3an 2307 | If ` x ` is not free in ` ... |
| sbi2 2308 | Introduction of implicatio... |
| sbim 2309 | Implication inside and out... |
| sbrim 2310 | Substitution in an implica... |
| sblim 2311 | Substitution in an implica... |
| sbor 2312 | Disjunction inside and out... |
| sbbi 2313 | Equivalence inside and out... |
| sblbis 2314 | Introduce left bicondition... |
| sbrbis 2315 | Introduce right biconditio... |
| sbrbif 2316 | Introduce right biconditio... |
| sbnf 2317 | Move nonfree predicate in ... |
| sbnfOLD 2318 | Obsolete version of ~ sbnf... |
| sbiev 2319 | Conversion of implicit sub... |
| sbievOLD 2320 | Obsolete version of ~ sbie... |
| sbiedw 2321 | Conversion of implicit sub... |
| axc7 2322 | Show that the original axi... |
| axc7e 2323 | Abbreviated version of ~ a... |
| modal-b 2324 | The analogue in our predic... |
| 19.9ht 2325 | A closed version of ~ 19.9... |
| axc4 2326 | Show that the original axi... |
| axc4i 2327 | Inference version of ~ axc... |
| nfal 2328 | If ` x ` is not free in ` ... |
| nfex 2329 | If ` x ` is not free in ` ... |
| hbex 2330 | If ` x ` is not free in ` ... |
| nfnf 2331 | If ` x ` is not free in ` ... |
| 19.12 2332 | Theorem 19.12 of [Margaris... |
| nfald 2333 | Deduction form of ~ nfal .... |
| nfexd 2334 | If ` x ` is not free in ` ... |
| nfsbv 2335 | If ` z ` is not free in ` ... |
| sbco2v 2336 | A composition law for subs... |
| aaan 2337 | Distribute universal quant... |
| eeor 2338 | Distribute existential qua... |
| cbv3v 2339 | Rule used to change bound ... |
| cbv1v 2340 | Rule used to change bound ... |
| cbv2w 2341 | Rule used to change bound ... |
| cbvaldw 2342 | Deduction used to change b... |
| cbvexdw 2343 | Deduction used to change b... |
| cbv3hv 2344 | Rule used to change bound ... |
| cbvalv1 2345 | Rule used to change bound ... |
| cbvexv1 2346 | Rule used to change bound ... |
| cbval2v 2347 | Rule used to change bound ... |
| cbvex2v 2348 | Rule used to change bound ... |
| dvelimhw 2349 | Proof of ~ dvelimh without... |
| pm11.53 2350 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2351 | Special case of ~ 19.12 wh... |
| eean 2352 | Distribute existential qua... |
| eeanv 2353 | Distribute a pair of exist... |
| eeeanv 2354 | Distribute three existenti... |
| ee4anv 2355 | Distribute two pairs of ex... |
| ee4anvOLD 2356 | Obsolete version of ~ ee4a... |
| sb8v 2357 | Substitution of variable i... |
| sb8f 2358 | Substitution of variable i... |
| sb8ef 2359 | Substitution of variable i... |
| 2sb8ef 2360 | An equivalent expression f... |
| sb6rfv 2361 | Reversed substitution. Ve... |
| sbnf2 2362 | Two ways of expressing " `... |
| exsb 2363 | An equivalent expression f... |
| 2exsb 2364 | An equivalent expression f... |
| sbbib 2365 | Reversal of substitution. ... |
| sbbibvv 2366 | Reversal of substitution. ... |
| cbvsbvf 2367 | Change the bound variable ... |
| cleljustALT 2368 | Alternate proof of ~ clelj... |
| cleljustALT2 2369 | Alternate proof of ~ clelj... |
| equs5aALT 2370 | Alternate proof of ~ equs5... |
| equs5eALT 2371 | Alternate proof of ~ equs5... |
| axc11r 2372 | Same as ~ axc11 but with r... |
| dral1v 2373 | Formula-building lemma for... |
| drex1v 2374 | Formula-building lemma for... |
| drnf1v 2375 | Formula-building lemma for... |
| ax13v 2377 | A weaker version of ~ ax-1... |
| ax13lem1 2378 | A version of ~ ax13v with ... |
| ax13 2379 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2380 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2381 | An equation between setvar... |
| dveeq2 2382 | Quantifier introduction wh... |
| nfeqf1 2383 | An equation between setvar... |
| dveeq1 2384 | Quantifier introduction wh... |
| nfeqf 2385 | A variable is effectively ... |
| axc9 2386 | Derive set.mm's original ~... |
| ax6e 2387 | At least one individual ex... |
| ax6 2388 | Theorem showing that ~ ax-... |
| axc10 2389 | Show that the original axi... |
| spimt 2390 | Closed theorem form of ~ s... |
| spim 2391 | Specialization, using impl... |
| spimed 2392 | Deduction version of ~ spi... |
| spime 2393 | Existential introduction, ... |
| spimv 2394 | A version of ~ spim with a... |
| spimvALT 2395 | Alternate proof of ~ spimv... |
| spimev 2396 | Distinct-variable version ... |
| spv 2397 | Specialization, using impl... |
| spei 2398 | Inference from existential... |
| chvar 2399 | Implicit substitution of `... |
| chvarv 2400 | Implicit substitution of `... |
| cbv3 2401 | Rule used to change bound ... |
| cbval 2402 | Rule used to change bound ... |
| cbvex 2403 | Rule used to change bound ... |
| cbvalv 2404 | Rule used to change bound ... |
| cbvexv 2405 | Rule used to change bound ... |
| cbv1 2406 | Rule used to change bound ... |
| cbv2 2407 | Rule used to change bound ... |
| cbv3h 2408 | Rule used to change bound ... |
| cbv1h 2409 | Rule used to change bound ... |
| cbv2h 2410 | Rule used to change bound ... |
| cbvald 2411 | Deduction used to change b... |
| cbvexd 2412 | Deduction used to change b... |
| cbvaldva 2413 | Rule used to change the bo... |
| cbvexdva 2414 | Rule used to change the bo... |
| cbval2 2415 | Rule used to change bound ... |
| cbvex2 2416 | Rule used to change bound ... |
| cbval2vv 2417 | Rule used to change bound ... |
| cbvex2vv 2418 | Rule used to change bound ... |
| cbvex4v 2419 | Rule used to change bound ... |
| equs4 2420 | Lemma used in proofs of im... |
| equsal 2421 | An equivalence related to ... |
| equsex 2422 | An equivalence related to ... |
| equsexALT 2423 | Alternate proof of ~ equse... |
| equsalh 2424 | An equivalence related to ... |
| equsexh 2425 | An equivalence related to ... |
| axc15 2426 | Derivation of set.mm's ori... |
| ax12 2427 | Rederivation of Axiom ~ ax... |
| ax12b 2428 | A bidirectional version of... |
| ax13ALT 2429 | Alternate proof of ~ ax13 ... |
| axc11n 2430 | Derive set.mm's original ~... |
| aecom 2431 | Commutation law for identi... |
| aecoms 2432 | A commutation rule for ide... |
| naecoms 2433 | A commutation rule for dis... |
| axc11 2434 | Show that ~ ax-c11 can be ... |
| hbae 2435 | All variables are effectiv... |
| hbnae 2436 | All variables are effectiv... |
| nfae 2437 | All variables are effectiv... |
| nfnae 2438 | All variables are effectiv... |
| hbnaes 2439 | Rule that applies ~ hbnae ... |
| axc16i 2440 | Inference with ~ axc16 as ... |
| axc16nfALT 2441 | Alternate proof of ~ axc16... |
| dral2 2442 | Formula-building lemma for... |
| dral1 2443 | Formula-building lemma for... |
| dral1ALT 2444 | Alternate proof of ~ dral1... |
| drex1 2445 | Formula-building lemma for... |
| drex2 2446 | Formula-building lemma for... |
| drnf1 2447 | Formula-building lemma for... |
| drnf2 2448 | Formula-building lemma for... |
| nfald2 2449 | Variation on ~ nfald which... |
| nfexd2 2450 | Variation on ~ nfexd which... |
| exdistrf 2451 | Distribution of existentia... |
| dvelimf 2452 | Version of ~ dvelimv witho... |
| dvelimdf 2453 | Deduction form of ~ dvelim... |
| dvelimh 2454 | Version of ~ dvelim withou... |
| dvelim 2455 | This theorem can be used t... |
| dvelimv 2456 | Similar to ~ dvelim with f... |
| dvelimnf 2457 | Version of ~ dvelim using ... |
| dveeq2ALT 2458 | Alternate proof of ~ dveeq... |
| equvini 2459 | A variable introduction la... |
| equvel 2460 | A variable elimination law... |
| equs5a 2461 | A property related to subs... |
| equs5e 2462 | A property related to subs... |
| equs45f 2463 | Two ways of expressing sub... |
| equs5 2464 | Lemma used in proofs of su... |
| dveel1 2465 | Quantifier introduction wh... |
| dveel2 2466 | Quantifier introduction wh... |
| axc14 2467 | Axiom ~ ax-c14 is redundan... |
| sb6x 2468 | Equivalence involving subs... |
| sbequ5 2469 | Substitution does not chan... |
| sbequ6 2470 | Substitution does not chan... |
| sb5rf 2471 | Reversed substitution. Us... |
| sb6rf 2472 | Reversed substitution. Fo... |
| ax12vALT 2473 | Alternate proof of ~ ax12v... |
| 2ax6elem 2474 | We can always find values ... |
| 2ax6e 2475 | We can always find values ... |
| 2sb5rf 2476 | Reversed double substituti... |
| 2sb6rf 2477 | Reversed double substituti... |
| sbel2x 2478 | Elimination of double subs... |
| sb4b 2479 | Simplified definition of s... |
| sb3b 2480 | Simplified definition of s... |
| sb3 2481 | One direction of a simplif... |
| sb1 2482 | One direction of a simplif... |
| sb2 2483 | One direction of a simplif... |
| sb4a 2484 | A version of one implicati... |
| dfsb1 2485 | Alternate definition of su... |
| hbsb2 2486 | Bound-variable hypothesis ... |
| nfsb2 2487 | Bound-variable hypothesis ... |
| hbsb2a 2488 | Special case of a bound-va... |
| sb4e 2489 | One direction of a simplif... |
| hbsb2e 2490 | Special case of a bound-va... |
| hbsb3 2491 | If ` y ` is not free in ` ... |
| nfs1 2492 | If ` y ` is not free in ` ... |
| axc16ALT 2493 | Alternate proof of ~ axc16... |
| axc16gALT 2494 | Alternate proof of ~ axc16... |
| equsb1 2495 | Substitution applied to an... |
| equsb2 2496 | Substitution applied to an... |
| dfsb2 2497 | An alternate definition of... |
| dfsb3 2498 | An alternate definition of... |
| drsb1 2499 | Formula-building lemma for... |
| sb2ae 2500 | In the case of two success... |
| sb6f 2501 | Equivalence for substituti... |
| sb5f 2502 | Equivalence for substituti... |
| nfsb4t 2503 | A variable not free in a p... |
| nfsb4 2504 | A variable not free in a p... |
| sbequ8 2505 | Elimination of equality fr... |
| sbie 2506 | Conversion of implicit sub... |
| sbied 2507 | Conversion of implicit sub... |
| sbiedv 2508 | Conversion of implicit sub... |
| 2sbiev 2509 | Conversion of double impli... |
| sbcom3 2510 | Substituting ` y ` for ` x... |
| sbco 2511 | A composition law for subs... |
| sbid2 2512 | An identity law for substi... |
| sbid2v 2513 | An identity law for substi... |
| sbidm 2514 | An idempotent law for subs... |
| sbco2 2515 | A composition law for subs... |
| sbco2d 2516 | A composition law for subs... |
| sbco3 2517 | A composition law for subs... |
| sbcom 2518 | A commutativity law for su... |
| sbtrt 2519 | Partially closed form of ~... |
| sbtr 2520 | A partial converse to ~ sb... |
| sb8 2521 | Substitution of variable i... |
| sb8e 2522 | Substitution of variable i... |
| sb9 2523 | Commutation of quantificat... |
| sb9i 2524 | Commutation of quantificat... |
| sbhb 2525 | Two ways of expressing " `... |
| nfsbd 2526 | Deduction version of ~ nfs... |
| nfsb 2527 | If ` z ` is not free in ` ... |
| hbsb 2528 | If ` z ` is not free in ` ... |
| sb7f 2529 | This version of ~ dfsb7 do... |
| sb7h 2530 | This version of ~ dfsb7 do... |
| sb10f 2531 | Hao Wang's identity axiom ... |
| sbal1 2532 | Check out ~ sbal for a ver... |
| sbal2 2533 | Move quantifier in and out... |
| 2sb8e 2534 | An equivalent expression f... |
| dfmoeu 2535 | An elementary proof of ~ m... |
| dfeumo 2536 | An elementary proof showin... |
| mojust 2538 | Soundness justification th... |
| dfmo 2540 | Simplify definition ~ df-m... |
| nexmo 2541 | Nonexistence implies uniqu... |
| exmo 2542 | Any proposition holds for ... |
| moabs 2543 | Absorption of existence co... |
| moim 2544 | The at-most-one quantifier... |
| moimi 2545 | The at-most-one quantifier... |
| moimdv 2546 | The at-most-one quantifier... |
| mobi 2547 | Equivalence theorem for th... |
| mobii 2548 | Formula-building rule for ... |
| mobidv 2549 | Formula-building rule for ... |
| mobid 2550 | Formula-building rule for ... |
| moa1 2551 | If an implication holds fo... |
| moan 2552 | "At most one" is still the... |
| moani 2553 | "At most one" is still tru... |
| moor 2554 | "At most one" is still the... |
| mooran1 2555 | "At most one" imports disj... |
| mooran2 2556 | "At most one" exports disj... |
| nfmo1 2557 | Bound-variable hypothesis ... |
| nfmod2 2558 | Bound-variable hypothesis ... |
| nfmodv 2559 | Bound-variable hypothesis ... |
| nfmov 2560 | Bound-variable hypothesis ... |
| nfmod 2561 | Bound-variable hypothesis ... |
| nfmo 2562 | Bound-variable hypothesis ... |
| mof 2563 | Version of ~ df-mo with di... |
| mo3 2564 | Alternate definition of th... |
| mo 2565 | Equivalent definitions of ... |
| mo4 2566 | At-most-one quantifier exp... |
| mo4f 2567 | At-most-one quantifier exp... |
| eu3v 2570 | An alternate way to expres... |
| eujust 2571 | Soundness justification th... |
| eujustALT 2572 | Alternate proof of ~ eujus... |
| eu6lem 2573 | Lemma of ~ eu6im . A diss... |
| eu6 2574 | Alternate definition of th... |
| eu6im 2575 | One direction of ~ eu6 nee... |
| euf 2576 | Version of ~ eu6 with disj... |
| euex 2577 | Existential uniqueness imp... |
| eumo 2578 | Existential uniqueness imp... |
| eumoi 2579 | Uniqueness inferred from e... |
| exmoeub 2580 | Existence implies that uni... |
| exmoeu 2581 | Existence is equivalent to... |
| moeuex 2582 | Uniqueness implies that ex... |
| moeu 2583 | Uniqueness is equivalent t... |
| eubi 2584 | Equivalence theorem for th... |
| eubii 2585 | Introduce unique existenti... |
| eubidv 2586 | Formula-building rule for ... |
| eubid 2587 | Formula-building rule for ... |
| nfeu1ALT 2588 | Alternate version of ~ nfe... |
| nfeu1 2589 | Bound-variable hypothesis ... |
| nfeud2 2590 | Bound-variable hypothesis ... |
| nfeudw 2591 | Bound-variable hypothesis ... |
| nfeud 2592 | Bound-variable hypothesis ... |
| nfeuw 2593 | Bound-variable hypothesis ... |
| nfeu 2594 | Bound-variable hypothesis ... |
| dfeu 2595 | Rederive ~ df-eu from the ... |
| dfmo2 2596 | Rederive ~ df-mo from the ... |
| euequ 2597 | There exists a unique set ... |
| sb8eulem 2598 | Lemma. Factor out the com... |
| sb8euv 2599 | Variable substitution in u... |
| sb8eu 2600 | Variable substitution in u... |
| sb8mo 2601 | Variable substitution for ... |
| cbvmovw 2602 | Change bound variable. Us... |
| cbvmow 2603 | Rule used to change bound ... |
| cbvmo 2604 | Rule used to change bound ... |
| cbveuvw 2605 | Change bound variable. Us... |
| cbveuw 2606 | Version of ~ cbveu with a ... |
| cbveu 2607 | Rule used to change bound ... |
| cbveuALT 2608 | Alternative proof of ~ cbv... |
| eu2 2609 | An alternate way of defini... |
| eu1 2610 | An alternate way to expres... |
| euor 2611 | Introduce a disjunct into ... |
| euorv 2612 | Introduce a disjunct into ... |
| euor2 2613 | Introduce or eliminate a d... |
| sbmo 2614 | Substitution into an at-mo... |
| eu4 2615 | Uniqueness using implicit ... |
| euimmo 2616 | Existential uniqueness imp... |
| euim 2617 | Add unique existential qua... |
| moanimlem 2618 | Factor out the common proo... |
| moanimv 2619 | Introduction of a conjunct... |
| moanim 2620 | Introduction of a conjunct... |
| euan 2621 | Introduction of a conjunct... |
| moanmo 2622 | Nested at-most-one quantif... |
| moaneu 2623 | Nested at-most-one and uni... |
| euanv 2624 | Introduction of a conjunct... |
| mopick 2625 | "At most one" picks a vari... |
| moexexlem 2626 | Factor out the proof skele... |
| 2moexv 2627 | Double quantification with... |
| moexexvw 2628 | "At most one" double quant... |
| 2moswapv 2629 | A condition allowing to sw... |
| 2euswapv 2630 | A condition allowing to sw... |
| 2euexv 2631 | Double quantification with... |
| 2exeuv 2632 | Double existential uniquen... |
| eupick 2633 | Existential uniqueness "pi... |
| eupicka 2634 | Version of ~ eupick with c... |
| eupickb 2635 | Existential uniqueness "pi... |
| eupickbi 2636 | Theorem *14.26 in [Whitehe... |
| mopick2 2637 | "At most one" can show the... |
| moexex 2638 | "At most one" double quant... |
| moexexv 2639 | "At most one" double quant... |
| 2moex 2640 | Double quantification with... |
| 2euex 2641 | Double quantification with... |
| 2eumo 2642 | Nested unique existential ... |
| 2eu2ex 2643 | Double existential uniquen... |
| 2moswap 2644 | A condition allowing to sw... |
| 2euswap 2645 | A condition allowing to sw... |
| 2exeu 2646 | Double existential uniquen... |
| 2mo2 2647 | Two ways of expressing "th... |
| 2mo 2648 | Two ways of expressing "th... |
| 2mos 2649 | Double "there exists at mo... |
| 2mosOLD 2650 | Obsolete version of ~ 2mos... |
| 2eu1 2651 | Double existential uniquen... |
| 2eu1v 2652 | Double existential uniquen... |
| 2eu2 2653 | Double existential uniquen... |
| 2eu3 2654 | Double existential uniquen... |
| 2eu4 2655 | This theorem provides us w... |
| 2eu5 2656 | An alternate definition of... |
| 2eu6 2657 | Two equivalent expressions... |
| 2eu7 2658 | Two equivalent expressions... |
| 2eu8 2659 | Two equivalent expressions... |
| euae 2660 | Two ways to express "exact... |
| exists1 2661 | Two ways to express "exact... |
| exists2 2662 | A condition implying that ... |
| barbara 2663 | "Barbara", one of the fund... |
| celarent 2664 | "Celarent", one of the syl... |
| darii 2665 | "Darii", one of the syllog... |
| dariiALT 2666 | Alternate proof of ~ darii... |
| ferio 2667 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2668 | Lemma for ~ barbari and th... |
| barbari 2669 | "Barbari", one of the syll... |
| barbariALT 2670 | Alternate proof of ~ barba... |
| celaront 2671 | "Celaront", one of the syl... |
| cesare 2672 | "Cesare", one of the syllo... |
| camestres 2673 | "Camestres", one of the sy... |
| festino 2674 | "Festino", one of the syll... |
| festinoALT 2675 | Alternate proof of ~ festi... |
| baroco 2676 | "Baroco", one of the syllo... |
| barocoALT 2677 | Alternate proof of ~ festi... |
| cesaro 2678 | "Cesaro", one of the syllo... |
| camestros 2679 | "Camestros", one of the sy... |
| datisi 2680 | "Datisi", one of the syllo... |
| disamis 2681 | "Disamis", one of the syll... |
| ferison 2682 | "Ferison", one of the syll... |
| bocardo 2683 | "Bocardo", one of the syll... |
| darapti 2684 | "Darapti", one of the syll... |
| daraptiALT 2685 | Alternate proof of ~ darap... |
| felapton 2686 | "Felapton", one of the syl... |
| calemes 2687 | "Calemes", one of the syll... |
| dimatis 2688 | "Dimatis", one of the syll... |
| fresison 2689 | "Fresison", one of the syl... |
| calemos 2690 | "Calemos", one of the syll... |
| fesapo 2691 | "Fesapo", one of the syllo... |
| bamalip 2692 | "Bamalip", one of the syll... |
| axia1 2693 | Left 'and' elimination (in... |
| axia2 2694 | Right 'and' elimination (i... |
| axia3 2695 | 'And' introduction (intuit... |
| axin1 2696 | 'Not' introduction (intuit... |
| axin2 2697 | 'Not' elimination (intuiti... |
| axio 2698 | Definition of 'or' (intuit... |
| axi4 2699 | Specialization (intuitioni... |
| axi5r 2700 | Converse of ~ axc4 (intuit... |
| axial 2701 | The setvar ` x ` is not fr... |
| axie1 2702 | The setvar ` x ` is not fr... |
| axie2 2703 | A key property of existent... |
| axi9 2704 | Axiom of existence (intuit... |
| axi10 2705 | Axiom of Quantifier Substi... |
| axi12 2706 | Axiom of Quantifier Introd... |
| axbnd 2707 | Axiom of Bundling (intuiti... |
| axexte 2709 | The axiom of extensionalit... |
| axextg 2710 | A generalization of the ax... |
| axextb 2711 | A bidirectional version of... |
| axextmo 2712 | There exists at most one s... |
| nulmo 2713 | There exists at most one e... |
| eleq1ab 2716 | Extension (in the sense of... |
| cleljustab 2717 | Extension of ~ cleljust fr... |
| abid 2718 | Simplification of class ab... |
| vexwt 2719 | A standard theorem of pred... |
| vexw 2720 | If ` ph ` is a theorem, th... |
| vextru 2721 | Every setvar is a member o... |
| nfsab1 2722 | Bound-variable hypothesis ... |
| hbab1 2723 | Bound-variable hypothesis ... |
| hbab 2724 | Bound-variable hypothesis ... |
| hbabg 2725 | Bound-variable hypothesis ... |
| nfsab 2726 | Bound-variable hypothesis ... |
| nfsabg 2727 | Bound-variable hypothesis ... |
| dfcleq 2729 | The defining characterizat... |
| cvjust 2730 | Every set is a class. Pro... |
| ax9ALT 2731 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2732 | A weaker version of ~ eleq... |
| eqriv 2733 | Infer equality of classes ... |
| eqrdv 2734 | Deduce equality of classes... |
| eqrdav 2735 | Deduce equality of classes... |
| eqid 2736 | Law of identity (reflexivi... |
| eqidd 2737 | Class identity law with an... |
| eqeq1d 2738 | Deduction from equality to... |
| eqeq1dALT 2739 | Alternate proof of ~ eqeq1... |
| eqeq1 2740 | Equality implies equivalen... |
| eqeq1i 2741 | Inference from equality to... |
| eqcomd 2742 | Deduction from commutative... |
| eqcom 2743 | Commutative law for class ... |
| eqcoms 2744 | Inference applying commuta... |
| eqcomi 2745 | Inference from commutative... |
| neqcomd 2746 | Commute an inequality. (C... |
| eqeq2d 2747 | Deduction from equality to... |
| eqeq2 2748 | Equality implies equivalen... |
| eqeq2i 2749 | Inference from equality to... |
| eqeqan12d 2750 | A useful inference for sub... |
| eqeqan12rd 2751 | A useful inference for sub... |
| eqeq12d 2752 | A useful inference for sub... |
| eqeq12 2753 | Equality relationship amon... |
| eqeq12i 2754 | A useful inference for sub... |
| eqeqan12dALT 2755 | Alternate proof of ~ eqeqa... |
| eqtr 2756 | Transitive law for class e... |
| eqtr2 2757 | A transitive law for class... |
| eqtr3 2758 | A transitive law for class... |
| eqtri 2759 | An equality transitivity i... |
| eqtr2i 2760 | An equality transitivity i... |
| eqtr3i 2761 | An equality transitivity i... |
| eqtr4i 2762 | An equality transitivity i... |
| 3eqtri 2763 | An inference from three ch... |
| 3eqtrri 2764 | An inference from three ch... |
| 3eqtr2i 2765 | An inference from three ch... |
| 3eqtr2ri 2766 | An inference from three ch... |
| 3eqtr3i 2767 | An inference from three ch... |
| 3eqtr3ri 2768 | An inference from three ch... |
| 3eqtr4i 2769 | An inference from three ch... |
| 3eqtr4ri 2770 | An inference from three ch... |
| eqtrd 2771 | An equality transitivity d... |
| eqtr2d 2772 | An equality transitivity d... |
| eqtr3d 2773 | An equality transitivity e... |
| eqtr4d 2774 | An equality transitivity e... |
| 3eqtrd 2775 | A deduction from three cha... |
| 3eqtrrd 2776 | A deduction from three cha... |
| 3eqtr2d 2777 | A deduction from three cha... |
| 3eqtr2rd 2778 | A deduction from three cha... |
| 3eqtr3d 2779 | A deduction from three cha... |
| 3eqtr3rd 2780 | A deduction from three cha... |
| 3eqtr4d 2781 | A deduction from three cha... |
| 3eqtr4rd 2782 | A deduction from three cha... |
| eqtrid 2783 | An equality transitivity d... |
| eqtr2id 2784 | An equality transitivity d... |
| eqtr3id 2785 | An equality transitivity d... |
| eqtr3di 2786 | An equality transitivity d... |
| eqtrdi 2787 | An equality transitivity d... |
| eqtr2di 2788 | An equality transitivity d... |
| eqtr4di 2789 | An equality transitivity d... |
| eqtr4id 2790 | An equality transitivity d... |
| sylan9eq 2791 | An equality transitivity d... |
| sylan9req 2792 | An equality transitivity d... |
| sylan9eqr 2793 | An equality transitivity d... |
| 3eqtr3g 2794 | A chained equality inferen... |
| 3eqtr3a 2795 | A chained equality inferen... |
| 3eqtr4g 2796 | A chained equality inferen... |
| 3eqtr4a 2797 | A chained equality inferen... |
| eq2tri 2798 | A compound transitive infe... |
| iseqsetvlem 2799 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2800 | Alternate proof of ~ iseqs... |
| abbi 2801 | Equivalent formulas yield ... |
| abbidv 2802 | Equivalent wff's yield equ... |
| abbii 2803 | Equivalent wff's yield equ... |
| abbid 2804 | Equivalent wff's yield equ... |
| abbib 2805 | Equal class abstractions r... |
| cbvabv 2806 | Rule used to change bound ... |
| cbvabw 2807 | Rule used to change bound ... |
| cbvab 2808 | Rule used to change bound ... |
| eqabbw 2809 | Version of ~ eqabb using i... |
| eqabcbw 2810 | Version of ~ eqabcb using ... |
| dfclel 2812 | Characterization of the el... |
| elex2 2813 | If a class contains anothe... |
| issettru 2814 | Weak version of ~ isset . ... |
| iseqsetv-clel 2815 | Alternate proof of ~ iseqs... |
| issetlem 2816 | Lemma for ~ elisset and ~ ... |
| elissetv 2817 | An element of a class exis... |
| elisset 2818 | An element of a class exis... |
| eleq1w 2819 | Weaker version of ~ eleq1 ... |
| eleq2w 2820 | Weaker version of ~ eleq2 ... |
| eleq1d 2821 | Deduction from equality to... |
| eleq2d 2822 | Deduction from equality to... |
| eleq2dALT 2823 | Alternate proof of ~ eleq2... |
| eleq1 2824 | Equality implies equivalen... |
| eleq2 2825 | Equality implies equivalen... |
| eleq12 2826 | Equality implies equivalen... |
| eleq1i 2827 | Inference from equality to... |
| eleq2i 2828 | Inference from equality to... |
| eleq12i 2829 | Inference from equality to... |
| eleq12d 2830 | Deduction from equality to... |
| eleq1a 2831 | A transitive-type law rela... |
| eqeltri 2832 | Substitution of equal clas... |
| eqeltrri 2833 | Substitution of equal clas... |
| eleqtri 2834 | Substitution of equal clas... |
| eleqtrri 2835 | Substitution of equal clas... |
| eqeltrd 2836 | Substitution of equal clas... |
| eqeltrrd 2837 | Deduction that substitutes... |
| eleqtrd 2838 | Deduction that substitutes... |
| eleqtrrd 2839 | Deduction that substitutes... |
| eqeltrid 2840 | A membership and equality ... |
| eqeltrrid 2841 | A membership and equality ... |
| eleqtrid 2842 | A membership and equality ... |
| eleqtrrid 2843 | A membership and equality ... |
| eqeltrdi 2844 | A membership and equality ... |
| eqeltrrdi 2845 | A membership and equality ... |
| eleqtrdi 2846 | A membership and equality ... |
| eleqtrrdi 2847 | A membership and equality ... |
| 3eltr3i 2848 | Substitution of equal clas... |
| 3eltr4i 2849 | Substitution of equal clas... |
| 3eltr3d 2850 | Substitution of equal clas... |
| 3eltr4d 2851 | Substitution of equal clas... |
| 3eltr3g 2852 | Substitution of equal clas... |
| 3eltr4g 2853 | Substitution of equal clas... |
| eleq2s 2854 | Substitution of equal clas... |
| eqneltri 2855 | If a class is not an eleme... |
| eqneltrd 2856 | If a class is not an eleme... |
| eqneltrrd 2857 | If a class is not an eleme... |
| neleqtrd 2858 | If a class is not an eleme... |
| neleqtrrd 2859 | If a class is not an eleme... |
| nelneq 2860 | A way of showing two class... |
| nelneq2 2861 | A way of showing two class... |
| eqsb1 2862 | Substitution for the left-... |
| clelsb1 2863 | Substitution for the first... |
| clelsb2 2864 | Substitution for the secon... |
| cleqh 2865 | Establish equality between... |
| hbxfreq 2866 | A utility lemma to transfe... |
| hblem 2867 | Change the free variable o... |
| hblemg 2868 | Change the free variable o... |
| eqabdv 2869 | Deduction from a wff to a ... |
| eqabcdv 2870 | Deduction from a wff to a ... |
| eqabi 2871 | Equality of a class variab... |
| abid1 2872 | Every class is equal to a ... |
| abid2 2873 | A simplification of class ... |
| eqab 2874 | One direction of ~ eqabb i... |
| eqabb 2875 | Equality of a class variab... |
| eqabcb 2876 | Equality of a class variab... |
| eqabrd 2877 | Equality of a class variab... |
| eqabri 2878 | Equality of a class variab... |
| eqabcri 2879 | Equality of a class variab... |
| clelab 2880 | Membership of a class vari... |
| clabel 2881 | Membership of a class abst... |
| sbab 2882 | The right-hand side of the... |
| nfcjust 2884 | Justification theorem for ... |
| nfci 2886 | Deduce that a class ` A ` ... |
| nfcii 2887 | Deduce that a class ` A ` ... |
| nfcr 2888 | Consequence of the not-fre... |
| nfcrALT 2889 | Alternate version of ~ nfc... |
| nfcri 2890 | Consequence of the not-fre... |
| nfcd 2891 | Deduce that a class ` A ` ... |
| nfcrd 2892 | Consequence of the not-fre... |
| nfcrii 2893 | Consequence of the not-fre... |
| nfceqdf 2894 | An equality theorem for ef... |
| nfceqi 2895 | Equality theorem for class... |
| nfcxfr 2896 | A utility lemma to transfe... |
| nfcxfrd 2897 | A utility lemma to transfe... |
| nfcv 2898 | If ` x ` is disjoint from ... |
| nfcvd 2899 | If ` x ` is disjoint from ... |
| nfab1 2900 | Bound-variable hypothesis ... |
| nfnfc1 2901 | The setvar ` x ` is bound ... |
| clelsb1fw 2902 | Substitution for the first... |
| clelsb1f 2903 | Substitution for the first... |
| nfab 2904 | Bound-variable hypothesis ... |
| nfabg 2905 | Bound-variable hypothesis ... |
| nfaba1 2906 | Bound-variable hypothesis ... |
| nfaba1OLD 2907 | Obsolete version of ~ nfab... |
| nfaba1g 2908 | Bound-variable hypothesis ... |
| nfeqd 2909 | Hypothesis builder for equ... |
| nfeld 2910 | Hypothesis builder for ele... |
| nfnfc 2911 | Hypothesis builder for ` F... |
| nfeq 2912 | Hypothesis builder for equ... |
| nfel 2913 | Hypothesis builder for ele... |
| nfeq1 2914 | Hypothesis builder for equ... |
| nfel1 2915 | Hypothesis builder for ele... |
| nfeq2 2916 | Hypothesis builder for equ... |
| nfel2 2917 | Hypothesis builder for ele... |
| drnfc1 2918 | Formula-building lemma for... |
| drnfc2 2919 | Formula-building lemma for... |
| nfabdw 2920 | Bound-variable hypothesis ... |
| nfabd 2921 | Bound-variable hypothesis ... |
| nfabd2 2922 | Bound-variable hypothesis ... |
| dvelimdc 2923 | Deduction form of ~ dvelim... |
| dvelimc 2924 | Version of ~ dvelim for cl... |
| nfcvf 2925 | If ` x ` and ` y ` are dis... |
| nfcvf2 2926 | If ` x ` and ` y ` are dis... |
| cleqf 2927 | Establish equality between... |
| eqabf 2928 | Equality of a class variab... |
| abid2f 2929 | A simplification of class ... |
| abid2fOLD 2930 | Obsolete version of ~ abid... |
| sbabel 2931 | Theorem to move a substitu... |
| neii 2934 | Inference associated with ... |
| neir 2935 | Inference associated with ... |
| nne 2936 | Negation of inequality. (... |
| neneqd 2937 | Deduction eliminating ineq... |
| neneq 2938 | From inequality to non-equ... |
| neqned 2939 | If it is not the case that... |
| neqne 2940 | From non-equality to inequ... |
| neirr 2941 | No class is unequal to its... |
| exmidne 2942 | Excluded middle with equal... |
| eqneqall 2943 | A contradiction concerning... |
| nonconne 2944 | Law of noncontradiction wi... |
| necon3ad 2945 | Contrapositive law deducti... |
| necon3bd 2946 | Contrapositive law deducti... |
| necon2ad 2947 | Contrapositive inference f... |
| necon2bd 2948 | Contrapositive inference f... |
| necon1ad 2949 | Contrapositive deduction f... |
| necon1bd 2950 | Contrapositive deduction f... |
| necon4ad 2951 | Contrapositive inference f... |
| necon4bd 2952 | Contrapositive inference f... |
| necon3d 2953 | Contrapositive law deducti... |
| necon1d 2954 | Contrapositive law deducti... |
| necon2d 2955 | Contrapositive inference f... |
| necon4d 2956 | Contrapositive inference f... |
| necon3ai 2957 | Contrapositive inference f... |
| necon3bi 2958 | Contrapositive inference f... |
| necon1ai 2959 | Contrapositive inference f... |
| necon1bi 2960 | Contrapositive inference f... |
| necon2ai 2961 | Contrapositive inference f... |
| necon2bi 2962 | Contrapositive inference f... |
| necon4ai 2963 | Contrapositive inference f... |
| necon3i 2964 | Contrapositive inference f... |
| necon1i 2965 | Contrapositive inference f... |
| necon2i 2966 | Contrapositive inference f... |
| necon4i 2967 | Contrapositive inference f... |
| necon3abid 2968 | Deduction from equality to... |
| necon3bbid 2969 | Deduction from equality to... |
| necon1abid 2970 | Contrapositive deduction f... |
| necon1bbid 2971 | Contrapositive inference f... |
| necon4abid 2972 | Contrapositive law deducti... |
| necon4bbid 2973 | Contrapositive law deducti... |
| necon2abid 2974 | Contrapositive deduction f... |
| necon2bbid 2975 | Contrapositive deduction f... |
| necon3bid 2976 | Deduction from equality to... |
| necon4bid 2977 | Contrapositive law deducti... |
| necon3abii 2978 | Deduction from equality to... |
| necon3bbii 2979 | Deduction from equality to... |
| necon1abii 2980 | Contrapositive inference f... |
| necon1bbii 2981 | Contrapositive inference f... |
| necon2abii 2982 | Contrapositive inference f... |
| necon2bbii 2983 | Contrapositive inference f... |
| necon3bii 2984 | Inference from equality to... |
| necom 2985 | Commutation of inequality.... |
| necomi 2986 | Inference from commutative... |
| necomd 2987 | Deduction from commutative... |
| nesym 2988 | Characterization of inequa... |
| nesymi 2989 | Inference associated with ... |
| nesymir 2990 | Inference associated with ... |
| neeq1d 2991 | Deduction for inequality. ... |
| neeq2d 2992 | Deduction for inequality. ... |
| neeq12d 2993 | Deduction for inequality. ... |
| neeq1 2994 | Equality theorem for inequ... |
| neeq2 2995 | Equality theorem for inequ... |
| neeq1i 2996 | Inference for inequality. ... |
| neeq2i 2997 | Inference for inequality. ... |
| neeq12i 2998 | Inference for inequality. ... |
| eqnetrd 2999 | Substitution of equal clas... |
| eqnetrrd 3000 | Substitution of equal clas... |
| neeqtrd 3001 | Substitution of equal clas... |
| eqnetri 3002 | Substitution of equal clas... |
| eqnetrri 3003 | Substitution of equal clas... |
| neeqtri 3004 | Substitution of equal clas... |
| neeqtrri 3005 | Substitution of equal clas... |
| neeqtrrd 3006 | Substitution of equal clas... |
| eqnetrrid 3007 | A chained equality inferen... |
| 3netr3d 3008 | Substitution of equality i... |
| 3netr4d 3009 | Substitution of equality i... |
| 3netr3g 3010 | Substitution of equality i... |
| 3netr4g 3011 | Substitution of equality i... |
| nebi 3012 | Contraposition law for ine... |
| pm13.18 3013 | Theorem *13.18 in [Whitehe... |
| pm13.181 3014 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3015 | Inference eliminating an i... |
| pm2.21ddne 3016 | A contradiction implies an... |
| pm2.61ne 3017 | Deduction eliminating an i... |
| pm2.61dne 3018 | Deduction eliminating an i... |
| pm2.61dane 3019 | Deduction eliminating an i... |
| pm2.61da2ne 3020 | Deduction eliminating two ... |
| pm2.61da3ne 3021 | Deduction eliminating thre... |
| pm2.61iine 3022 | Equality version of ~ pm2.... |
| mteqand 3023 | A modus tollens deduction ... |
| neor 3024 | Logical OR with an equalit... |
| neanior 3025 | A De Morgan's law for ineq... |
| ne3anior 3026 | A De Morgan's law for ineq... |
| neorian 3027 | A De Morgan's law for ineq... |
| nemtbir 3028 | An inference from an inequ... |
| nelne1 3029 | Two classes are different ... |
| nelne2 3030 | Two classes are different ... |
| nelelne 3031 | Two classes are different ... |
| neneor 3032 | If two classes are differe... |
| nfne 3033 | Bound-variable hypothesis ... |
| nfned 3034 | Bound-variable hypothesis ... |
| nabbib 3035 | Not equivalent wff's corre... |
| neli 3038 | Inference associated with ... |
| nelir 3039 | Inference associated with ... |
| nelcon3d 3040 | Contrapositive law deducti... |
| neleq12d 3041 | Equality theorem for negat... |
| neleq1 3042 | Equality theorem for negat... |
| neleq2 3043 | Equality theorem for negat... |
| nfnel 3044 | Bound-variable hypothesis ... |
| nfneld 3045 | Bound-variable hypothesis ... |
| nnel 3046 | Negation of negated member... |
| elnelne1 3047 | Two classes are different ... |
| elnelne2 3048 | Two classes are different ... |
| pm2.24nel 3049 | A contradiction concerning... |
| pm2.61danel 3050 | Deduction eliminating an e... |
| rgen 3053 | Generalization rule for re... |
| ralel 3054 | All elements of a class ar... |
| rgenw 3055 | Generalization rule for re... |
| rgen2w 3056 | Generalization rule for re... |
| mprg 3057 | Modus ponens combined with... |
| mprgbir 3058 | Modus ponens on biconditio... |
| raln 3059 | Restricted universally qua... |
| ralnex 3062 | Relationship between restr... |
| dfrex2 3063 | Relationship between restr... |
| nrex 3064 | Inference adding restricte... |
| alral 3065 | Universal quantification i... |
| rexex 3066 | Restricted existence impli... |
| rextru 3067 | Two ways of expressing tha... |
| ralimi2 3068 | Inference quantifying both... |
| reximi2 3069 | Inference quantifying both... |
| ralimia 3070 | Inference quantifying both... |
| reximia 3071 | Inference quantifying both... |
| ralimiaa 3072 | Inference quantifying both... |
| ralimi 3073 | Inference quantifying both... |
| reximi 3074 | Inference quantifying both... |
| ral2imi 3075 | Inference quantifying ante... |
| ralim 3076 | Distribution of restricted... |
| rexim 3077 | Theorem 19.22 of [Margaris... |
| ralbii2 3078 | Inference adding different... |
| rexbii2 3079 | Inference adding different... |
| ralbiia 3080 | Inference adding restricte... |
| rexbiia 3081 | Inference adding restricte... |
| ralbii 3082 | Inference adding restricte... |
| rexbii 3083 | Inference adding restricte... |
| ralanid 3084 | Cancellation law for restr... |
| rexanid 3085 | Cancellation law for restr... |
| ralcom3 3086 | A commutation law for rest... |
| dfral2 3087 | Relationship between restr... |
| rexnal 3088 | Relationship between restr... |
| ralinexa 3089 | A transformation of restri... |
| rexanali 3090 | A transformation of restri... |
| ralbi 3091 | Distribute a restricted un... |
| rexbi 3092 | Distribute restricted quan... |
| ralrexbid 3093 | Formula-building rule for ... |
| r19.35 3094 | Restricted quantifier vers... |
| r19.26m 3095 | Version of ~ 19.26 and ~ r... |
| r19.26 3096 | Restricted quantifier vers... |
| r19.26-3 3097 | Version of ~ r19.26 with t... |
| ralbiim 3098 | Split a biconditional and ... |
| r19.29 3099 | Restricted quantifier vers... |
| r19.29r 3100 | Restricted quantifier vers... |
| r19.29imd 3101 | Theorem 19.29 of [Margaris... |
| r19.40 3102 | Restricted quantifier vers... |
| r19.30 3103 | Restricted quantifier vers... |
| r19.43 3104 | Restricted quantifier vers... |
| 3r19.43 3105 | Restricted quantifier vers... |
| 2ralimi 3106 | Inference quantifying both... |
| 3ralimi 3107 | Inference quantifying both... |
| 4ralimi 3108 | Inference quantifying both... |
| 5ralimi 3109 | Inference quantifying both... |
| 6ralimi 3110 | Inference quantifying both... |
| 2ralbii 3111 | Inference adding two restr... |
| 2rexbii 3112 | Inference adding two restr... |
| 3ralbii 3113 | Inference adding three res... |
| 4ralbii 3114 | Inference adding four rest... |
| 2ralbiim 3115 | Split a biconditional and ... |
| ralnex2 3116 | Relationship between two r... |
| ralnex3 3117 | Relationship between three... |
| rexnal2 3118 | Relationship between two r... |
| rexnal3 3119 | Relationship between three... |
| nrexralim 3120 | Negation of a complex pred... |
| r19.26-2 3121 | Restricted quantifier vers... |
| 2r19.29 3122 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3123 | Theorem 19.29 of [Margaris... |
| r2allem 3124 | Lemma factoring out common... |
| r2exlem 3125 | Lemma factoring out common... |
| hbralrimi 3126 | Inference from Theorem 19.... |
| ralrimiv 3127 | Inference from Theorem 19.... |
| ralrimiva 3128 | Inference from Theorem 19.... |
| rexlimiva 3129 | Inference from Theorem 19.... |
| rexlimiv 3130 | Inference from Theorem 19.... |
| nrexdv 3131 | Deduction adding restricte... |
| ralrimivw 3132 | Inference from Theorem 19.... |
| rexlimivw 3133 | Weaker version of ~ rexlim... |
| ralrimdv 3134 | Inference from Theorem 19.... |
| rexlimdv 3135 | Inference from Theorem 19.... |
| ralrimdva 3136 | Inference from Theorem 19.... |
| rexlimdva 3137 | Inference from Theorem 19.... |
| rexlimdvaa 3138 | Inference from Theorem 19.... |
| rexlimdva2 3139 | Inference from Theorem 19.... |
| r19.29an 3140 | A commonly used pattern in... |
| rexlimdv3a 3141 | Inference from Theorem 19.... |
| rexlimdvw 3142 | Inference from Theorem 19.... |
| rexlimddv 3143 | Restricted existential eli... |
| r19.29a 3144 | A commonly used pattern in... |
| ralimdv2 3145 | Inference quantifying both... |
| reximdv2 3146 | Deduction quantifying both... |
| reximdvai 3147 | Deduction quantifying both... |
| ralimdva 3148 | Deduction quantifying both... |
| reximdva 3149 | Deduction quantifying both... |
| ralimdv 3150 | Deduction quantifying both... |
| reximdv 3151 | Deduction from Theorem 19.... |
| reximddv 3152 | Deduction from Theorem 19.... |
| reximddv3 3153 | Deduction from Theorem 19.... |
| reximssdv 3154 | Derivation of a restricted... |
| ralbidv2 3155 | Formula-building rule for ... |
| rexbidv2 3156 | Formula-building rule for ... |
| ralbidva 3157 | Formula-building rule for ... |
| rexbidva 3158 | Formula-building rule for ... |
| ralbidv 3159 | Formula-building rule for ... |
| rexbidv 3160 | Formula-building rule for ... |
| r19.21v 3161 | Restricted quantifier vers... |
| r19.37v 3162 | Restricted quantifier vers... |
| r19.23v 3163 | Restricted quantifier vers... |
| r19.36v 3164 | Restricted quantifier vers... |
| r19.27v 3165 | Restricted quantitifer ver... |
| r19.41v 3166 | Restricted quantifier vers... |
| r19.28v 3167 | Restricted quantifier vers... |
| r19.42v 3168 | Restricted quantifier vers... |
| r19.32v 3169 | Restricted quantifier vers... |
| r19.45v 3170 | Restricted quantifier vers... |
| r19.44v 3171 | One direction of a restric... |
| r2al 3172 | Double restricted universa... |
| r2ex 3173 | Double restricted existent... |
| r3al 3174 | Triple restricted universa... |
| r3ex 3175 | Triple existential quantif... |
| rgen2 3176 | Generalization rule for re... |
| ralrimivv 3177 | Inference from Theorem 19.... |
| rexlimivv 3178 | Inference from Theorem 19.... |
| ralrimivva 3179 | Inference from Theorem 19.... |
| ralrimdvv 3180 | Inference from Theorem 19.... |
| rgen3 3181 | Generalization rule for re... |
| ralrimivvva 3182 | Inference from Theorem 19.... |
| ralimdvva 3183 | Deduction doubly quantifyi... |
| reximdvva 3184 | Deduction doubly quantifyi... |
| ralimdvv 3185 | Deduction doubly quantifyi... |
| ralimdvvOLD 3186 | Obsolete version of ~ rali... |
| ralimd4v 3187 | Deduction quadrupally quan... |
| ralimd4vOLD 3188 | Obsolete version of ~ rali... |
| ralimd6v 3189 | Deduction sextupally quant... |
| ralimd6vOLD 3190 | Obsolete version of ~ rali... |
| ralrimdvva 3191 | Inference from Theorem 19.... |
| rexlimdvv 3192 | Inference from Theorem 19.... |
| rexlimdvva 3193 | Inference from Theorem 19.... |
| rexlimdvvva 3194 | Inference from Theorem 19.... |
| reximddv2 3195 | Double deduction from Theo... |
| r19.29vva 3196 | A commonly used pattern ba... |
| 2rexbiia 3197 | Inference adding two restr... |
| 2ralbidva 3198 | Formula-building rule for ... |
| 2rexbidva 3199 | Formula-building rule for ... |
| 2ralbidv 3200 | Formula-building rule for ... |
| 2rexbidv 3201 | Formula-building rule for ... |
| rexralbidv 3202 | Formula-building rule for ... |
| 3ralbidv 3203 | Formula-building rule for ... |
| 4ralbidv 3204 | Formula-building rule for ... |
| 6ralbidv 3205 | Formula-building rule for ... |
| r19.41vv 3206 | Version of ~ r19.41v with ... |
| reeanlem 3207 | Lemma factoring out common... |
| reeanv 3208 | Rearrange restricted exist... |
| 3reeanv 3209 | Rearrange three restricted... |
| 2ralor 3210 | Distribute restricted univ... |
| risset 3211 | Two ways to say " ` A ` be... |
| nelb 3212 | A definition of ` -. A e. ... |
| rspw 3213 | Restricted specialization.... |
| cbvralvw 3214 | Change the bound variable ... |
| cbvrexvw 3215 | Change the bound variable ... |
| cbvraldva 3216 | Rule used to change the bo... |
| cbvrexdva 3217 | Rule used to change the bo... |
| cbvral2vw 3218 | Change bound variables of ... |
| cbvrex2vw 3219 | Change bound variables of ... |
| cbvral3vw 3220 | Change bound variables of ... |
| cbvral4vw 3221 | Change bound variables of ... |
| cbvral6vw 3222 | Change bound variables of ... |
| cbvral8vw 3223 | Change bound variables of ... |
| rsp 3224 | Restricted specialization.... |
| rspa 3225 | Restricted specialization.... |
| rspe 3226 | Restricted specialization.... |
| rspec 3227 | Specialization rule for re... |
| r19.21bi 3228 | Inference from Theorem 19.... |
| r19.21be 3229 | Inference from Theorem 19.... |
| r19.21t 3230 | Restricted quantifier vers... |
| r19.21 3231 | Restricted quantifier vers... |
| r19.23t 3232 | Closed theorem form of ~ r... |
| r19.23 3233 | Restricted quantifier vers... |
| ralrimi 3234 | Inference from Theorem 19.... |
| ralrimia 3235 | Inference from Theorem 19.... |
| rexlimi 3236 | Restricted quantifier vers... |
| ralimdaa 3237 | Deduction quantifying both... |
| reximdai 3238 | Deduction from Theorem 19.... |
| r19.37 3239 | Restricted quantifier vers... |
| r19.41 3240 | Restricted quantifier vers... |
| ralrimd 3241 | Inference from Theorem 19.... |
| rexlimd2 3242 | Version of ~ rexlimd with ... |
| rexlimd 3243 | Deduction form of ~ rexlim... |
| r19.29af2 3244 | A commonly used pattern ba... |
| r19.29af 3245 | A commonly used pattern ba... |
| reximd2a 3246 | Deduction quantifying both... |
| ralbida 3247 | Formula-building rule for ... |
| rexbida 3248 | Formula-building rule for ... |
| ralbid 3249 | Formula-building rule for ... |
| rexbid 3250 | Formula-building rule for ... |
| rexbidvALT 3251 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3252 | Alternate proof of ~ rexbi... |
| rsp2 3253 | Restricted specialization,... |
| rsp2e 3254 | Restricted specialization.... |
| rspec2 3255 | Specialization rule for re... |
| rspec3 3256 | Specialization rule for re... |
| r2alf 3257 | Double restricted universa... |
| r2exf 3258 | Double restricted existent... |
| 2ralbida 3259 | Formula-building rule for ... |
| nfra1 3260 | The setvar ` x ` is not fr... |
| nfre1 3261 | The setvar ` x ` is not fr... |
| ralcom4 3262 | Commutation of restricted ... |
| rexcom4 3263 | Commutation of restricted ... |
| ralcom 3264 | Commutation of restricted ... |
| rexcom 3265 | Commutation of restricted ... |
| rexcom4a 3266 | Specialized existential co... |
| ralrot3 3267 | Rotate three restricted un... |
| ralcom13 3268 | Swap first and third restr... |
| rexcom13 3269 | Swap first and third restr... |
| rexrot4 3270 | Rotate four restricted exi... |
| 2ex2rexrot 3271 | Rotate two existential qua... |
| nfra2w 3272 | Similar to Lemma 24 of [Mo... |
| hbra1 3273 | The setvar ` x ` is not fr... |
| ralcomf 3274 | Commutation of restricted ... |
| rexcomf 3275 | Commutation of restricted ... |
| cbvralfw 3276 | Rule used to change bound ... |
| cbvrexfw 3277 | Rule used to change bound ... |
| cbvralw 3278 | Rule used to change bound ... |
| cbvrexw 3279 | Rule used to change bound ... |
| hbral 3280 | Bound-variable hypothesis ... |
| nfraldw 3281 | Deduction version of ~ nfr... |
| nfrexdw 3282 | Deduction version of ~ nfr... |
| nfralw 3283 | Bound-variable hypothesis ... |
| nfrexw 3284 | Bound-variable hypothesis ... |
| r19.12 3285 | Restricted quantifier vers... |
| reean 3286 | Rearrange restricted exist... |
| cbvralsvw 3287 | Change bound variable by u... |
| cbvrexsvw 3288 | Change bound variable by u... |
| cbvralsvwOLD 3289 | Obsolete version of ~ cbvr... |
| cbvralsvwOLDOLD 3290 | Obsolete version of ~ cbvr... |
| cbvrexsvwOLD 3291 | Obsolete version of ~ cbvr... |
| rexeq 3292 | Equality theorem for restr... |
| raleq 3293 | Equality theorem for restr... |
| raleqi 3294 | Equality inference for res... |
| rexeqi 3295 | Equality inference for res... |
| raleqdv 3296 | Equality deduction for res... |
| rexeqdv 3297 | Equality deduction for res... |
| raleqtrdv 3298 | Substitution of equal clas... |
| rexeqtrdv 3299 | Substitution of equal clas... |
| raleqtrrdv 3300 | Substitution of equal clas... |
| rexeqtrrdv 3301 | Substitution of equal clas... |
| raleqbidva 3302 | Equality deduction for res... |
| rexeqbidva 3303 | Equality deduction for res... |
| raleqbidvv 3304 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3305 | Obsolete version of ~ rale... |
| rexeqbidvv 3306 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3307 | Obsolete version of ~ rexe... |
| raleqbi1dv 3308 | Equality deduction for res... |
| rexeqbi1dv 3309 | Equality deduction for res... |
| raleqOLD 3310 | Obsolete version of ~ rale... |
| rexeqOLD 3311 | Obsolete version of ~ rale... |
| raleleq 3312 | All elements of a class ar... |
| raleleqOLD 3313 | Obsolete version of ~ rale... |
| raleqbii 3314 | Equality deduction for res... |
| rexeqbii 3315 | Equality deduction for res... |
| raleqbidv 3316 | Equality deduction for res... |
| rexeqbidv 3317 | Equality deduction for res... |
| cbvraldva2 3318 | Rule used to change the bo... |
| cbvrexdva2 3319 | Rule used to change the bo... |
| cbvraldvaOLD 3320 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3321 | Obsolete version of ~ cbvr... |
| sbralie 3322 | Implicit to explicit subst... |
| sbralieALT 3323 | Alternative shorter proof ... |
| sbralieOLD 3324 | Obsolete version of ~ sbra... |
| raleqf 3325 | Equality theorem for restr... |
| rexeqf 3326 | Equality theorem for restr... |
| rexeqfOLD 3327 | Obsolete version of ~ rexe... |
| raleqbid 3328 | Equality deduction for res... |
| rexeqbid 3329 | Equality deduction for res... |
| cbvralf 3330 | Rule used to change bound ... |
| cbvrexf 3331 | Rule used to change bound ... |
| cbvral 3332 | Rule used to change bound ... |
| cbvrex 3333 | Rule used to change bound ... |
| cbvralv 3334 | Change the bound variable ... |
| cbvrexv 3335 | Change the bound variable ... |
| cbvralsv 3336 | Change bound variable by u... |
| cbvrexsv 3337 | Change bound variable by u... |
| cbvral2v 3338 | Change bound variables of ... |
| cbvrex2v 3339 | Change bound variables of ... |
| cbvral3v 3340 | Change bound variables of ... |
| rgen2a 3341 | Generalization rule for re... |
| nfrald 3342 | Deduction version of ~ nfr... |
| nfrexd 3343 | Deduction version of ~ nfr... |
| nfral 3344 | Bound-variable hypothesis ... |
| nfrex 3345 | Bound-variable hypothesis ... |
| nfra2 3346 | Similar to Lemma 24 of [Mo... |
| ralcom2 3347 | Commutation of restricted ... |
| reu5 3352 | Restricted uniqueness in t... |
| reurmo 3353 | Restricted existential uni... |
| reurex 3354 | Restricted unique existenc... |
| mormo 3355 | Unrestricted "at most one"... |
| rmobiia 3356 | Formula-building rule for ... |
| reubiia 3357 | Formula-building rule for ... |
| rmobii 3358 | Formula-building rule for ... |
| reubii 3359 | Formula-building rule for ... |
| rmoanid 3360 | Cancellation law for restr... |
| reuanid 3361 | Cancellation law for restr... |
| 2reu2rex 3362 | Double restricted existent... |
| rmobidva 3363 | Formula-building rule for ... |
| reubidva 3364 | Formula-building rule for ... |
| rmobidv 3365 | Formula-building rule for ... |
| reubidv 3366 | Formula-building rule for ... |
| reueubd 3367 | Restricted existential uni... |
| rmo5 3368 | Restricted "at most one" i... |
| nrexrmo 3369 | Nonexistence implies restr... |
| moel 3370 | "At most one" element in a... |
| cbvrmovw 3371 | Change the bound variable ... |
| cbvreuvw 3372 | Change the bound variable ... |
| rmobida 3373 | Formula-building rule for ... |
| reubida 3374 | Formula-building rule for ... |
| cbvrmow 3375 | Change the bound variable ... |
| cbvreuw 3376 | Change the bound variable ... |
| nfrmo1 3377 | The setvar ` x ` is not fr... |
| nfreu1 3378 | The setvar ` x ` is not fr... |
| nfrmow 3379 | Bound-variable hypothesis ... |
| nfreuw 3380 | Bound-variable hypothesis ... |
| rmoeq1 3381 | Equality theorem for restr... |
| reueq1 3382 | Equality theorem for restr... |
| rmoeq1OLD 3383 | Obsolete version of ~ rmoe... |
| reueq1OLD 3384 | Obsolete version of ~ reue... |
| rmoeqd 3385 | Equality deduction for res... |
| reueqd 3386 | Equality deduction for res... |
| reueqdv 3387 | Formula-building rule for ... |
| reueqbidv 3388 | Formula-building rule for ... |
| rmoeq1f 3389 | Equality theorem for restr... |
| reueq1f 3390 | Equality theorem for restr... |
| cbvreu 3391 | Change the bound variable ... |
| cbvrmo 3392 | Change the bound variable ... |
| cbvrmov 3393 | Change the bound variable ... |
| cbvreuv 3394 | Change the bound variable ... |
| nfrmod 3395 | Deduction version of ~ nfr... |
| nfreud 3396 | Deduction version of ~ nfr... |
| nfrmo 3397 | Bound-variable hypothesis ... |
| nfreu 3398 | Bound-variable hypothesis ... |
| rabbidva2 3401 | Equivalent wff's yield equ... |
| rabbia2 3402 | Equivalent wff's yield equ... |
| rabbiia 3403 | Equivalent formulas yield ... |
| rabbii 3404 | Equivalent wff's correspon... |
| rabbidva 3405 | Equivalent wff's yield equ... |
| rabbidv 3406 | Equivalent wff's yield equ... |
| rabbieq 3407 | Equivalent wff's correspon... |
| rabswap 3408 | Swap with a membership rel... |
| cbvrabv 3409 | Rule to change the bound v... |
| rabeqcda 3410 | When ` ps ` is always true... |
| rabeqc 3411 | A restricted class abstrac... |
| rabeqi 3412 | Equality theorem for restr... |
| rabeq 3413 | Equality theorem for restr... |
| rabeqdv 3414 | Equality of restricted cla... |
| rabeqbidva 3415 | Equality of restricted cla... |
| rabeqbidvaOLD 3416 | Obsolete version of ~ rabe... |
| rabeqbidv 3417 | Equality of restricted cla... |
| rabrabi 3418 | Abstract builder restricte... |
| nfrab1 3419 | The abstraction variable i... |
| rabid 3420 | An "identity" law of concr... |
| rabidim1 3421 | Membership in a restricted... |
| reqabi 3422 | Inference from equality of... |
| rabrab 3423 | Abstract builder restricte... |
| rabbida4 3424 | Version of ~ rabbidva2 wit... |
| rabbida 3425 | Equivalent wff's yield equ... |
| rabbid 3426 | Version of ~ rabbidv with ... |
| rabeqd 3427 | Deduction form of ~ rabeq ... |
| rabeqbida 3428 | Version of ~ rabeqbidva wi... |
| rabbi 3429 | Equivalent wff's correspon... |
| rabid2f 3430 | An "identity" law for rest... |
| rabid2im 3431 | One direction of ~ rabid2 ... |
| rabid2 3432 | An "identity" law for rest... |
| rabeqf 3433 | Equality theorem for restr... |
| cbvrabw 3434 | Rule to change the bound v... |
| cbvrabwOLD 3435 | Obsolete version of ~ cbvr... |
| nfrabw 3436 | A variable not free in a w... |
| rabbidaOLD 3437 | Obsolete version of ~ rabb... |
| nfrab 3438 | A variable not free in a w... |
| cbvrab 3439 | Rule to change the bound v... |
| vjust 3441 | Justification theorem for ... |
| dfv2 3443 | Alternate definition of th... |
| vex 3444 | All setvar variables are s... |
| elv 3445 | If a proposition is implie... |
| elvd 3446 | If a proposition is implie... |
| el2v 3447 | If a proposition is implie... |
| el3v 3448 | If a proposition is implie... |
| el3v3 3449 | If a proposition is implie... |
| eqv 3450 | The universe contains ever... |
| eqvf 3451 | The universe contains ever... |
| abv 3452 | The class of sets verifyin... |
| abvALT 3453 | Alternate proof of ~ abv ,... |
| isset 3454 | Two ways to express that "... |
| cbvexeqsetf 3455 | The expression ` E. x x = ... |
| issetft 3456 | Closed theorem form of ~ i... |
| issetf 3457 | A version of ~ isset that ... |
| isseti 3458 | A way to say " ` A ` is a ... |
| issetri 3459 | A way to say " ` A ` is a ... |
| eqvisset 3460 | A class equal to a variabl... |
| elex 3461 | If a class is a member of ... |
| elexOLD 3462 | Obsolete version of ~ elex... |
| elexi 3463 | If a class is a member of ... |
| elexd 3464 | If a class is a member of ... |
| elex22 3465 | If two classes each contai... |
| prcnel 3466 | A proper class doesn't bel... |
| ralv 3467 | A universal quantifier res... |
| rexv 3468 | An existential quantifier ... |
| reuv 3469 | A unique existential quant... |
| rmov 3470 | An at-most-one quantifier ... |
| rabab 3471 | A class abstraction restri... |
| rexcom4b 3472 | Specialized existential co... |
| ceqsal1t 3473 | One direction of ~ ceqsalt... |
| ceqsalt 3474 | Closed theorem version of ... |
| ceqsralt 3475 | Restricted quantifier vers... |
| ceqsalg 3476 | A representation of explic... |
| ceqsalgALT 3477 | Alternate proof of ~ ceqsa... |
| ceqsal 3478 | A representation of explic... |
| ceqsalALT 3479 | A representation of explic... |
| ceqsalv 3480 | A representation of explic... |
| ceqsralv 3481 | Restricted quantifier vers... |
| gencl 3482 | Implicit substitution for ... |
| 2gencl 3483 | Implicit substitution for ... |
| 3gencl 3484 | Implicit substitution for ... |
| cgsexg 3485 | Implicit substitution infe... |
| cgsex2g 3486 | Implicit substitution infe... |
| cgsex4g 3487 | An implicit substitution i... |
| cgsex4gOLD 3488 | Obsolete version of ~ cgse... |
| ceqsex 3489 | Elimination of an existent... |
| ceqsexv 3490 | Elimination of an existent... |
| ceqsexv2d 3491 | Elimination of an existent... |
| ceqsexv2dOLD 3492 | Obsolete version of ~ ceqs... |
| ceqsex2 3493 | Elimination of two existen... |
| ceqsex2v 3494 | Elimination of two existen... |
| ceqsex3v 3495 | Elimination of three exist... |
| ceqsex4v 3496 | Elimination of four existe... |
| ceqsex6v 3497 | Elimination of six existen... |
| ceqsex8v 3498 | Elimination of eight exist... |
| gencbvex 3499 | Change of bound variable u... |
| gencbvex2 3500 | Restatement of ~ gencbvex ... |
| gencbval 3501 | Change of bound variable u... |
| sbhypf 3502 | Introduce an explicit subs... |
| spcimgft 3503 | Closed theorem form of ~ s... |
| spcimgfi1 3504 | A closed version of ~ spci... |
| spcimgfi1OLD 3505 | Obsolete version of ~ spci... |
| spcgft 3506 | A closed version of ~ spcg... |
| spcimgf 3507 | Rule of specialization, us... |
| spcimegf 3508 | Existential specialization... |
| vtoclgft 3509 | Closed theorem form of ~ v... |
| vtocleg 3510 | Implicit substitution of a... |
| vtoclg 3511 | Implicit substitution of a... |
| vtocle 3512 | Implicit substitution of a... |
| vtocleOLD 3513 | Obsolete version of ~ vtoc... |
| vtoclbg 3514 | Implicit substitution of a... |
| vtocl 3515 | Implicit substitution of a... |
| vtoclOLD 3516 | Obsolete version of ~ vtoc... |
| vtocldf 3517 | Implicit substitution of a... |
| vtocld 3518 | Implicit substitution of a... |
| vtocl2d 3519 | Implicit substitution of t... |
| vtoclef 3520 | Implicit substitution of a... |
| vtoclf 3521 | Implicit substitution of a... |
| vtocl2 3522 | Implicit substitution of c... |
| vtocl3 3523 | Implicit substitution of c... |
| vtoclb 3524 | Implicit substitution of a... |
| vtoclgf 3525 | Implicit substitution of a... |
| vtoclg1f 3526 | Version of ~ vtoclgf with ... |
| vtocl2gf 3527 | Implicit substitution of a... |
| vtocl3gf 3528 | Implicit substitution of a... |
| vtocl2g 3529 | Implicit substitution of 2... |
| vtocl3g 3530 | Implicit substitution of a... |
| vtoclgaf 3531 | Implicit substitution of a... |
| vtoclga 3532 | Implicit substitution of a... |
| vtocl2ga 3533 | Implicit substitution of 2... |
| vtocl2gaf 3534 | Implicit substitution of 2... |
| vtocl2gafOLD 3535 | Obsolete version of ~ vtoc... |
| vtocl3gaf 3536 | Implicit substitution of 3... |
| vtocl3gafOLD 3537 | Obsolete version of ~ vtoc... |
| vtocl3ga 3538 | Implicit substitution of 3... |
| vtocl3gaOLD 3539 | Obsolete version of ~ vtoc... |
| vtocl4g 3540 | Implicit substitution of 4... |
| vtocl4ga 3541 | Implicit substitution of 4... |
| vtocl4gaOLD 3542 | Obsolete version of ~ vtoc... |
| vtoclegft 3543 | Implicit substitution of a... |
| vtoclri 3544 | Implicit substitution of a... |
| spcgf 3545 | Rule of specialization, us... |
| spcegf 3546 | Existential specialization... |
| spcimdv 3547 | Restricted specialization,... |
| spcdv 3548 | Rule of specialization, us... |
| spcimedv 3549 | Restricted existential spe... |
| spcgv 3550 | Rule of specialization, us... |
| spcegv 3551 | Existential specialization... |
| spcedv 3552 | Existential specialization... |
| spc2egv 3553 | Existential specialization... |
| spc2gv 3554 | Specialization with two qu... |
| spc2ed 3555 | Existential specialization... |
| spc2d 3556 | Specialization with 2 quan... |
| spc3egv 3557 | Existential specialization... |
| spc3gv 3558 | Specialization with three ... |
| spcv 3559 | Rule of specialization, us... |
| spcev 3560 | Existential specialization... |
| spc2ev 3561 | Existential specialization... |
| rspct 3562 | A closed version of ~ rspc... |
| rspcdf 3563 | Restricted specialization,... |
| rspc 3564 | Restricted specialization,... |
| rspce 3565 | Restricted existential spe... |
| rspcimdv 3566 | Restricted specialization,... |
| rspcimedv 3567 | Restricted existential spe... |
| rspcdv 3568 | Restricted specialization,... |
| rspcedv 3569 | Restricted existential spe... |
| rspcebdv 3570 | Restricted existential spe... |
| rspcdv2 3571 | Restricted specialization,... |
| rspcv 3572 | Restricted specialization,... |
| rspccv 3573 | Restricted specialization,... |
| rspcva 3574 | Restricted specialization,... |
| rspccva 3575 | Restricted specialization,... |
| rspcev 3576 | Restricted existential spe... |
| rspcdva 3577 | Restricted specialization,... |
| rspcedvd 3578 | Restricted existential spe... |
| rspcedvdw 3579 | Version of ~ rspcedvd wher... |
| rspceb2dv 3580 | Restricted existential spe... |
| rspcime 3581 | Prove a restricted existen... |
| rspceaimv 3582 | Restricted existential spe... |
| rspcedeq1vd 3583 | Restricted existential spe... |
| rspcedeq2vd 3584 | Restricted existential spe... |
| rspc2 3585 | Restricted specialization ... |
| rspc2gv 3586 | Restricted specialization ... |
| rspc2v 3587 | 2-variable restricted spec... |
| rspc2va 3588 | 2-variable restricted spec... |
| rspc2ev 3589 | 2-variable restricted exis... |
| 2rspcedvdw 3590 | Double application of ~ rs... |
| rspc2dv 3591 | 2-variable restricted spec... |
| rspc3v 3592 | 3-variable restricted spec... |
| rspc3ev 3593 | 3-variable restricted exis... |
| 3rspcedvdw 3594 | Triple application of ~ rs... |
| rspc3dv 3595 | 3-variable restricted spec... |
| rspc4v 3596 | 4-variable restricted spec... |
| rspc6v 3597 | 6-variable restricted spec... |
| rspc8v 3598 | 8-variable restricted spec... |
| rspceeqv 3599 | Restricted existential spe... |
| ralxpxfr2d 3600 | Transfer a universal quant... |
| rexraleqim 3601 | Statement following from e... |
| eqvincg 3602 | A variable introduction la... |
| eqvinc 3603 | A variable introduction la... |
| eqvincf 3604 | A variable introduction la... |
| alexeqg 3605 | Two ways to express substi... |
| ceqex 3606 | Equality implies equivalen... |
| ceqsexg 3607 | A representation of explic... |
| ceqsexgv 3608 | Elimination of an existent... |
| ceqsrexv 3609 | Elimination of a restricte... |
| ceqsrexbv 3610 | Elimination of a restricte... |
| ceqsralbv 3611 | Elimination of a restricte... |
| ceqsrex2v 3612 | Elimination of a restricte... |
| clel2g 3613 | Alternate definition of me... |
| clel2 3614 | Alternate definition of me... |
| clel3g 3615 | Alternate definition of me... |
| clel3 3616 | Alternate definition of me... |
| clel4g 3617 | Alternate definition of me... |
| clel4 3618 | Alternate definition of me... |
| clel5 3619 | Alternate definition of cl... |
| pm13.183 3620 | Compare theorem *13.183 in... |
| rr19.3v 3621 | Restricted quantifier vers... |
| rr19.28v 3622 | Restricted quantifier vers... |
| elab6g 3623 | Membership in a class abst... |
| elabd2 3624 | Membership in a class abst... |
| elabd3 3625 | Membership in a class abst... |
| elabgt 3626 | Membership in a class abst... |
| elabgtOLD 3627 | Obsolete version of ~ elab... |
| elabgtOLDOLD 3628 | Obsolete version of ~ elab... |
| elabgf 3629 | Membership in a class abst... |
| elabf 3630 | Membership in a class abst... |
| elabg 3631 | Membership in a class abst... |
| elabgw 3632 | Membership in a class abst... |
| elab2gw 3633 | Membership in a class abst... |
| elab 3634 | Membership in a class abst... |
| elab2g 3635 | Membership in a class abst... |
| elabd 3636 | Explicit demonstration the... |
| elab2 3637 | Membership in a class abst... |
| elab4g 3638 | Membership in a class abst... |
| elab3gf 3639 | Membership in a class abst... |
| elab3g 3640 | Membership in a class abst... |
| elab3 3641 | Membership in a class abst... |
| elrabi 3642 | Implication for the member... |
| elrabf 3643 | Membership in a restricted... |
| rabtru 3644 | Abstract builder using the... |
| elrab3t 3645 | Membership in a restricted... |
| elrab 3646 | Membership in a restricted... |
| elrab3 3647 | Membership in a restricted... |
| elrabd 3648 | Membership in a restricted... |
| elrab2 3649 | Membership in a restricted... |
| elrab2w 3650 | Membership in a restricted... |
| ralab 3651 | Universal quantification o... |
| ralrab 3652 | Universal quantification o... |
| rexab 3653 | Existential quantification... |
| rexrab 3654 | Existential quantification... |
| ralab2 3655 | Universal quantification o... |
| ralrab2 3656 | Universal quantification o... |
| rexab2 3657 | Existential quantification... |
| rexrab2 3658 | Existential quantification... |
| reurab 3659 | Restricted existential uni... |
| abidnf 3660 | Identity used to create cl... |
| dedhb 3661 | A deduction theorem for co... |
| class2seteq 3662 | Writing a set as a class a... |
| nelrdva 3663 | Deduce negative membership... |
| eqeu 3664 | A condition which implies ... |
| moeq 3665 | There exists at most one s... |
| eueq 3666 | A class is a set if and on... |
| eueqi 3667 | There exists a unique set ... |
| eueq2 3668 | Equality has existential u... |
| eueq3 3669 | Equality has existential u... |
| moeq3 3670 | "At most one" property of ... |
| mosub 3671 | "At most one" remains true... |
| mo2icl 3672 | Theorem for inferring "at ... |
| mob2 3673 | Consequence of "at most on... |
| moi2 3674 | Consequence of "at most on... |
| mob 3675 | Equality implied by "at mo... |
| moi 3676 | Equality implied by "at mo... |
| morex 3677 | Derive membership from uni... |
| euxfr2w 3678 | Transfer existential uniqu... |
| euxfrw 3679 | Transfer existential uniqu... |
| euxfr2 3680 | Transfer existential uniqu... |
| euxfr 3681 | Transfer existential uniqu... |
| euind 3682 | Existential uniqueness via... |
| reu2 3683 | A way to express restricte... |
| reu6 3684 | A way to express restricte... |
| reu3 3685 | A way to express restricte... |
| reu6i 3686 | A condition which implies ... |
| eqreu 3687 | A condition which implies ... |
| rmo4 3688 | Restricted "at most one" u... |
| reu4 3689 | Restricted uniqueness usin... |
| reu7 3690 | Restricted uniqueness usin... |
| reu8 3691 | Restricted uniqueness usin... |
| rmo3f 3692 | Restricted "at most one" u... |
| rmo4f 3693 | Restricted "at most one" u... |
| reu2eqd 3694 | Deduce equality from restr... |
| reueq 3695 | Equality has existential u... |
| rmoeq 3696 | Equality's restricted exis... |
| rmoan 3697 | Restricted "at most one" s... |
| rmoim 3698 | Restricted "at most one" i... |
| rmoimia 3699 | Restricted "at most one" i... |
| rmoimi 3700 | Restricted "at most one" i... |
| rmoimi2 3701 | Restricted "at most one" i... |
| 2reu5a 3702 | Double restricted existent... |
| reuimrmo 3703 | Restricted uniqueness impl... |
| 2reuswap 3704 | A condition allowing swap ... |
| 2reuswap2 3705 | A condition allowing swap ... |
| reuxfrd 3706 | Transfer existential uniqu... |
| reuxfr 3707 | Transfer existential uniqu... |
| reuxfr1d 3708 | Transfer existential uniqu... |
| reuxfr1ds 3709 | Transfer existential uniqu... |
| reuxfr1 3710 | Transfer existential uniqu... |
| reuind 3711 | Existential uniqueness via... |
| 2rmorex 3712 | Double restricted quantifi... |
| 2reu5lem1 3713 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3714 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3715 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3716 | Double restricted existent... |
| 2reurmo 3717 | Double restricted quantifi... |
| 2reurex 3718 | Double restricted quantifi... |
| 2rmoswap 3719 | A condition allowing to sw... |
| 2rexreu 3720 | Double restricted existent... |
| cdeqi 3723 | Deduce conditional equalit... |
| cdeqri 3724 | Property of conditional eq... |
| cdeqth 3725 | Deduce conditional equalit... |
| cdeqnot 3726 | Distribute conditional equ... |
| cdeqal 3727 | Distribute conditional equ... |
| cdeqab 3728 | Distribute conditional equ... |
| cdeqal1 3729 | Distribute conditional equ... |
| cdeqab1 3730 | Distribute conditional equ... |
| cdeqim 3731 | Distribute conditional equ... |
| cdeqcv 3732 | Conditional equality for s... |
| cdeqeq 3733 | Distribute conditional equ... |
| cdeqel 3734 | Distribute conditional equ... |
| nfcdeq 3735 | If we have a conditional e... |
| nfccdeq 3736 | Variation of ~ nfcdeq for ... |
| rru 3737 | Relative version of Russel... |
| ru 3738 | Russell's Paradox. Propos... |
| ruOLD 3739 | Obsolete version of ~ ru a... |
| dfsbcq 3742 | Proper substitution of a c... |
| dfsbcq2 3743 | This theorem, which is sim... |
| sbsbc 3744 | Show that ~ df-sb and ~ df... |
| sbceq1d 3745 | Equality theorem for class... |
| sbceq1dd 3746 | Equality theorem for class... |
| sbceqbid 3747 | Equality theorem for class... |
| sbc8g 3748 | This is the closest we can... |
| sbc2or 3749 | The disjunction of two equ... |
| sbcex 3750 | By our definition of prope... |
| sbceq1a 3751 | Equality theorem for class... |
| sbceq2a 3752 | Equality theorem for class... |
| spsbc 3753 | Specialization: if a formu... |
| spsbcd 3754 | Specialization: if a formu... |
| sbcth 3755 | A substitution into a theo... |
| sbcthdv 3756 | Deduction version of ~ sbc... |
| sbcid 3757 | An identity theorem for su... |
| nfsbc1d 3758 | Deduction version of ~ nfs... |
| nfsbc1 3759 | Bound-variable hypothesis ... |
| nfsbc1v 3760 | Bound-variable hypothesis ... |
| nfsbcdw 3761 | Deduction version of ~ nfs... |
| nfsbcw 3762 | Bound-variable hypothesis ... |
| sbccow 3763 | A composition law for clas... |
| nfsbcd 3764 | Deduction version of ~ nfs... |
| nfsbc 3765 | Bound-variable hypothesis ... |
| sbcco 3766 | A composition law for clas... |
| sbcco2 3767 | A composition law for clas... |
| sbc5 3768 | An equivalence for class s... |
| sbc5ALT 3769 | Alternate proof of ~ sbc5 ... |
| sbc6g 3770 | An equivalence for class s... |
| sbc6 3771 | An equivalence for class s... |
| sbc7 3772 | An equivalence for class s... |
| cbvsbcw 3773 | Change bound variables in ... |
| cbvsbcvw 3774 | Change the bound variable ... |
| cbvsbc 3775 | Change bound variables in ... |
| cbvsbcv 3776 | Change the bound variable ... |
| sbciegft 3777 | Conversion of implicit sub... |
| sbciegftOLD 3778 | Obsolete version of ~ sbci... |
| sbciegf 3779 | Conversion of implicit sub... |
| sbcieg 3780 | Conversion of implicit sub... |
| sbcie2g 3781 | Conversion of implicit sub... |
| sbcie 3782 | Conversion of implicit sub... |
| sbciedf 3783 | Conversion of implicit sub... |
| sbcied 3784 | Conversion of implicit sub... |
| sbcied2 3785 | Conversion of implicit sub... |
| elrabsf 3786 | Membership in a restricted... |
| eqsbc1 3787 | Substitution for the left-... |
| sbcng 3788 | Move negation in and out o... |
| sbcimg 3789 | Distribution of class subs... |
| sbcan 3790 | Distribution of class subs... |
| sbcor 3791 | Distribution of class subs... |
| sbcbig 3792 | Distribution of class subs... |
| sbcn1 3793 | Move negation in and out o... |
| sbcim1 3794 | Distribution of class subs... |
| sbcbid 3795 | Formula-building deduction... |
| sbcbidv 3796 | Formula-building deduction... |
| sbcbii 3797 | Formula-building inference... |
| sbcbi1 3798 | Distribution of class subs... |
| sbcbi2 3799 | Substituting into equivale... |
| sbcal 3800 | Move universal quantifier ... |
| sbcex2 3801 | Move existential quantifie... |
| sbceqal 3802 | Class version of one impli... |
| sbeqalb 3803 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3804 | Substitution for the right... |
| sbc3an 3805 | Distribution of class subs... |
| sbcel1v 3806 | Class substitution into a ... |
| sbcel2gv 3807 | Class substitution into a ... |
| sbcel21v 3808 | Class substitution into a ... |
| sbcimdv 3809 | Substitution analogue of T... |
| sbctt 3810 | Substitution for a variabl... |
| sbcgf 3811 | Substitution for a variabl... |
| sbc19.21g 3812 | Substitution for a variabl... |
| sbcg 3813 | Substitution for a variabl... |
| sbcgfi 3814 | Substitution for a variabl... |
| sbc2iegf 3815 | Conversion of implicit sub... |
| sbc2ie 3816 | Conversion of implicit sub... |
| sbc2iedv 3817 | Conversion of implicit sub... |
| sbc3ie 3818 | Conversion of implicit sub... |
| sbccomlem 3819 | Lemma for ~ sbccom . (Con... |
| sbccomlemOLD 3820 | Obsolete version of ~ sbcc... |
| sbccom 3821 | Commutative law for double... |
| sbcralt 3822 | Interchange class substitu... |
| sbcrext 3823 | Interchange class substitu... |
| sbcralg 3824 | Interchange class substitu... |
| sbcrex 3825 | Interchange class substitu... |
| sbcreu 3826 | Interchange class substitu... |
| reu8nf 3827 | Restricted uniqueness usin... |
| sbcabel 3828 | Interchange class substitu... |
| rspsbc 3829 | Restricted quantifier vers... |
| rspsbca 3830 | Restricted quantifier vers... |
| rspesbca 3831 | Existence form of ~ rspsbc... |
| spesbc 3832 | Existence form of ~ spsbc ... |
| spesbcd 3833 | form of ~ spsbc . (Contri... |
| sbcth2 3834 | A substitution into a theo... |
| ra4v 3835 | Version of ~ ra4 with a di... |
| ra4 3836 | Restricted quantifier vers... |
| rmo2 3837 | Alternate definition of re... |
| rmo2i 3838 | Condition implying restric... |
| rmo3 3839 | Restricted "at most one" u... |
| rmob 3840 | Consequence of "at most on... |
| rmoi 3841 | Consequence of "at most on... |
| rmob2 3842 | Consequence of "restricted... |
| rmoi2 3843 | Consequence of "restricted... |
| rmoanim 3844 | Introduction of a conjunct... |
| rmoanimALT 3845 | Alternate proof of ~ rmoan... |
| reuan 3846 | Introduction of a conjunct... |
| 2reu1 3847 | Double restricted existent... |
| 2reu2 3848 | Double restricted existent... |
| csb2 3851 | Alternate expression for t... |
| csbeq1 3852 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3853 | Equality deduction for pro... |
| csbeq2 3854 | Substituting into equivale... |
| csbeq2d 3855 | Formula-building deduction... |
| csbeq2dv 3856 | Formula-building deduction... |
| csbeq2i 3857 | Formula-building inference... |
| csbeq12dv 3858 | Formula-building inference... |
| cbvcsbw 3859 | Change bound variables in ... |
| cbvcsb 3860 | Change bound variables in ... |
| cbvcsbv 3861 | Change the bound variable ... |
| csbid 3862 | Analogue of ~ sbid for pro... |
| csbeq1a 3863 | Equality theorem for prope... |
| csbcow 3864 | Composition law for chaine... |
| csbco 3865 | Composition law for chaine... |
| csbtt 3866 | Substitution doesn't affec... |
| csbconstgf 3867 | Substitution doesn't affec... |
| csbconstg 3868 | Substitution doesn't affec... |
| csbgfi 3869 | Substitution for a variabl... |
| csbconstgi 3870 | The proper substitution of... |
| nfcsb1d 3871 | Bound-variable hypothesis ... |
| nfcsb1 3872 | Bound-variable hypothesis ... |
| nfcsb1v 3873 | Bound-variable hypothesis ... |
| nfcsbd 3874 | Deduction version of ~ nfc... |
| nfcsbw 3875 | Bound-variable hypothesis ... |
| nfcsb 3876 | Bound-variable hypothesis ... |
| csbhypf 3877 | Introduce an explicit subs... |
| csbiebt 3878 | Conversion of implicit sub... |
| csbiedf 3879 | Conversion of implicit sub... |
| csbieb 3880 | Bidirectional conversion b... |
| csbiebg 3881 | Bidirectional conversion b... |
| csbiegf 3882 | Conversion of implicit sub... |
| csbief 3883 | Conversion of implicit sub... |
| csbie 3884 | Conversion of implicit sub... |
| csbied 3885 | Conversion of implicit sub... |
| csbied2 3886 | Conversion of implicit sub... |
| csbie2t 3887 | Conversion of implicit sub... |
| csbie2 3888 | Conversion of implicit sub... |
| csbie2g 3889 | Conversion of implicit sub... |
| cbvrabcsfw 3890 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3891 | A more general version of ... |
| cbvrexcsf 3892 | A more general version of ... |
| cbvreucsf 3893 | A more general version of ... |
| cbvrabcsf 3894 | A more general version of ... |
| cbvralv2 3895 | Rule used to change the bo... |
| cbvrexv2 3896 | Rule used to change the bo... |
| rspc2vd 3897 | Deduction version of 2-var... |
| difjust 3903 | Soundness justification th... |
| unjust 3905 | Soundness justification th... |
| injust 3907 | Soundness justification th... |
| dfin5 3909 | Alternate definition for t... |
| dfdif2 3910 | Alternate definition of cl... |
| eldif 3911 | Expansion of membership in... |
| eldifd 3912 | If a class is in one class... |
| eldifad 3913 | If a class is in the diffe... |
| eldifbd 3914 | If a class is in the diffe... |
| elneeldif 3915 | The elements of a set diff... |
| velcomp 3916 | Characterization of setvar... |
| elin 3917 | Expansion of membership in... |
| dfss2 3919 | Alternate definition of th... |
| dfss 3920 | Variant of subclass defini... |
| dfss3 3922 | Alternate definition of su... |
| dfss6 3923 | Alternate definition of su... |
| dfssf 3924 | Equivalence for subclass r... |
| dfss3f 3925 | Equivalence for subclass r... |
| nfss 3926 | If ` x ` is not free in ` ... |
| ssel 3927 | Membership relationships f... |
| ssel2 3928 | Membership relationships f... |
| sseli 3929 | Membership implication fro... |
| sselii 3930 | Membership inference from ... |
| sselid 3931 | Membership inference from ... |
| sseld 3932 | Membership deduction from ... |
| sselda 3933 | Membership deduction from ... |
| sseldd 3934 | Membership inference from ... |
| ssneld 3935 | If a class is not in anoth... |
| ssneldd 3936 | If an element is not in a ... |
| ssriv 3937 | Inference based on subclas... |
| ssrd 3938 | Deduction based on subclas... |
| ssrdv 3939 | Deduction based on subclas... |
| sstr2 3940 | Transitivity of subclass r... |
| sstr2OLD 3941 | Obsolete version of ~ sstr... |
| sstr 3942 | Transitivity of subclass r... |
| sstri 3943 | Subclass transitivity infe... |
| sstrd 3944 | Subclass transitivity dedu... |
| sstrid 3945 | Subclass transitivity dedu... |
| sstrdi 3946 | Subclass transitivity dedu... |
| sylan9ss 3947 | A subclass transitivity de... |
| sylan9ssr 3948 | A subclass transitivity de... |
| eqss 3949 | The subclass relationship ... |
| eqssi 3950 | Infer equality from two su... |
| eqssd 3951 | Equality deduction from tw... |
| sssseq 3952 | If a class is a subclass o... |
| eqrd 3953 | Deduce equality of classes... |
| eqri 3954 | Infer equality of classes ... |
| eqelssd 3955 | Equality deduction from su... |
| ssid 3956 | Any class is a subclass of... |
| ssidd 3957 | Weakening of ~ ssid . (Co... |
| ssv 3958 | Any class is a subclass of... |
| sseq1 3959 | Equality theorem for subcl... |
| sseq2 3960 | Equality theorem for the s... |
| sseq12 3961 | Equality theorem for the s... |
| sseq1i 3962 | An equality inference for ... |
| sseq2i 3963 | An equality inference for ... |
| sseq12i 3964 | An equality inference for ... |
| sseq1d 3965 | An equality deduction for ... |
| sseq2d 3966 | An equality deduction for ... |
| sseq12d 3967 | An equality deduction for ... |
| eqsstrd 3968 | Substitution of equality i... |
| eqsstrrd 3969 | Substitution of equality i... |
| sseqtrd 3970 | Substitution of equality i... |
| sseqtrrd 3971 | Substitution of equality i... |
| eqsstrid 3972 | A chained subclass and equ... |
| eqsstrrid 3973 | A chained subclass and equ... |
| sseqtrdi 3974 | A chained subclass and equ... |
| sseqtrrdi 3975 | A chained subclass and equ... |
| sseqtrid 3976 | Subclass transitivity dedu... |
| sseqtrrid 3977 | Subclass transitivity dedu... |
| eqsstrdi 3978 | A chained subclass and equ... |
| eqsstrrdi 3979 | A chained subclass and equ... |
| eqsstri 3980 | Substitution of equality i... |
| eqsstrri 3981 | Substitution of equality i... |
| sseqtri 3982 | Substitution of equality i... |
| sseqtrri 3983 | Substitution of equality i... |
| 3sstr3i 3984 | Substitution of equality i... |
| 3sstr4i 3985 | Substitution of equality i... |
| 3sstr3g 3986 | Substitution of equality i... |
| 3sstr4g 3987 | Substitution of equality i... |
| 3sstr3d 3988 | Substitution of equality i... |
| 3sstr4d 3989 | Substitution of equality i... |
| eqimssd 3990 | Equality implies inclusion... |
| eqimsscd 3991 | Equality implies inclusion... |
| eqimss 3992 | Equality implies inclusion... |
| eqimss2 3993 | Equality implies inclusion... |
| eqimssi 3994 | Infer subclass relationshi... |
| eqimss2i 3995 | Infer subclass relationshi... |
| nssne1 3996 | Two classes are different ... |
| nssne2 3997 | Two classes are different ... |
| nss 3998 | Negation of subclass relat... |
| nelss 3999 | Demonstrate by witnesses t... |
| ssrexf 4000 | Restricted existential qua... |
| ssrmof 4001 | "At most one" existential ... |
| ssralv 4002 | Quantification restricted ... |
| ssrexv 4003 | Existential quantification... |
| ss2ralv 4004 | Two quantifications restri... |
| ss2rexv 4005 | Two existential quantifica... |
| ssralvOLD 4006 | Obsolete version of ~ ssra... |
| ssrexvOLD 4007 | Obsolete version of ~ ssre... |
| ralss 4008 | Restricted universal quant... |
| rexss 4009 | Restricted existential qua... |
| ralssOLD 4010 | Obsolete version of ~ rals... |
| rexssOLD 4011 | Obsolete version of ~ rexs... |
| ss2abim 4012 | Class abstractions in a su... |
| ss2ab 4013 | Class abstractions in a su... |
| abss 4014 | Class abstraction in a sub... |
| ssab 4015 | Subclass of a class abstra... |
| ssabral 4016 | The relation for a subclas... |
| ss2abdv 4017 | Deduction of abstraction s... |
| ss2abi 4018 | Inference of abstraction s... |
| abssdv 4019 | Deduction of abstraction s... |
| abssi 4020 | Inference of abstraction s... |
| ss2rab 4021 | Restricted abstraction cla... |
| rabss 4022 | Restricted class abstracti... |
| ssrab 4023 | Subclass of a restricted c... |
| ss2rabd 4024 | Subclass of a restricted c... |
| ssrabdv 4025 | Subclass of a restricted c... |
| rabssdv 4026 | Subclass of a restricted c... |
| ss2rabdv 4027 | Deduction of restricted ab... |
| ss2rabi 4028 | Inference of restricted ab... |
| rabss2 4029 | Subclass law for restricte... |
| rabss2OLD 4030 | Obsolete version of ~ rabs... |
| ssab2 4031 | Subclass relation for the ... |
| ssrab2 4032 | Subclass relation for a re... |
| rabss3d 4033 | Subclass law for restricte... |
| ssrab3 4034 | Subclass relation for a re... |
| rabssrabd 4035 | Subclass of a restricted c... |
| ssrabeq 4036 | If the restricting class o... |
| rabssab 4037 | A restricted class is a su... |
| eqrrabd 4038 | Deduce equality with a res... |
| uniiunlem 4039 | A subset relationship usef... |
| dfpss2 4040 | Alternate definition of pr... |
| dfpss3 4041 | Alternate definition of pr... |
| psseq1 4042 | Equality theorem for prope... |
| psseq2 4043 | Equality theorem for prope... |
| psseq1i 4044 | An equality inference for ... |
| psseq2i 4045 | An equality inference for ... |
| psseq12i 4046 | An equality inference for ... |
| psseq1d 4047 | An equality deduction for ... |
| psseq2d 4048 | An equality deduction for ... |
| psseq12d 4049 | An equality deduction for ... |
| pssss 4050 | A proper subclass is a sub... |
| pssne 4051 | Two classes in a proper su... |
| pssssd 4052 | Deduce subclass from prope... |
| pssned 4053 | Proper subclasses are uneq... |
| sspss 4054 | Subclass in terms of prope... |
| pssirr 4055 | Proper subclass is irrefle... |
| pssn2lp 4056 | Proper subclass has no 2-c... |
| sspsstri 4057 | Two ways of stating tricho... |
| ssnpss 4058 | Partial trichotomy law for... |
| psstr 4059 | Transitive law for proper ... |
| sspsstr 4060 | Transitive law for subclas... |
| psssstr 4061 | Transitive law for subclas... |
| psstrd 4062 | Proper subclass inclusion ... |
| sspsstrd 4063 | Transitivity involving sub... |
| psssstrd 4064 | Transitivity involving sub... |
| npss 4065 | A class is not a proper su... |
| ssnelpss 4066 | A subclass missing a membe... |
| ssnelpssd 4067 | Subclass inclusion with on... |
| ssexnelpss 4068 | If there is an element of ... |
| dfdif3 4069 | Alternate definition of cl... |
| dfdif3OLD 4070 | Obsolete version of ~ dfdi... |
| difeq1 4071 | Equality theorem for class... |
| difeq2 4072 | Equality theorem for class... |
| difeq12 4073 | Equality theorem for class... |
| difeq1i 4074 | Inference adding differenc... |
| difeq2i 4075 | Inference adding differenc... |
| difeq12i 4076 | Equality inference for cla... |
| difeq1d 4077 | Deduction adding differenc... |
| difeq2d 4078 | Deduction adding differenc... |
| difeq12d 4079 | Equality deduction for cla... |
| difeqri 4080 | Inference from membership ... |
| nfdif 4081 | Bound-variable hypothesis ... |
| nfdifOLD 4082 | Obsolete version of ~ nfdi... |
| eldifi 4083 | Implication of membership ... |
| eldifn 4084 | Implication of membership ... |
| elndif 4085 | A set does not belong to a... |
| neldif 4086 | Implication of membership ... |
| difdif 4087 | Double class difference. ... |
| difss 4088 | Subclass relationship for ... |
| difssd 4089 | A difference of two classe... |
| difss2 4090 | If a class is contained in... |
| difss2d 4091 | If a class is contained in... |
| ssdifss 4092 | Preservation of a subclass... |
| ddif 4093 | Double complement under un... |
| ssconb 4094 | Contraposition law for sub... |
| sscon 4095 | Contraposition law for sub... |
| ssdif 4096 | Difference law for subsets... |
| ssdifd 4097 | If ` A ` is contained in `... |
| sscond 4098 | If ` A ` is contained in `... |
| ssdifssd 4099 | If ` A ` is contained in `... |
| ssdif2d 4100 | If ` A ` is contained in `... |
| raldifb 4101 | Restricted universal quant... |
| rexdifi 4102 | Restricted existential qua... |
| complss 4103 | Complementation reverses i... |
| compleq 4104 | Two classes are equal if a... |
| elun 4105 | Expansion of membership in... |
| elunnel1 4106 | A member of a union that i... |
| elunnel2 4107 | A member of a union that i... |
| uneqri 4108 | Inference from membership ... |
| unidm 4109 | Idempotent law for union o... |
| uncom 4110 | Commutative law for union ... |
| equncom 4111 | If a class equals the unio... |
| equncomi 4112 | Inference form of ~ equnco... |
| uneq1 4113 | Equality theorem for the u... |
| uneq2 4114 | Equality theorem for the u... |
| uneq12 4115 | Equality theorem for the u... |
| uneq1i 4116 | Inference adding union to ... |
| uneq2i 4117 | Inference adding union to ... |
| uneq12i 4118 | Equality inference for the... |
| uneq1d 4119 | Deduction adding union to ... |
| uneq2d 4120 | Deduction adding union to ... |
| uneq12d 4121 | Equality deduction for the... |
| nfun 4122 | Bound-variable hypothesis ... |
| nfunOLD 4123 | Obsolete version of ~ nfun... |
| unass 4124 | Associative law for union ... |
| un12 4125 | A rearrangement of union. ... |
| un23 4126 | A rearrangement of union. ... |
| un4 4127 | A rearrangement of the uni... |
| unundi 4128 | Union distributes over its... |
| unundir 4129 | Union distributes over its... |
| ssun1 4130 | Subclass relationship for ... |
| ssun2 4131 | Subclass relationship for ... |
| ssun3 4132 | Subclass law for union of ... |
| ssun4 4133 | Subclass law for union of ... |
| elun1 4134 | Membership law for union o... |
| elun2 4135 | Membership law for union o... |
| elunant 4136 | A statement is true for ev... |
| unss1 4137 | Subclass law for union of ... |
| ssequn1 4138 | A relationship between sub... |
| unss2 4139 | Subclass law for union of ... |
| unss12 4140 | Subclass law for union of ... |
| ssequn2 4141 | A relationship between sub... |
| unss 4142 | The union of two subclasse... |
| unssi 4143 | An inference showing the u... |
| unssd 4144 | A deduction showing the un... |
| unssad 4145 | If ` ( A u. B ) ` is conta... |
| unssbd 4146 | If ` ( A u. B ) ` is conta... |
| ssun 4147 | A condition that implies i... |
| rexun 4148 | Restricted existential qua... |
| ralunb 4149 | Restricted quantification ... |
| ralun 4150 | Restricted quantification ... |
| elini 4151 | Membership in an intersect... |
| elind 4152 | Deduce membership in an in... |
| elinel1 4153 | Membership in an intersect... |
| elinel2 4154 | Membership in an intersect... |
| elin2 4155 | Membership in a class defi... |
| elin1d 4156 | Elementhood in the first s... |
| elin2d 4157 | Elementhood in the first s... |
| elin3 4158 | Membership in a class defi... |
| nel1nelin 4159 | Membership in an intersect... |
| nel2nelin 4160 | Membership in an intersect... |
| incom 4161 | Commutative law for inters... |
| ineqcom 4162 | Two ways of expressing tha... |
| ineqcomi 4163 | Two ways of expressing tha... |
| ineqri 4164 | Inference from membership ... |
| ineq1 4165 | Equality theorem for inter... |
| ineq2 4166 | Equality theorem for inter... |
| ineq12 4167 | Equality theorem for inter... |
| ineq1i 4168 | Equality inference for int... |
| ineq2i 4169 | Equality inference for int... |
| ineq12i 4170 | Equality inference for int... |
| ineq1d 4171 | Equality deduction for int... |
| ineq2d 4172 | Equality deduction for int... |
| ineq12d 4173 | Equality deduction for int... |
| ineqan12d 4174 | Equality deduction for int... |
| sseqin2 4175 | A relationship between sub... |
| nfin 4176 | Bound-variable hypothesis ... |
| nfinOLD 4177 | Obsolete version of ~ nfin... |
| rabbi2dva 4178 | Deduction from a wff to a ... |
| inidm 4179 | Idempotent law for interse... |
| inass 4180 | Associative law for inters... |
| in12 4181 | A rearrangement of interse... |
| in32 4182 | A rearrangement of interse... |
| in13 4183 | A rearrangement of interse... |
| in31 4184 | A rearrangement of interse... |
| inrot 4185 | Rotate the intersection of... |
| in4 4186 | Rearrangement of intersect... |
| inindi 4187 | Intersection distributes o... |
| inindir 4188 | Intersection distributes o... |
| inss1 4189 | The intersection of two cl... |
| inss2 4190 | The intersection of two cl... |
| ssin 4191 | Subclass of intersection. ... |
| ssini 4192 | An inference showing that ... |
| ssind 4193 | A deduction showing that a... |
| ssrin 4194 | Add right intersection to ... |
| sslin 4195 | Add left intersection to s... |
| ssrind 4196 | Add right intersection to ... |
| ss2in 4197 | Intersection of subclasses... |
| ssinss1 4198 | Intersection preserves sub... |
| ssinss1d 4199 | Intersection preserves sub... |
| inss 4200 | Inclusion of an intersecti... |
| ralin 4201 | Restricted universal quant... |
| rexin 4202 | Restricted existential qua... |
| dfss7 4203 | Alternate definition of su... |
| symdifcom 4206 | Symmetric difference commu... |
| symdifeq1 4207 | Equality theorem for symme... |
| symdifeq2 4208 | Equality theorem for symme... |
| nfsymdif 4209 | Hypothesis builder for sym... |
| elsymdif 4210 | Membership in a symmetric ... |
| dfsymdif4 4211 | Alternate definition of th... |
| elsymdifxor 4212 | Membership in a symmetric ... |
| dfsymdif2 4213 | Alternate definition of th... |
| symdifass 4214 | Symmetric difference is as... |
| difsssymdif 4215 | The symmetric difference c... |
| difsymssdifssd 4216 | If the symmetric differenc... |
| unabs 4217 | Absorption law for union. ... |
| inabs 4218 | Absorption law for interse... |
| nssinpss 4219 | Negation of subclass expre... |
| nsspssun 4220 | Negation of subclass expre... |
| dfss4 4221 | Subclass defined in terms ... |
| dfun2 4222 | An alternate definition of... |
| dfin2 4223 | An alternate definition of... |
| difin 4224 | Difference with intersecti... |
| ssdifim 4225 | Implication of a class dif... |
| ssdifsym 4226 | Symmetric class difference... |
| dfss5 4227 | Alternate definition of su... |
| dfun3 4228 | Union defined in terms of ... |
| dfin3 4229 | Intersection defined in te... |
| dfin4 4230 | Alternate definition of th... |
| invdif 4231 | Intersection with universa... |
| indif 4232 | Intersection with class di... |
| indif2 4233 | Bring an intersection in a... |
| indif1 4234 | Bring an intersection in a... |
| indifcom 4235 | Commutation law for inters... |
| indi 4236 | Distributive law for inter... |
| undi 4237 | Distributive law for union... |
| indir 4238 | Distributive law for inter... |
| undir 4239 | Distributive law for union... |
| unineq 4240 | Infer equality from equali... |
| uneqin 4241 | Equality of union and inte... |
| difundi 4242 | Distributive law for class... |
| difundir 4243 | Distributive law for class... |
| difindi 4244 | Distributive law for class... |
| difindir 4245 | Distributive law for class... |
| indifdi 4246 | Distribute intersection ov... |
| indifdir 4247 | Distribute intersection ov... |
| difdif2 4248 | Class difference by a clas... |
| undm 4249 | De Morgan's law for union.... |
| indm 4250 | De Morgan's law for inters... |
| difun1 4251 | A relationship involving d... |
| undif3 4252 | An equality involving clas... |
| difin2 4253 | Represent a class differen... |
| dif32 4254 | Swap second and third argu... |
| difabs 4255 | Absorption-like law for cl... |
| sscon34b 4256 | Relative complementation r... |
| rcompleq 4257 | Two subclasses are equal i... |
| dfsymdif3 4258 | Alternate definition of th... |
| unabw 4259 | Union of two class abstrac... |
| unab 4260 | Union of two class abstrac... |
| inab 4261 | Intersection of two class ... |
| difab 4262 | Difference of two class ab... |
| abanssl 4263 | A class abstraction with a... |
| abanssr 4264 | A class abstraction with a... |
| notabw 4265 | A class abstraction define... |
| notab 4266 | A class abstraction define... |
| unrab 4267 | Union of two restricted cl... |
| inrab 4268 | Intersection of two restri... |
| inrab2 4269 | Intersection with a restri... |
| difrab 4270 | Difference of two restrict... |
| dfrab3 4271 | Alternate definition of re... |
| dfrab2 4272 | Alternate definition of re... |
| rabdif 4273 | Move difference in and out... |
| notrab 4274 | Complementation of restric... |
| dfrab3ss 4275 | Restricted class abstracti... |
| rabun2 4276 | Abstraction restricted to ... |
| reuun2 4277 | Transfer uniqueness to a s... |
| reuss2 4278 | Transfer uniqueness to a s... |
| reuss 4279 | Transfer uniqueness to a s... |
| reuun1 4280 | Transfer uniqueness to a s... |
| reupick 4281 | Restricted uniqueness "pic... |
| reupick3 4282 | Restricted uniqueness "pic... |
| reupick2 4283 | Restricted uniqueness "pic... |
| euelss 4284 | Transfer uniqueness of an ... |
| dfnul4 4287 | Alternate definition of th... |
| dfnul2 4288 | Alternate definition of th... |
| dfnul3 4289 | Alternate definition of th... |
| noel 4290 | The empty set has no eleme... |
| nel02 4291 | The empty set has no eleme... |
| n0i 4292 | If a class has elements, t... |
| ne0i 4293 | If a class has elements, t... |
| ne0d 4294 | Deduction form of ~ ne0i .... |
| n0ii 4295 | If a class has elements, t... |
| ne0ii 4296 | If a class has elements, t... |
| vn0 4297 | The universal class is not... |
| vn0ALT 4298 | Alternate proof of ~ vn0 .... |
| eq0f 4299 | A class is equal to the em... |
| neq0f 4300 | A class is not empty if an... |
| n0f 4301 | A class is nonempty if and... |
| eq0 4302 | A class is equal to the em... |
| eq0ALT 4303 | Alternate proof of ~ eq0 .... |
| neq0 4304 | A class is not empty if an... |
| n0 4305 | A class is nonempty if and... |
| nel0 4306 | From the general negation ... |
| reximdva0 4307 | Restricted existence deduc... |
| rspn0 4308 | Specialization for restric... |
| n0rex 4309 | There is an element in a n... |
| ssn0rex 4310 | There is an element in a c... |
| n0moeu 4311 | A case of equivalence of "... |
| rex0 4312 | Vacuous restricted existen... |
| reu0 4313 | Vacuous restricted uniquen... |
| rmo0 4314 | Vacuous restricted at-most... |
| 0el 4315 | Membership of the empty se... |
| n0el 4316 | Negated membership of the ... |
| eqeuel 4317 | A condition which implies ... |
| ssdif0 4318 | Subclass expressed in term... |
| difn0 4319 | If the difference of two s... |
| pssdifn0 4320 | A proper subclass has a no... |
| pssdif 4321 | A proper subclass has a no... |
| ndisj 4322 | Express that an intersecti... |
| inn0f 4323 | A nonempty intersection. ... |
| inn0 4324 | A nonempty intersection. ... |
| difin0ss 4325 | Difference, intersection, ... |
| inssdif0 4326 | Intersection, subclass, an... |
| inindif 4327 | The intersection and class... |
| difid 4328 | The difference between a c... |
| difidALT 4329 | Alternate proof of ~ difid... |
| dif0 4330 | The difference between a c... |
| ab0w 4331 | The class of sets verifyin... |
| ab0 4332 | The class of sets verifyin... |
| ab0ALT 4333 | Alternate proof of ~ ab0 ,... |
| dfnf5 4334 | Characterization of nonfre... |
| ab0orv 4335 | The class abstraction defi... |
| ab0orvALT 4336 | Alternate proof of ~ ab0or... |
| abn0 4337 | Nonempty class abstraction... |
| rab0 4338 | Any restricted class abstr... |
| rabeq0w 4339 | Condition for a restricted... |
| rabeq0 4340 | Condition for a restricted... |
| rabn0 4341 | Nonempty restricted class ... |
| rabxm 4342 | Law of excluded middle, in... |
| rabnc 4343 | Law of noncontradiction, i... |
| elneldisj 4344 | The set of elements ` s ` ... |
| elnelun 4345 | The union of the set of el... |
| un0 4346 | The union of a class with ... |
| in0 4347 | The intersection of a clas... |
| 0un 4348 | The union of the empty set... |
| 0in 4349 | The intersection of the em... |
| inv1 4350 | The intersection of a clas... |
| unv 4351 | The union of a class with ... |
| 0ss 4352 | The null set is a subset o... |
| ss0b 4353 | Any subset of the empty se... |
| ss0 4354 | Any subset of the empty se... |
| sseq0 4355 | A subclass of an empty cla... |
| ssn0 4356 | A class with a nonempty su... |
| 0dif 4357 | The difference between the... |
| abf 4358 | A class abstraction determ... |
| eq0rdv 4359 | Deduction for equality to ... |
| eq0rdvALT 4360 | Alternate proof of ~ eq0rd... |
| csbprc 4361 | The proper substitution of... |
| csb0 4362 | The proper substitution of... |
| sbcel12 4363 | Distribute proper substitu... |
| sbceqg 4364 | Distribute proper substitu... |
| sbceqi 4365 | Distribution of class subs... |
| sbcnel12g 4366 | Distribute proper substitu... |
| sbcne12 4367 | Distribute proper substitu... |
| sbcel1g 4368 | Move proper substitution i... |
| sbceq1g 4369 | Move proper substitution t... |
| sbcel2 4370 | Move proper substitution i... |
| sbceq2g 4371 | Move proper substitution t... |
| csbcom 4372 | Commutative law for double... |
| sbcnestgfw 4373 | Nest the composition of tw... |
| csbnestgfw 4374 | Nest the composition of tw... |
| sbcnestgw 4375 | Nest the composition of tw... |
| csbnestgw 4376 | Nest the composition of tw... |
| sbcco3gw 4377 | Composition of two substit... |
| sbcnestgf 4378 | Nest the composition of tw... |
| csbnestgf 4379 | Nest the composition of tw... |
| sbcnestg 4380 | Nest the composition of tw... |
| csbnestg 4381 | Nest the composition of tw... |
| sbcco3g 4382 | Composition of two substit... |
| csbco3g 4383 | Composition of two class s... |
| csbnest1g 4384 | Nest the composition of tw... |
| csbidm 4385 | Idempotent law for class s... |
| csbvarg 4386 | The proper substitution of... |
| csbvargi 4387 | The proper substitution of... |
| sbccsb 4388 | Substitution into a wff ex... |
| sbccsb2 4389 | Substitution into a wff ex... |
| rspcsbela 4390 | Special case related to ~ ... |
| sbnfc2 4391 | Two ways of expressing " `... |
| csbab 4392 | Move substitution into a c... |
| csbun 4393 | Distribution of class subs... |
| csbin 4394 | Distribute proper substitu... |
| csbie2df 4395 | Conversion of implicit sub... |
| 2nreu 4396 | If there are two different... |
| un00 4397 | Two classes are empty iff ... |
| vss 4398 | Only the universal class h... |
| 0pss 4399 | The null set is a proper s... |
| npss0 4400 | No set is a proper subset ... |
| pssv 4401 | Any non-universal class is... |
| disj 4402 | Two ways of saying that tw... |
| disjr 4403 | Two ways of saying that tw... |
| disj1 4404 | Two ways of saying that tw... |
| reldisj 4405 | Two ways of saying that tw... |
| disj3 4406 | Two ways of saying that tw... |
| disjne 4407 | Members of disjoint sets a... |
| disjeq0 4408 | Two disjoint sets are equa... |
| disjel 4409 | A set can't belong to both... |
| disj2 4410 | Two ways of saying that tw... |
| disj4 4411 | Two ways of saying that tw... |
| ssdisj 4412 | Intersection with a subcla... |
| disjpss 4413 | A class is a proper subset... |
| undisj1 4414 | The union of disjoint clas... |
| undisj2 4415 | The union of disjoint clas... |
| ssindif0 4416 | Subclass expressed in term... |
| inelcm 4417 | The intersection of classe... |
| minel 4418 | A minimum element of a cla... |
| undif4 4419 | Distribute union over diff... |
| disjssun 4420 | Subset relation for disjoi... |
| vdif0 4421 | Universal class equality i... |
| difrab0eq 4422 | If the difference between ... |
| pssnel 4423 | A proper subclass has a me... |
| disjdif 4424 | A class and its relative c... |
| disjdifr 4425 | A class and its relative c... |
| difin0 4426 | The difference of a class ... |
| unvdif 4427 | The union of a class and i... |
| undif1 4428 | Absorption of difference b... |
| undif2 4429 | Absorption of difference b... |
| undifabs 4430 | Absorption of difference b... |
| inundif 4431 | The intersection and class... |
| disjdif2 4432 | The difference of a class ... |
| difun2 4433 | Absorption of union by dif... |
| undif 4434 | Union of complementary par... |
| undifr 4435 | Union of complementary par... |
| undifrOLD 4436 | Obsolete version of ~ undi... |
| undif5 4437 | An equality involving clas... |
| ssdifin0 4438 | A subset of a difference d... |
| ssdifeq0 4439 | A class is a subclass of i... |
| ssundif 4440 | A condition equivalent to ... |
| difcom 4441 | Swap the arguments of a cl... |
| pssdifcom1 4442 | Two ways to express overla... |
| pssdifcom2 4443 | Two ways to express non-co... |
| difdifdir 4444 | Distributive law for class... |
| uneqdifeq 4445 | Two ways to say that ` A `... |
| raldifeq 4446 | Equality theorem for restr... |
| rzal 4447 | Vacuous quantification is ... |
| rzalALT 4448 | Alternate proof of ~ rzal ... |
| rexn0 4449 | Restricted existential qua... |
| ralf0 4450 | The quantification of a fa... |
| ral0 4451 | Vacuous universal quantifi... |
| r19.2z 4452 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4453 | A response to the notion t... |
| r19.3rz 4454 | Restricted quantification ... |
| r19.28z 4455 | Restricted quantifier vers... |
| r19.3rzv 4456 | Restricted quantification ... |
| r19.3rzvOLD 4457 | Obsolete version of ~ r19.... |
| r19.9rzv 4458 | Restricted quantification ... |
| r19.28zv 4459 | Restricted quantifier vers... |
| r19.37zv 4460 | Restricted quantifier vers... |
| r19.45zv 4461 | Restricted version of Theo... |
| r19.44zv 4462 | Restricted version of Theo... |
| r19.27z 4463 | Restricted quantifier vers... |
| r19.27zv 4464 | Restricted quantifier vers... |
| r19.36zv 4465 | Restricted quantifier vers... |
| ralnralall 4466 | A contradiction concerning... |
| falseral0 4467 | A false statement can only... |
| falseral0OLD 4468 | Obsolete version of ~ fals... |
| ralidmw 4469 | Idempotent law for restric... |
| ralidm 4470 | Idempotent law for restric... |
| raaan 4471 | Rearrange restricted quant... |
| raaanv 4472 | Rearrange restricted quant... |
| sbss 4473 | Set substitution into the ... |
| sbcssg 4474 | Distribute proper substitu... |
| raaan2 4475 | Rearrange restricted quant... |
| 2reu4lem 4476 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4477 | Definition of double restr... |
| csbdif 4478 | Distribution of class subs... |
| dfif2 4481 | An alternate definition of... |
| dfif6 4482 | An alternate definition of... |
| ifeq1 4483 | Equality theorem for condi... |
| ifeq2 4484 | Equality theorem for condi... |
| iftrue 4485 | Value of the conditional o... |
| iftruei 4486 | Inference associated with ... |
| iftrued 4487 | Value of the conditional o... |
| iffalse 4488 | Value of the conditional o... |
| iffalsei 4489 | Inference associated with ... |
| iffalsed 4490 | Value of the conditional o... |
| ifnefalse 4491 | When values are unequal, b... |
| iftrueb 4492 | When the branches are not ... |
| ifsb 4493 | Distribute a function over... |
| dfif3 4494 | Alternate definition of th... |
| dfif4 4495 | Alternate definition of th... |
| dfif5 4496 | Alternate definition of th... |
| ifssun 4497 | A conditional class is inc... |
| ifeq12 4498 | Equality theorem for condi... |
| ifeq1d 4499 | Equality deduction for con... |
| ifeq2d 4500 | Equality deduction for con... |
| ifeq12d 4501 | Equality deduction for con... |
| ifbi 4502 | Equivalence theorem for co... |
| ifbid 4503 | Equivalence deduction for ... |
| ifbieq1d 4504 | Equivalence/equality deduc... |
| ifbieq2i 4505 | Equivalence/equality infer... |
| ifbieq2d 4506 | Equivalence/equality deduc... |
| ifbieq12i 4507 | Equivalence deduction for ... |
| ifbieq12d 4508 | Equivalence deduction for ... |
| nfifd 4509 | Deduction form of ~ nfif .... |
| nfif 4510 | Bound-variable hypothesis ... |
| ifeq1da 4511 | Conditional equality. (Co... |
| ifeq2da 4512 | Conditional equality. (Co... |
| ifeq12da 4513 | Equivalence deduction for ... |
| ifbieq12d2 4514 | Equivalence deduction for ... |
| ifclda 4515 | Conditional closure. (Con... |
| ifeqda 4516 | Separation of the values o... |
| elimif 4517 | Elimination of a condition... |
| ifbothda 4518 | A wff ` th ` containing a ... |
| ifboth 4519 | A wff ` th ` containing a ... |
| ifid 4520 | Identical true and false a... |
| eqif 4521 | Expansion of an equality w... |
| ifval 4522 | Another expression of the ... |
| elif 4523 | Membership in a conditiona... |
| ifel 4524 | Membership of a conditiona... |
| ifcl 4525 | Membership (closure) of a ... |
| ifcld 4526 | Membership (closure) of a ... |
| ifcli 4527 | Inference associated with ... |
| ifexd 4528 | Existence of the condition... |
| ifexg 4529 | Existence of the condition... |
| ifex 4530 | Existence of the condition... |
| ifeqor 4531 | The possible values of a c... |
| ifnot 4532 | Negating the first argumen... |
| ifan 4533 | Rewrite a conjunction in a... |
| ifor 4534 | Rewrite a disjunction in a... |
| 2if2 4535 | Resolve two nested conditi... |
| ifcomnan 4536 | Commute the conditions in ... |
| csbif 4537 | Distribute proper substitu... |
| dedth 4538 | Weak deduction theorem tha... |
| dedth2h 4539 | Weak deduction theorem eli... |
| dedth3h 4540 | Weak deduction theorem eli... |
| dedth4h 4541 | Weak deduction theorem eli... |
| dedth2v 4542 | Weak deduction theorem for... |
| dedth3v 4543 | Weak deduction theorem for... |
| dedth4v 4544 | Weak deduction theorem for... |
| elimhyp 4545 | Eliminate a hypothesis con... |
| elimhyp2v 4546 | Eliminate a hypothesis con... |
| elimhyp3v 4547 | Eliminate a hypothesis con... |
| elimhyp4v 4548 | Eliminate a hypothesis con... |
| elimel 4549 | Eliminate a membership hyp... |
| elimdhyp 4550 | Version of ~ elimhyp where... |
| keephyp 4551 | Transform a hypothesis ` p... |
| keephyp2v 4552 | Keep a hypothesis containi... |
| keephyp3v 4553 | Keep a hypothesis containi... |
| pwjust 4555 | Soundness justification th... |
| elpwg 4557 | Membership in a power clas... |
| elpw 4558 | Membership in a power clas... |
| velpw 4559 | Setvar variable membership... |
| elpwd 4560 | Membership in a power clas... |
| elpwi 4561 | Subset relation implied by... |
| elpwb 4562 | Characterization of the el... |
| elpwid 4563 | An element of a power clas... |
| elelpwi 4564 | If ` A ` belongs to a part... |
| sspw 4565 | The powerclass preserves i... |
| sspwi 4566 | The powerclass preserves i... |
| sspwd 4567 | The powerclass preserves i... |
| pweq 4568 | Equality theorem for power... |
| pweqALT 4569 | Alternate proof of ~ pweq ... |
| pweqi 4570 | Equality inference for pow... |
| pweqd 4571 | Equality deduction for pow... |
| pwunss 4572 | The power class of the uni... |
| nfpw 4573 | Bound-variable hypothesis ... |
| pwidg 4574 | A set is an element of its... |
| pwidb 4575 | A class is an element of i... |
| pwid 4576 | A set is a member of its p... |
| pwss 4577 | Subclass relationship for ... |
| pwundif 4578 | Break up the power class o... |
| snjust 4579 | Soundness justification th... |
| sneq 4590 | Equality theorem for singl... |
| sneqi 4591 | Equality inference for sin... |
| sneqd 4592 | Equality deduction for sin... |
| dfsn2 4593 | Alternate definition of si... |
| elsng 4594 | There is exactly one eleme... |
| elsn 4595 | There is exactly one eleme... |
| velsn 4596 | There is only one element ... |
| elsni 4597 | There is at most one eleme... |
| elsnd 4598 | There is at most one eleme... |
| rabsneq 4599 | Equality of class abstract... |
| absn 4600 | Condition for a class abst... |
| dfpr2 4601 | Alternate definition of a ... |
| dfsn2ALT 4602 | Alternate definition of si... |
| elprg 4603 | A member of a pair of clas... |
| elpri 4604 | If a class is an element o... |
| elpr 4605 | A member of a pair of clas... |
| elpr2g 4606 | A member of a pair of sets... |
| elpr2 4607 | A member of a pair of sets... |
| elprn1 4608 | A member of an unordered p... |
| elprn2 4609 | A member of an unordered p... |
| nelpr2 4610 | If a class is not an eleme... |
| nelpr1 4611 | If a class is not an eleme... |
| nelpri 4612 | If an element doesn't matc... |
| prneli 4613 | If an element doesn't matc... |
| nelprd 4614 | If an element doesn't matc... |
| eldifpr 4615 | Membership in a set with t... |
| rexdifpr 4616 | Restricted existential qua... |
| snidg 4617 | A set is a member of its s... |
| snidb 4618 | A class is a set iff it is... |
| snid 4619 | A set is a member of its s... |
| vsnid 4620 | A setvar variable is a mem... |
| elsn2g 4621 | There is exactly one eleme... |
| elsn2 4622 | There is exactly one eleme... |
| nelsn 4623 | If a class is not equal to... |
| rabeqsn 4624 | Conditions for a restricte... |
| rabsssn 4625 | Conditions for a restricte... |
| rabeqsnd 4626 | Conditions for a restricte... |
| ralsnsg 4627 | Substitution expressed in ... |
| rexsns 4628 | Restricted existential qua... |
| rexsngf 4629 | Restricted existential qua... |
| ralsngf 4630 | Restricted universal quant... |
| reusngf 4631 | Restricted existential uni... |
| ralsng 4632 | Substitution expressed in ... |
| rexsng 4633 | Restricted existential qua... |
| reusng 4634 | Restricted existential uni... |
| 2ralsng 4635 | Substitution expressed in ... |
| rexreusng 4636 | Restricted existential uni... |
| exsnrex 4637 | There is a set being the e... |
| ralsn 4638 | Convert a universal quanti... |
| rexsn 4639 | Convert an existential qua... |
| elunsn 4640 | Elementhood in a union wit... |
| elpwunsn 4641 | Membership in an extension... |
| eqoreldif 4642 | An element of a set is eit... |
| eltpg 4643 | Members of an unordered tr... |
| eldiftp 4644 | Membership in a set with t... |
| eltpi 4645 | A member of an unordered t... |
| eltp 4646 | A member of an unordered t... |
| el7g 4647 | Members of a set with seve... |
| dftp2 4648 | Alternate definition of un... |
| nfpr 4649 | Bound-variable hypothesis ... |
| ifpr 4650 | Membership of a conditiona... |
| ralprgf 4651 | Convert a restricted unive... |
| rexprgf 4652 | Convert a restricted exist... |
| ralprg 4653 | Convert a restricted unive... |
| rexprg 4654 | Convert a restricted exist... |
| raltpg 4655 | Convert a restricted unive... |
| rextpg 4656 | Convert a restricted exist... |
| ralpr 4657 | Convert a restricted unive... |
| rexpr 4658 | Convert a restricted exist... |
| reuprg0 4659 | Convert a restricted exist... |
| reuprg 4660 | Convert a restricted exist... |
| reurexprg 4661 | Convert a restricted exist... |
| raltp 4662 | Convert a universal quanti... |
| rextp 4663 | Convert an existential qua... |
| nfsn 4664 | Bound-variable hypothesis ... |
| csbsng 4665 | Distribute proper substitu... |
| csbprg 4666 | Distribute proper substitu... |
| elinsn 4667 | If the intersection of two... |
| disjsn 4668 | Intersection with the sing... |
| disjsn2 4669 | Two distinct singletons ar... |
| disjpr2 4670 | Two completely distinct un... |
| disjprsn 4671 | The disjoint intersection ... |
| disjtpsn 4672 | The disjoint intersection ... |
| disjtp2 4673 | Two completely distinct un... |
| snprc 4674 | The singleton of a proper ... |
| snnzb 4675 | A singleton is nonempty if... |
| rmosn 4676 | A restricted at-most-one q... |
| r19.12sn 4677 | Special case of ~ r19.12 w... |
| rabsn 4678 | Condition where a restrict... |
| rabsnifsb 4679 | A restricted class abstrac... |
| rabsnif 4680 | A restricted class abstrac... |
| rabrsn 4681 | A restricted class abstrac... |
| euabsn2 4682 | Another way to express exi... |
| euabsn 4683 | Another way to express exi... |
| reusn 4684 | A way to express restricte... |
| absneu 4685 | Restricted existential uni... |
| rabsneu 4686 | Restricted existential uni... |
| eusn 4687 | Two ways to express " ` A ... |
| rabsnt 4688 | Truth implied by equality ... |
| prcom 4689 | Commutative law for unorde... |
| preq1 4690 | Equality theorem for unord... |
| preq2 4691 | Equality theorem for unord... |
| preq12 4692 | Equality theorem for unord... |
| preq1i 4693 | Equality inference for uno... |
| preq2i 4694 | Equality inference for uno... |
| preq12i 4695 | Equality inference for uno... |
| preq1d 4696 | Equality deduction for uno... |
| preq2d 4697 | Equality deduction for uno... |
| preq12d 4698 | Equality deduction for uno... |
| tpeq1 4699 | Equality theorem for unord... |
| tpeq2 4700 | Equality theorem for unord... |
| tpeq3 4701 | Equality theorem for unord... |
| tpeq1d 4702 | Equality theorem for unord... |
| tpeq2d 4703 | Equality theorem for unord... |
| tpeq3d 4704 | Equality theorem for unord... |
| tpeq123d 4705 | Equality theorem for unord... |
| tprot 4706 | Rotation of the elements o... |
| tpcoma 4707 | Swap 1st and 2nd members o... |
| tpcomb 4708 | Swap 2nd and 3rd members o... |
| tpass 4709 | Split off the first elemen... |
| qdass 4710 | Two ways to write an unord... |
| qdassr 4711 | Two ways to write an unord... |
| tpidm12 4712 | Unordered triple ` { A , A... |
| tpidm13 4713 | Unordered triple ` { A , B... |
| tpidm23 4714 | Unordered triple ` { A , B... |
| tpidm 4715 | Unordered triple ` { A , A... |
| tppreq3 4716 | An unordered triple is an ... |
| prid1g 4717 | An unordered pair contains... |
| prid2g 4718 | An unordered pair contains... |
| prid1 4719 | An unordered pair contains... |
| prid2 4720 | An unordered pair contains... |
| ifpprsnss 4721 | An unordered pair is a sin... |
| prprc1 4722 | A proper class vanishes in... |
| prprc2 4723 | A proper class vanishes in... |
| prprc 4724 | An unordered pair containi... |
| tpid1 4725 | One of the three elements ... |
| tpid1g 4726 | Closed theorem form of ~ t... |
| tpid2 4727 | One of the three elements ... |
| tpid2g 4728 | Closed theorem form of ~ t... |
| tpid3g 4729 | Closed theorem form of ~ t... |
| tpid3 4730 | One of the three elements ... |
| snnzg 4731 | The singleton of a set is ... |
| snn0d 4732 | The singleton of a set is ... |
| snnz 4733 | The singleton of a set is ... |
| prnz 4734 | A pair containing a set is... |
| prnzg 4735 | A pair containing a set is... |
| tpnz 4736 | An unordered triple contai... |
| tpnzd 4737 | An unordered triple contai... |
| raltpd 4738 | Convert a universal quanti... |
| snssb 4739 | Characterization of the in... |
| snssg 4740 | The singleton formed on a ... |
| snss 4741 | The singleton of an elemen... |
| eldifsn 4742 | Membership in a set with a... |
| eldifsnd 4743 | Membership in a set with a... |
| ssdifsn 4744 | Subset of a set with an el... |
| elpwdifsn 4745 | A subset of a set is an el... |
| eldifsni 4746 | Membership in a set with a... |
| eldifsnneq 4747 | An element of a difference... |
| neldifsn 4748 | The class ` A ` is not in ... |
| neldifsnd 4749 | The class ` A ` is not in ... |
| rexdifsn 4750 | Restricted existential qua... |
| raldifsni 4751 | Rearrangement of a propert... |
| raldifsnb 4752 | Restricted universal quant... |
| eldifvsn 4753 | A set is an element of the... |
| difsn 4754 | An element not in a set ca... |
| difprsnss 4755 | Removal of a singleton fro... |
| difprsn1 4756 | Removal of a singleton fro... |
| difprsn2 4757 | Removal of a singleton fro... |
| diftpsn3 4758 | Removal of a singleton fro... |
| difpr 4759 | Removing two elements as p... |
| tpprceq3 4760 | An unordered triple is an ... |
| tppreqb 4761 | An unordered triple is an ... |
| difsnb 4762 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4763 | ` ( B \ { A } ) ` is a pro... |
| snssi 4764 | The singleton of an elemen... |
| snssd 4765 | The singleton of an elemen... |
| difsnid 4766 | If we remove a single elem... |
| eldifeldifsn 4767 | An element of a difference... |
| pw0 4768 | Compute the power set of t... |
| pwpw0 4769 | Compute the power set of t... |
| snsspr1 4770 | A singleton is a subset of... |
| snsspr2 4771 | A singleton is a subset of... |
| snsstp1 4772 | A singleton is a subset of... |
| snsstp2 4773 | A singleton is a subset of... |
| snsstp3 4774 | A singleton is a subset of... |
| prssg 4775 | A pair of elements of a cl... |
| prss 4776 | A pair of elements of a cl... |
| prssi 4777 | A pair of elements of a cl... |
| prssd 4778 | Deduction version of ~ prs... |
| prsspwg 4779 | An unordered pair belongs ... |
| ssprss 4780 | A pair as subset of a pair... |
| ssprsseq 4781 | A proper pair is a subset ... |
| sssn 4782 | The subsets of a singleton... |
| ssunsn2 4783 | The property of being sand... |
| ssunsn 4784 | Possible values for a set ... |
| eqsn 4785 | Two ways to express that a... |
| eqsnd 4786 | Deduce that a set is a sin... |
| eqsndOLD 4787 | Obsolete version of ~ eqsn... |
| issn 4788 | A sufficient condition for... |
| n0snor2el 4789 | A nonempty set is either a... |
| ssunpr 4790 | Possible values for a set ... |
| sspr 4791 | The subsets of a pair. (C... |
| sstp 4792 | The subsets of an unordere... |
| tpss 4793 | An unordered triple of ele... |
| tpssi 4794 | An unordered triple of ele... |
| sneqrg 4795 | Closed form of ~ sneqr . ... |
| sneqr 4796 | If the singletons of two s... |
| snsssn 4797 | If a singleton is a subset... |
| mosneq 4798 | There exists at most one s... |
| sneqbg 4799 | Two singletons of sets are... |
| snsspw 4800 | The singleton of a class i... |
| prsspw 4801 | An unordered pair belongs ... |
| preq1b 4802 | Biconditional equality lem... |
| preq2b 4803 | Biconditional equality lem... |
| preqr1 4804 | Reverse equality lemma for... |
| preqr2 4805 | Reverse equality lemma for... |
| preq12b 4806 | Equality relationship for ... |
| opthpr 4807 | An unordered pair has the ... |
| preqr1g 4808 | Reverse equality lemma for... |
| preq12bg 4809 | Closed form of ~ preq12b .... |
| prneimg 4810 | Two pairs are not equal if... |
| prneimg2 4811 | Two pairs are not equal if... |
| prnebg 4812 | A (proper) pair is not equ... |
| pr1eqbg 4813 | A (proper) pair is equal t... |
| pr1nebg 4814 | A (proper) pair is not equ... |
| preqsnd 4815 | Equivalence for a pair equ... |
| prnesn 4816 | A proper unordered pair is... |
| prneprprc 4817 | A proper unordered pair is... |
| preqsn 4818 | Equivalence for a pair equ... |
| preq12nebg 4819 | Equality relationship for ... |
| prel12g 4820 | Equality of two unordered ... |
| opthprneg 4821 | An unordered pair has the ... |
| elpreqprlem 4822 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4823 | Equality and membership ru... |
| elpreqprb 4824 | A set is an element of an ... |
| elpr2elpr 4825 | For an element ` A ` of an... |
| dfopif 4826 | Rewrite ~ df-op using ` if... |
| dfopg 4827 | Value of the ordered pair ... |
| dfop 4828 | Value of an ordered pair w... |
| opeq1 4829 | Equality theorem for order... |
| opeq2 4830 | Equality theorem for order... |
| opeq12 4831 | Equality theorem for order... |
| opeq1i 4832 | Equality inference for ord... |
| opeq2i 4833 | Equality inference for ord... |
| opeq12i 4834 | Equality inference for ord... |
| opeq1d 4835 | Equality deduction for ord... |
| opeq2d 4836 | Equality deduction for ord... |
| opeq12d 4837 | Equality deduction for ord... |
| oteq1 4838 | Equality theorem for order... |
| oteq2 4839 | Equality theorem for order... |
| oteq3 4840 | Equality theorem for order... |
| oteq1d 4841 | Equality deduction for ord... |
| oteq2d 4842 | Equality deduction for ord... |
| oteq3d 4843 | Equality deduction for ord... |
| oteq123d 4844 | Equality deduction for ord... |
| nfop 4845 | Bound-variable hypothesis ... |
| nfopd 4846 | Deduction version of bound... |
| csbopg 4847 | Distribution of class subs... |
| opidg 4848 | The ordered pair ` <. A , ... |
| opid 4849 | The ordered pair ` <. A , ... |
| ralunsn 4850 | Restricted quantification ... |
| 2ralunsn 4851 | Double restricted quantifi... |
| opprc 4852 | Expansion of an ordered pa... |
| opprc1 4853 | Expansion of an ordered pa... |
| opprc2 4854 | Expansion of an ordered pa... |
| oprcl 4855 | If an ordered pair has an ... |
| pwsn 4856 | The power set of a singlet... |
| pwpr 4857 | The power set of an unorde... |
| pwtp 4858 | The power set of an unorde... |
| pwpwpw0 4859 | Compute the power set of t... |
| pwv 4860 | The power class of the uni... |
| prproe 4861 | For an element of a proper... |
| 3elpr2eq 4862 | If there are three element... |
| dfuni2 4865 | Alternate definition of cl... |
| eluni 4866 | Membership in class union.... |
| eluni2 4867 | Membership in class union.... |
| elunii 4868 | Membership in class union.... |
| nfunid 4869 | Deduction version of ~ nfu... |
| nfuni 4870 | Bound-variable hypothesis ... |
| uniss 4871 | Subclass relationship for ... |
| unissi 4872 | Subclass relationship for ... |
| unissd 4873 | Subclass relationship for ... |
| unieq 4874 | Equality theorem for class... |
| unieqi 4875 | Inference of equality of t... |
| unieqd 4876 | Deduction of equality of t... |
| eluniab 4877 | Membership in union of a c... |
| elunirab 4878 | Membership in union of a c... |
| uniprg 4879 | The union of a pair is the... |
| unipr 4880 | The union of a pair is the... |
| unisng 4881 | A set equals the union of ... |
| unisn 4882 | A set equals the union of ... |
| unisnv 4883 | A set equals the union of ... |
| unisn3 4884 | Union of a singleton in th... |
| dfnfc2 4885 | An alternative statement o... |
| uniun 4886 | The class union of the uni... |
| uniin 4887 | The class union of the int... |
| ssuni 4888 | Subclass relationship for ... |
| uni0b 4889 | The union of a set is empt... |
| uni0c 4890 | The union of a set is empt... |
| uni0 4891 | The union of the empty set... |
| uni0OLD 4892 | Obsolete version of ~ uni0... |
| csbuni 4893 | Distribute proper substitu... |
| elssuni 4894 | An element of a class is a... |
| unissel 4895 | Condition turning a subcla... |
| unissb 4896 | Relationship involving mem... |
| uniss2 4897 | A subclass condition on th... |
| unidif 4898 | If the difference ` A \ B ... |
| ssunieq 4899 | Relationship implying unio... |
| unimax 4900 | Any member of a class is t... |
| pwuni 4901 | A class is a subclass of t... |
| dfint2 4904 | Alternate definition of cl... |
| inteq 4905 | Equality law for intersect... |
| inteqi 4906 | Equality inference for cla... |
| inteqd 4907 | Equality deduction for cla... |
| elint 4908 | Membership in class inters... |
| elint2 4909 | Membership in class inters... |
| elintg 4910 | Membership in class inters... |
| elinti 4911 | Membership in class inters... |
| nfint 4912 | Bound-variable hypothesis ... |
| elintabg 4913 | Two ways of saying a set i... |
| elintab 4914 | Membership in the intersec... |
| elintrab 4915 | Membership in the intersec... |
| elintrabg 4916 | Membership in the intersec... |
| int0 4917 | The intersection of the em... |
| intss1 4918 | An element of a class incl... |
| ssint 4919 | Subclass of a class inters... |
| ssintab 4920 | Subclass of the intersecti... |
| ssintub 4921 | Subclass of the least uppe... |
| ssmin 4922 | Subclass of the minimum va... |
| intmin 4923 | Any member of a class is t... |
| intss 4924 | Intersection of subclasses... |
| intssuni 4925 | The intersection of a none... |
| ssintrab 4926 | Subclass of the intersecti... |
| unissint 4927 | If the union of a class is... |
| intssuni2 4928 | Subclass relationship for ... |
| intminss 4929 | Under subset ordering, the... |
| intmin2 4930 | Any set is the smallest of... |
| intmin3 4931 | Under subset ordering, the... |
| intmin4 4932 | Elimination of a conjunct ... |
| intab 4933 | The intersection of a spec... |
| int0el 4934 | The intersection of a clas... |
| intun 4935 | The class intersection of ... |
| intprg 4936 | The intersection of a pair... |
| intpr 4937 | The intersection of a pair... |
| intsng 4938 | Intersection of a singleto... |
| intsn 4939 | The intersection of a sing... |
| uniintsn 4940 | Two ways to express " ` A ... |
| uniintab 4941 | The union and the intersec... |
| intunsn 4942 | Theorem joining a singleto... |
| rint0 4943 | Relative intersection of a... |
| elrint 4944 | Membership in a restricted... |
| elrint2 4945 | Membership in a restricted... |
| eliun 4950 | Membership in indexed unio... |
| eliin 4951 | Membership in indexed inte... |
| eliuni 4952 | Membership in an indexed u... |
| eliund 4953 | Membership in indexed unio... |
| iuncom 4954 | Commutation of indexed uni... |
| iuncom4 4955 | Commutation of union with ... |
| iunconst 4956 | Indexed union of a constan... |
| iinconst 4957 | Indexed intersection of a ... |
| iuneqconst 4958 | Indexed union of identical... |
| iuniin 4959 | Law combining indexed unio... |
| iinssiun 4960 | An indexed intersection is... |
| iunss1 4961 | Subclass theorem for index... |
| iinss1 4962 | Subclass theorem for index... |
| iuneq1 4963 | Equality theorem for index... |
| iineq1 4964 | Equality theorem for index... |
| ss2iun 4965 | Subclass theorem for index... |
| iuneq2 4966 | Equality theorem for index... |
| iineq2 4967 | Equality theorem for index... |
| iuneq2i 4968 | Equality inference for ind... |
| iineq2i 4969 | Equality inference for ind... |
| iineq2d 4970 | Equality deduction for ind... |
| iuneq2dv 4971 | Equality deduction for ind... |
| iineq2dv 4972 | Equality deduction for ind... |
| iuneq12df 4973 | Equality deduction for ind... |
| iuneq1d 4974 | Equality theorem for index... |
| iuneq12dOLD 4975 | Obsolete version of ~ iune... |
| iuneq12d 4976 | Equality deduction for ind... |
| iuneq2d 4977 | Equality deduction for ind... |
| nfiun 4978 | Bound-variable hypothesis ... |
| nfiin 4979 | Bound-variable hypothesis ... |
| nfiung 4980 | Bound-variable hypothesis ... |
| nfiing 4981 | Bound-variable hypothesis ... |
| nfiu1 4982 | Bound-variable hypothesis ... |
| nfiu1OLD 4983 | Obsolete version of ~ nfiu... |
| nfii1 4984 | Bound-variable hypothesis ... |
| dfiun2g 4985 | Alternate definition of in... |
| dfiin2g 4986 | Alternate definition of in... |
| dfiun2 4987 | Alternate definition of in... |
| dfiin2 4988 | Alternate definition of in... |
| dfiunv2 4989 | Define double indexed unio... |
| cbviun 4990 | Rule used to change the bo... |
| cbviin 4991 | Change bound variables in ... |
| cbviung 4992 | Rule used to change the bo... |
| cbviing 4993 | Change bound variables in ... |
| cbviunv 4994 | Rule used to change the bo... |
| cbviinv 4995 | Change bound variables in ... |
| cbviunvg 4996 | Rule used to change the bo... |
| cbviinvg 4997 | Change bound variables in ... |
| iunssf 4998 | Subset theorem for an inde... |
| iunssfOLD 4999 | Obsolete version of ~ iuns... |
| iunss 5000 | Subset theorem for an inde... |
| iunssOLD 5001 | Obsolete version of ~ iuns... |
| ssiun 5002 | Subset implication for an ... |
| ssiun2 5003 | Identity law for subset of... |
| ssiun2s 5004 | Subset relationship for an... |
| iunss2 5005 | A subclass condition on th... |
| iunssd 5006 | Subset theorem for an inde... |
| iunab 5007 | The indexed union of a cla... |
| iunrab 5008 | The indexed union of a res... |
| iunxdif2 5009 | Indexed union with a class... |
| ssiinf 5010 | Subset theorem for an inde... |
| ssiin 5011 | Subset theorem for an inde... |
| iinss 5012 | Subset implication for an ... |
| iinss2 5013 | An indexed intersection is... |
| uniiun 5014 | Class union in terms of in... |
| intiin 5015 | Class intersection in term... |
| iunid 5016 | An indexed union of single... |
| iun0 5017 | An indexed union of the em... |
| 0iun 5018 | An empty indexed union is ... |
| 0iin 5019 | An empty indexed intersect... |
| viin 5020 | Indexed intersection with ... |
| iunsn 5021 | Indexed union of a singlet... |
| iunn0 5022 | There is a nonempty class ... |
| iinab 5023 | Indexed intersection of a ... |
| iinrab 5024 | Indexed intersection of a ... |
| iinrab2 5025 | Indexed intersection of a ... |
| iunin2 5026 | Indexed union of intersect... |
| iunin1 5027 | Indexed union of intersect... |
| iinun2 5028 | Indexed intersection of un... |
| iundif2 5029 | Indexed union of class dif... |
| iindif1 5030 | Indexed intersection of cl... |
| 2iunin 5031 | Rearrange indexed unions o... |
| iindif2 5032 | Indexed intersection of cl... |
| iinin2 5033 | Indexed intersection of in... |
| iinin1 5034 | Indexed intersection of in... |
| iinvdif 5035 | The indexed intersection o... |
| elriin 5036 | Elementhood in a relative ... |
| riin0 5037 | Relative intersection of a... |
| riinn0 5038 | Relative intersection of a... |
| riinrab 5039 | Relative intersection of a... |
| symdif0 5040 | Symmetric difference with ... |
| symdifv 5041 | The symmetric difference w... |
| symdifid 5042 | The symmetric difference o... |
| iinxsng 5043 | A singleton index picks ou... |
| iinxprg 5044 | Indexed intersection with ... |
| iunxsng 5045 | A singleton index picks ou... |
| iunxsn 5046 | A singleton index picks ou... |
| iunxsngf 5047 | A singleton index picks ou... |
| iunun 5048 | Separate a union in an ind... |
| iunxun 5049 | Separate a union in the in... |
| iunxdif3 5050 | An indexed union where som... |
| iunxprg 5051 | A pair index picks out two... |
| iunxiun 5052 | Separate an indexed union ... |
| iinuni 5053 | A relationship involving u... |
| iununi 5054 | A relationship involving u... |
| sspwuni 5055 | Subclass relationship for ... |
| pwssb 5056 | Two ways to express a coll... |
| elpwpw 5057 | Characterization of the el... |
| pwpwab 5058 | The double power class wri... |
| pwpwssunieq 5059 | The class of sets whose un... |
| elpwuni 5060 | Relationship for power cla... |
| iinpw 5061 | The power class of an inte... |
| iunpwss 5062 | Inclusion of an indexed un... |
| intss2 5063 | A nonempty intersection of... |
| rintn0 5064 | Relative intersection of a... |
| dfdisj2 5067 | Alternate definition for d... |
| disjss2 5068 | If each element of a colle... |
| disjeq2 5069 | Equality theorem for disjo... |
| disjeq2dv 5070 | Equality deduction for dis... |
| disjss1 5071 | A subset of a disjoint col... |
| disjeq1 5072 | Equality theorem for disjo... |
| disjeq1d 5073 | Equality theorem for disjo... |
| disjeq12d 5074 | Equality theorem for disjo... |
| cbvdisj 5075 | Change bound variables in ... |
| cbvdisjv 5076 | Change bound variables in ... |
| nfdisjw 5077 | Bound-variable hypothesis ... |
| nfdisj 5078 | Bound-variable hypothesis ... |
| nfdisj1 5079 | Bound-variable hypothesis ... |
| disjor 5080 | Two ways to say that a col... |
| disjors 5081 | Two ways to say that a col... |
| disji2 5082 | Property of a disjoint col... |
| disji 5083 | Property of a disjoint col... |
| invdisj 5084 | If there is a function ` C... |
| invdisjrab 5085 | The restricted class abstr... |
| disjiun 5086 | A disjoint collection yiel... |
| disjord 5087 | Conditions for a collectio... |
| disjiunb 5088 | Two ways to say that a col... |
| disjiund 5089 | Conditions for a collectio... |
| sndisj 5090 | Any collection of singleto... |
| 0disj 5091 | Any collection of empty se... |
| disjxsn 5092 | A singleton collection is ... |
| disjx0 5093 | An empty collection is dis... |
| disjprg 5094 | A pair collection is disjo... |
| disjxiun 5095 | An indexed union of a disj... |
| disjxun 5096 | The union of two disjoint ... |
| disjss3 5097 | Expand a disjoint collecti... |
| breq 5100 | Equality theorem for binar... |
| breq1 5101 | Equality theorem for a bin... |
| breq2 5102 | Equality theorem for a bin... |
| breq12 5103 | Equality theorem for a bin... |
| breqi 5104 | Equality inference for bin... |
| breq1i 5105 | Equality inference for a b... |
| breq2i 5106 | Equality inference for a b... |
| breq12i 5107 | Equality inference for a b... |
| breq1d 5108 | Equality deduction for a b... |
| breqd 5109 | Equality deduction for a b... |
| breq2d 5110 | Equality deduction for a b... |
| breq12d 5111 | Equality deduction for a b... |
| breq123d 5112 | Equality deduction for a b... |
| breqdi 5113 | Equality deduction for a b... |
| breqan12d 5114 | Equality deduction for a b... |
| breqan12rd 5115 | Equality deduction for a b... |
| eqnbrtrd 5116 | Substitution of equal clas... |
| nbrne1 5117 | Two classes are different ... |
| nbrne2 5118 | Two classes are different ... |
| eqbrtri 5119 | Substitution of equal clas... |
| eqbrtrd 5120 | Substitution of equal clas... |
| eqbrtrri 5121 | Substitution of equal clas... |
| eqbrtrrd 5122 | Substitution of equal clas... |
| breqtri 5123 | Substitution of equal clas... |
| breqtrd 5124 | Substitution of equal clas... |
| breqtrri 5125 | Substitution of equal clas... |
| breqtrrd 5126 | Substitution of equal clas... |
| 3brtr3i 5127 | Substitution of equality i... |
| 3brtr4i 5128 | Substitution of equality i... |
| 3brtr3d 5129 | Substitution of equality i... |
| 3brtr4d 5130 | Substitution of equality i... |
| 3brtr3g 5131 | Substitution of equality i... |
| 3brtr4g 5132 | Substitution of equality i... |
| eqbrtrid 5133 | A chained equality inferen... |
| eqbrtrrid 5134 | A chained equality inferen... |
| breqtrid 5135 | A chained equality inferen... |
| breqtrrid 5136 | A chained equality inferen... |
| eqbrtrdi 5137 | A chained equality inferen... |
| eqbrtrrdi 5138 | A chained equality inferen... |
| breqtrdi 5139 | A chained equality inferen... |
| breqtrrdi 5140 | A chained equality inferen... |
| ssbrd 5141 | Deduction from a subclass ... |
| ssbr 5142 | Implication from a subclas... |
| ssbri 5143 | Inference from a subclass ... |
| nfbrd 5144 | Deduction version of bound... |
| nfbr 5145 | Bound-variable hypothesis ... |
| brab1 5146 | Relationship between a bin... |
| br0 5147 | The empty binary relation ... |
| brne0 5148 | If two sets are in a binar... |
| brun 5149 | The union of two binary re... |
| brin 5150 | The intersection of two re... |
| brdif 5151 | The difference of two bina... |
| sbcbr123 5152 | Move substitution in and o... |
| sbcbr 5153 | Move substitution in and o... |
| sbcbr12g 5154 | Move substitution in and o... |
| sbcbr1g 5155 | Move substitution in and o... |
| sbcbr2g 5156 | Move substitution in and o... |
| brsymdif 5157 | Characterization of the sy... |
| brralrspcev 5158 | Restricted existential spe... |
| brimralrspcev 5159 | Restricted existential spe... |
| opabss 5162 | The collection of ordered ... |
| opabbid 5163 | Equivalent wff's yield equ... |
| opabbidv 5164 | Equivalent wff's yield equ... |
| opabbii 5165 | Equivalent wff's yield equ... |
| nfopabd 5166 | Bound-variable hypothesis ... |
| nfopab 5167 | Bound-variable hypothesis ... |
| nfopab1 5168 | The first abstraction vari... |
| nfopab2 5169 | The second abstraction var... |
| cbvopab 5170 | Rule used to change bound ... |
| cbvopabv 5171 | Rule used to change bound ... |
| cbvopab1 5172 | Change first bound variabl... |
| cbvopab1g 5173 | Change first bound variabl... |
| cbvopab2 5174 | Change second bound variab... |
| cbvopab1s 5175 | Change first bound variabl... |
| cbvopab1v 5176 | Rule used to change the fi... |
| cbvopab2v 5177 | Rule used to change the se... |
| unopab 5178 | Union of two ordered pair ... |
| mpteq12da 5181 | An equality inference for ... |
| mpteq12df 5182 | An equality inference for ... |
| mpteq12f 5183 | An equality theorem for th... |
| mpteq12dva 5184 | An equality inference for ... |
| mpteq12dv 5185 | An equality inference for ... |
| mpteq12 5186 | An equality theorem for th... |
| mpteq1 5187 | An equality theorem for th... |
| mpteq1d 5188 | An equality theorem for th... |
| mpteq1i 5189 | An equality theorem for th... |
| mpteq2da 5190 | Slightly more general equa... |
| mpteq2dva 5191 | Slightly more general equa... |
| mpteq2dv 5192 | An equality inference for ... |
| mpteq2ia 5193 | An equality inference for ... |
| mpteq2i 5194 | An equality inference for ... |
| mpteq12i 5195 | An equality inference for ... |
| nfmpt 5196 | Bound-variable hypothesis ... |
| nfmpt1 5197 | Bound-variable hypothesis ... |
| cbvmptf 5198 | Rule to change the bound v... |
| cbvmptfg 5199 | Rule to change the bound v... |
| cbvmpt 5200 | Rule to change the bound v... |
| cbvmptg 5201 | Rule to change the bound v... |
| cbvmptv 5202 | Rule to change the bound v... |
| cbvmptvg 5203 | Rule to change the bound v... |
| mptv 5204 | Function with universal do... |
| dftr2 5207 | An alternate way of defini... |
| dftr2c 5208 | Variant of ~ dftr2 with co... |
| dftr5 5209 | An alternate way of defini... |
| dftr3 5210 | An alternate way of defini... |
| dftr4 5211 | An alternate way of defini... |
| treq 5212 | Equality theorem for the t... |
| trel 5213 | In a transitive class, the... |
| trel3 5214 | In a transitive class, the... |
| trss 5215 | An element of a transitive... |
| trin 5216 | The intersection of transi... |
| tr0 5217 | The empty set is transitiv... |
| trv 5218 | The universe is transitive... |
| triun 5219 | An indexed union of a clas... |
| truni 5220 | The union of a class of tr... |
| triin 5221 | An indexed intersection of... |
| trint 5222 | The intersection of a clas... |
| trintss 5223 | Any nonempty transitive cl... |
| axrep1 5225 | The version of the Axiom o... |
| axreplem 5226 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5227 | Axiom of Replacement expre... |
| axrep3 5228 | Axiom of Replacement sligh... |
| axrep4v 5229 | Version of ~ axrep4 with a... |
| axrep4 5230 | A more traditional version... |
| axrep4OLD 5231 | Obsolete version of ~ axre... |
| axrep5 5232 | Axiom of Replacement (simi... |
| axrep6 5233 | A condensed form of ~ ax-r... |
| axrep6OLD 5234 | Obsolete version of ~ axre... |
| axrep6g 5235 | ~ axrep6 in class notation... |
| zfrepclf 5236 | An inference based on the ... |
| zfrep3cl 5237 | An inference based on the ... |
| zfrep4 5238 | A version of Replacement u... |
| axsepgfromrep 5239 | A more general version ~ a... |
| axsep 5240 | Axiom scheme of separation... |
| axsepg 5242 | A more general version of ... |
| zfauscl 5243 | Separation Scheme (Aussond... |
| sepexlem 5244 | Lemma for ~ sepex . Use ~... |
| sepex 5245 | Convert implication to equ... |
| sepexi 5246 | Convert implication to equ... |
| bm1.3iiOLD 5247 | Obsolete version of ~ sepe... |
| ax6vsep 5248 | Derive ~ ax6v (a weakened ... |
| axnulALT 5249 | Alternate proof of ~ axnul... |
| axnul 5250 | The Null Set Axiom of ZF s... |
| 0ex 5252 | The Null Set Axiom of ZF s... |
| al0ssb 5253 | The empty set is the uniqu... |
| sseliALT 5254 | Alternate proof of ~ sseli... |
| csbexg 5255 | The existence of proper su... |
| csbex 5256 | The existence of proper su... |
| unisn2 5257 | A version of ~ unisn witho... |
| nalset 5258 | No set contains all sets. ... |
| vnex 5259 | The universal class does n... |
| vprc 5260 | The universal class is not... |
| nvel 5261 | The universal class does n... |
| inex1 5262 | Separation Scheme (Aussond... |
| inex2 5263 | Separation Scheme (Aussond... |
| inex1g 5264 | Closed-form, generalized S... |
| inex2g 5265 | Sufficient condition for a... |
| ssex 5266 | The subset of a set is als... |
| ssexi 5267 | The subset of a set is als... |
| ssexg 5268 | The subset of a set is als... |
| ssexd 5269 | A subclass of a set is a s... |
| abexd 5270 | Conditions for a class abs... |
| abex 5271 | Conditions for a class abs... |
| prcssprc 5272 | The superclass of a proper... |
| sselpwd 5273 | Elementhood to a power set... |
| difexg 5274 | Existence of a difference.... |
| difexi 5275 | Existence of a difference,... |
| difexd 5276 | Existence of a difference.... |
| zfausab 5277 | Separation Scheme (Aussond... |
| elpw2g 5278 | Membership in a power clas... |
| elpw2 5279 | Membership in a power clas... |
| elpwi2 5280 | Membership in a power clas... |
| rabelpw 5281 | A restricted class abstrac... |
| rabexg 5282 | Separation Scheme in terms... |
| rabexgOLD 5283 | Obsolete version of ~ rabe... |
| rabex 5284 | Separation Scheme in terms... |
| rabexd 5285 | Separation Scheme in terms... |
| rabex2 5286 | Separation Scheme in terms... |
| rab2ex 5287 | A class abstraction based ... |
| elssabg 5288 | Membership in a class abst... |
| intex 5289 | The intersection of a none... |
| intnex 5290 | If a class intersection is... |
| intexab 5291 | The intersection of a none... |
| intexrab 5292 | The intersection of a none... |
| iinexg 5293 | The existence of a class i... |
| intabs 5294 | Absorption of a redundant ... |
| inuni 5295 | The intersection of a unio... |
| axpweq 5296 | Two equivalent ways to exp... |
| pwnss 5297 | The power set of a set is ... |
| pwne 5298 | No set equals its power se... |
| difelpw 5299 | A difference is an element... |
| class2set 5300 | The class of elements of `... |
| 0elpw 5301 | Every power class contains... |
| pwne0 5302 | A power class is never emp... |
| 0nep0 5303 | The empty set and its powe... |
| 0inp0 5304 | Something cannot be equal ... |
| unidif0 5305 | The removal of the empty s... |
| eqsnuniex 5306 | If a class is equal to the... |
| iin0 5307 | An indexed intersection of... |
| notzfaus 5308 | In the Separation Scheme ~... |
| intv 5309 | The intersection of the un... |
| zfpow 5311 | Axiom of Power Sets expres... |
| axpow2 5312 | A variant of the Axiom of ... |
| axpow3 5313 | A variant of the Axiom of ... |
| elALT2 5314 | Alternate proof of ~ el us... |
| dtruALT2 5315 | Alternate proof of ~ dtru ... |
| dtrucor 5316 | Corollary of ~ dtru . Thi... |
| dtrucor2 5317 | The theorem form of the de... |
| dvdemo1 5318 | Demonstration of a theorem... |
| dvdemo2 5319 | Demonstration of a theorem... |
| nfnid 5320 | A setvar variable is not f... |
| nfcvb 5321 | The "distinctor" expressio... |
| vpwex 5322 | Power set axiom: the power... |
| pwexg 5323 | Power set axiom expressed ... |
| pwexd 5324 | Deduction version of the p... |
| pwex 5325 | Power set axiom expressed ... |
| pwel 5326 | Quantitative version of ~ ... |
| abssexg 5327 | Existence of a class of su... |
| snexALT 5328 | Alternate proof of ~ snex ... |
| p0ex 5329 | The power set of the empty... |
| p0exALT 5330 | Alternate proof of ~ p0ex ... |
| pp0ex 5331 | The power set of the power... |
| ord3ex 5332 | The ordinal number 3 is a ... |
| dtruALT 5333 | Alternate proof of ~ dtru ... |
| axc16b 5334 | This theorem shows that Ax... |
| eunex 5335 | Existential uniqueness imp... |
| eusv1 5336 | Two ways to express single... |
| eusvnf 5337 | Even if ` x ` is free in `... |
| eusvnfb 5338 | Two ways to say that ` A (... |
| eusv2i 5339 | Two ways to express single... |
| eusv2nf 5340 | Two ways to express single... |
| eusv2 5341 | Two ways to express single... |
| reusv1 5342 | Two ways to express single... |
| reusv2lem1 5343 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5344 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5345 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5346 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5347 | Lemma for ~ reusv2 . (Con... |
| reusv2 5348 | Two ways to express single... |
| reusv3i 5349 | Two ways of expressing exi... |
| reusv3 5350 | Two ways to express single... |
| eusv4 5351 | Two ways to express single... |
| alxfr 5352 | Transfer universal quantif... |
| ralxfrd 5353 | Transfer universal quantif... |
| rexxfrd 5354 | Transfer existential quant... |
| ralxfr2d 5355 | Transfer universal quantif... |
| rexxfr2d 5356 | Transfer existential quant... |
| ralxfrd2 5357 | Transfer universal quantif... |
| rexxfrd2 5358 | Transfer existence from a ... |
| ralxfr 5359 | Transfer universal quantif... |
| ralxfrALT 5360 | Alternate proof of ~ ralxf... |
| rexxfr 5361 | Transfer existence from a ... |
| rabxfrd 5362 | Membership in a restricted... |
| rabxfr 5363 | Membership in a restricted... |
| reuhypd 5364 | A theorem useful for elimi... |
| reuhyp 5365 | A theorem useful for elimi... |
| zfpair 5366 | The Axiom of Pairing of Ze... |
| axprALT 5367 | Alternate proof of ~ axpr ... |
| axprlem1 5368 | Lemma for ~ axpr . There ... |
| axprlem2 5369 | Lemma for ~ axpr . There ... |
| axprlem3 5370 | Lemma for ~ axpr . Elimin... |
| axprlem4 5371 | Lemma for ~ axpr . If an ... |
| axpr 5372 | Unabbreviated version of t... |
| axprlem3OLD 5373 | Obsolete version of ~ axpr... |
| axprlem4OLD 5374 | Obsolete version of ~ axpr... |
| axprlem5OLD 5375 | Obsolete version of ~ axpr... |
| axprOLD 5376 | Obsolete version of ~ axpr... |
| zfpair2 5378 | Derive the abbreviated ver... |
| vsnex 5379 | A singleton built on a set... |
| snexg 5380 | A singleton built on a set... |
| snex 5381 | A singleton is a set. The... |
| prex 5382 | The Axiom of Pairing using... |
| exel 5383 | There exist two sets, one ... |
| exexneq 5384 | There exist two different ... |
| exneq 5385 | Given any set (the " ` y `... |
| dtru 5386 | Given any set (the " ` y `... |
| el 5387 | Any set is an element of s... |
| sels 5388 | If a class is a set, then ... |
| selsALT 5389 | Alternate proof of ~ sels ... |
| elALT 5390 | Alternate proof of ~ el , ... |
| snelpwg 5391 | A singleton of a set is a ... |
| snelpwi 5392 | If a set is a member of a ... |
| snelpw 5393 | A singleton of a set is a ... |
| prelpw 5394 | An unordered pair of two s... |
| prelpwi 5395 | If two sets are members of... |
| rext 5396 | A theorem similar to exten... |
| sspwb 5397 | The powerclass constructio... |
| unipw 5398 | A class equals the union o... |
| univ 5399 | The union of the universe ... |
| pwtr 5400 | A class is transitive iff ... |
| ssextss 5401 | An extensionality-like pri... |
| ssext 5402 | An extensionality-like pri... |
| nssss 5403 | Negation of subclass relat... |
| pweqb 5404 | Classes are equal if and o... |
| intidg 5405 | The intersection of all se... |
| moabex 5406 | "At most one" existence im... |
| moabexOLD 5407 | Obsolete version of ~ moab... |
| rmorabex 5408 | Restricted "at most one" e... |
| euabex 5409 | The abstraction of a wff w... |
| nnullss 5410 | A nonempty class (even if ... |
| exss 5411 | Restricted existence in a ... |
| opex 5412 | An ordered pair of classes... |
| otex 5413 | An ordered triple of class... |
| elopg 5414 | Characterization of the el... |
| elop 5415 | Characterization of the el... |
| opi1 5416 | One of the two elements in... |
| opi2 5417 | One of the two elements of... |
| opeluu 5418 | Each member of an ordered ... |
| op1stb 5419 | Extract the first member o... |
| brv 5420 | Two classes are always in ... |
| opnz 5421 | An ordered pair is nonempt... |
| opnzi 5422 | An ordered pair is nonempt... |
| opth1 5423 | Equality of the first memb... |
| opth 5424 | The ordered pair theorem. ... |
| opthg 5425 | Ordered pair theorem. ` C ... |
| opth1g 5426 | Equality of the first memb... |
| opthg2 5427 | Ordered pair theorem. (Co... |
| opth2 5428 | Ordered pair theorem. (Co... |
| opthneg 5429 | Two ordered pairs are not ... |
| opthne 5430 | Two ordered pairs are not ... |
| otth2 5431 | Ordered triple theorem, wi... |
| otth 5432 | Ordered triple theorem. (... |
| otthg 5433 | Ordered triple theorem, cl... |
| otthne 5434 | Contrapositive of the orde... |
| eqvinop 5435 | A variable introduction la... |
| sbcop1 5436 | The proper substitution of... |
| sbcop 5437 | The proper substitution of... |
| copsexgw 5438 | Version of ~ copsexg with ... |
| copsexg 5439 | Substitution of class ` A ... |
| copsex2t 5440 | Closed theorem form of ~ c... |
| copsex2g 5441 | Implicit substitution infe... |
| copsex2dv 5442 | Implicit substitution dedu... |
| copsex4g 5443 | An implicit substitution i... |
| 0nelop 5444 | A property of ordered pair... |
| opwo0id 5445 | An ordered pair is equal t... |
| opeqex 5446 | Equivalence of existence i... |
| oteqex2 5447 | Equivalence of existence i... |
| oteqex 5448 | Equivalence of existence i... |
| opcom 5449 | An ordered pair commutes i... |
| moop2 5450 | "At most one" property of ... |
| opeqsng 5451 | Equivalence for an ordered... |
| opeqsn 5452 | Equivalence for an ordered... |
| opeqpr 5453 | Equivalence for an ordered... |
| snopeqop 5454 | Equivalence for an ordered... |
| propeqop 5455 | Equivalence for an ordered... |
| propssopi 5456 | If a pair of ordered pairs... |
| snopeqopsnid 5457 | Equivalence for an ordered... |
| mosubopt 5458 | "At most one" remains true... |
| mosubop 5459 | "At most one" remains true... |
| euop2 5460 | Transfer existential uniqu... |
| euotd 5461 | Prove existential uniquene... |
| opthwiener 5462 | Justification theorem for ... |
| uniop 5463 | The union of an ordered pa... |
| uniopel 5464 | Ordered pair membership is... |
| opthhausdorff 5465 | Justification theorem for ... |
| opthhausdorff0 5466 | Justification theorem for ... |
| otsndisj 5467 | The singletons consisting ... |
| otiunsndisj 5468 | The union of singletons co... |
| iunopeqop 5469 | Implication of an ordered ... |
| brsnop 5470 | Binary relation for an ord... |
| brtp 5471 | A necessary and sufficient... |
| opabidw 5472 | The law of concretion. Sp... |
| opabid 5473 | The law of concretion. Sp... |
| elopabw 5474 | Membership in a class abst... |
| elopab 5475 | Membership in a class abst... |
| rexopabb 5476 | Restricted existential qua... |
| vopelopabsb 5477 | The law of concretion in t... |
| opelopabsb 5478 | The law of concretion in t... |
| brabsb 5479 | The law of concretion in t... |
| opelopabt 5480 | Closed theorem form of ~ o... |
| opelopabga 5481 | The law of concretion. Th... |
| brabga 5482 | The law of concretion for ... |
| opelopab2a 5483 | Ordered pair membership in... |
| opelopaba 5484 | The law of concretion. Th... |
| braba 5485 | The law of concretion for ... |
| opelopabg 5486 | The law of concretion. Th... |
| brabg 5487 | The law of concretion for ... |
| opelopabgf 5488 | The law of concretion. Th... |
| opelopab2 5489 | Ordered pair membership in... |
| opelopab 5490 | The law of concretion. Th... |
| brab 5491 | The law of concretion for ... |
| opelopabaf 5492 | The law of concretion. Th... |
| opelopabf 5493 | The law of concretion. Th... |
| ssopab2 5494 | Equivalence of ordered pai... |
| ssopab2bw 5495 | Equivalence of ordered pai... |
| eqopab2bw 5496 | Equivalence of ordered pai... |
| ssopab2b 5497 | Equivalence of ordered pai... |
| ssopab2i 5498 | Inference of ordered pair ... |
| ssopab2dv 5499 | Inference of ordered pair ... |
| eqopab2b 5500 | Equivalence of ordered pai... |
| opabn0 5501 | Nonempty ordered pair clas... |
| opab0 5502 | Empty ordered pair class a... |
| csbopab 5503 | Move substitution into a c... |
| csbopabgALT 5504 | Move substitution into a c... |
| csbmpt12 5505 | Move substitution into a m... |
| csbmpt2 5506 | Move substitution into the... |
| iunopab 5507 | Move indexed union inside ... |
| elopabr 5508 | Membership in an ordered-p... |
| elopabran 5509 | Membership in an ordered-p... |
| rbropapd 5510 | Properties of a pair in an... |
| rbropap 5511 | Properties of a pair in a ... |
| 2rbropap 5512 | Properties of a pair in a ... |
| 0nelopab 5513 | The empty set is never an ... |
| brabv 5514 | If two classes are in a re... |
| pwin 5515 | The power class of the int... |
| pwssun 5516 | The power class of the uni... |
| pwun 5517 | The power class of the uni... |
| dfid4 5520 | The identity function expr... |
| dfid2 5521 | Alternate definition of th... |
| dfid3 5522 | A stronger version of ~ df... |
| epelg 5525 | The membership relation an... |
| epeli 5526 | The membership relation an... |
| epel 5527 | The membership relation an... |
| 0sn0ep 5528 | An example for the members... |
| epn0 5529 | The membership relation is... |
| poss 5534 | Subset theorem for the par... |
| poeq1 5535 | Equality theorem for parti... |
| poeq2 5536 | Equality theorem for parti... |
| poeq12d 5537 | Equality deduction for par... |
| nfpo 5538 | Bound-variable hypothesis ... |
| nfso 5539 | Bound-variable hypothesis ... |
| pocl 5540 | Characteristic properties ... |
| ispod 5541 | Sufficient conditions for ... |
| swopolem 5542 | Perform the substitutions ... |
| swopo 5543 | A strict weak order is a p... |
| poirr 5544 | A partial order is irrefle... |
| potr 5545 | A partial order is a trans... |
| po2nr 5546 | A partial order has no 2-c... |
| po3nr 5547 | A partial order has no 3-c... |
| po2ne 5548 | Two sets related by a part... |
| po0 5549 | Any relation is a partial ... |
| pofun 5550 | The inverse image of a par... |
| sopo 5551 | A strict linear order is a... |
| soss 5552 | Subset theorem for the str... |
| soeq1 5553 | Equality theorem for the s... |
| soeq2 5554 | Equality theorem for the s... |
| soeq12d 5555 | Equality deduction for tot... |
| sonr 5556 | A strict order relation is... |
| sotr 5557 | A strict order relation is... |
| sotrd 5558 | Transitivity law for stric... |
| solin 5559 | A strict order relation is... |
| so2nr 5560 | A strict order relation ha... |
| so3nr 5561 | A strict order relation ha... |
| sotric 5562 | A strict order relation sa... |
| sotrieq 5563 | Trichotomy law for strict ... |
| sotrieq2 5564 | Trichotomy law for strict ... |
| soasym 5565 | Asymmetry law for strict o... |
| sotr2 5566 | A transitivity relation. ... |
| issod 5567 | An irreflexive, transitive... |
| issoi 5568 | An irreflexive, transitive... |
| isso2i 5569 | Deduce strict ordering fro... |
| so0 5570 | Any relation is a strict o... |
| somo 5571 | A totally ordered set has ... |
| sotrine 5572 | Trichotomy law for strict ... |
| sotr3 5573 | Transitivity law for stric... |
| dffr6 5580 | Alternate definition of ~ ... |
| frd 5581 | A nonempty subset of an ` ... |
| fri 5582 | A nonempty subset of an ` ... |
| seex 5583 | The ` R ` -preimage of an ... |
| exse 5584 | Any relation on a set is s... |
| dffr2 5585 | Alternate definition of we... |
| dffr2ALT 5586 | Alternate proof of ~ dffr2... |
| frc 5587 | Property of well-founded r... |
| frss 5588 | Subset theorem for the wel... |
| sess1 5589 | Subset theorem for the set... |
| sess2 5590 | Subset theorem for the set... |
| freq1 5591 | Equality theorem for the w... |
| freq2 5592 | Equality theorem for the w... |
| freq12d 5593 | Equality deduction for wel... |
| seeq1 5594 | Equality theorem for the s... |
| seeq2 5595 | Equality theorem for the s... |
| seeq12d 5596 | Equality deduction for the... |
| nffr 5597 | Bound-variable hypothesis ... |
| nfse 5598 | Bound-variable hypothesis ... |
| nfwe 5599 | Bound-variable hypothesis ... |
| frirr 5600 | A well-founded relation is... |
| fr2nr 5601 | A well-founded relation ha... |
| fr0 5602 | Any relation is well-found... |
| frminex 5603 | If an element of a well-fo... |
| efrirr 5604 | A well-founded class does ... |
| efrn2lp 5605 | A well-founded class conta... |
| epse 5606 | The membership relation is... |
| tz7.2 5607 | Similar to Theorem 7.2 of ... |
| dfepfr 5608 | An alternate way of saying... |
| epfrc 5609 | A subset of a well-founded... |
| wess 5610 | Subset theorem for the wel... |
| weeq1 5611 | Equality theorem for the w... |
| weeq2 5612 | Equality theorem for the w... |
| weeq12d 5613 | Equality deduction for wel... |
| wefr 5614 | A well-ordering is well-fo... |
| weso 5615 | A well-ordering is a stric... |
| wecmpep 5616 | The elements of a class we... |
| wetrep 5617 | On a class well-ordered by... |
| wefrc 5618 | A nonempty subclass of a c... |
| we0 5619 | Any relation is a well-ord... |
| wereu 5620 | A nonempty subset of an ` ... |
| wereu2 5621 | A nonempty subclass of an ... |
| xpeq1 5638 | Equality theorem for Carte... |
| xpss12 5639 | Subset theorem for Cartesi... |
| xpss 5640 | A Cartesian product is inc... |
| inxpssres 5641 | Intersection with a Cartes... |
| relxp 5642 | A Cartesian product is a r... |
| xpss1 5643 | Subset relation for Cartes... |
| xpss2 5644 | Subset relation for Cartes... |
| xpeq2 5645 | Equality theorem for Carte... |
| elxpi 5646 | Membership in a Cartesian ... |
| elxp 5647 | Membership in a Cartesian ... |
| elxp2 5648 | Membership in a Cartesian ... |
| xpeq12 5649 | Equality theorem for Carte... |
| xpeq1i 5650 | Equality inference for Car... |
| xpeq2i 5651 | Equality inference for Car... |
| xpeq12i 5652 | Equality inference for Car... |
| xpeq1d 5653 | Equality deduction for Car... |
| xpeq2d 5654 | Equality deduction for Car... |
| xpeq12d 5655 | Equality deduction for Car... |
| sqxpeqd 5656 | Equality deduction for a C... |
| nfxp 5657 | Bound-variable hypothesis ... |
| 0nelxp 5658 | The empty set is not a mem... |
| 0nelelxp 5659 | A member of a Cartesian pr... |
| opelxp 5660 | Ordered pair membership in... |
| opelxpi 5661 | Ordered pair membership in... |
| opelxpii 5662 | Ordered pair membership in... |
| opelxpd 5663 | Ordered pair membership in... |
| opelvv 5664 | Ordered pair membership in... |
| opelvvg 5665 | Ordered pair membership in... |
| opelxp1 5666 | The first member of an ord... |
| opelxp2 5667 | The second member of an or... |
| otelxp 5668 | Ordered triple membership ... |
| otelxp1 5669 | The first member of an ord... |
| otel3xp 5670 | An ordered triple is an el... |
| opabssxpd 5671 | An ordered-pair class abst... |
| rabxp 5672 | Class abstraction restrict... |
| brxp 5673 | Binary relation on a Carte... |
| pwvrel 5674 | A set is a binary relation... |
| pwvabrel 5675 | The powerclass of the cart... |
| brrelex12 5676 | Two classes related by a b... |
| brrelex1 5677 | If two classes are related... |
| brrelex2 5678 | If two classes are related... |
| brrelex12i 5679 | Two classes that are relat... |
| brrelex1i 5680 | The first argument of a bi... |
| brrelex2i 5681 | The second argument of a b... |
| nprrel12 5682 | Proper classes are not rel... |
| nprrel 5683 | No proper class is related... |
| 0nelrel0 5684 | A binary relation does not... |
| 0nelrel 5685 | A binary relation does not... |
| fconstmpt 5686 | Representation of a consta... |
| vtoclr 5687 | Variable to class conversi... |
| opthprc 5688 | Justification theorem for ... |
| brel 5689 | Two things in a binary rel... |
| elxp3 5690 | Membership in a Cartesian ... |
| opeliunxp 5691 | Membership in a union of C... |
| opeliun2xp 5692 | Membership of an ordered p... |
| xpundi 5693 | Distributive law for Carte... |
| xpundir 5694 | Distributive law for Carte... |
| xpiundi 5695 | Distributive law for Carte... |
| xpiundir 5696 | Distributive law for Carte... |
| iunxpconst 5697 | Membership in a union of C... |
| xpun 5698 | The Cartesian product of t... |
| elvv 5699 | Membership in universal cl... |
| elvvv 5700 | Membership in universal cl... |
| elvvuni 5701 | An ordered pair contains i... |
| brinxp2 5702 | Intersection of binary rel... |
| brinxp 5703 | Intersection of binary rel... |
| opelinxp 5704 | Ordered pair element in an... |
| poinxp 5705 | Intersection of partial or... |
| soinxp 5706 | Intersection of total orde... |
| frinxp 5707 | Intersection of well-found... |
| seinxp 5708 | Intersection of set-like r... |
| weinxp 5709 | Intersection of well-order... |
| posn 5710 | Partial ordering of a sing... |
| sosn 5711 | Strict ordering on a singl... |
| frsn 5712 | Founded relation on a sing... |
| wesn 5713 | Well-ordering of a singlet... |
| elopaelxp 5714 | Membership in an ordered-p... |
| bropaex12 5715 | Two classes related by an ... |
| opabssxp 5716 | An abstraction relation is... |
| brab2a 5717 | The law of concretion for ... |
| optocl 5718 | Implicit substitution of c... |
| optoclOLD 5719 | Obsolete version of ~ opto... |
| 2optocl 5720 | Implicit substitution of c... |
| 3optocl 5721 | Implicit substitution of c... |
| opbrop 5722 | Ordered pair membership in... |
| 0xp 5723 | The Cartesian product with... |
| xp0 5724 | The Cartesian product with... |
| csbxp 5725 | Distribute proper substitu... |
| releq 5726 | Equality theorem for the r... |
| releqi 5727 | Equality inference for the... |
| releqd 5728 | Equality deduction for the... |
| nfrel 5729 | Bound-variable hypothesis ... |
| sbcrel 5730 | Distribute proper substitu... |
| relss 5731 | Subclass theorem for relat... |
| ssrel 5732 | A subclass relationship de... |
| eqrel 5733 | Extensionality principle f... |
| ssrel2 5734 | A subclass relationship de... |
| ssrel3 5735 | Subclass relation in anoth... |
| relssi 5736 | Inference from subclass pr... |
| relssdv 5737 | Deduction from subclass pr... |
| eqrelriv 5738 | Inference from extensional... |
| eqrelriiv 5739 | Inference from extensional... |
| eqbrriv 5740 | Inference from extensional... |
| eqrelrdv 5741 | Deduce equality of relatio... |
| eqbrrdv 5742 | Deduction from extensional... |
| eqbrrdiv 5743 | Deduction from extensional... |
| eqrelrdv2 5744 | A version of ~ eqrelrdv . ... |
| ssrelrel 5745 | A subclass relationship de... |
| eqrelrel 5746 | Extensionality principle f... |
| elrel 5747 | A member of a relation is ... |
| rel0 5748 | The empty set is a relatio... |
| nrelv 5749 | The universal class is not... |
| relsng 5750 | A singleton is a relation ... |
| relsnb 5751 | An at-most-singleton is a ... |
| relsnopg 5752 | A singleton of an ordered ... |
| relsn 5753 | A singleton is a relation ... |
| relsnop 5754 | A singleton of an ordered ... |
| copsex2gb 5755 | Implicit substitution infe... |
| copsex2ga 5756 | Implicit substitution infe... |
| elopaba 5757 | Membership in an ordered-p... |
| xpsspw 5758 | A Cartesian product is inc... |
| unixpss 5759 | The double class union of ... |
| relun 5760 | The union of two relations... |
| relin1 5761 | The intersection with a re... |
| relin2 5762 | The intersection with a re... |
| relinxp 5763 | Intersection with a Cartes... |
| reldif 5764 | A difference cutting down ... |
| reliun 5765 | An indexed union is a rela... |
| reliin 5766 | An indexed intersection is... |
| reluni 5767 | The union of a class is a ... |
| relint 5768 | The intersection of a clas... |
| relopabiv 5769 | A class of ordered pairs i... |
| relopabv 5770 | A class of ordered pairs i... |
| relopabi 5771 | A class of ordered pairs i... |
| relopabiALT 5772 | Alternate proof of ~ relop... |
| relopab 5773 | A class of ordered pairs i... |
| mptrel 5774 | The maps-to notation alway... |
| reli 5775 | The identity relation is a... |
| rele 5776 | The membership relation is... |
| opabid2 5777 | A relation expressed as an... |
| inopab 5778 | Intersection of two ordere... |
| difopab 5779 | Difference of two ordered-... |
| inxp 5780 | Intersection of two Cartes... |
| inxpOLD 5781 | Obsolete version of ~ inxp... |
| xpindi 5782 | Distributive law for Carte... |
| xpindir 5783 | Distributive law for Carte... |
| xpiindi 5784 | Distributive law for Carte... |
| xpriindi 5785 | Distributive law for Carte... |
| eliunxp 5786 | Membership in a union of C... |
| opeliunxp2 5787 | Membership in a union of C... |
| raliunxp 5788 | Write a double restricted ... |
| rexiunxp 5789 | Write a double restricted ... |
| ralxp 5790 | Universal quantification r... |
| rexxp 5791 | Existential quantification... |
| exopxfr 5792 | Transfer ordered-pair exis... |
| exopxfr2 5793 | Transfer ordered-pair exis... |
| djussxp 5794 | Disjoint union is a subset... |
| ralxpf 5795 | Version of ~ ralxp with bo... |
| rexxpf 5796 | Version of ~ rexxp with bo... |
| iunxpf 5797 | Indexed union on a Cartesi... |
| opabbi2dv 5798 | Deduce equality of a relat... |
| relop 5799 | A necessary and sufficient... |
| ideqg 5800 | For sets, the identity rel... |
| ideq 5801 | For sets, the identity rel... |
| ididg 5802 | A set is identical to itse... |
| issetid 5803 | Two ways of expressing set... |
| coss1 5804 | Subclass theorem for compo... |
| coss2 5805 | Subclass theorem for compo... |
| coeq1 5806 | Equality theorem for compo... |
| coeq2 5807 | Equality theorem for compo... |
| coeq1i 5808 | Equality inference for com... |
| coeq2i 5809 | Equality inference for com... |
| coeq1d 5810 | Equality deduction for com... |
| coeq2d 5811 | Equality deduction for com... |
| coeq12i 5812 | Equality inference for com... |
| coeq12d 5813 | Equality deduction for com... |
| nfco 5814 | Bound-variable hypothesis ... |
| brcog 5815 | Ordered pair membership in... |
| opelco2g 5816 | Ordered pair membership in... |
| brcogw 5817 | Ordered pair membership in... |
| eqbrrdva 5818 | Deduction from extensional... |
| brco 5819 | Binary relation on a compo... |
| opelco 5820 | Ordered pair membership in... |
| cnvss 5821 | Subset theorem for convers... |
| cnveq 5822 | Equality theorem for conve... |
| cnveqi 5823 | Equality inference for con... |
| cnveqd 5824 | Equality deduction for con... |
| elcnv 5825 | Membership in a converse r... |
| elcnv2 5826 | Membership in a converse r... |
| nfcnv 5827 | Bound-variable hypothesis ... |
| brcnvg 5828 | The converse of a binary r... |
| opelcnvg 5829 | Ordered-pair membership in... |
| opelcnv 5830 | Ordered-pair membership in... |
| brcnv 5831 | The converse of a binary r... |
| csbcnv 5832 | Move class substitution in... |
| csbcnvgALT 5833 | Move class substitution in... |
| cnvco 5834 | Distributive law of conver... |
| cnvuni 5835 | The converse of a class un... |
| dfdm3 5836 | Alternate definition of do... |
| dfrn2 5837 | Alternate definition of ra... |
| dfrn3 5838 | Alternate definition of ra... |
| elrn2g 5839 | Membership in a range. (C... |
| elrng 5840 | Membership in a range. (C... |
| elrn2 5841 | Membership in a range. (C... |
| elrn 5842 | Membership in a range. (C... |
| ssrelrn 5843 | If a relation is a subset ... |
| dfdm4 5844 | Alternate definition of do... |
| dfdmf 5845 | Definition of domain, usin... |
| csbdm 5846 | Distribute proper substitu... |
| eldmg 5847 | Domain membership. Theore... |
| eldm2g 5848 | Domain membership. Theore... |
| eldm 5849 | Membership in a domain. T... |
| eldm2 5850 | Membership in a domain. T... |
| dmss 5851 | Subset theorem for domain.... |
| dmeq 5852 | Equality theorem for domai... |
| dmeqi 5853 | Equality inference for dom... |
| dmeqd 5854 | Equality deduction for dom... |
| opeldmd 5855 | Membership of first of an ... |
| opeldm 5856 | Membership of first of an ... |
| breldm 5857 | Membership of first of a b... |
| breldmg 5858 | Membership of first of a b... |
| dmun 5859 | The domain of a union is t... |
| dmin 5860 | The domain of an intersect... |
| breldmd 5861 | Membership of first of a b... |
| dmiun 5862 | The domain of an indexed u... |
| dmuni 5863 | The domain of a union. Pa... |
| dmopab 5864 | The domain of a class of o... |
| dmopabelb 5865 | A set is an element of the... |
| dmopab2rex 5866 | The domain of an ordered p... |
| dmopabss 5867 | Upper bound for the domain... |
| dmopab3 5868 | The domain of a restricted... |
| dm0 5869 | The domain of the empty se... |
| dmi 5870 | The domain of the identity... |
| dmv 5871 | The domain of the universe... |
| dmep 5872 | The domain of the membersh... |
| dm0rn0 5873 | An empty domain is equival... |
| dm0rn0OLD 5874 | Obsolete version of ~ dm0r... |
| rn0 5875 | The range of the empty set... |
| rnep 5876 | The range of the membershi... |
| reldm0 5877 | A relation is empty iff it... |
| dmxp 5878 | The domain of a Cartesian ... |
| dmxpid 5879 | The domain of a Cartesian ... |
| dmxpin 5880 | The domain of the intersec... |
| xpid11 5881 | The Cartesian square is a ... |
| dmcnvcnv 5882 | The domain of the double c... |
| rncnvcnv 5883 | The range of the double co... |
| elreldm 5884 | The first member of an ord... |
| rneq 5885 | Equality theorem for range... |
| rneqi 5886 | Equality inference for ran... |
| rneqd 5887 | Equality deduction for ran... |
| rnss 5888 | Subset theorem for range. ... |
| rnssi 5889 | Subclass inference for ran... |
| brelrng 5890 | The second argument of a b... |
| brelrn 5891 | The second argument of a b... |
| opelrn 5892 | Membership of second membe... |
| releldm 5893 | The first argument of a bi... |
| relelrn 5894 | The second argument of a b... |
| releldmb 5895 | Membership in a domain. (... |
| relelrnb 5896 | Membership in a range. (C... |
| releldmi 5897 | The first argument of a bi... |
| relelrni 5898 | The second argument of a b... |
| dfrnf 5899 | Definition of range, using... |
| nfdm 5900 | Bound-variable hypothesis ... |
| nfrn 5901 | Bound-variable hypothesis ... |
| dmiin 5902 | Domain of an intersection.... |
| rnopab 5903 | The range of a class of or... |
| rnopabss 5904 | Upper bound for the range ... |
| rnopab3 5905 | The range of a restricted ... |
| rnmpt 5906 | The range of a function in... |
| elrnmpt 5907 | The range of a function in... |
| elrnmpt1s 5908 | Elementhood in an image se... |
| elrnmpt1 5909 | Elementhood in an image se... |
| elrnmptg 5910 | Membership in the range of... |
| elrnmpti 5911 | Membership in the range of... |
| elrnmptd 5912 | The range of a function in... |
| elrnmpt1d 5913 | Elementhood in an image se... |
| elrnmptdv 5914 | Elementhood in the range o... |
| elrnmpt2d 5915 | Elementhood in the range o... |
| nelrnmpt 5916 | Non-membership in the rang... |
| dfiun3g 5917 | Alternate definition of in... |
| dfiin3g 5918 | Alternate definition of in... |
| dfiun3 5919 | Alternate definition of in... |
| dfiin3 5920 | Alternate definition of in... |
| riinint 5921 | Express a relative indexed... |
| relrn0 5922 | A relation is empty iff it... |
| dmrnssfld 5923 | The domain and range of a ... |
| dmcoss 5924 | Domain of a composition. ... |
| dmcossOLD 5925 | Obsolete version of ~ dmco... |
| rncoss 5926 | Range of a composition. (... |
| dmcosseq 5927 | Domain of a composition. ... |
| dmcosseqOLD 5928 | Obsolete version of ~ dmco... |
| dmcosseqOLDOLD 5929 | Obsolete version of ~ dmco... |
| dmcoeq 5930 | Domain of a composition. ... |
| rncoeq 5931 | Range of a composition. (... |
| reseq1 5932 | Equality theorem for restr... |
| reseq2 5933 | Equality theorem for restr... |
| reseq1i 5934 | Equality inference for res... |
| reseq2i 5935 | Equality inference for res... |
| reseq12i 5936 | Equality inference for res... |
| reseq1d 5937 | Equality deduction for res... |
| reseq2d 5938 | Equality deduction for res... |
| reseq12d 5939 | Equality deduction for res... |
| nfres 5940 | Bound-variable hypothesis ... |
| csbres 5941 | Distribute proper substitu... |
| res0 5942 | A restriction to the empty... |
| dfres3 5943 | Alternate definition of re... |
| opelres 5944 | Ordered pair elementhood i... |
| brres 5945 | Binary relation on a restr... |
| opelresi 5946 | Ordered pair membership in... |
| brresi 5947 | Binary relation on a restr... |
| opres 5948 | Ordered pair membership in... |
| resieq 5949 | A restricted identity rela... |
| opelidres 5950 | ` <. A , A >. ` belongs to... |
| resres 5951 | The restriction of a restr... |
| resundi 5952 | Distributive law for restr... |
| resundir 5953 | Distributive law for restr... |
| resindi 5954 | Class restriction distribu... |
| resindir 5955 | Class restriction distribu... |
| inres 5956 | Move intersection into cla... |
| resdifcom 5957 | Commutative law for restri... |
| resiun1 5958 | Distribution of restrictio... |
| resiun2 5959 | Distribution of restrictio... |
| resss 5960 | A class includes its restr... |
| rescom 5961 | Commutative law for restri... |
| ssres 5962 | Subclass theorem for restr... |
| ssres2 5963 | Subclass theorem for restr... |
| relres 5964 | A restriction is a relatio... |
| resabs1 5965 | Absorption law for restric... |
| resabs1i 5966 | Absorption law for restric... |
| resabs1d 5967 | Absorption law for restric... |
| resabs2 5968 | Absorption law for restric... |
| residm 5969 | Idempotent law for restric... |
| dmresss 5970 | The domain of a restrictio... |
| dmres 5971 | The domain of a restrictio... |
| ssdmres 5972 | A domain restricted to a s... |
| dmresexg 5973 | The domain of a restrictio... |
| resima 5974 | A restriction to an image.... |
| resima2 5975 | Image under a restricted c... |
| rnresss 5976 | The range of a restriction... |
| xpssres 5977 | Restriction of a constant ... |
| elinxp 5978 | Membership in an intersect... |
| elres 5979 | Membership in a restrictio... |
| elsnres 5980 | Membership in restriction ... |
| relssres 5981 | Simplification law for res... |
| dmressnsn 5982 | The domain of a restrictio... |
| eldmressnsn 5983 | The element of the domain ... |
| eldmeldmressn 5984 | An element of the domain (... |
| resdm 5985 | A relation restricted to i... |
| resexg 5986 | The restriction of a set i... |
| resexd 5987 | The restriction of a set i... |
| resex 5988 | The restriction of a set i... |
| resindm 5989 | When restricting a relatio... |
| resdmdfsn 5990 | Restricting a relation to ... |
| reldisjun 5991 | Split a relation into two ... |
| relresdm1 5992 | Restriction of a disjoint ... |
| resopab 5993 | Restriction of a class abs... |
| iss 5994 | A subclass of the identity... |
| resopab2 5995 | Restriction of a class abs... |
| resmpt 5996 | Restriction of the mapping... |
| resmpt3 5997 | Unconditional restriction ... |
| resmptf 5998 | Restriction of the mapping... |
| resmptd 5999 | Restriction of the mapping... |
| dfres2 6000 | Alternate definition of th... |
| mptss 6001 | Sufficient condition for i... |
| elimampt 6002 | Membership in the image of... |
| elidinxp 6003 | Characterization of the el... |
| elidinxpid 6004 | Characterization of the el... |
| elrid 6005 | Characterization of the el... |
| idinxpres 6006 | The intersection of the id... |
| idinxpresid 6007 | The intersection of the id... |
| idssxp 6008 | A diagonal set as a subset... |
| opabresid 6009 | The restricted identity re... |
| mptresid 6010 | The restricted identity re... |
| dmresi 6011 | The domain of a restricted... |
| restidsing 6012 | Restriction of the identit... |
| iresn0n0 6013 | The identity function rest... |
| imaeq1 6014 | Equality theorem for image... |
| imaeq2 6015 | Equality theorem for image... |
| imaeq1i 6016 | Equality theorem for image... |
| imaeq2i 6017 | Equality theorem for image... |
| imaeq1d 6018 | Equality theorem for image... |
| imaeq2d 6019 | Equality theorem for image... |
| imaeq12d 6020 | Equality theorem for image... |
| dfima2 6021 | Alternate definition of im... |
| dfima3 6022 | Alternate definition of im... |
| elimag 6023 | Membership in an image. T... |
| elima 6024 | Membership in an image. T... |
| elima2 6025 | Membership in an image. T... |
| elima3 6026 | Membership in an image. T... |
| nfima 6027 | Bound-variable hypothesis ... |
| nfimad 6028 | Deduction version of bound... |
| imadmrn 6029 | The image of the domain of... |
| imassrn 6030 | The image of a class is a ... |
| mptima 6031 | Image of a function in map... |
| mptimass 6032 | Image of a function in map... |
| imai 6033 | Image under the identity r... |
| rnresi 6034 | The range of the restricte... |
| resiima 6035 | The image of a restriction... |
| ima0 6036 | Image of the empty set. T... |
| 0ima 6037 | Image under the empty rela... |
| csbima12 6038 | Move class substitution in... |
| imadisj 6039 | A class whose image under ... |
| imadisjlnd 6040 | Deduction form of one nega... |
| cnvimass 6041 | A preimage under any class... |
| cnvimarndm 6042 | The preimage of the range ... |
| imasng 6043 | The image of a singleton. ... |
| relimasn 6044 | The image of a singleton. ... |
| elrelimasn 6045 | Elementhood in the image o... |
| elimasng1 6046 | Membership in an image of ... |
| elimasn1 6047 | Membership in an image of ... |
| elimasng 6048 | Membership in an image of ... |
| elimasn 6049 | Membership in an image of ... |
| elimasni 6050 | Membership in an image of ... |
| args 6051 | Two ways to express the cl... |
| elinisegg 6052 | Membership in the inverse ... |
| eliniseg 6053 | Membership in the inverse ... |
| epin 6054 | Any set is equal to its pr... |
| epini 6055 | Any set is equal to its pr... |
| iniseg 6056 | An idiom that signifies an... |
| inisegn0 6057 | Nonemptiness of an initial... |
| dffr3 6058 | Alternate definition of we... |
| dfse2 6059 | Alternate definition of se... |
| imass1 6060 | Subset theorem for image. ... |
| imass2 6061 | Subset theorem for image. ... |
| ndmima 6062 | The image of a singleton o... |
| relcnv 6063 | A converse is a relation. ... |
| relbrcnvg 6064 | When ` R ` is a relation, ... |
| eliniseg2 6065 | Eliminate the class existe... |
| relbrcnv 6066 | When ` R ` is a relation, ... |
| relco 6067 | A composition is a relatio... |
| cotrg 6068 | Two ways of saying that th... |
| cotr 6069 | Two ways of saying a relat... |
| idrefALT 6070 | Alternate proof of ~ idref... |
| cnvsym 6071 | Two ways of saying a relat... |
| intasym 6072 | Two ways of saying a relat... |
| asymref 6073 | Two ways of saying a relat... |
| asymref2 6074 | Two ways of saying a relat... |
| intirr 6075 | Two ways of saying a relat... |
| brcodir 6076 | Two ways of saying that tw... |
| codir 6077 | Two ways of saying a relat... |
| qfto 6078 | A quantifier-free way of e... |
| xpidtr 6079 | A Cartesian square is a tr... |
| trin2 6080 | The intersection of two tr... |
| poirr2 6081 | A partial order is irrefle... |
| trinxp 6082 | The relation induced by a ... |
| soirri 6083 | A strict order relation is... |
| sotri 6084 | A strict order relation is... |
| son2lpi 6085 | A strict order relation ha... |
| sotri2 6086 | A transitivity relation. ... |
| sotri3 6087 | A transitivity relation. ... |
| poleloe 6088 | Express "less than or equa... |
| poltletr 6089 | Transitive law for general... |
| somin1 6090 | Property of a minimum in a... |
| somincom 6091 | Commutativity of minimum i... |
| somin2 6092 | Property of a minimum in a... |
| soltmin 6093 | Being less than a minimum,... |
| cnvopab 6094 | The converse of a class ab... |
| cnvopabOLD 6095 | Obsolete version of ~ cnvo... |
| mptcnv 6096 | The converse of a mapping ... |
| cnv0 6097 | The converse of the empty ... |
| cnv0OLD 6098 | Obsolete version of ~ cnv0... |
| cnvi 6099 | The converse of the identi... |
| cnvun 6100 | The converse of a union is... |
| cnvdif 6101 | Distributive law for conve... |
| cnvin 6102 | Distributive law for conve... |
| rnun 6103 | Distributive law for range... |
| rnin 6104 | The range of an intersecti... |
| rniun 6105 | The range of an indexed un... |
| rnuni 6106 | The range of a union. Par... |
| imaundi 6107 | Distributive law for image... |
| imaundir 6108 | The image of a union. (Co... |
| imadifssran 6109 | Condition for the range of... |
| cnvimassrndm 6110 | The preimage of a superset... |
| dminss 6111 | An upper bound for interse... |
| imainss 6112 | An upper bound for interse... |
| inimass 6113 | The image of an intersecti... |
| inimasn 6114 | The intersection of the im... |
| cnvxp 6115 | The converse of a Cartesia... |
| xp0OLD 6116 | Obsolete version of ~ xp0 ... |
| xpnz 6117 | The Cartesian product of n... |
| xpeq0 6118 | At least one member of an ... |
| xpdisj1 6119 | Cartesian products with di... |
| xpdisj2 6120 | Cartesian products with di... |
| xpsndisj 6121 | Cartesian products with tw... |
| difxp 6122 | Difference of Cartesian pr... |
| difxp1 6123 | Difference law for Cartesi... |
| difxp2 6124 | Difference law for Cartesi... |
| djudisj 6125 | Disjoint unions with disjo... |
| xpdifid 6126 | The set of distinct couple... |
| resdisj 6127 | A double restriction to di... |
| rnxp 6128 | The range of a Cartesian p... |
| dmxpss 6129 | The domain of a Cartesian ... |
| rnxpss 6130 | The range of a Cartesian p... |
| rnxpid 6131 | The range of a Cartesian s... |
| ssxpb 6132 | A Cartesian product subcla... |
| xp11 6133 | The Cartesian product of n... |
| xpcan 6134 | Cancellation law for Carte... |
| xpcan2 6135 | Cancellation law for Carte... |
| ssrnres 6136 | Two ways to express surjec... |
| rninxp 6137 | Two ways to express surjec... |
| dminxp 6138 | Two ways to express totali... |
| imainrect 6139 | Image by a restricted and ... |
| xpima 6140 | Direct image by a Cartesia... |
| xpima1 6141 | Direct image by a Cartesia... |
| xpima2 6142 | Direct image by a Cartesia... |
| xpimasn 6143 | Direct image of a singleto... |
| sossfld 6144 | The base set of a strict o... |
| sofld 6145 | The base set of a nonempty... |
| cnvcnv3 6146 | The set of all ordered pai... |
| dfrel2 6147 | Alternate definition of re... |
| dfrel4v 6148 | A relation can be expresse... |
| dfrel4 6149 | A relation can be expresse... |
| cnvcnv 6150 | The double converse of a c... |
| cnvcnv2 6151 | The double converse of a c... |
| cnvcnvss 6152 | The double converse of a c... |
| cnvrescnv 6153 | Two ways to express the co... |
| cnveqb 6154 | Equality theorem for conve... |
| cnveq0 6155 | A relation empty iff its c... |
| dfrel3 6156 | Alternate definition of re... |
| elid 6157 | Characterization of the el... |
| dmresv 6158 | The domain of a universal ... |
| rnresv 6159 | The range of a universal r... |
| dfrn4 6160 | Range defined in terms of ... |
| csbrn 6161 | Distribute proper substitu... |
| rescnvcnv 6162 | The restriction of the dou... |
| cnvcnvres 6163 | The double converse of the... |
| imacnvcnv 6164 | The image of the double co... |
| dmsnn0 6165 | The domain of a singleton ... |
| rnsnn0 6166 | The range of a singleton i... |
| dmsn0 6167 | The domain of the singleto... |
| cnvsn0 6168 | The converse of the single... |
| dmsn0el 6169 | The domain of a singleton ... |
| relsn2 6170 | A singleton is a relation ... |
| dmsnopg 6171 | The domain of a singleton ... |
| dmsnopss 6172 | The domain of a singleton ... |
| dmpropg 6173 | The domain of an unordered... |
| dmsnop 6174 | The domain of a singleton ... |
| dmprop 6175 | The domain of an unordered... |
| dmtpop 6176 | The domain of an unordered... |
| cnvcnvsn 6177 | Double converse of a singl... |
| dmsnsnsn 6178 | The domain of the singleto... |
| rnsnopg 6179 | The range of a singleton o... |
| rnpropg 6180 | The range of a pair of ord... |
| cnvsng 6181 | Converse of a singleton of... |
| rnsnop 6182 | The range of a singleton o... |
| op1sta 6183 | Extract the first member o... |
| cnvsn 6184 | Converse of a singleton of... |
| op2ndb 6185 | Extract the second member ... |
| op2nda 6186 | Extract the second member ... |
| opswap 6187 | Swap the members of an ord... |
| cnvresima 6188 | An image under the convers... |
| resdm2 6189 | A class restricted to its ... |
| resdmres 6190 | Restriction to the domain ... |
| resresdm 6191 | A restriction by an arbitr... |
| imadmres 6192 | The image of the domain of... |
| resdmss 6193 | Subset relationship for th... |
| resdifdi 6194 | Distributive law for restr... |
| resdifdir 6195 | Distributive law for restr... |
| mptpreima 6196 | The preimage of a function... |
| mptiniseg 6197 | Converse singleton image o... |
| dmmpt 6198 | The domain of the mapping ... |
| dmmptss 6199 | The domain of a mapping is... |
| dmmptg 6200 | The domain of the mapping ... |
| rnmpt0f 6201 | The range of a function in... |
| rnmptn0 6202 | The range of a function in... |
| dfco2 6203 | Alternate definition of a ... |
| dfco2a 6204 | Generalization of ~ dfco2 ... |
| coundi 6205 | Class composition distribu... |
| coundir 6206 | Class composition distribu... |
| cores 6207 | Restricted first member of... |
| resco 6208 | Associative law for the re... |
| imaco 6209 | Image of the composition o... |
| rnco 6210 | The range of the compositi... |
| rncoOLD 6211 | Obsolete version of ~ rnco... |
| rnco2 6212 | The range of the compositi... |
| dmco 6213 | The domain of a compositio... |
| coeq0 6214 | A composition of two relat... |
| coiun 6215 | Composition with an indexe... |
| cocnvcnv1 6216 | A composition is not affec... |
| cocnvcnv2 6217 | A composition is not affec... |
| cores2 6218 | Absorption of a reverse (p... |
| co02 6219 | Composition with the empty... |
| co01 6220 | Composition with the empty... |
| coi1 6221 | Composition with the ident... |
| coi2 6222 | Composition with the ident... |
| coires1 6223 | Composition with a restric... |
| coass 6224 | Associative law for class ... |
| relcnvtrg 6225 | General form of ~ relcnvtr... |
| relcnvtr 6226 | A relation is transitive i... |
| relssdmrn 6227 | A relation is included in ... |
| resssxp 6228 | If the ` R ` -image of a c... |
| cnvssrndm 6229 | The converse is a subset o... |
| cossxp 6230 | Composition as a subset of... |
| relrelss 6231 | Two ways to describe the s... |
| unielrel 6232 | The membership relation fo... |
| relfld 6233 | The double union of a rela... |
| relresfld 6234 | Restriction of a relation ... |
| relcoi2 6235 | Composition with the ident... |
| relcoi1 6236 | Composition with the ident... |
| unidmrn 6237 | The double union of the co... |
| relcnvfld 6238 | if ` R ` is a relation, it... |
| dfdm2 6239 | Alternate definition of do... |
| unixp 6240 | The double class union of ... |
| unixp0 6241 | A Cartesian product is emp... |
| unixpid 6242 | Field of a Cartesian squar... |
| ressn 6243 | Restriction of a class to ... |
| cnviin 6244 | The converse of an interse... |
| cnvpo 6245 | The converse of a partial ... |
| cnvso 6246 | The converse of a strict o... |
| xpco 6247 | Composition of two Cartesi... |
| xpcoid 6248 | Composition of two Cartesi... |
| elsnxp 6249 | Membership in a Cartesian ... |
| reu3op 6250 | There is a unique ordered ... |
| reuop 6251 | There is a unique ordered ... |
| opreu2reurex 6252 | There is a unique ordered ... |
| opreu2reu 6253 | If there is a unique order... |
| dfpo2 6254 | Quantifier-free definition... |
| csbcog 6255 | Distribute proper substitu... |
| snres0 6256 | Condition for restriction ... |
| imaindm 6257 | The image is unaffected by... |
| predeq123 6260 | Equality theorem for the p... |
| predeq1 6261 | Equality theorem for the p... |
| predeq2 6262 | Equality theorem for the p... |
| predeq3 6263 | Equality theorem for the p... |
| nfpred 6264 | Bound-variable hypothesis ... |
| csbpredg 6265 | Move class substitution in... |
| predpredss 6266 | If ` A ` is a subset of ` ... |
| predss 6267 | The predecessor class of `... |
| sspred 6268 | Another subset/predecessor... |
| dfpred2 6269 | An alternate definition of... |
| dfpred3 6270 | An alternate definition of... |
| dfpred3g 6271 | An alternate definition of... |
| elpredgg 6272 | Membership in a predecesso... |
| elpredg 6273 | Membership in a predecesso... |
| elpredimg 6274 | Membership in a predecesso... |
| elpredim 6275 | Membership in a predecesso... |
| elpred 6276 | Membership in a predecesso... |
| predexg 6277 | The predecessor class exis... |
| dffr4 6278 | Alternate definition of we... |
| predel 6279 | Membership in the predeces... |
| predtrss 6280 | If ` R ` is transitive ove... |
| predpo 6281 | Property of the predecesso... |
| predso 6282 | Property of the predecesso... |
| setlikespec 6283 | If ` R ` is set-like in ` ... |
| predidm 6284 | Idempotent law for the pre... |
| predin 6285 | Intersection law for prede... |
| predun 6286 | Union law for predecessor ... |
| preddif 6287 | Difference law for predece... |
| predep 6288 | The predecessor under the ... |
| trpred 6289 | The class of predecessors ... |
| preddowncl 6290 | A property of classes that... |
| predpoirr 6291 | Given a partial ordering, ... |
| predfrirr 6292 | Given a well-founded relat... |
| pred0 6293 | The predecessor class over... |
| dfse3 6294 | Alternate definition of se... |
| predrelss 6295 | Subset carries from relati... |
| predprc 6296 | The predecessor of a prope... |
| predres 6297 | Predecessor class is unaff... |
| frpomin 6298 | Every nonempty (possibly p... |
| frpomin2 6299 | Every nonempty (possibly p... |
| frpoind 6300 | The principle of well-foun... |
| frpoinsg 6301 | Well-Founded Induction Sch... |
| frpoins2fg 6302 | Well-Founded Induction sch... |
| frpoins2g 6303 | Well-Founded Induction sch... |
| frpoins3g 6304 | Well-Founded Induction sch... |
| tz6.26 6305 | All nonempty subclasses of... |
| tz6.26i 6306 | All nonempty subclasses of... |
| wfi 6307 | The Principle of Well-Orde... |
| wfii 6308 | The Principle of Well-Orde... |
| wfisg 6309 | Well-Ordered Induction Sch... |
| wfis 6310 | Well-Ordered Induction Sch... |
| wfis2fg 6311 | Well-Ordered Induction Sch... |
| wfis2f 6312 | Well-Ordered Induction sch... |
| wfis2g 6313 | Well-Ordered Induction Sch... |
| wfis2 6314 | Well-Ordered Induction sch... |
| wfis3 6315 | Well-Ordered Induction sch... |
| ordeq 6324 | Equality theorem for the o... |
| elong 6325 | An ordinal number is an or... |
| elon 6326 | An ordinal number is an or... |
| eloni 6327 | An ordinal number has the ... |
| elon2 6328 | An ordinal number is an or... |
| limeq 6329 | Equality theorem for the l... |
| ordwe 6330 | Membership well-orders eve... |
| ordtr 6331 | An ordinal class is transi... |
| ordfr 6332 | Membership is well-founded... |
| ordelss 6333 | An element of an ordinal c... |
| trssord 6334 | A transitive subclass of a... |
| ordirr 6335 | No ordinal class is a memb... |
| nordeq 6336 | A member of an ordinal cla... |
| ordn2lp 6337 | An ordinal class cannot be... |
| tz7.5 6338 | A nonempty subclass of an ... |
| ordelord 6339 | An element of an ordinal c... |
| tron 6340 | The class of all ordinal n... |
| ordelon 6341 | An element of an ordinal c... |
| onelon 6342 | An element of an ordinal n... |
| tz7.7 6343 | A transitive class belongs... |
| ordelssne 6344 | For ordinal classes, membe... |
| ordelpss 6345 | For ordinal classes, membe... |
| ordsseleq 6346 | For ordinal classes, inclu... |
| ordin 6347 | The intersection of two or... |
| onin 6348 | The intersection of two or... |
| ordtri3or 6349 | A trichotomy law for ordin... |
| ordtri1 6350 | A trichotomy law for ordin... |
| ontri1 6351 | A trichotomy law for ordin... |
| ordtri2 6352 | A trichotomy law for ordin... |
| ordtri3 6353 | A trichotomy law for ordin... |
| ordtri4 6354 | A trichotomy law for ordin... |
| orddisj 6355 | An ordinal class and its s... |
| onfr 6356 | The ordinal class is well-... |
| onelpss 6357 | Relationship between membe... |
| onsseleq 6358 | Relationship between subse... |
| onelss 6359 | An element of an ordinal n... |
| oneltri 6360 | The elementhood relation o... |
| ordtr1 6361 | Transitive law for ordinal... |
| ordtr2 6362 | Transitive law for ordinal... |
| ordtr3 6363 | Transitive law for ordinal... |
| ontr1 6364 | Transitive law for ordinal... |
| ontr2 6365 | Transitive law for ordinal... |
| onelssex 6366 | Ordinal less than is equiv... |
| ordunidif 6367 | The union of an ordinal st... |
| ordintdif 6368 | If ` B ` is smaller than `... |
| onintss 6369 | If a property is true for ... |
| oneqmini 6370 | A way to show that an ordi... |
| ord0 6371 | The empty set is an ordina... |
| 0elon 6372 | The empty set is an ordina... |
| ord0eln0 6373 | A nonempty ordinal contain... |
| on0eln0 6374 | An ordinal number contains... |
| dflim2 6375 | An alternate definition of... |
| inton 6376 | The intersection of the cl... |
| nlim0 6377 | The empty set is not a lim... |
| limord 6378 | A limit ordinal is ordinal... |
| limuni 6379 | A limit ordinal is its own... |
| limuni2 6380 | The union of a limit ordin... |
| 0ellim 6381 | A limit ordinal contains t... |
| limelon 6382 | A limit ordinal class that... |
| onn0 6383 | The class of all ordinal n... |
| suceqd 6384 | Deduction associated with ... |
| suceq 6385 | Equality of successors. (... |
| elsuci 6386 | Membership in a successor.... |
| elsucg 6387 | Membership in a successor.... |
| elsuc2g 6388 | Variant of membership in a... |
| elsuc 6389 | Membership in a successor.... |
| elsuc2 6390 | Membership in a successor.... |
| nfsuc 6391 | Bound-variable hypothesis ... |
| elelsuc 6392 | Membership in a successor.... |
| sucel 6393 | Membership of a successor ... |
| suc0 6394 | The successor of the empty... |
| sucprc 6395 | A proper class is its own ... |
| unisucs 6396 | The union of the successor... |
| unisucg 6397 | A transitive class is equa... |
| unisuc 6398 | A transitive class is equa... |
| sssucid 6399 | A class is included in its... |
| sucidg 6400 | Part of Proposition 7.23 o... |
| sucid 6401 | A set belongs to its succe... |
| nsuceq0 6402 | No successor is empty. (C... |
| eqelsuc 6403 | A set belongs to the succe... |
| iunsuc 6404 | Inductive definition for t... |
| suctr 6405 | The successor of a transit... |
| trsuc 6406 | A set whose successor belo... |
| trsucss 6407 | A member of the successor ... |
| ordsssuc 6408 | An ordinal is a subset of ... |
| onsssuc 6409 | A subset of an ordinal num... |
| ordsssuc2 6410 | An ordinal subset of an or... |
| onmindif 6411 | When its successor is subt... |
| ordnbtwn 6412 | There is no set between an... |
| onnbtwn 6413 | There is no set between an... |
| sucssel 6414 | A set whose successor is a... |
| orddif 6415 | Ordinal derived from its s... |
| orduniss 6416 | An ordinal class includes ... |
| ordtri2or 6417 | A trichotomy law for ordin... |
| ordtri2or2 6418 | A trichotomy law for ordin... |
| ordtri2or3 6419 | A consequence of total ord... |
| ordelinel 6420 | The intersection of two or... |
| ordssun 6421 | Property of a subclass of ... |
| ordequn 6422 | The maximum (i.e. union) o... |
| ordun 6423 | The maximum (i.e., union) ... |
| onunel 6424 | The union of two ordinals ... |
| ordunisssuc 6425 | A subclass relationship fo... |
| suc11 6426 | The successor operation be... |
| onun2 6427 | The union of two ordinals ... |
| ontr 6428 | An ordinal number is a tra... |
| onunisuc 6429 | An ordinal number is equal... |
| onordi 6430 | An ordinal number is an or... |
| onirri 6431 | An ordinal number is not a... |
| oneli 6432 | A member of an ordinal num... |
| onelssi 6433 | A member of an ordinal num... |
| onssneli 6434 | An ordering law for ordina... |
| onssnel2i 6435 | An ordering law for ordina... |
| onelini 6436 | An element of an ordinal n... |
| oneluni 6437 | An ordinal number equals i... |
| onunisuci 6438 | An ordinal number is equal... |
| onsseli 6439 | Subset is equivalent to me... |
| onun2i 6440 | The union of two ordinal n... |
| unizlim 6441 | An ordinal equal to its ow... |
| on0eqel 6442 | An ordinal number either e... |
| snsn0non 6443 | The singleton of the singl... |
| onxpdisj 6444 | Ordinal numbers and ordere... |
| onnev 6445 | The class of ordinal numbe... |
| iotajust 6447 | Soundness justification th... |
| dfiota2 6449 | Alternate definition for d... |
| nfiota1 6450 | Bound-variable hypothesis ... |
| nfiotadw 6451 | Deduction version of ~ nfi... |
| nfiotaw 6452 | Bound-variable hypothesis ... |
| nfiotad 6453 | Deduction version of ~ nfi... |
| nfiota 6454 | Bound-variable hypothesis ... |
| cbviotaw 6455 | Change bound variables in ... |
| cbviotavw 6456 | Change bound variables in ... |
| cbviota 6457 | Change bound variables in ... |
| cbviotav 6458 | Change bound variables in ... |
| sb8iota 6459 | Variable substitution in d... |
| iotaeq 6460 | Equality theorem for descr... |
| iotabi 6461 | Equivalence theorem for de... |
| uniabio 6462 | Part of Theorem 8.17 in [Q... |
| iotaval2 6463 | Version of ~ iotaval using... |
| iotauni2 6464 | Version of ~ iotauni using... |
| iotanul2 6465 | Version of ~ iotanul using... |
| iotaval 6466 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6467 | The ` iota ` class is a su... |
| iotaex 6468 | Theorem 8.23 in [Quine] p.... |
| iotauni 6469 | Equivalence between two di... |
| iotaint 6470 | Equivalence between two di... |
| iota1 6471 | Property of iota. (Contri... |
| iotanul 6472 | Theorem 8.22 in [Quine] p.... |
| iota4 6473 | Theorem *14.22 in [Whitehe... |
| iota4an 6474 | Theorem *14.23 in [Whitehe... |
| iota5 6475 | A method for computing iot... |
| iotabidv 6476 | Formula-building deduction... |
| iotabii 6477 | Formula-building deduction... |
| iotacl 6478 | Membership law for descrip... |
| iota2df 6479 | A condition that allows to... |
| iota2d 6480 | A condition that allows to... |
| iota2 6481 | The unique element such th... |
| iotan0 6482 | Representation of "the uni... |
| sniota 6483 | A class abstraction with a... |
| dfiota4 6484 | The ` iota ` operation usi... |
| csbiota 6485 | Class substitution within ... |
| dffun2 6502 | Alternate definition of a ... |
| dffun6 6503 | Alternate definition of a ... |
| dffun3 6504 | Alternate definition of fu... |
| dffun4 6505 | Alternate definition of a ... |
| dffun5 6506 | Alternate definition of fu... |
| dffun6f 6507 | Definition of function, us... |
| funmo 6508 | A function has at most one... |
| funrel 6509 | A function is a relation. ... |
| 0nelfun 6510 | A function does not contai... |
| funss 6511 | Subclass theorem for funct... |
| funeq 6512 | Equality theorem for funct... |
| funeqi 6513 | Equality inference for the... |
| funeqd 6514 | Equality deduction for the... |
| nffun 6515 | Bound-variable hypothesis ... |
| sbcfung 6516 | Distribute proper substitu... |
| funeu 6517 | There is exactly one value... |
| funeu2 6518 | There is exactly one value... |
| dffun7 6519 | Alternate definition of a ... |
| dffun8 6520 | Alternate definition of a ... |
| dffun9 6521 | Alternate definition of a ... |
| funfn 6522 | A class is a function if a... |
| funfnd 6523 | A function is a function o... |
| funi 6524 | The identity relation is a... |
| nfunv 6525 | The universal class is not... |
| funopg 6526 | A Kuratowski ordered pair ... |
| funopab 6527 | A class of ordered pairs i... |
| funopabeq 6528 | A class of ordered pairs o... |
| funopab4 6529 | A class of ordered pairs o... |
| funmpt 6530 | A function in maps-to nota... |
| funmpt2 6531 | Functionality of a class g... |
| funco 6532 | The composition of two fun... |
| funresfunco 6533 | Composition of two functio... |
| funres 6534 | A restriction of a functio... |
| funresd 6535 | A restriction of a functio... |
| funssres 6536 | The restriction of a funct... |
| fun2ssres 6537 | Equality of restrictions o... |
| funun 6538 | The union of functions wit... |
| fununmo 6539 | If the union of classes is... |
| fununfun 6540 | If the union of classes is... |
| fundif 6541 | A function with removed el... |
| funcnvsn 6542 | The converse singleton of ... |
| funsng 6543 | A singleton of an ordered ... |
| fnsng 6544 | Functionality and domain o... |
| funsn 6545 | A singleton of an ordered ... |
| funprg 6546 | A set of two pairs is a fu... |
| funtpg 6547 | A set of three pairs is a ... |
| funpr 6548 | A function with a domain o... |
| funtp 6549 | A function with a domain o... |
| fnsn 6550 | Functionality and domain o... |
| fnprg 6551 | Function with a domain of ... |
| fntpg 6552 | Function with a domain of ... |
| fntp 6553 | A function with a domain o... |
| funcnvpr 6554 | The converse pair of order... |
| funcnvtp 6555 | The converse triple of ord... |
| funcnvqp 6556 | The converse quadruple of ... |
| fun0 6557 | The empty set is a functio... |
| funcnv0 6558 | The converse of the empty ... |
| funcnvcnv 6559 | The double converse of a f... |
| funcnv2 6560 | A simpler equivalence for ... |
| funcnv 6561 | The converse of a class is... |
| funcnv3 6562 | A condition showing a clas... |
| fun2cnv 6563 | The double converse of a c... |
| svrelfun 6564 | A single-valued relation i... |
| fncnv 6565 | Single-rootedness (see ~ f... |
| fun11 6566 | Two ways of stating that `... |
| fununi 6567 | The union of a chain (with... |
| funin 6568 | The intersection with a fu... |
| funres11 6569 | The restriction of a one-t... |
| funcnvres 6570 | The converse of a restrict... |
| cnvresid 6571 | Converse of a restricted i... |
| funcnvres2 6572 | The converse of a restrict... |
| funimacnv 6573 | The image of the preimage ... |
| funimass1 6574 | A kind of contraposition l... |
| funimass2 6575 | A kind of contraposition l... |
| imadif 6576 | The image of a difference ... |
| imain 6577 | The image of an intersecti... |
| f1imadifssran 6578 | Condition for the range of... |
| funimaexg 6579 | Axiom of Replacement using... |
| funimaex 6580 | The image of a set under a... |
| isarep1 6581 | Part of a study of the Axi... |
| isarep2 6582 | Part of a study of the Axi... |
| fneq1 6583 | Equality theorem for funct... |
| fneq2 6584 | Equality theorem for funct... |
| fneq1d 6585 | Equality deduction for fun... |
| fneq2d 6586 | Equality deduction for fun... |
| fneq12d 6587 | Equality deduction for fun... |
| fneq12 6588 | Equality theorem for funct... |
| fneq1i 6589 | Equality inference for fun... |
| fneq2i 6590 | Equality inference for fun... |
| nffn 6591 | Bound-variable hypothesis ... |
| fnfun 6592 | A function with domain is ... |
| fnfund 6593 | A function with domain is ... |
| fnrel 6594 | A function with domain is ... |
| fndm 6595 | The domain of a function. ... |
| fndmi 6596 | The domain of a function. ... |
| fndmd 6597 | The domain of a function. ... |
| funfni 6598 | Inference to convert a fun... |
| fndmu 6599 | A function has a unique do... |
| fnbr 6600 | The first argument of bina... |
| fnop 6601 | The first argument of an o... |
| fneu 6602 | There is exactly one value... |
| fneu2 6603 | There is exactly one value... |
| fnunres1 6604 | Restriction of a disjoint ... |
| fnunres2 6605 | Restriction of a disjoint ... |
| fnun 6606 | The union of two functions... |
| fnund 6607 | The union of two functions... |
| fnunop 6608 | Extension of a function wi... |
| fncofn 6609 | Composition of a function ... |
| fnco 6610 | Composition of two functio... |
| fnresdm 6611 | A function does not change... |
| fnresdisj 6612 | A function restricted to a... |
| 2elresin 6613 | Membership in two function... |
| fnssresb 6614 | Restriction of a function ... |
| fnssres 6615 | Restriction of a function ... |
| fnssresd 6616 | Restriction of a function ... |
| fnresin1 6617 | Restriction of a function'... |
| fnresin2 6618 | Restriction of a function'... |
| fnres 6619 | An equivalence for functio... |
| idfn 6620 | The identity relation is a... |
| fnresi 6621 | The restricted identity re... |
| fnima 6622 | The image of a function's ... |
| fn0 6623 | A function with empty doma... |
| fnimadisj 6624 | A class that is disjoint w... |
| fnimaeq0 6625 | Images under a function ne... |
| dfmpt3 6626 | Alternate definition for t... |
| mptfnf 6627 | The maps-to notation defin... |
| fnmptf 6628 | The maps-to notation defin... |
| fnopabg 6629 | Functionality and domain o... |
| fnopab 6630 | Functionality and domain o... |
| mptfng 6631 | The maps-to notation defin... |
| fnmpt 6632 | The maps-to notation defin... |
| fnmptd 6633 | The maps-to notation defin... |
| mpt0 6634 | A mapping operation with e... |
| fnmpti 6635 | Functionality and domain o... |
| dmmpti 6636 | Domain of the mapping oper... |
| dmmptd 6637 | The domain of the mapping ... |
| mptun 6638 | Union of mappings which ar... |
| partfun 6639 | Rewrite a function defined... |
| feq1 6640 | Equality theorem for funct... |
| feq2 6641 | Equality theorem for funct... |
| feq3 6642 | Equality theorem for funct... |
| feq23 6643 | Equality theorem for funct... |
| feq1d 6644 | Equality deduction for fun... |
| feq1dd 6645 | Equality deduction for fun... |
| feq2d 6646 | Equality deduction for fun... |
| feq3d 6647 | Equality deduction for fun... |
| feq2dd 6648 | Equality deduction for fun... |
| feq3dd 6649 | Equality deduction for fun... |
| feq12d 6650 | Equality deduction for fun... |
| feq123d 6651 | Equality deduction for fun... |
| feq123 6652 | Equality theorem for funct... |
| feq1i 6653 | Equality inference for fun... |
| feq2i 6654 | Equality inference for fun... |
| feq12i 6655 | Equality inference for fun... |
| feq23i 6656 | Equality inference for fun... |
| feq23d 6657 | Equality deduction for fun... |
| nff 6658 | Bound-variable hypothesis ... |
| sbcfng 6659 | Distribute proper substitu... |
| sbcfg 6660 | Distribute proper substitu... |
| elimf 6661 | Eliminate a mapping hypoth... |
| ffn 6662 | A mapping is a function wi... |
| ffnd 6663 | A mapping is a function wi... |
| dffn2 6664 | Any function is a mapping ... |
| ffun 6665 | A mapping is a function. ... |
| ffund 6666 | A mapping is a function, d... |
| frel 6667 | A mapping is a relation. ... |
| freld 6668 | A mapping is a relation. ... |
| frn 6669 | The range of a mapping. (... |
| frnd 6670 | Deduction form of ~ frn . ... |
| fdm 6671 | The domain of a mapping. ... |
| fdmd 6672 | Deduction form of ~ fdm . ... |
| fdmi 6673 | Inference associated with ... |
| dffn3 6674 | A function maps to its ran... |
| ffrn 6675 | A function maps to its ran... |
| ffrnb 6676 | Characterization of a func... |
| ffrnbd 6677 | A function maps to its ran... |
| fss 6678 | Expanding the codomain of ... |
| fssd 6679 | Expanding the codomain of ... |
| fssdmd 6680 | Expressing that a class is... |
| fssdm 6681 | Expressing that a class is... |
| fimass 6682 | The image of a class under... |
| fimassd 6683 | The image of a class is a ... |
| fimacnv 6684 | The preimage of the codoma... |
| fcof 6685 | Composition of a function ... |
| fco 6686 | Composition of two functio... |
| fcod 6687 | Composition of two mapping... |
| fco2 6688 | Functionality of a composi... |
| fssxp 6689 | A mapping is a class of or... |
| funssxp 6690 | Two ways of specifying a p... |
| ffdm 6691 | A mapping is a partial fun... |
| ffdmd 6692 | The domain of a function. ... |
| fdmrn 6693 | A different way to write `... |
| funcofd 6694 | Composition of two functio... |
| opelf 6695 | The members of an ordered ... |
| fun 6696 | The union of two functions... |
| fun2 6697 | The union of two functions... |
| fun2d 6698 | The union of functions wit... |
| fnfco 6699 | Composition of two functio... |
| fssres 6700 | Restriction of a function ... |
| fssresd 6701 | Restriction of a function ... |
| fssres2 6702 | Restriction of a restricte... |
| fresin 6703 | An identity for the mappin... |
| resasplit 6704 | If two functions agree on ... |
| fresaun 6705 | The union of two functions... |
| fresaunres2 6706 | From the union of two func... |
| fresaunres1 6707 | From the union of two func... |
| fcoi1 6708 | Composition of a mapping a... |
| fcoi2 6709 | Composition of restricted ... |
| feu 6710 | There is exactly one value... |
| fcnvres 6711 | The converse of a restrict... |
| fimacnvdisj 6712 | The preimage of a class di... |
| fint 6713 | Function into an intersect... |
| fin 6714 | Mapping into an intersecti... |
| f0 6715 | The empty function. (Cont... |
| f00 6716 | A class is a function with... |
| f0bi 6717 | A function with empty doma... |
| f0dom0 6718 | A function is empty iff it... |
| f0rn0 6719 | If there is no element in ... |
| fconst 6720 | A Cartesian product with a... |
| fconstg 6721 | A Cartesian product with a... |
| fnconstg 6722 | A Cartesian product with a... |
| fconst6g 6723 | Constant function with loo... |
| fconst6 6724 | A constant function as a m... |
| f1eq1 6725 | Equality theorem for one-t... |
| f1eq2 6726 | Equality theorem for one-t... |
| f1eq3 6727 | Equality theorem for one-t... |
| nff1 6728 | Bound-variable hypothesis ... |
| dff12 6729 | Alternate definition of a ... |
| f1f 6730 | A one-to-one mapping is a ... |
| f1fn 6731 | A one-to-one mapping is a ... |
| f1fun 6732 | A one-to-one mapping is a ... |
| f1rel 6733 | A one-to-one onto mapping ... |
| f1dm 6734 | The domain of a one-to-one... |
| f1ss 6735 | A function that is one-to-... |
| f1ssr 6736 | A function that is one-to-... |
| f1ssres 6737 | A function that is one-to-... |
| f1resf1 6738 | The restriction of an inje... |
| f1cnvcnv 6739 | Two ways to express that a... |
| f1cof1 6740 | Composition of two one-to-... |
| f1co 6741 | Composition of one-to-one ... |
| foeq1 6742 | Equality theorem for onto ... |
| foeq2 6743 | Equality theorem for onto ... |
| foeq3 6744 | Equality theorem for onto ... |
| nffo 6745 | Bound-variable hypothesis ... |
| fof 6746 | An onto mapping is a mappi... |
| fofun 6747 | An onto mapping is a funct... |
| fofn 6748 | An onto mapping is a funct... |
| forn 6749 | The codomain of an onto fu... |
| dffo2 6750 | Alternate definition of an... |
| foima 6751 | The image of the domain of... |
| dffn4 6752 | A function maps onto its r... |
| funforn 6753 | A function maps its domain... |
| fodmrnu 6754 | An onto function has uniqu... |
| fimadmfo 6755 | A function is a function o... |
| fores 6756 | Restriction of an onto fun... |
| fimadmfoALT 6757 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6758 | The preimage of the codoma... |
| focofo 6759 | Composition of onto functi... |
| foco 6760 | Composition of onto functi... |
| foconst 6761 | A nonzero constant functio... |
| f1oeq1 6762 | Equality theorem for one-t... |
| f1oeq2 6763 | Equality theorem for one-t... |
| f1oeq3 6764 | Equality theorem for one-t... |
| f1oeq23 6765 | Equality theorem for one-t... |
| f1eq123d 6766 | Equality deduction for one... |
| foeq123d 6767 | Equality deduction for ont... |
| f1oeq123d 6768 | Equality deduction for one... |
| f1oeq1d 6769 | Equality deduction for one... |
| f1oeq2d 6770 | Equality deduction for one... |
| f1oeq3d 6771 | Equality deduction for one... |
| nff1o 6772 | Bound-variable hypothesis ... |
| f1of1 6773 | A one-to-one onto mapping ... |
| f1of 6774 | A one-to-one onto mapping ... |
| f1ofn 6775 | A one-to-one onto mapping ... |
| f1ofun 6776 | A one-to-one onto mapping ... |
| f1orel 6777 | A one-to-one onto mapping ... |
| f1odm 6778 | The domain of a one-to-one... |
| dff1o2 6779 | Alternate definition of on... |
| dff1o3 6780 | Alternate definition of on... |
| f1ofo 6781 | A one-to-one onto function... |
| dff1o4 6782 | Alternate definition of on... |
| dff1o5 6783 | Alternate definition of on... |
| f1orn 6784 | A one-to-one function maps... |
| f1f1orn 6785 | A one-to-one function maps... |
| f1ocnv 6786 | The converse of a one-to-o... |
| f1ocnvb 6787 | A relation is a one-to-one... |
| f1ores 6788 | The restriction of a one-t... |
| f1orescnv 6789 | The converse of a one-to-o... |
| f1imacnv 6790 | Preimage of an image. (Co... |
| foimacnv 6791 | A reverse version of ~ f1i... |
| foun 6792 | The union of two onto func... |
| f1oun 6793 | The union of two one-to-on... |
| f1un 6794 | The union of two one-to-on... |
| resdif 6795 | The restriction of a one-t... |
| resin 6796 | The restriction of a one-t... |
| f1oco 6797 | Composition of one-to-one ... |
| f1cnv 6798 | The converse of an injecti... |
| funcocnv2 6799 | Composition with the conve... |
| fococnv2 6800 | The composition of an onto... |
| f1ococnv2 6801 | The composition of a one-t... |
| f1cocnv2 6802 | Composition of an injectiv... |
| f1ococnv1 6803 | The composition of a one-t... |
| f1cocnv1 6804 | Composition of an injectiv... |
| funcoeqres 6805 | Express a constraint on a ... |
| f1ssf1 6806 | A subset of an injective f... |
| f10 6807 | The empty set maps one-to-... |
| f10d 6808 | The empty set maps one-to-... |
| f1o00 6809 | One-to-one onto mapping of... |
| fo00 6810 | Onto mapping of the empty ... |
| f1o0 6811 | One-to-one onto mapping of... |
| f1oi 6812 | A restriction of the ident... |
| f1oiOLD 6813 | Obsolete version of ~ f1oi... |
| f1ovi 6814 | The identity relation is a... |
| f1osn 6815 | A singleton of an ordered ... |
| f1osng 6816 | A singleton of an ordered ... |
| f1sng 6817 | A singleton of an ordered ... |
| fsnd 6818 | A singleton of an ordered ... |
| f1oprswap 6819 | A two-element swap is a bi... |
| f1oprg 6820 | An unordered pair of order... |
| tz6.12-2 6821 | Function value when ` F ` ... |
| tz6.12-2OLD 6822 | Obsolete version of ~ tz6.... |
| fveu 6823 | The value of a function at... |
| brprcneu 6824 | If ` A ` is a proper class... |
| brprcneuALT 6825 | Alternate proof of ~ brprc... |
| fvprc 6826 | A function's value at a pr... |
| fvprcALT 6827 | Alternate proof of ~ fvprc... |
| rnfvprc 6828 | The range of a function va... |
| fv2 6829 | Alternate definition of fu... |
| dffv3 6830 | A definition of function v... |
| dffv4 6831 | The previous definition of... |
| elfv 6832 | Membership in a function v... |
| fveq1 6833 | Equality theorem for funct... |
| fveq2 6834 | Equality theorem for funct... |
| fveq1i 6835 | Equality inference for fun... |
| fveq1d 6836 | Equality deduction for fun... |
| fveq2i 6837 | Equality inference for fun... |
| fveq2d 6838 | Equality deduction for fun... |
| 2fveq3 6839 | Equality theorem for neste... |
| fveq12i 6840 | Equality deduction for fun... |
| fveq12d 6841 | Equality deduction for fun... |
| fveqeq2d 6842 | Equality deduction for fun... |
| fveqeq2 6843 | Equality deduction for fun... |
| nffv 6844 | Bound-variable hypothesis ... |
| nffvmpt1 6845 | Bound-variable hypothesis ... |
| nffvd 6846 | Deduction version of bound... |
| fvex 6847 | The value of a class exist... |
| fvexi 6848 | The value of a class exist... |
| fvexd 6849 | The value of a class exist... |
| fvif 6850 | Move a conditional outside... |
| iffv 6851 | Move a conditional outside... |
| fv3 6852 | Alternate definition of th... |
| fvres 6853 | The value of a restricted ... |
| fvresd 6854 | The value of a restricted ... |
| funssfv 6855 | The value of a member of t... |
| tz6.12c 6856 | Corollary of Theorem 6.12(... |
| tz6.12-1 6857 | Function value. Theorem 6... |
| tz6.12 6858 | Function value. Theorem 6... |
| tz6.12f 6859 | Function value, using boun... |
| tz6.12i 6860 | Corollary of Theorem 6.12(... |
| fvbr0 6861 | Two possibilities for the ... |
| fvrn0 6862 | A function value is a memb... |
| fvn0fvelrn 6863 | If the value of a function... |
| elfvunirn 6864 | A function value is a subs... |
| fvssunirn 6865 | The result of a function v... |
| ndmfv 6866 | The value of a class outsi... |
| ndmfvrcl 6867 | Reverse closure law for fu... |
| elfvdm 6868 | If a function value has a ... |
| elfvex 6869 | If a function value has a ... |
| elfvexd 6870 | If a function value has a ... |
| eliman0 6871 | A nonempty function value ... |
| nfvres 6872 | The value of a non-member ... |
| nfunsn 6873 | If the restriction of a cl... |
| fvfundmfvn0 6874 | If the "value of a class" ... |
| 0fv 6875 | Function value of the empt... |
| fv2prc 6876 | A function value of a func... |
| elfv2ex 6877 | If a function value of a f... |
| fveqres 6878 | Equal values imply equal v... |
| csbfv12 6879 | Move class substitution in... |
| csbfv2g 6880 | Move class substitution in... |
| csbfv 6881 | Substitution for a functio... |
| funbrfv 6882 | The second argument of a b... |
| funopfv 6883 | The second element in an o... |
| fnbrfvb 6884 | Equivalence of function va... |
| fnopfvb 6885 | Equivalence of function va... |
| fvelima2 6886 | Function value in an image... |
| funbrfvb 6887 | Equivalence of function va... |
| funopfvb 6888 | Equivalence of function va... |
| fnbrfvb2 6889 | Version of ~ fnbrfvb for f... |
| fdmeu 6890 | There is exactly one codom... |
| funbrfv2b 6891 | Function value in terms of... |
| dffn5 6892 | Representation of a functi... |
| fnrnfv 6893 | The range of a function ex... |
| fvelrnb 6894 | A member of a function's r... |
| foelcdmi 6895 | A member of a surjective f... |
| dfimafn 6896 | Alternate definition of th... |
| dfimafn2 6897 | Alternate definition of th... |
| funimass4 6898 | Membership relation for th... |
| fvelima 6899 | Function value in an image... |
| funimassd 6900 | Sufficient condition for t... |
| fvelimad 6901 | Function value in an image... |
| feqmptd 6902 | Deduction form of ~ dffn5 ... |
| feqresmpt 6903 | Express a restricted funct... |
| feqmptdf 6904 | Deduction form of ~ dffn5f... |
| dffn5f 6905 | Representation of a functi... |
| fvelimab 6906 | Function value in an image... |
| fvelimabd 6907 | Deduction form of ~ fvelim... |
| fimarab 6908 | Expressing the image of a ... |
| unima 6909 | Image of a union. (Contri... |
| fvi 6910 | The value of the identity ... |
| fviss 6911 | The value of the identity ... |
| fniinfv 6912 | The indexed intersection o... |
| fnsnfv 6913 | Singleton of function valu... |
| opabiotafun 6914 | Define a function whose va... |
| opabiotadm 6915 | Define a function whose va... |
| opabiota 6916 | Define a function whose va... |
| fnimapr 6917 | The image of a pair under ... |
| fnimatpd 6918 | The image of an unordered ... |
| ssimaex 6919 | The existence of a subimag... |
| ssimaexg 6920 | The existence of a subimag... |
| funfv 6921 | A simplified expression fo... |
| funfv2 6922 | The value of a function. ... |
| funfv2f 6923 | The value of a function. ... |
| fvun 6924 | Value of the union of two ... |
| fvun1 6925 | The value of a union when ... |
| fvun2 6926 | The value of a union when ... |
| fvun1d 6927 | The value of a union when ... |
| fvun2d 6928 | The value of a union when ... |
| dffv2 6929 | Alternate definition of fu... |
| dmfco 6930 | Domains of a function comp... |
| fvco2 6931 | Value of a function compos... |
| fvco 6932 | Value of a function compos... |
| fvco3 6933 | Value of a function compos... |
| fvco3d 6934 | Value of a function compos... |
| fvco4i 6935 | Conditions for a compositi... |
| fvopab3g 6936 | Value of a function given ... |
| fvopab3ig 6937 | Value of a function given ... |
| brfvopabrbr 6938 | The binary relation of a f... |
| fvmptg 6939 | Value of a function given ... |
| fvmpti 6940 | Value of a function given ... |
| fvmpt 6941 | Value of a function given ... |
| fvmpt2f 6942 | Value of a function given ... |
| fvtresfn 6943 | Functionality of a tuple-r... |
| fvmpts 6944 | Value of a function given ... |
| fvmpt3 6945 | Value of a function given ... |
| fvmpt3i 6946 | Value of a function given ... |
| fvmptdf 6947 | Deduction version of ~ fvm... |
| fvmptd 6948 | Deduction version of ~ fvm... |
| fvmptd2 6949 | Deduction version of ~ fvm... |
| mptrcl 6950 | Reverse closure for a mapp... |
| fvmpt2i 6951 | Value of a function given ... |
| fvmpt2 6952 | Value of a function given ... |
| fvmptss 6953 | If all the values of the m... |
| fvmpt2d 6954 | Deduction version of ~ fvm... |
| fvmptex 6955 | Express a function ` F ` w... |
| fvmptd3f 6956 | Alternate deduction versio... |
| fvmptd2f 6957 | Alternate deduction versio... |
| fvmptdv 6958 | Alternate deduction versio... |
| fvmptdv2 6959 | Alternate deduction versio... |
| mpteqb 6960 | Bidirectional equality the... |
| fvmptt 6961 | Closed theorem form of ~ f... |
| fvmptf 6962 | Value of a function given ... |
| fvmptnf 6963 | The value of a function gi... |
| fvmptd3 6964 | Deduction version of ~ fvm... |
| fvmptd4 6965 | Deduction version of ~ fvm... |
| fvmptn 6966 | This somewhat non-intuitiv... |
| fvmptss2 6967 | A mapping always evaluates... |
| elfvmptrab1w 6968 | Implications for the value... |
| elfvmptrab1 6969 | Implications for the value... |
| elfvmptrab 6970 | Implications for the value... |
| fvopab4ndm 6971 | Value of a function given ... |
| fvmptndm 6972 | Value of a function given ... |
| fvmptrabfv 6973 | Value of a function mappin... |
| fvopab5 6974 | The value of a function th... |
| fvopab6 6975 | Value of a function given ... |
| eqfnfv 6976 | Equality of functions is d... |
| eqfnfv2 6977 | Equality of functions is d... |
| eqfnfv3 6978 | Derive equality of functio... |
| eqfnfvd 6979 | Deduction for equality of ... |
| eqfnfv2f 6980 | Equality of functions is d... |
| eqfunfv 6981 | Equality of functions is d... |
| eqfnun 6982 | Two functions on ` A u. B ... |
| fvreseq0 6983 | Equality of restricted fun... |
| fvreseq1 6984 | Equality of a function res... |
| fvreseq 6985 | Equality of restricted fun... |
| fnmptfvd 6986 | A function with a given do... |
| fndmdif 6987 | Two ways to express the lo... |
| fndmdifcom 6988 | The difference set between... |
| fndmdifeq0 6989 | The difference set of two ... |
| fndmin 6990 | Two ways to express the lo... |
| fneqeql 6991 | Two functions are equal if... |
| fneqeql2 6992 | Two functions are equal if... |
| fnreseql 6993 | Two functions are equal on... |
| chfnrn 6994 | The range of a choice func... |
| funfvop 6995 | Ordered pair with function... |
| funfvbrb 6996 | Two ways to say that ` A `... |
| fvimacnvi 6997 | A member of a preimage is ... |
| fvimacnv 6998 | The argument of a function... |
| funimass3 6999 | A kind of contraposition l... |
| funimass5 7000 | A subclass of a preimage i... |
| funconstss 7001 | Two ways of specifying tha... |
| fvimacnvALT 7002 | Alternate proof of ~ fvima... |
| elpreima 7003 | Membership in the preimage... |
| elpreimad 7004 | Membership in the preimage... |
| fniniseg 7005 | Membership in the preimage... |
| fncnvima2 7006 | Inverse images under funct... |
| fniniseg2 7007 | Inverse point images under... |
| unpreima 7008 | Preimage of a union. (Con... |
| inpreima 7009 | Preimage of an intersectio... |
| difpreima 7010 | Preimage of a difference. ... |
| respreima 7011 | The preimage of a restrict... |
| cnvimainrn 7012 | The preimage of the inters... |
| sspreima 7013 | The preimage of a subset i... |
| iinpreima 7014 | Preimage of an intersectio... |
| intpreima 7015 | Preimage of an intersectio... |
| fimacnvinrn 7016 | Taking the converse image ... |
| fimacnvinrn2 7017 | Taking the converse image ... |
| rescnvimafod 7018 | The restriction of a funct... |
| fvn0ssdmfun 7019 | If a class' function value... |
| fnopfv 7020 | Ordered pair with function... |
| fvelrn 7021 | A function's value belongs... |
| nelrnfvne 7022 | A function value cannot be... |
| fveqdmss 7023 | If the empty set is not co... |
| fveqressseq 7024 | If the empty set is not co... |
| fnfvelrn 7025 | A function's value belongs... |
| ffvelcdm 7026 | A function's value belongs... |
| fnfvelrnd 7027 | A function's value belongs... |
| ffvelcdmi 7028 | A function's value belongs... |
| ffvelcdmda 7029 | A function's value belongs... |
| ffvelcdmd 7030 | A function's value belongs... |
| feldmfvelcdm 7031 | A class is an element of t... |
| rexrn 7032 | Restricted existential qua... |
| ralrn 7033 | Restricted universal quant... |
| elrnrexdm 7034 | For any element in the ran... |
| elrnrexdmb 7035 | For any element in the ran... |
| eldmrexrn 7036 | For any element in the dom... |
| eldmrexrnb 7037 | For any element in the dom... |
| fvcofneq 7038 | The values of two function... |
| ralrnmptw 7039 | A restricted quantifier ov... |
| rexrnmptw 7040 | A restricted quantifier ov... |
| ralrnmpt 7041 | A restricted quantifier ov... |
| rexrnmpt 7042 | A restricted quantifier ov... |
| f0cli 7043 | Unconditional closure of a... |
| dff2 7044 | Alternate definition of a ... |
| dff3 7045 | Alternate definition of a ... |
| dff4 7046 | Alternate definition of a ... |
| dffo3 7047 | An onto mapping expressed ... |
| dffo4 7048 | Alternate definition of an... |
| dffo5 7049 | Alternate definition of an... |
| exfo 7050 | A relation equivalent to t... |
| dffo3f 7051 | An onto mapping expressed ... |
| foelrn 7052 | Property of a surjective f... |
| foelrnf 7053 | Property of a surjective f... |
| foco2 7054 | If a composition of two fu... |
| fmpt 7055 | Functionality of the mappi... |
| f1ompt 7056 | Express bijection for a ma... |
| fmpti 7057 | Functionality of the mappi... |
| fvmptelcdm 7058 | The value of a function at... |
| fmptd 7059 | Domain and codomain of the... |
| fmpttd 7060 | Version of ~ fmptd with in... |
| fmpt3d 7061 | Domain and codomain of the... |
| fmptdf 7062 | A version of ~ fmptd using... |
| fompt 7063 | Express being onto for a m... |
| ffnfv 7064 | A function maps to a class... |
| ffnfvf 7065 | A function maps to a class... |
| fnfvrnss 7066 | An upper bound for range d... |
| fcdmssb 7067 | A function is a function i... |
| rnmptss 7068 | The range of an operation ... |
| fmpt2d 7069 | Domain and codomain of the... |
| ffvresb 7070 | A necessary and sufficient... |
| fssrescdmd 7071 | Restriction of a function ... |
| f1oresrab 7072 | Build a bijection between ... |
| f1ossf1o 7073 | Restricting a bijection, w... |
| fmptco 7074 | Composition of two functio... |
| fmptcof 7075 | Version of ~ fmptco where ... |
| fmptcos 7076 | Composition of two functio... |
| cofmpt 7077 | Express composition of a m... |
| fcompt 7078 | Express composition of two... |
| fcoconst 7079 | Composition with a constan... |
| fsn 7080 | A function maps a singleto... |
| fsn2 7081 | A function that maps a sin... |
| fsng 7082 | A function maps a singleto... |
| fsn2g 7083 | A function that maps a sin... |
| xpsng 7084 | The Cartesian product of t... |
| xpprsng 7085 | The Cartesian product of a... |
| xpsn 7086 | The Cartesian product of t... |
| f1o2sn 7087 | A singleton consisting in ... |
| residpr 7088 | Restriction of the identit... |
| dfmpt 7089 | Alternate definition for t... |
| fnasrn 7090 | A function expressed as th... |
| idref 7091 | Two ways to state that a r... |
| funiun 7092 | A function is a union of s... |
| funopsn 7093 | If a function is an ordere... |
| funop 7094 | An ordered pair is a funct... |
| funopdmsn 7095 | The domain of a function w... |
| funsndifnop 7096 | A singleton of an ordered ... |
| funsneqopb 7097 | A singleton of an ordered ... |
| ressnop0 7098 | If ` A ` is not in ` C ` ,... |
| fpr 7099 | A function with a domain o... |
| fprg 7100 | A function with a domain o... |
| ftpg 7101 | A function with a domain o... |
| ftp 7102 | A function with a domain o... |
| fnressn 7103 | A function restricted to a... |
| funressn 7104 | A function restricted to a... |
| fressnfv 7105 | The value of a function re... |
| fvrnressn 7106 | If the value of a function... |
| fvressn 7107 | The value of a function re... |
| fvconst 7108 | The value of a constant fu... |
| fnsnr 7109 | If a class belongs to a fu... |
| fnsnbg 7110 | A function's domain is a s... |
| fnsnb 7111 | A function whose domain is... |
| fnsnbOLD 7112 | Obsolete version of ~ fnsn... |
| fmptsn 7113 | Express a singleton functi... |
| fmptsng 7114 | Express a singleton functi... |
| fmptsnd 7115 | Express a singleton functi... |
| fmptap 7116 | Append an additional value... |
| fmptapd 7117 | Append an additional value... |
| fmptpr 7118 | Express a pair function in... |
| fvresi 7119 | The value of a restricted ... |
| fninfp 7120 | Express the class of fixed... |
| fnelfp 7121 | Property of a fixed point ... |
| fndifnfp 7122 | Express the class of non-f... |
| fnelnfp 7123 | Property of a non-fixed po... |
| fnnfpeq0 7124 | A function is the identity... |
| fvunsn 7125 | Remove an ordered pair not... |
| fvsng 7126 | The value of a singleton o... |
| fvsn 7127 | The value of a singleton o... |
| fvsnun1 7128 | The value of a function wi... |
| fvsnun2 7129 | The value of a function wi... |
| fnsnsplit 7130 | Split a function into a si... |
| fsnunf 7131 | Adjoining a point to a fun... |
| fsnunf2 7132 | Adjoining a point to a pun... |
| fsnunfv 7133 | Recover the added point fr... |
| fsnunres 7134 | Recover the original funct... |
| funresdfunsn 7135 | Restricting a function to ... |
| fvpr1g 7136 | The value of a function wi... |
| fvpr2g 7137 | The value of a function wi... |
| fvpr1 7138 | The value of a function wi... |
| fvpr2 7139 | The value of a function wi... |
| fprb 7140 | A condition for functionho... |
| fvtp1 7141 | The first value of a funct... |
| fvtp2 7142 | The second value of a func... |
| fvtp3 7143 | The third value of a funct... |
| fvtp1g 7144 | The value of a function wi... |
| fvtp2g 7145 | The value of a function wi... |
| fvtp3g 7146 | The value of a function wi... |
| tpres 7147 | An unordered triple of ord... |
| fvconst2g 7148 | The value of a constant fu... |
| fconst2g 7149 | A constant function expres... |
| fvconst2 7150 | The value of a constant fu... |
| fconst2 7151 | A constant function expres... |
| fconst5 7152 | Two ways to express that a... |
| rnmptc 7153 | Range of a constant functi... |
| fnprb 7154 | A function whose domain ha... |
| fntpb 7155 | A function whose domain ha... |
| fnpr2g 7156 | A function whose domain ha... |
| fpr2g 7157 | A function that maps a pai... |
| fconstfv 7158 | A constant function expres... |
| fconst3 7159 | Two ways to express a cons... |
| fconst4 7160 | Two ways to express a cons... |
| resfunexg 7161 | The restriction of a funct... |
| resiexd 7162 | The restriction of the ide... |
| fnex 7163 | If the domain of a functio... |
| fnexd 7164 | If the domain of a functio... |
| funex 7165 | If the domain of a functio... |
| opabex 7166 | Existence of a function ex... |
| mptexg 7167 | If the domain of a functio... |
| mptexgf 7168 | If the domain of a functio... |
| mptex 7169 | If the domain of a functio... |
| mptexd 7170 | If the domain of a functio... |
| mptrabex 7171 | If the domain of a functio... |
| fex 7172 | If the domain of a mapping... |
| fexd 7173 | If the domain of a mapping... |
| mptfvmpt 7174 | A function in maps-to nota... |
| eufnfv 7175 | A function is uniquely det... |
| funfvima 7176 | A function's value in a pr... |
| funfvima2 7177 | A function's value in an i... |
| funfvima2d 7178 | A function's value in a pr... |
| fnfvima 7179 | The function value of an o... |
| fnfvimad 7180 | A function's value belongs... |
| resfvresima 7181 | The value of the function ... |
| funfvima3 7182 | A class including a functi... |
| ralima 7183 | Universal quantification u... |
| rexima 7184 | Existential quantification... |
| reximaOLD 7185 | Obsolete version of ~ rexi... |
| ralimaOLD 7186 | Obsolete version of ~ rali... |
| fvclss 7187 | Upper bound for the class ... |
| elabrex 7188 | Elementhood in an image se... |
| elabrexg 7189 | Elementhood in an image se... |
| abrexco 7190 | Composition of two image m... |
| imaiun 7191 | The image of an indexed un... |
| imauni 7192 | The image of a union is th... |
| fniunfv 7193 | The indexed union of a fun... |
| funiunfv 7194 | The indexed union of a fun... |
| funiunfvf 7195 | The indexed union of a fun... |
| eluniima 7196 | Membership in the union of... |
| elunirn 7197 | Membership in the union of... |
| elunirnALT 7198 | Alternate proof of ~ eluni... |
| fnunirn 7199 | Membership in a union of s... |
| dff13 7200 | A one-to-one function in t... |
| dff13f 7201 | A one-to-one function in t... |
| f1veqaeq 7202 | If the values of a one-to-... |
| f1cofveqaeq 7203 | If the values of a composi... |
| f1cofveqaeqALT 7204 | Alternate proof of ~ f1cof... |
| dff14i 7205 | A one-to-one function maps... |
| 2f1fvneq 7206 | If two one-to-one function... |
| f1mpt 7207 | Express injection for a ma... |
| f1fveq 7208 | Equality of function value... |
| f1elima 7209 | Membership in the image of... |
| f1imass 7210 | Taking images under a one-... |
| f1imaeq 7211 | Taking images under a one-... |
| f1imapss 7212 | Taking images under a one-... |
| fpropnf1 7213 | A function, given by an un... |
| f1dom3fv3dif 7214 | The function values for a ... |
| f1dom3el3dif 7215 | The codomain of a 1-1 func... |
| dff14a 7216 | A one-to-one function in t... |
| dff14b 7217 | A one-to-one function in t... |
| f1ounsn 7218 | Extension of a bijection b... |
| f12dfv 7219 | A one-to-one function with... |
| f13dfv 7220 | A one-to-one function with... |
| dff1o6 7221 | A one-to-one onto function... |
| f1ocnvfv1 7222 | The converse value of the ... |
| f1ocnvfv2 7223 | The value of the converse ... |
| f1ocnvfv 7224 | Relationship between the v... |
| f1ocnvfvb 7225 | Relationship between the v... |
| nvof1o 7226 | An involution is a bijecti... |
| nvocnv 7227 | The converse of an involut... |
| f1cdmsn 7228 | If a one-to-one function w... |
| fsnex 7229 | Relate a function with a s... |
| f1prex 7230 | Relate a one-to-one functi... |
| f1ocnvdm 7231 | The value of the converse ... |
| f1ocnvfvrneq 7232 | If the values of a one-to-... |
| fcof1 7233 | An application is injectiv... |
| fcofo 7234 | An application is surjecti... |
| cbvfo 7235 | Change bound variable betw... |
| cbvexfo 7236 | Change bound variable betw... |
| cocan1 7237 | An injection is left-cance... |
| cocan2 7238 | A surjection is right-canc... |
| fcof1oinvd 7239 | Show that a function is th... |
| fcof1od 7240 | A function is bijective if... |
| 2fcoidinvd 7241 | Show that a function is th... |
| fcof1o 7242 | Show that two functions ar... |
| 2fvcoidd 7243 | Show that the composition ... |
| 2fvidf1od 7244 | A function is bijective if... |
| 2fvidinvd 7245 | Show that two functions ar... |
| foeqcnvco 7246 | Condition for function equ... |
| f1eqcocnv 7247 | Condition for function equ... |
| fveqf1o 7248 | Given a bijection ` F ` , ... |
| f1ocoima 7249 | The composition of two bij... |
| nf1const 7250 | A constant function from a... |
| nf1oconst 7251 | A constant function from a... |
| f1ofvswap 7252 | Swapping two values in a b... |
| fvf1pr 7253 | Values of a one-to-one fun... |
| fliftrel 7254 | ` F ` , a function lift, i... |
| fliftel 7255 | Elementhood in the relatio... |
| fliftel1 7256 | Elementhood in the relatio... |
| fliftcnv 7257 | Converse of the relation `... |
| fliftfun 7258 | The function ` F ` is the ... |
| fliftfund 7259 | The function ` F ` is the ... |
| fliftfuns 7260 | The function ` F ` is the ... |
| fliftf 7261 | The domain and range of th... |
| fliftval 7262 | The value of the function ... |
| isoeq1 7263 | Equality theorem for isomo... |
| isoeq2 7264 | Equality theorem for isomo... |
| isoeq3 7265 | Equality theorem for isomo... |
| isoeq4 7266 | Equality theorem for isomo... |
| isoeq5 7267 | Equality theorem for isomo... |
| nfiso 7268 | Bound-variable hypothesis ... |
| isof1o 7269 | An isomorphism is a one-to... |
| isof1oidb 7270 | A function is a bijection ... |
| isof1oopb 7271 | A function is a bijection ... |
| isorel 7272 | An isomorphism connects bi... |
| soisores 7273 | Express the condition of i... |
| soisoi 7274 | Infer isomorphism from one... |
| isoid 7275 | Identity law for isomorphi... |
| isocnv 7276 | Converse law for isomorphi... |
| isocnv2 7277 | Converse law for isomorphi... |
| isocnv3 7278 | Complementation law for is... |
| isores2 7279 | An isomorphism from one we... |
| isores1 7280 | An isomorphism from one we... |
| isores3 7281 | Induced isomorphism on a s... |
| isotr 7282 | Composition (transitive) l... |
| isomin 7283 | Isomorphisms preserve mini... |
| isoini 7284 | Isomorphisms preserve init... |
| isoini2 7285 | Isomorphisms are isomorphi... |
| isofrlem 7286 | Lemma for ~ isofr . (Cont... |
| isoselem 7287 | Lemma for ~ isose . (Cont... |
| isofr 7288 | An isomorphism preserves w... |
| isose 7289 | An isomorphism preserves s... |
| isofr2 7290 | A weak form of ~ isofr tha... |
| isopolem 7291 | Lemma for ~ isopo . (Cont... |
| isopo 7292 | An isomorphism preserves t... |
| isosolem 7293 | Lemma for ~ isoso . (Cont... |
| isoso 7294 | An isomorphism preserves t... |
| isowe 7295 | An isomorphism preserves t... |
| isowe2 7296 | A weak form of ~ isowe tha... |
| f1oiso 7297 | Any one-to-one onto functi... |
| f1oiso2 7298 | Any one-to-one onto functi... |
| f1owe 7299 | Well-ordering of isomorphi... |
| weniso 7300 | A set-like well-ordering h... |
| weisoeq 7301 | Thus, there is at most one... |
| weisoeq2 7302 | Thus, there is at most one... |
| knatar 7303 | The Knaster-Tarski theorem... |
| fvresval 7304 | The value of a restricted ... |
| funeldmb 7305 | If ` (/) ` is not part of ... |
| eqfunresadj 7306 | Law for adjoining an eleme... |
| eqfunressuc 7307 | Law for equality of restri... |
| fnssintima 7308 | Condition for subset of an... |
| imaeqsexvOLD 7309 | Obsolete version of ~ rexi... |
| imaeqsalvOLD 7310 | Obsolete version of ~ rali... |
| fnimasnd 7311 | The image of a function by... |
| canth 7312 | No set ` A ` is equinumero... |
| ncanth 7313 | Cantor's theorem fails for... |
| riotaeqdv 7316 | Formula-building deduction... |
| riotabidv 7317 | Formula-building deduction... |
| riotaeqbidv 7318 | Equality deduction for res... |
| riotaex 7319 | Restricted iota is a set. ... |
| riotav 7320 | An iota restricted to the ... |
| riotauni 7321 | Restricted iota in terms o... |
| nfriota1 7322 | The abstraction variable i... |
| nfriotadw 7323 | Deduction version of ~ nfr... |
| cbvriotaw 7324 | Change bound variable in a... |
| cbvriotavw 7325 | Change bound variable in a... |
| nfriotad 7326 | Deduction version of ~ nfr... |
| nfriota 7327 | A variable not free in a w... |
| cbvriota 7328 | Change bound variable in a... |
| cbvriotav 7329 | Change bound variable in a... |
| csbriota 7330 | Interchange class substitu... |
| riotacl2 7331 | Membership law for "the un... |
| riotacl 7332 | Closure of restricted iota... |
| riotasbc 7333 | Substitution law for descr... |
| riotabidva 7334 | Equivalent wff's yield equ... |
| riotabiia 7335 | Equivalent wff's yield equ... |
| riota1 7336 | Property of restricted iot... |
| riota1a 7337 | Property of iota. (Contri... |
| riota2df 7338 | A deduction version of ~ r... |
| riota2f 7339 | This theorem shows a condi... |
| riota2 7340 | This theorem shows a condi... |
| riotaeqimp 7341 | If two restricted iota des... |
| riotaprop 7342 | Properties of a restricted... |
| riota5f 7343 | A method for computing res... |
| riota5 7344 | A method for computing res... |
| riotass2 7345 | Restriction of a unique el... |
| riotass 7346 | Restriction of a unique el... |
| moriotass 7347 | Restriction of a unique el... |
| snriota 7348 | A restricted class abstrac... |
| riotaxfrd 7349 | Change the variable ` x ` ... |
| eusvobj2 7350 | Specify the same property ... |
| eusvobj1 7351 | Specify the same object in... |
| f1ofveu 7352 | There is one domain elemen... |
| f1ocnvfv3 7353 | Value of the converse of a... |
| riotaund 7354 | Restricted iota equals the... |
| riotassuni 7355 | The restricted iota class ... |
| riotaclb 7356 | Bidirectional closure of r... |
| riotarab 7357 | Restricted iota of a restr... |
| oveq 7364 | Equality theorem for opera... |
| oveq1 7365 | Equality theorem for opera... |
| oveq2 7366 | Equality theorem for opera... |
| oveq12 7367 | Equality theorem for opera... |
| oveq1i 7368 | Equality inference for ope... |
| oveq2i 7369 | Equality inference for ope... |
| oveq12i 7370 | Equality inference for ope... |
| oveqi 7371 | Equality inference for ope... |
| oveq123i 7372 | Equality inference for ope... |
| oveq1d 7373 | Equality deduction for ope... |
| oveq2d 7374 | Equality deduction for ope... |
| oveqd 7375 | Equality deduction for ope... |
| oveq12d 7376 | Equality deduction for ope... |
| oveqan12d 7377 | Equality deduction for ope... |
| oveqan12rd 7378 | Equality deduction for ope... |
| oveq123d 7379 | Equality deduction for ope... |
| fvoveq1d 7380 | Equality deduction for nes... |
| fvoveq1 7381 | Equality theorem for neste... |
| ovanraleqv 7382 | Equality theorem for a con... |
| imbrov2fvoveq 7383 | Equality theorem for neste... |
| ovrspc2v 7384 | If an operation value is a... |
| oveqrspc2v 7385 | Restricted specialization ... |
| oveqdr 7386 | Equality of two operations... |
| nfovd 7387 | Deduction version of bound... |
| nfov 7388 | Bound-variable hypothesis ... |
| oprabidw 7389 | The law of concretion. Sp... |
| oprabid 7390 | The law of concretion. Sp... |
| ovex 7391 | The result of an operation... |
| ovexi 7392 | The result of an operation... |
| ovexd 7393 | The result of an operation... |
| ovssunirn 7394 | The result of an operation... |
| 0ov 7395 | Operation value of the emp... |
| ovprc 7396 | The value of an operation ... |
| ovprc1 7397 | The value of an operation ... |
| ovprc2 7398 | The value of an operation ... |
| ovrcl 7399 | Reverse closure for an ope... |
| elfvov1 7400 | Utility theorem: reverse c... |
| elfvov2 7401 | Utility theorem: reverse c... |
| csbov123 7402 | Move class substitution in... |
| csbov 7403 | Move class substitution in... |
| csbov12g 7404 | Move class substitution in... |
| csbov1g 7405 | Move class substitution in... |
| csbov2g 7406 | Move class substitution in... |
| rspceov 7407 | A frequently used special ... |
| elovimad 7408 | Elementhood of the image s... |
| fnbrovb 7409 | Value of a binary operatio... |
| fnotovb 7410 | Equivalence of operation v... |
| opabbrex 7411 | A collection of ordered pa... |
| opabresex2 7412 | Restrictions of a collecti... |
| fvmptopab 7413 | The function value of a ma... |
| f1opr 7414 | Condition for an operation... |
| brfvopab 7415 | The classes involved in a ... |
| dfoprab2 7416 | Class abstraction for oper... |
| reloprab 7417 | An operation class abstrac... |
| oprabv 7418 | If a pair and a class are ... |
| nfoprab1 7419 | The abstraction variables ... |
| nfoprab2 7420 | The abstraction variables ... |
| nfoprab3 7421 | The abstraction variables ... |
| nfoprab 7422 | Bound-variable hypothesis ... |
| oprabbid 7423 | Equivalent wff's yield equ... |
| oprabbidv 7424 | Equivalent wff's yield equ... |
| oprabbii 7425 | Equivalent wff's yield equ... |
| ssoprab2 7426 | Equivalence of ordered pai... |
| ssoprab2b 7427 | Equivalence of ordered pai... |
| eqoprab2bw 7428 | Equivalence of ordered pai... |
| eqoprab2b 7429 | Equivalence of ordered pai... |
| mpoeq123 7430 | An equality theorem for th... |
| mpoeq12 7431 | An equality theorem for th... |
| mpoeq123dva 7432 | An equality deduction for ... |
| mpoeq123dv 7433 | An equality deduction for ... |
| mpoeq123i 7434 | An equality inference for ... |
| mpoeq3dva 7435 | Slightly more general equa... |
| mpoeq3ia 7436 | An equality inference for ... |
| mpoeq3dv 7437 | An equality deduction for ... |
| nfmpo1 7438 | Bound-variable hypothesis ... |
| nfmpo2 7439 | Bound-variable hypothesis ... |
| nfmpo 7440 | Bound-variable hypothesis ... |
| 0mpo0 7441 | A mapping operation with e... |
| mpo0v 7442 | A mapping operation with e... |
| mpo0 7443 | A mapping operation with e... |
| oprab4 7444 | Two ways to state the doma... |
| cbvoprab1 7445 | Rule used to change first ... |
| cbvoprab2 7446 | Change the second bound va... |
| cbvoprab12 7447 | Rule used to change first ... |
| cbvoprab12v 7448 | Rule used to change first ... |
| cbvoprab3 7449 | Rule used to change the th... |
| cbvoprab3v 7450 | Rule used to change the th... |
| cbvmpox 7451 | Rule to change the bound v... |
| cbvmpo 7452 | Rule to change the bound v... |
| cbvmpov 7453 | Rule to change the bound v... |
| elimdelov 7454 | Eliminate a hypothesis whi... |
| brif1 7455 | Move a relation inside and... |
| ovif 7456 | Move a conditional outside... |
| ovif2 7457 | Move a conditional outside... |
| ovif12 7458 | Move a conditional outside... |
| ifov 7459 | Move a conditional outside... |
| ifmpt2v 7460 | Move a conditional inside ... |
| dmoprab 7461 | The domain of an operation... |
| dmoprabss 7462 | The domain of an operation... |
| rnoprab 7463 | The range of an operation ... |
| rnoprab2 7464 | The range of a restricted ... |
| reldmoprab 7465 | The domain of an operation... |
| oprabss 7466 | Structure of an operation ... |
| eloprabga 7467 | The law of concretion for ... |
| eloprabg 7468 | The law of concretion for ... |
| ssoprab2i 7469 | Inference of operation cla... |
| mpov 7470 | Operation with universal d... |
| mpomptx 7471 | Express a two-argument fun... |
| mpompt 7472 | Express a two-argument fun... |
| mpodifsnif 7473 | A mapping with two argumen... |
| mposnif 7474 | A mapping with two argumen... |
| fconstmpo 7475 | Representation of a consta... |
| resoprab 7476 | Restriction of an operatio... |
| resoprab2 7477 | Restriction of an operator... |
| resmpo 7478 | Restriction of the mapping... |
| funoprabg 7479 | "At most one" is a suffici... |
| funoprab 7480 | "At most one" is a suffici... |
| fnoprabg 7481 | Functionality and domain o... |
| mpofun 7482 | The maps-to notation for a... |
| fnoprab 7483 | Functionality and domain o... |
| ffnov 7484 | An operation maps to a cla... |
| fovcld 7485 | Closure law for an operati... |
| fovcl 7486 | Closure law for an operati... |
| eqfnov 7487 | Equality of two operations... |
| eqfnov2 7488 | Two operators with the sam... |
| fnov 7489 | Representation of a functi... |
| mpo2eqb 7490 | Bidirectional equality the... |
| rnmpo 7491 | The range of an operation ... |
| reldmmpo 7492 | The domain of an operation... |
| elrnmpog 7493 | Membership in the range of... |
| elrnmpo 7494 | Membership in the range of... |
| elimampo 7495 | Membership in the image of... |
| elrnmpores 7496 | Membership in the range of... |
| ralrnmpo 7497 | A restricted quantifier ov... |
| rexrnmpo 7498 | A restricted quantifier ov... |
| ovid 7499 | The value of an operation ... |
| ovidig 7500 | The value of an operation ... |
| ovidi 7501 | The value of an operation ... |
| ov 7502 | The value of an operation ... |
| ovigg 7503 | The value of an operation ... |
| ovig 7504 | The value of an operation ... |
| ovmpt4g 7505 | Value of a function given ... |
| ovmpos 7506 | Value of a function given ... |
| ov2gf 7507 | The value of an operation ... |
| ovmpodxf 7508 | Value of an operation give... |
| ovmpodx 7509 | Value of an operation give... |
| ovmpod 7510 | Value of an operation give... |
| ovmpox 7511 | The value of an operation ... |
| ovmpoga 7512 | Value of an operation give... |
| ovmpoa 7513 | Value of an operation give... |
| ovmpodf 7514 | Alternate deduction versio... |
| ovmpodv 7515 | Alternate deduction versio... |
| ovmpodv2 7516 | Alternate deduction versio... |
| ovmpog 7517 | Value of an operation give... |
| ovmpo 7518 | Value of an operation give... |
| ovmpot 7519 | The value of an operation ... |
| fvmpopr2d 7520 | Value of an operation give... |
| ov3 7521 | The value of an operation ... |
| ov6g 7522 | The value of an operation ... |
| ovg 7523 | The value of an operation ... |
| ovres 7524 | The value of a restricted ... |
| ovresd 7525 | Lemma for converting metri... |
| oprres 7526 | The restriction of an oper... |
| oprssov 7527 | The value of a member of t... |
| fovcdm 7528 | An operation's value belon... |
| fovcdmda 7529 | An operation's value belon... |
| fovcdmd 7530 | An operation's value belon... |
| fnrnov 7531 | The range of an operation ... |
| foov 7532 | An onto mapping of an oper... |
| fnovrn 7533 | An operation's value belon... |
| ovelrn 7534 | A member of an operation's... |
| funimassov 7535 | Membership relation for th... |
| ovelimab 7536 | Operation value in an imag... |
| ovima0 7537 | An operation value is a me... |
| ovconst2 7538 | The value of a constant op... |
| oprssdm 7539 | Domain of closure of an op... |
| nssdmovg 7540 | The value of an operation ... |
| ndmovg 7541 | The value of an operation ... |
| ndmov 7542 | The value of an operation ... |
| ndmovcl 7543 | The closure of an operatio... |
| ndmovrcl 7544 | Reverse closure law, when ... |
| ndmovcom 7545 | Any operation is commutati... |
| ndmovass 7546 | Any operation is associati... |
| ndmovdistr 7547 | Any operation is distribut... |
| ndmovord 7548 | Elimination of redundant a... |
| ndmovordi 7549 | Elimination of redundant a... |
| caovclg 7550 | Convert an operation closu... |
| caovcld 7551 | Convert an operation closu... |
| caovcl 7552 | Convert an operation closu... |
| caovcomg 7553 | Convert an operation commu... |
| caovcomd 7554 | Convert an operation commu... |
| caovcom 7555 | Convert an operation commu... |
| caovassg 7556 | Convert an operation assoc... |
| caovassd 7557 | Convert an operation assoc... |
| caovass 7558 | Convert an operation assoc... |
| caovcang 7559 | Convert an operation cance... |
| caovcand 7560 | Convert an operation cance... |
| caovcanrd 7561 | Commute the arguments of a... |
| caovcan 7562 | Convert an operation cance... |
| caovordig 7563 | Convert an operation order... |
| caovordid 7564 | Convert an operation order... |
| caovordg 7565 | Convert an operation order... |
| caovordd 7566 | Convert an operation order... |
| caovord2d 7567 | Operation ordering law wit... |
| caovord3d 7568 | Ordering law. (Contribute... |
| caovord 7569 | Convert an operation order... |
| caovord2 7570 | Operation ordering law wit... |
| caovord3 7571 | Ordering law. (Contribute... |
| caovdig 7572 | Convert an operation distr... |
| caovdid 7573 | Convert an operation distr... |
| caovdir2d 7574 | Convert an operation distr... |
| caovdirg 7575 | Convert an operation rever... |
| caovdird 7576 | Convert an operation distr... |
| caovdi 7577 | Convert an operation distr... |
| caov32d 7578 | Rearrange arguments in a c... |
| caov12d 7579 | Rearrange arguments in a c... |
| caov31d 7580 | Rearrange arguments in a c... |
| caov13d 7581 | Rearrange arguments in a c... |
| caov4d 7582 | Rearrange arguments in a c... |
| caov411d 7583 | Rearrange arguments in a c... |
| caov42d 7584 | Rearrange arguments in a c... |
| caov32 7585 | Rearrange arguments in a c... |
| caov12 7586 | Rearrange arguments in a c... |
| caov31 7587 | Rearrange arguments in a c... |
| caov13 7588 | Rearrange arguments in a c... |
| caov4 7589 | Rearrange arguments in a c... |
| caov411 7590 | Rearrange arguments in a c... |
| caov42 7591 | Rearrange arguments in a c... |
| caovdir 7592 | Reverse distributive law. ... |
| caovdilem 7593 | Lemma used by real number ... |
| caovlem2 7594 | Lemma used in real number ... |
| caovmo 7595 | Uniqueness of inverse elem... |
| imaeqexov 7596 | Substitute an operation va... |
| imaeqalov 7597 | Substitute an operation va... |
| mpondm0 7598 | The value of an operation ... |
| elmpocl 7599 | If a two-parameter class i... |
| elmpocl1 7600 | If a two-parameter class i... |
| elmpocl2 7601 | If a two-parameter class i... |
| elovmpod 7602 | Utility lemma for two-para... |
| elovmpo 7603 | Utility lemma for two-para... |
| elovmporab 7604 | Implications for the value... |
| elovmporab1w 7605 | Implications for the value... |
| elovmporab1 7606 | Implications for the value... |
| 2mpo0 7607 | If the operation value of ... |
| relmptopab 7608 | Any function to sets of or... |
| f1ocnvd 7609 | Describe an implicit one-t... |
| f1od 7610 | Describe an implicit one-t... |
| f1ocnv2d 7611 | Describe an implicit one-t... |
| f1o2d 7612 | Describe an implicit one-t... |
| f1opw2 7613 | A one-to-one mapping induc... |
| f1opw 7614 | A one-to-one mapping induc... |
| elovmpt3imp 7615 | If the value of a function... |
| ovmpt3rab1 7616 | The value of an operation ... |
| ovmpt3rabdm 7617 | If the value of a function... |
| elovmpt3rab1 7618 | Implications for the value... |
| elovmpt3rab 7619 | Implications for the value... |
| ofeqd 7624 | Equality theorem for funct... |
| ofeq 7625 | Equality theorem for funct... |
| ofreq 7626 | Equality theorem for funct... |
| ofexg 7627 | A function operation restr... |
| nfof 7628 | Hypothesis builder for fun... |
| nfofr 7629 | Hypothesis builder for fun... |
| ofrfvalg 7630 | Value of a relation applie... |
| offval 7631 | Value of an operation appl... |
| ofrfval 7632 | Value of a relation applie... |
| ofval 7633 | Evaluate a function operat... |
| ofrval 7634 | Exhibit a function relatio... |
| offn 7635 | The function operation pro... |
| offun 7636 | The function operation pro... |
| offval2f 7637 | The function operation exp... |
| ofmresval 7638 | Value of a restriction of ... |
| fnfvof 7639 | Function value of a pointw... |
| off 7640 | The function operation pro... |
| ofres 7641 | Restrict the operands of a... |
| offval2 7642 | The function operation exp... |
| ofrfval2 7643 | The function relation acti... |
| offvalfv 7644 | The function operation exp... |
| ofmpteq 7645 | Value of a pointwise opera... |
| coof 7646 | The composition of a _homo... |
| ofco 7647 | The composition of a funct... |
| offveq 7648 | Convert an identity of the... |
| offveqb 7649 | Equivalent expressions for... |
| ofc1 7650 | Left operation by a consta... |
| ofc2 7651 | Right operation by a const... |
| ofc12 7652 | Function operation on two ... |
| caofref 7653 | Transfer a reflexive law t... |
| caofinvl 7654 | Transfer a left inverse la... |
| caofid0l 7655 | Transfer a left identity l... |
| caofid0r 7656 | Transfer a right identity ... |
| caofid1 7657 | Transfer a right absorptio... |
| caofid2 7658 | Transfer a right absorptio... |
| caofcom 7659 | Transfer a commutative law... |
| caofidlcan 7660 | Transfer a cancellation/id... |
| caofrss 7661 | Transfer a relation subset... |
| caofass 7662 | Transfer an associative la... |
| caoftrn 7663 | Transfer a transitivity la... |
| caofdi 7664 | Transfer a distributive la... |
| caofdir 7665 | Transfer a reverse distrib... |
| caonncan 7666 | Transfer ~ nncan -shaped l... |
| relrpss 7669 | The proper subset relation... |
| brrpssg 7670 | The proper subset relation... |
| brrpss 7671 | The proper subset relation... |
| porpss 7672 | Every class is partially o... |
| sorpss 7673 | Express strict ordering un... |
| sorpssi 7674 | Property of a chain of set... |
| sorpssun 7675 | A chain of sets is closed ... |
| sorpssin 7676 | A chain of sets is closed ... |
| sorpssuni 7677 | In a chain of sets, a maxi... |
| sorpssint 7678 | In a chain of sets, a mini... |
| sorpsscmpl 7679 | The componentwise compleme... |
| zfun 7681 | Axiom of Union expressed w... |
| axun2 7682 | A variant of the Axiom of ... |
| uniex2 7683 | The Axiom of Union using t... |
| vuniex 7684 | The union of a setvar is a... |
| uniexg 7685 | The ZF Axiom of Union in c... |
| uniex 7686 | The Axiom of Union in clas... |
| uniexd 7687 | Deduction version of the Z... |
| unexg 7688 | The union of two sets is a... |
| unex 7689 | The union of two sets is a... |
| unexOLD 7690 | Obsolete version of ~ unex... |
| tpex 7691 | An unordered triple of cla... |
| unexb 7692 | Existence of union is equi... |
| unexbOLD 7693 | Obsolete version of ~ unex... |
| unexgOLD 7694 | Obsolete version of ~ unex... |
| xpexg 7695 | The Cartesian product of t... |
| xpexd 7696 | The Cartesian product of t... |
| 3xpexg 7697 | The Cartesian product of t... |
| xpex 7698 | The Cartesian product of t... |
| unexd 7699 | The union of two sets is a... |
| sqxpexg 7700 | The Cartesian square of a ... |
| abnexg 7701 | Sufficient condition for a... |
| abnex 7702 | Sufficient condition for a... |
| snnex 7703 | The class of all singleton... |
| pwnex 7704 | The class of all power set... |
| difex2 7705 | If the subtrahend of a cla... |
| difsnexi 7706 | If the difference of a cla... |
| uniuni 7707 | Expression for double unio... |
| uniexr 7708 | Converse of the Axiom of U... |
| uniexb 7709 | The Axiom of Union and its... |
| pwexr 7710 | Converse of the Axiom of P... |
| pwexb 7711 | The Axiom of Power Sets an... |
| elpwpwel 7712 | A class belongs to a doubl... |
| eldifpw 7713 | Membership in a power clas... |
| elpwun 7714 | Membership in the power cl... |
| pwuncl 7715 | Power classes are closed u... |
| iunpw 7716 | An indexed union of a powe... |
| fr3nr 7717 | A well-founded relation ha... |
| epne3 7718 | A well-founded class conta... |
| dfwe2 7719 | Alternate definition of we... |
| epweon 7720 | The membership relation we... |
| epweonALT 7721 | Alternate proof of ~ epweo... |
| ordon 7722 | The class of all ordinal n... |
| onprc 7723 | No set contains all ordina... |
| ssorduni 7724 | The union of a class of or... |
| ssonuni 7725 | The union of a set of ordi... |
| ssonunii 7726 | The union of a set of ordi... |
| ordeleqon 7727 | A way to express the ordin... |
| ordsson 7728 | Any ordinal class is a sub... |
| dford5 7729 | A class is ordinal iff it ... |
| onss 7730 | An ordinal number is a sub... |
| predon 7731 | The predecessor of an ordi... |
| ssonprc 7732 | Two ways of saying a class... |
| onuni 7733 | The union of an ordinal nu... |
| orduni 7734 | The union of an ordinal cl... |
| onint 7735 | The intersection (infimum)... |
| onint0 7736 | The intersection of a clas... |
| onssmin 7737 | A nonempty class of ordina... |
| onminesb 7738 | If a property is true for ... |
| onminsb 7739 | If a property is true for ... |
| oninton 7740 | The intersection of a none... |
| onintrab 7741 | The intersection of a clas... |
| onintrab2 7742 | An existence condition equ... |
| onnmin 7743 | No member of a set of ordi... |
| onnminsb 7744 | An ordinal number smaller ... |
| oneqmin 7745 | A way to show that an ordi... |
| uniordint 7746 | The union of a set of ordi... |
| onminex 7747 | If a wff is true for an or... |
| sucon 7748 | The class of all ordinal n... |
| sucexb 7749 | A successor exists iff its... |
| sucexg 7750 | The successor of a set is ... |
| sucex 7751 | The successor of a set is ... |
| onmindif2 7752 | The minimum of a class of ... |
| ordsuci 7753 | The successor of an ordina... |
| sucexeloni 7754 | If the successor of an ord... |
| onsuc 7755 | The successor of an ordina... |
| ordsuc 7756 | A class is ordinal if and ... |
| ordpwsuc 7757 | The collection of ordinals... |
| onpwsuc 7758 | The collection of ordinal ... |
| onsucb 7759 | A class is an ordinal numb... |
| ordsucss 7760 | The successor of an elemen... |
| onpsssuc 7761 | An ordinal number is a pro... |
| ordelsuc 7762 | A set belongs to an ordina... |
| onsucmin 7763 | The successor of an ordina... |
| ordsucelsuc 7764 | Membership is inherited by... |
| ordsucsssuc 7765 | The subclass relationship ... |
| ordsucuniel 7766 | Given an element ` A ` of ... |
| ordsucun 7767 | The successor of the maxim... |
| ordunpr 7768 | The maximum of two ordinal... |
| ordunel 7769 | The maximum of two ordinal... |
| onsucuni 7770 | A class of ordinal numbers... |
| ordsucuni 7771 | An ordinal class is a subc... |
| orduniorsuc 7772 | An ordinal class is either... |
| unon 7773 | The class of all ordinal n... |
| ordunisuc 7774 | An ordinal class is equal ... |
| orduniss2 7775 | The union of the ordinal s... |
| onsucuni2 7776 | A successor ordinal is the... |
| 0elsuc 7777 | The successor of an ordina... |
| limon 7778 | The class of ordinal numbe... |
| onuniorsuc 7779 | An ordinal number is eithe... |
| onssi 7780 | An ordinal number is a sub... |
| onsuci 7781 | The successor of an ordina... |
| onuninsuci 7782 | An ordinal is equal to its... |
| onsucssi 7783 | A set belongs to an ordina... |
| nlimsucg 7784 | A successor is not a limit... |
| orduninsuc 7785 | An ordinal class is equal ... |
| ordunisuc2 7786 | An ordinal equal to its un... |
| ordzsl 7787 | An ordinal is zero, a succ... |
| onzsl 7788 | An ordinal number is zero,... |
| dflim3 7789 | An alternate definition of... |
| dflim4 7790 | An alternate definition of... |
| limsuc 7791 | The successor of a member ... |
| limsssuc 7792 | A class includes a limit o... |
| nlimon 7793 | Two ways to express the cl... |
| limuni3 7794 | The union of a nonempty cl... |
| tfi 7795 | The Principle of Transfini... |
| tfisg 7796 | A closed form of ~ tfis . ... |
| tfis 7797 | Transfinite Induction Sche... |
| tfis2f 7798 | Transfinite Induction Sche... |
| tfis2 7799 | Transfinite Induction Sche... |
| tfis3 7800 | Transfinite Induction Sche... |
| tfisi 7801 | A transfinite induction sc... |
| tfinds 7802 | Principle of Transfinite I... |
| tfindsg 7803 | Transfinite Induction (inf... |
| tfindsg2 7804 | Transfinite Induction (inf... |
| tfindes 7805 | Transfinite Induction with... |
| tfinds2 7806 | Transfinite Induction (inf... |
| tfinds3 7807 | Principle of Transfinite I... |
| dfom2 7810 | An alternate definition of... |
| elom 7811 | Membership in omega. The ... |
| omsson 7812 | Omega is a subset of ` On ... |
| limomss 7813 | The class of natural numbe... |
| nnon 7814 | A natural number is an ord... |
| nnoni 7815 | A natural number is an ord... |
| nnord 7816 | A natural number is ordina... |
| trom 7817 | The class of finite ordina... |
| ordom 7818 | The class of finite ordina... |
| elnn 7819 | A member of a natural numb... |
| omon 7820 | The class of natural numbe... |
| omelon2 7821 | Omega is an ordinal number... |
| nnlim 7822 | A natural number is not a ... |
| omssnlim 7823 | The class of natural numbe... |
| limom 7824 | Omega is a limit ordinal. ... |
| peano2b 7825 | A class belongs to omega i... |
| nnsuc 7826 | A nonzero natural number i... |
| omsucne 7827 | A natural number is not th... |
| ssnlim 7828 | An ordinal subclass of non... |
| omsinds 7829 | Strong (or "total") induct... |
| omun 7830 | The union of two finite or... |
| peano1 7831 | Zero is a natural number. ... |
| peano2 7832 | The successor of any natur... |
| peano3 7833 | The successor of any natur... |
| peano4 7834 | Two natural numbers are eq... |
| peano5 7835 | The induction postulate: a... |
| nn0suc 7836 | A natural number is either... |
| find 7837 | The Principle of Finite In... |
| finds 7838 | Principle of Finite Induct... |
| findsg 7839 | Principle of Finite Induct... |
| finds2 7840 | Principle of Finite Induct... |
| finds1 7841 | Principle of Finite Induct... |
| findes 7842 | Finite induction with expl... |
| dmexg 7843 | The domain of a set is a s... |
| rnexg 7844 | The range of a set is a se... |
| dmexd 7845 | The domain of a set is a s... |
| fndmexd 7846 | If a function is a set, it... |
| dmfex 7847 | If a mapping is a set, its... |
| fndmexb 7848 | The domain of a function i... |
| fdmexb 7849 | The domain of a function i... |
| dmfexALT 7850 | Alternate proof of ~ dmfex... |
| dmex 7851 | The domain of a set is a s... |
| rnex 7852 | The range of a set is a se... |
| iprc 7853 | The identity function is a... |
| resiexg 7854 | The existence of a restric... |
| imaexg 7855 | The image of a set is a se... |
| imaex 7856 | The image of a set is a se... |
| rnexd 7857 | The range of a set is a se... |
| imaexd 7858 | The image of a set is a se... |
| exse2 7859 | Any set relation is set-li... |
| xpexr 7860 | If a Cartesian product is ... |
| xpexr2 7861 | If a nonempty Cartesian pr... |
| xpexcnv 7862 | A condition where the conv... |
| soex 7863 | If the relation in a stric... |
| elxp4 7864 | Membership in a Cartesian ... |
| elxp5 7865 | Membership in a Cartesian ... |
| cnvexg 7866 | The converse of a set is a... |
| cnvex 7867 | The converse of a set is a... |
| relcnvexb 7868 | A relation is a set iff it... |
| f1oexrnex 7869 | If the range of a 1-1 onto... |
| f1oexbi 7870 | There is a one-to-one onto... |
| coexg 7871 | The composition of two set... |
| coex 7872 | The composition of two set... |
| coexd 7873 | The composition of two set... |
| funcnvuni 7874 | The union of a chain (with... |
| fun11uni 7875 | The union of a chain (with... |
| resf1extb 7876 | Extension of an injection ... |
| resf1ext2b 7877 | Extension of an injection ... |
| fex2 7878 | A function with bounded do... |
| fabexd 7879 | Existence of a set of func... |
| fabexg 7880 | Existence of a set of func... |
| fabexgOLD 7881 | Obsolete version of ~ fabe... |
| fabex 7882 | Existence of a set of func... |
| mapex 7883 | The class of all functions... |
| f1oabexg 7884 | The class of all 1-1-onto ... |
| f1oabexgOLD 7885 | Obsolete version of ~ f1oa... |
| fiunlem 7886 | Lemma for ~ fiun and ~ f1i... |
| fiun 7887 | The union of a chain (with... |
| f1iun 7888 | The union of a chain (with... |
| fviunfun 7889 | The function value of an i... |
| ffoss 7890 | Relationship between a map... |
| f11o 7891 | Relationship between one-t... |
| resfunexgALT 7892 | Alternate proof of ~ resfu... |
| cofunexg 7893 | Existence of a composition... |
| cofunex2g 7894 | Existence of a composition... |
| fnexALT 7895 | Alternate proof of ~ fnex ... |
| funexw 7896 | Weak version of ~ funex th... |
| mptexw 7897 | Weak version of ~ mptex th... |
| funrnex 7898 | If the domain of a functio... |
| zfrep6 7899 | A version of the Axiom of ... |
| focdmex 7900 | If the domain of an onto f... |
| f1dmex 7901 | If the codomain of a one-t... |
| f1ovv 7902 | The codomain/range of a 1-... |
| fvclex 7903 | Existence of the class of ... |
| fvresex 7904 | Existence of the class of ... |
| abrexexg 7905 | Existence of a class abstr... |
| abrexex 7906 | Existence of a class abstr... |
| iunexg 7907 | The existence of an indexe... |
| abrexex2g 7908 | Existence of an existentia... |
| opabex3d 7909 | Existence of an ordered pa... |
| opabex3rd 7910 | Existence of an ordered pa... |
| opabex3 7911 | Existence of an ordered pa... |
| iunex 7912 | The existence of an indexe... |
| abrexex2 7913 | Existence of an existentia... |
| abexssex 7914 | Existence of a class abstr... |
| abexex 7915 | A condition where a class ... |
| f1oweALT 7916 | Alternate proof of ~ f1owe... |
| wemoiso 7917 | Thus, there is at most one... |
| wemoiso2 7918 | Thus, there is at most one... |
| oprabexd 7919 | Existence of an operator a... |
| oprabex 7920 | Existence of an operation ... |
| oprabex3 7921 | Existence of an operation ... |
| oprabrexex2 7922 | Existence of an existentia... |
| ab2rexex 7923 | Existence of a class abstr... |
| ab2rexex2 7924 | Existence of an existentia... |
| xpexgALT 7925 | Alternate proof of ~ xpexg... |
| offval3 7926 | General value of ` ( F oF ... |
| offres 7927 | Pointwise combination comm... |
| ofmres 7928 | Equivalent expressions for... |
| ofmresex 7929 | Existence of a restriction... |
| mptcnfimad 7930 | The converse of a mapping ... |
| 1stval 7935 | The value of the function ... |
| 2ndval 7936 | The value of the function ... |
| 1stnpr 7937 | Value of the first-member ... |
| 2ndnpr 7938 | Value of the second-member... |
| 1st0 7939 | The value of the first-mem... |
| 2nd0 7940 | The value of the second-me... |
| op1st 7941 | Extract the first member o... |
| op2nd 7942 | Extract the second member ... |
| op1std 7943 | Extract the first member o... |
| op2ndd 7944 | Extract the second member ... |
| op1stg 7945 | Extract the first member o... |
| op2ndg 7946 | Extract the second member ... |
| ot1stg 7947 | Extract the first member o... |
| ot2ndg 7948 | Extract the second member ... |
| ot3rdg 7949 | Extract the third member o... |
| 1stval2 7950 | Alternate value of the fun... |
| 2ndval2 7951 | Alternate value of the fun... |
| oteqimp 7952 | The components of an order... |
| fo1st 7953 | The ` 1st ` function maps ... |
| fo2nd 7954 | The ` 2nd ` function maps ... |
| br1steqg 7955 | Uniqueness condition for t... |
| br2ndeqg 7956 | Uniqueness condition for t... |
| f1stres 7957 | Mapping of a restriction o... |
| f2ndres 7958 | Mapping of a restriction o... |
| fo1stres 7959 | Onto mapping of a restrict... |
| fo2ndres 7960 | Onto mapping of a restrict... |
| 1st2val 7961 | Value of an alternate defi... |
| 2nd2val 7962 | Value of an alternate defi... |
| 1stcof 7963 | Composition of the first m... |
| 2ndcof 7964 | Composition of the second ... |
| xp1st 7965 | Location of the first elem... |
| xp2nd 7966 | Location of the second ele... |
| elxp6 7967 | Membership in a Cartesian ... |
| elxp7 7968 | Membership in a Cartesian ... |
| eqopi 7969 | Equality with an ordered p... |
| xp2 7970 | Representation of Cartesia... |
| unielxp 7971 | The membership relation fo... |
| 1st2nd2 7972 | Reconstruction of a member... |
| 1st2ndb 7973 | Reconstruction of an order... |
| xpopth 7974 | An ordered pair theorem fo... |
| eqop 7975 | Two ways to express equali... |
| eqop2 7976 | Two ways to express equali... |
| op1steq 7977 | Two ways of expressing tha... |
| opreuopreu 7978 | There is a unique ordered ... |
| el2xptp 7979 | A member of a nested Carte... |
| el2xptp0 7980 | A member of a nested Carte... |
| el2xpss 7981 | Version of ~ elrel for tri... |
| 2nd1st 7982 | Swap the members of an ord... |
| 1st2nd 7983 | Reconstruction of a member... |
| 1stdm 7984 | The first ordered pair com... |
| 2ndrn 7985 | The second ordered pair co... |
| 1st2ndbr 7986 | Express an element of a re... |
| releldm2 7987 | Two ways of expressing mem... |
| reldm 7988 | An expression for the doma... |
| releldmdifi 7989 | One way of expressing memb... |
| funfv1st2nd 7990 | The function value for the... |
| funelss 7991 | If the first component of ... |
| funeldmdif 7992 | Two ways of expressing mem... |
| sbcopeq1a 7993 | Equality theorem for subst... |
| csbopeq1a 7994 | Equality theorem for subst... |
| sbcoteq1a 7995 | Equality theorem for subst... |
| dfopab2 7996 | A way to define an ordered... |
| dfoprab3s 7997 | A way to define an operati... |
| dfoprab3 7998 | Operation class abstractio... |
| dfoprab4 7999 | Operation class abstractio... |
| dfoprab4f 8000 | Operation class abstractio... |
| opabex2 8001 | Condition for an operation... |
| opabn1stprc 8002 | An ordered-pair class abst... |
| opiota 8003 | The property of a uniquely... |
| cnvoprab 8004 | The converse of a class ab... |
| dfxp3 8005 | Define the Cartesian produ... |
| elopabi 8006 | A consequence of membershi... |
| eloprabi 8007 | A consequence of membershi... |
| mpomptsx 8008 | Express a two-argument fun... |
| mpompts 8009 | Express a two-argument fun... |
| dmmpossx 8010 | The domain of a mapping is... |
| fmpox 8011 | Functionality, domain and ... |
| fmpo 8012 | Functionality, domain and ... |
| fnmpo 8013 | Functionality and domain o... |
| fnmpoi 8014 | Functionality and domain o... |
| dmmpo 8015 | Domain of a class given by... |
| ovmpoelrn 8016 | An operation's value belon... |
| dmmpoga 8017 | Domain of an operation giv... |
| dmmpog 8018 | Domain of an operation giv... |
| mpoexxg 8019 | Existence of an operation ... |
| mpoexg 8020 | Existence of an operation ... |
| mpoexga 8021 | If the domain of an operat... |
| mpoexw 8022 | Weak version of ~ mpoex th... |
| mpoex 8023 | If the domain of an operat... |
| mptmpoopabbrd 8024 | The operation value of a f... |
| mptmpoopabbrdOLD 8025 | Obsolete version of ~ mptm... |
| mptmpoopabovd 8026 | The operation value of a f... |
| el2mpocsbcl 8027 | If the operation value of ... |
| el2mpocl 8028 | If the operation value of ... |
| fnmpoovd 8029 | A function with a Cartesia... |
| offval22 8030 | The function operation exp... |
| brovpreldm 8031 | If a binary relation holds... |
| bropopvvv 8032 | If a binary relation holds... |
| bropfvvvvlem 8033 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8034 | If a binary relation holds... |
| ovmptss 8035 | If all the values of the m... |
| relmpoopab 8036 | Any function to sets of or... |
| fmpoco 8037 | Composition of two functio... |
| oprabco 8038 | Composition of a function ... |
| oprab2co 8039 | Composition of operator ab... |
| df1st2 8040 | An alternate possible defi... |
| df2nd2 8041 | An alternate possible defi... |
| 1stconst 8042 | The mapping of a restricti... |
| 2ndconst 8043 | The mapping of a restricti... |
| dfmpo 8044 | Alternate definition for t... |
| mposn 8045 | An operation (in maps-to n... |
| curry1 8046 | Composition with ` ``' ( 2... |
| curry1val 8047 | The value of a curried fun... |
| curry1f 8048 | Functionality of a curried... |
| curry2 8049 | Composition with ` ``' ( 1... |
| curry2f 8050 | Functionality of a curried... |
| curry2val 8051 | The value of a curried fun... |
| cnvf1olem 8052 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8053 | Describe a function that m... |
| fparlem1 8054 | Lemma for ~ fpar . (Contr... |
| fparlem2 8055 | Lemma for ~ fpar . (Contr... |
| fparlem3 8056 | Lemma for ~ fpar . (Contr... |
| fparlem4 8057 | Lemma for ~ fpar . (Contr... |
| fpar 8058 | Merge two functions in par... |
| fsplit 8059 | A function that can be use... |
| fsplitfpar 8060 | Merge two functions with a... |
| offsplitfpar 8061 | Express the function opera... |
| f2ndf 8062 | The ` 2nd ` (second compon... |
| fo2ndf 8063 | The ` 2nd ` (second compon... |
| f1o2ndf1 8064 | The ` 2nd ` (second compon... |
| opco1 8065 | Value of an operation prec... |
| opco2 8066 | Value of an operation prec... |
| opco1i 8067 | Inference form of ~ opco1 ... |
| frxp 8068 | A lexicographical ordering... |
| xporderlem 8069 | Lemma for lexicographical ... |
| poxp 8070 | A lexicographical ordering... |
| soxp 8071 | A lexicographical ordering... |
| wexp 8072 | A lexicographical ordering... |
| fnwelem 8073 | Lemma for ~ fnwe . (Contr... |
| fnwe 8074 | A variant on lexicographic... |
| fnse 8075 | Condition for the well-ord... |
| fvproj 8076 | Value of a function on ord... |
| fimaproj 8077 | Image of a cartesian produ... |
| ralxpes 8078 | A version of ~ ralxp with ... |
| ralxp3f 8079 | Restricted for all over a ... |
| ralxp3 8080 | Restricted for all over a ... |
| ralxp3es 8081 | Restricted for-all over a ... |
| frpoins3xpg 8082 | Special case of founded pa... |
| frpoins3xp3g 8083 | Special case of founded pa... |
| xpord2lem 8084 | Lemma for Cartesian produc... |
| poxp2 8085 | Another way of partially o... |
| frxp2 8086 | Another way of giving a we... |
| xpord2pred 8087 | Calculate the predecessor ... |
| sexp2 8088 | Condition for the relation... |
| xpord2indlem 8089 | Induction over the Cartesi... |
| xpord2ind 8090 | Induction over the Cartesi... |
| xpord3lem 8091 | Lemma for triple ordering.... |
| poxp3 8092 | Triple Cartesian product p... |
| frxp3 8093 | Give well-foundedness over... |
| xpord3pred 8094 | Calculate the predecsessor... |
| sexp3 8095 | Show that the triple order... |
| xpord3inddlem 8096 | Induction over the triple ... |
| xpord3indd 8097 | Induction over the triple ... |
| xpord3ind 8098 | Induction over the triple ... |
| orderseqlem 8099 | Lemma for ~ poseq and ~ so... |
| poseq 8100 | A partial ordering of ordi... |
| soseq 8101 | A linear ordering of ordin... |
| suppval 8104 | The value of the operation... |
| supp0prc 8105 | The support of a class is ... |
| suppvalbr 8106 | The value of the operation... |
| supp0 8107 | The support of the empty s... |
| suppval1 8108 | The value of the operation... |
| suppvalfng 8109 | The value of the operation... |
| suppvalfn 8110 | The value of the operation... |
| elsuppfng 8111 | An element of the support ... |
| elsuppfn 8112 | An element of the support ... |
| fvdifsupp 8113 | Function value is zero out... |
| cnvimadfsn 8114 | The support of functions "... |
| suppimacnvss 8115 | The support of functions "... |
| suppimacnv 8116 | Support sets of functions ... |
| fsuppeq 8117 | Two ways of writing the su... |
| fsuppeqg 8118 | Version of ~ fsuppeq avoid... |
| suppssdm 8119 | The support of a function ... |
| suppsnop 8120 | The support of a singleton... |
| snopsuppss 8121 | The support of a singleton... |
| fvn0elsupp 8122 | If the function value for ... |
| fvn0elsuppb 8123 | The function value for a g... |
| rexsupp 8124 | Existential quantification... |
| ressuppss 8125 | The support of the restric... |
| suppun 8126 | The support of a class/fun... |
| ressuppssdif 8127 | The support of the restric... |
| mptsuppdifd 8128 | The support of a function ... |
| mptsuppd 8129 | The support of a function ... |
| extmptsuppeq 8130 | The support of an extended... |
| suppfnss 8131 | The support of a function ... |
| funsssuppss 8132 | The support of a function ... |
| fnsuppres 8133 | Two ways to express restri... |
| fnsuppeq0 8134 | The support of a function ... |
| fczsupp0 8135 | The support of a constant ... |
| suppss 8136 | Show that the support of a... |
| suppssr 8137 | A function is zero outside... |
| suppssrg 8138 | A function is zero outside... |
| suppssov1 8139 | Formula building theorem f... |
| suppssov2 8140 | Formula building theorem f... |
| suppssof1 8141 | Formula building theorem f... |
| suppss2 8142 | Show that the support of a... |
| suppsssn 8143 | Show that the support of a... |
| suppssfv 8144 | Formula building theorem f... |
| suppofssd 8145 | Condition for the support ... |
| suppofss1d 8146 | Condition for the support ... |
| suppofss2d 8147 | Condition for the support ... |
| suppco 8148 | The support of the composi... |
| suppcoss 8149 | The support of the composi... |
| supp0cosupp0 8150 | The support of the composi... |
| imacosupp 8151 | The image of the support o... |
| opeliunxp2f 8152 | Membership in a union of C... |
| mpoxeldm 8153 | If there is an element of ... |
| mpoxneldm 8154 | If the first argument of a... |
| mpoxopn0yelv 8155 | If there is an element of ... |
| mpoxopynvov0g 8156 | If the second argument of ... |
| mpoxopxnop0 8157 | If the first argument of a... |
| mpoxopx0ov0 8158 | If the first argument of a... |
| mpoxopxprcov0 8159 | If the components of the f... |
| mpoxopynvov0 8160 | If the second argument of ... |
| mpoxopoveq 8161 | Value of an operation give... |
| mpoxopovel 8162 | Element of the value of an... |
| mpoxopoveqd 8163 | Value of an operation give... |
| brovex 8164 | A binary relation of the v... |
| brovmpoex 8165 | A binary relation of the v... |
| sprmpod 8166 | The extension of a binary ... |
| tposss 8169 | Subset theorem for transpo... |
| tposeq 8170 | Equality theorem for trans... |
| tposeqd 8171 | Equality theorem for trans... |
| tposssxp 8172 | The transposition is a sub... |
| reltpos 8173 | The transposition is a rel... |
| brtpos2 8174 | Value of the transposition... |
| brtpos0 8175 | The behavior of ` tpos ` w... |
| reldmtpos 8176 | Necessary and sufficient c... |
| brtpos 8177 | The transposition swaps ar... |
| ottpos 8178 | The transposition swaps th... |
| relbrtpos 8179 | The transposition swaps ar... |
| dmtpos 8180 | The domain of ` tpos F ` w... |
| rntpos 8181 | The range of ` tpos F ` wh... |
| tposexg 8182 | The transposition of a set... |
| ovtpos 8183 | The transposition swaps th... |
| tposfun 8184 | The transposition of a fun... |
| dftpos2 8185 | Alternate definition of ` ... |
| dftpos3 8186 | Alternate definition of ` ... |
| dftpos4 8187 | Alternate definition of ` ... |
| tpostpos 8188 | Value of the double transp... |
| tpostpos2 8189 | Value of the double transp... |
| tposfn2 8190 | The domain of a transposit... |
| tposfo2 8191 | Condition for a surjective... |
| tposf2 8192 | The domain and codomain of... |
| tposf12 8193 | Condition for an injective... |
| tposf1o2 8194 | Condition of a bijective t... |
| tposfo 8195 | The domain and codomain/ra... |
| tposf 8196 | The domain and codomain of... |
| tposfn 8197 | Functionality of a transpo... |
| tpos0 8198 | Transposition of the empty... |
| tposco 8199 | Transposition of a composi... |
| tpossym 8200 | Two ways to say a function... |
| tposeqi 8201 | Equality theorem for trans... |
| tposex 8202 | A transposition is a set. ... |
| nftpos 8203 | Hypothesis builder for tra... |
| tposoprab 8204 | Transposition of a class o... |
| tposmpo 8205 | Transposition of a two-arg... |
| tposconst 8206 | The transposition of a con... |
| mpocurryd 8211 | The currying of an operati... |
| mpocurryvald 8212 | The value of a curried ope... |
| fvmpocurryd 8213 | The value of the value of ... |
| pwuninel2 8216 | Proof of ~ pwuninel under ... |
| pwuninel 8217 | The powerclass of the unio... |
| undefval 8218 | Value of the undefined val... |
| undefnel2 8219 | The undefined value genera... |
| undefnel 8220 | The undefined value genera... |
| undefne0 8221 | The undefined value genera... |
| frecseq123 8224 | Equality theorem for the w... |
| nffrecs 8225 | Bound-variable hypothesis ... |
| csbfrecsg 8226 | Move class substitution in... |
| fpr3g 8227 | Functions defined by well-... |
| frrlem1 8228 | Lemma for well-founded rec... |
| frrlem2 8229 | Lemma for well-founded rec... |
| frrlem3 8230 | Lemma for well-founded rec... |
| frrlem4 8231 | Lemma for well-founded rec... |
| frrlem5 8232 | Lemma for well-founded rec... |
| frrlem6 8233 | Lemma for well-founded rec... |
| frrlem7 8234 | Lemma for well-founded rec... |
| frrlem8 8235 | Lemma for well-founded rec... |
| frrlem9 8236 | Lemma for well-founded rec... |
| frrlem10 8237 | Lemma for well-founded rec... |
| frrlem11 8238 | Lemma for well-founded rec... |
| frrlem12 8239 | Lemma for well-founded rec... |
| frrlem13 8240 | Lemma for well-founded rec... |
| frrlem14 8241 | Lemma for well-founded rec... |
| fprlem1 8242 | Lemma for well-founded rec... |
| fprlem2 8243 | Lemma for well-founded rec... |
| fpr2a 8244 | Weak version of ~ fpr2 whi... |
| fpr1 8245 | Law of well-founded recurs... |
| fpr2 8246 | Law of well-founded recurs... |
| fpr3 8247 | Law of well-founded recurs... |
| frrrel 8248 | Show without using the axi... |
| frrdmss 8249 | Show without using the axi... |
| frrdmcl 8250 | Show without using the axi... |
| fprfung 8251 | A "function" defined by we... |
| fprresex 8252 | The restriction of a funct... |
| wrecseq123 8255 | General equality theorem f... |
| nfwrecs 8256 | Bound-variable hypothesis ... |
| wrecseq1 8257 | Equality theorem for the w... |
| wrecseq2 8258 | Equality theorem for the w... |
| wrecseq3 8259 | Equality theorem for the w... |
| csbwrecsg 8260 | Move class substitution in... |
| wfr3g 8261 | Functions defined by well-... |
| wfrrel 8262 | The well-ordered recursion... |
| wfrdmss 8263 | The domain of the well-ord... |
| wfrdmcl 8264 | The predecessor class of a... |
| wfrfun 8265 | The "function" generated b... |
| wfrresex 8266 | Show without using the axi... |
| wfr2a 8267 | A weak version of ~ wfr2 w... |
| wfr1 8268 | The Principle of Well-Orde... |
| wfr2 8269 | The Principle of Well-Orde... |
| wfr3 8270 | The principle of Well-Orde... |
| iunon 8271 | The indexed union of a set... |
| iinon 8272 | The nonempty indexed inter... |
| onfununi 8273 | A property of functions on... |
| onovuni 8274 | A variant of ~ onfununi fo... |
| onoviun 8275 | A variant of ~ onovuni wit... |
| onnseq 8276 | There are no length ` _om ... |
| dfsmo2 8279 | Alternate definition of a ... |
| issmo 8280 | Conditions for which ` A `... |
| issmo2 8281 | Alternate definition of a ... |
| smoeq 8282 | Equality theorem for stric... |
| smodm 8283 | The domain of a strictly m... |
| smores 8284 | A strictly monotone functi... |
| smores3 8285 | A strictly monotone functi... |
| smores2 8286 | A strictly monotone ordina... |
| smodm2 8287 | The domain of a strictly m... |
| smofvon2 8288 | The function values of a s... |
| iordsmo 8289 | The identity relation rest... |
| smo0 8290 | The null set is a strictly... |
| smofvon 8291 | If ` B ` is a strictly mon... |
| smoel 8292 | If ` x ` is less than ` y ... |
| smoiun 8293 | The value of a strictly mo... |
| smoiso 8294 | If ` F ` is an isomorphism... |
| smoel2 8295 | A strictly monotone ordina... |
| smo11 8296 | A strictly monotone ordina... |
| smoord 8297 | A strictly monotone ordina... |
| smoword 8298 | A strictly monotone ordina... |
| smogt 8299 | A strictly monotone ordina... |
| smocdmdom 8300 | The codomain of a strictly... |
| smoiso2 8301 | The strictly monotone ordi... |
| dfrecs3 8304 | The old definition of tran... |
| recseq 8305 | Equality theorem for ` rec... |
| nfrecs 8306 | Bound-variable hypothesis ... |
| tfrlem1 8307 | A technical lemma for tran... |
| tfrlem3a 8308 | Lemma for transfinite recu... |
| tfrlem3 8309 | Lemma for transfinite recu... |
| tfrlem4 8310 | Lemma for transfinite recu... |
| tfrlem5 8311 | Lemma for transfinite recu... |
| recsfval 8312 | Lemma for transfinite recu... |
| tfrlem6 8313 | Lemma for transfinite recu... |
| tfrlem7 8314 | Lemma for transfinite recu... |
| tfrlem8 8315 | Lemma for transfinite recu... |
| tfrlem9 8316 | Lemma for transfinite recu... |
| tfrlem9a 8317 | Lemma for transfinite recu... |
| tfrlem10 8318 | Lemma for transfinite recu... |
| tfrlem11 8319 | Lemma for transfinite recu... |
| tfrlem12 8320 | Lemma for transfinite recu... |
| tfrlem13 8321 | Lemma for transfinite recu... |
| tfrlem14 8322 | Lemma for transfinite recu... |
| tfrlem15 8323 | Lemma for transfinite recu... |
| tfrlem16 8324 | Lemma for finite recursion... |
| tfr1a 8325 | A weak version of ~ tfr1 w... |
| tfr2a 8326 | A weak version of ~ tfr2 w... |
| tfr2b 8327 | Without assuming ~ ax-rep ... |
| tfr1 8328 | Principle of Transfinite R... |
| tfr2 8329 | Principle of Transfinite R... |
| tfr3 8330 | Principle of Transfinite R... |
| tfr1ALT 8331 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8332 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8333 | Alternate proof of ~ tfr3 ... |
| recsfnon 8334 | Strong transfinite recursi... |
| recsval 8335 | Strong transfinite recursi... |
| tz7.44lem1 8336 | The ordered pair abstracti... |
| tz7.44-1 8337 | The value of ` F ` at ` (/... |
| tz7.44-2 8338 | The value of ` F ` at a su... |
| tz7.44-3 8339 | The value of ` F ` at a li... |
| rdgeq1 8342 | Equality theorem for the r... |
| rdgeq2 8343 | Equality theorem for the r... |
| rdgeq12 8344 | Equality theorem for the r... |
| nfrdg 8345 | Bound-variable hypothesis ... |
| rdglem1 8346 | Lemma used with the recurs... |
| rdgfun 8347 | The recursive definition g... |
| rdgdmlim 8348 | The domain of the recursiv... |
| rdgfnon 8349 | The recursive definition g... |
| rdgvalg 8350 | Value of the recursive def... |
| rdgval 8351 | Value of the recursive def... |
| rdg0 8352 | The initial value of the r... |
| rdgseg 8353 | The initial segments of th... |
| rdgsucg 8354 | The value of the recursive... |
| rdgsuc 8355 | The value of the recursive... |
| rdglimg 8356 | The value of the recursive... |
| rdglim 8357 | The value of the recursive... |
| rdg0g 8358 | The initial value of the r... |
| rdgsucmptf 8359 | The value of the recursive... |
| rdgsucmptnf 8360 | The value of the recursive... |
| rdgsucmpt2 8361 | This version of ~ rdgsucmp... |
| rdgsucmpt 8362 | The value of the recursive... |
| rdglim2 8363 | The value of the recursive... |
| rdglim2a 8364 | The value of the recursive... |
| rdg0n 8365 | If ` A ` is a proper class... |
| frfnom 8366 | The function generated by ... |
| fr0g 8367 | The initial value resultin... |
| frsuc 8368 | The successor value result... |
| frsucmpt 8369 | The successor value result... |
| frsucmptn 8370 | The value of the finite re... |
| frsucmpt2 8371 | The successor value result... |
| tz7.48lem 8372 | A way of showing an ordina... |
| tz7.48-2 8373 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8374 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8375 | Proposition 7.48(3) of [Ta... |
| tz7.49 8376 | Proposition 7.49 of [Takeu... |
| tz7.49c 8377 | Corollary of Proposition 7... |
| seqomlem0 8380 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8381 | Lemma for ` seqom ` . The... |
| seqomlem2 8382 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8383 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8384 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8385 | Equality theorem for ` seq... |
| fnseqom 8386 | An index-aware recursive d... |
| seqom0g 8387 | Value of an index-aware re... |
| seqomsuc 8388 | Value of an index-aware re... |
| omsucelsucb 8389 | Membership is inherited by... |
| df1o2 8404 | Expanded value of the ordi... |
| df2o3 8405 | Expanded value of the ordi... |
| df2o2 8406 | Expanded value of the ordi... |
| 1oex 8407 | Ordinal 1 is a set. (Cont... |
| 2oex 8408 | ` 2o ` is a set. (Contrib... |
| 1on 8409 | Ordinal 1 is an ordinal nu... |
| 2on 8410 | Ordinal 2 is an ordinal nu... |
| 2on0 8411 | Ordinal two is not zero. ... |
| ord3 8412 | Ordinal 3 is an ordinal cl... |
| 3on 8413 | Ordinal 3 is an ordinal nu... |
| 4on 8414 | Ordinal 4 is an ordinal nu... |
| 1n0 8415 | Ordinal one is not equal t... |
| nlim1 8416 | 1 is not a limit ordinal. ... |
| nlim2 8417 | 2 is not a limit ordinal. ... |
| xp01disj 8418 | Cartesian products with th... |
| xp01disjl 8419 | Cartesian products with th... |
| ordgt0ge1 8420 | Two ways to express that a... |
| ordge1n0 8421 | An ordinal greater than or... |
| el1o 8422 | Membership in ordinal one.... |
| ord1eln01 8423 | An ordinal that is not 0 o... |
| ord2eln012 8424 | An ordinal that is not 0, ... |
| 1ellim 8425 | A limit ordinal contains 1... |
| 2ellim 8426 | A limit ordinal contains 2... |
| dif1o 8427 | Two ways to say that ` A `... |
| ondif1 8428 | Two ways to say that ` A `... |
| ondif2 8429 | Two ways to say that ` A `... |
| 2oconcl 8430 | Closure of the pair swappi... |
| 0lt1o 8431 | Ordinal zero is less than ... |
| dif20el 8432 | An ordinal greater than on... |
| 0we1 8433 | The empty set is a well-or... |
| brwitnlem 8434 | Lemma for relations which ... |
| fnoa 8435 | Functionality and domain o... |
| fnom 8436 | Functionality and domain o... |
| fnoe 8437 | Functionality and domain o... |
| oav 8438 | Value of ordinal addition.... |
| omv 8439 | Value of ordinal multiplic... |
| oe0lem 8440 | A helper lemma for ~ oe0 a... |
| oev 8441 | Value of ordinal exponenti... |
| oevn0 8442 | Value of ordinal exponenti... |
| oa0 8443 | Addition with zero. Propo... |
| om0 8444 | Ordinal multiplication wit... |
| oe0m 8445 | Value of zero raised to an... |
| om0x 8446 | Ordinal multiplication wit... |
| oe0m0 8447 | Ordinal exponentiation wit... |
| oe0m1 8448 | Ordinal exponentiation wit... |
| oe0 8449 | Ordinal exponentiation wit... |
| oev2 8450 | Alternate value of ordinal... |
| oasuc 8451 | Addition with successor. ... |
| oesuclem 8452 | Lemma for ~ oesuc . (Cont... |
| omsuc 8453 | Multiplication with succes... |
| oesuc 8454 | Ordinal exponentiation wit... |
| onasuc 8455 | Addition with successor. ... |
| onmsuc 8456 | Multiplication with succes... |
| onesuc 8457 | Exponentiation with a succ... |
| oa1suc 8458 | Addition with 1 is same as... |
| oalim 8459 | Ordinal addition with a li... |
| omlim 8460 | Ordinal multiplication wit... |
| oelim 8461 | Ordinal exponentiation wit... |
| oacl 8462 | Closure law for ordinal ad... |
| omcl 8463 | Closure law for ordinal mu... |
| oecl 8464 | Closure law for ordinal ex... |
| oa0r 8465 | Ordinal addition with zero... |
| om0r 8466 | Ordinal multiplication wit... |
| o1p1e2 8467 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8468 | 2 + 2 = 4 for ordinal numb... |
| om1 8469 | Ordinal multiplication wit... |
| om1r 8470 | Ordinal multiplication wit... |
| oe1 8471 | Ordinal exponentiation wit... |
| oe1m 8472 | Ordinal exponentiation wit... |
| oaordi 8473 | Ordering property of ordin... |
| oaord 8474 | Ordering property of ordin... |
| oacan 8475 | Left cancellation law for ... |
| oaword 8476 | Weak ordering property of ... |
| oawordri 8477 | Weak ordering property of ... |
| oaord1 8478 | An ordinal is less than it... |
| oaword1 8479 | An ordinal is less than or... |
| oaword2 8480 | An ordinal is less than or... |
| oawordeulem 8481 | Lemma for ~ oawordex . (C... |
| oawordeu 8482 | Existence theorem for weak... |
| oawordexr 8483 | Existence theorem for weak... |
| oawordex 8484 | Existence theorem for weak... |
| oaordex 8485 | Existence theorem for orde... |
| oa00 8486 | An ordinal sum is zero iff... |
| oalimcl 8487 | The ordinal sum with a lim... |
| oaass 8488 | Ordinal addition is associ... |
| oarec 8489 | Recursive definition of or... |
| oaf1o 8490 | Left addition by a constan... |
| oacomf1olem 8491 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8492 | Define a bijection from ` ... |
| omordi 8493 | Ordering property of ordin... |
| omord2 8494 | Ordering property of ordin... |
| omord 8495 | Ordering property of ordin... |
| omcan 8496 | Left cancellation law for ... |
| omword 8497 | Weak ordering property of ... |
| omwordi 8498 | Weak ordering property of ... |
| omwordri 8499 | Weak ordering property of ... |
| omword1 8500 | An ordinal is less than or... |
| omword2 8501 | An ordinal is less than or... |
| om00 8502 | The product of two ordinal... |
| om00el 8503 | The product of two nonzero... |
| omordlim 8504 | Ordering involving the pro... |
| omlimcl 8505 | The product of any nonzero... |
| odi 8506 | Distributive law for ordin... |
| omass 8507 | Multiplication of ordinal ... |
| oneo 8508 | If an ordinal number is ev... |
| omeulem1 8509 | Lemma for ~ omeu : existen... |
| omeulem2 8510 | Lemma for ~ omeu : uniquen... |
| omopth2 8511 | An ordered pair-like theor... |
| omeu 8512 | The division algorithm for... |
| om2 8513 | Two ways to double an ordi... |
| oen0 8514 | Ordinal exponentiation wit... |
| oeordi 8515 | Ordering law for ordinal e... |
| oeord 8516 | Ordering property of ordin... |
| oecan 8517 | Left cancellation law for ... |
| oeword 8518 | Weak ordering property of ... |
| oewordi 8519 | Weak ordering property of ... |
| oewordri 8520 | Weak ordering property of ... |
| oeworde 8521 | Ordinal exponentiation com... |
| oeordsuc 8522 | Ordering property of ordin... |
| oelim2 8523 | Ordinal exponentiation wit... |
| oeoalem 8524 | Lemma for ~ oeoa . (Contr... |
| oeoa 8525 | Sum of exponents law for o... |
| oeoelem 8526 | Lemma for ~ oeoe . (Contr... |
| oeoe 8527 | Product of exponents law f... |
| oelimcl 8528 | The ordinal exponential wi... |
| oeeulem 8529 | Lemma for ~ oeeu . (Contr... |
| oeeui 8530 | The division algorithm for... |
| oeeu 8531 | The division algorithm for... |
| nna0 8532 | Addition with zero. Theor... |
| nnm0 8533 | Multiplication with zero. ... |
| nnasuc 8534 | Addition with successor. ... |
| nnmsuc 8535 | Multiplication with succes... |
| nnesuc 8536 | Exponentiation with a succ... |
| nna0r 8537 | Addition to zero. Remark ... |
| nnm0r 8538 | Multiplication with zero. ... |
| nnacl 8539 | Closure of addition of nat... |
| nnmcl 8540 | Closure of multiplication ... |
| nnecl 8541 | Closure of exponentiation ... |
| nnacli 8542 | ` _om ` is closed under ad... |
| nnmcli 8543 | ` _om ` is closed under mu... |
| nnarcl 8544 | Reverse closure law for ad... |
| nnacom 8545 | Addition of natural number... |
| nnaordi 8546 | Ordering property of addit... |
| nnaord 8547 | Ordering property of addit... |
| nnaordr 8548 | Ordering property of addit... |
| nnawordi 8549 | Adding to both sides of an... |
| nnaass 8550 | Addition of natural number... |
| nndi 8551 | Distributive law for natur... |
| nnmass 8552 | Multiplication of natural ... |
| nnmsucr 8553 | Multiplication with succes... |
| nnmcom 8554 | Multiplication of natural ... |
| nnaword 8555 | Weak ordering property of ... |
| nnacan 8556 | Cancellation law for addit... |
| nnaword1 8557 | Weak ordering property of ... |
| nnaword2 8558 | Weak ordering property of ... |
| nnmordi 8559 | Ordering property of multi... |
| nnmord 8560 | Ordering property of multi... |
| nnmword 8561 | Weak ordering property of ... |
| nnmcan 8562 | Cancellation law for multi... |
| nnmwordi 8563 | Weak ordering property of ... |
| nnmwordri 8564 | Weak ordering property of ... |
| nnawordex 8565 | Equivalence for weak order... |
| nnaordex 8566 | Equivalence for ordering. ... |
| nnaordex2 8567 | Equivalence for ordering. ... |
| 1onn 8568 | The ordinal 1 is a natural... |
| 1onnALT 8569 | Shorter proof of ~ 1onn us... |
| 2onn 8570 | The ordinal 2 is a natural... |
| 2onnALT 8571 | Shorter proof of ~ 2onn us... |
| 3onn 8572 | The ordinal 3 is a natural... |
| 4onn 8573 | The ordinal 4 is a natural... |
| 1one2o 8574 | Ordinal one is not ordinal... |
| oaabslem 8575 | Lemma for ~ oaabs . (Cont... |
| oaabs 8576 | Ordinal addition absorbs a... |
| oaabs2 8577 | The absorption law ~ oaabs... |
| omabslem 8578 | Lemma for ~ omabs . (Cont... |
| omabs 8579 | Ordinal multiplication is ... |
| nnm1 8580 | Multiply an element of ` _... |
| nnm2 8581 | Multiply an element of ` _... |
| nn2m 8582 | Multiply an element of ` _... |
| nnneo 8583 | If a natural number is eve... |
| nneob 8584 | A natural number is even i... |
| omsmolem 8585 | Lemma for ~ omsmo . (Cont... |
| omsmo 8586 | A strictly monotonic ordin... |
| omopthlem1 8587 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8588 | Lemma for ~ omopthi . (Co... |
| omopthi 8589 | An ordered pair theorem fo... |
| omopth 8590 | An ordered pair theorem fo... |
| nnasmo 8591 | There is at most one left ... |
| eldifsucnn 8592 | Condition for membership i... |
| on2recsfn 8595 | Show that double recursion... |
| on2recsov 8596 | Calculate the value of the... |
| on2ind 8597 | Double induction over ordi... |
| on3ind 8598 | Triple induction over ordi... |
| coflton 8599 | Cofinality theorem for ord... |
| cofon1 8600 | Cofinality theorem for ord... |
| cofon2 8601 | Cofinality theorem for ord... |
| cofonr 8602 | Inverse cofinality law for... |
| naddfn 8603 | Natural addition is a func... |
| naddcllem 8604 | Lemma for ordinal addition... |
| naddcl 8605 | Closure law for natural ad... |
| naddov 8606 | The value of natural addit... |
| naddov2 8607 | Alternate expression for n... |
| naddov3 8608 | Alternate expression for n... |
| naddf 8609 | Function statement for nat... |
| naddcom 8610 | Natural addition commutes.... |
| naddrid 8611 | Ordinal zero is the additi... |
| naddlid 8612 | Ordinal zero is the additi... |
| naddssim 8613 | Ordinal less-than-or-equal... |
| naddelim 8614 | Ordinal less-than is prese... |
| naddel1 8615 | Ordinal less-than is not a... |
| naddel2 8616 | Ordinal less-than is not a... |
| naddss1 8617 | Ordinal less-than-or-equal... |
| naddss2 8618 | Ordinal less-than-or-equal... |
| naddword1 8619 | Weak-ordering principle fo... |
| naddword2 8620 | Weak-ordering principle fo... |
| naddunif 8621 | Uniformity theorem for nat... |
| naddasslem1 8622 | Lemma for ~ naddass . Exp... |
| naddasslem2 8623 | Lemma for ~ naddass . Exp... |
| naddass 8624 | Natural ordinal addition i... |
| nadd32 8625 | Commutative/associative la... |
| nadd4 8626 | Rearragement of terms in a... |
| nadd42 8627 | Rearragement of terms in a... |
| naddel12 8628 | Natural addition to both s... |
| naddsuc2 8629 | Natural addition with succ... |
| naddoa 8630 | Natural addition of a natu... |
| omnaddcl 8631 | The naturals are closed un... |
| dfer2 8636 | Alternate definition of eq... |
| dfec2 8638 | Alternate definition of ` ... |
| ecexg 8639 | An equivalence class modul... |
| ecexr 8640 | A nonempty equivalence cla... |
| dfqs2 8642 | Alternate definition of qu... |
| ereq1 8643 | Equality theorem for equiv... |
| ereq2 8644 | Equality theorem for equiv... |
| errel 8645 | An equivalence relation is... |
| erdm 8646 | The domain of an equivalen... |
| ercl 8647 | Elementhood in the field o... |
| ersym 8648 | An equivalence relation is... |
| ercl2 8649 | Elementhood in the field o... |
| ersymb 8650 | An equivalence relation is... |
| ertr 8651 | An equivalence relation is... |
| ertrd 8652 | A transitivity relation fo... |
| ertr2d 8653 | A transitivity relation fo... |
| ertr3d 8654 | A transitivity relation fo... |
| ertr4d 8655 | A transitivity relation fo... |
| erref 8656 | An equivalence relation is... |
| ercnv 8657 | The converse of an equival... |
| errn 8658 | The range and domain of an... |
| erssxp 8659 | An equivalence relation is... |
| erex 8660 | An equivalence relation is... |
| erexb 8661 | An equivalence relation is... |
| iserd 8662 | A reflexive, symmetric, tr... |
| iseri 8663 | A reflexive, symmetric, tr... |
| iseriALT 8664 | Alternate proof of ~ iseri... |
| brinxper 8665 | Conditions for a reflexive... |
| brdifun 8666 | Evaluate the incomparabili... |
| swoer 8667 | Incomparability under a st... |
| swoord1 8668 | The incomparability equiva... |
| swoord2 8669 | The incomparability equiva... |
| swoso 8670 | If the incomparability rel... |
| eqerlem 8671 | Lemma for ~ eqer . (Contr... |
| eqer 8672 | Equivalence relation invol... |
| ider 8673 | The identity relation is a... |
| 0er 8674 | The empty set is an equiva... |
| eceq1 8675 | Equality theorem for equiv... |
| eceq1d 8676 | Equality theorem for equiv... |
| eceq2 8677 | Equality theorem for equiv... |
| eceq2i 8678 | Equality theorem for the `... |
| eceq2d 8679 | Equality theorem for the `... |
| elecg 8680 | Membership in an equivalen... |
| ecref 8681 | All elements are in their ... |
| elec 8682 | Membership in an equivalen... |
| relelec 8683 | Membership in an equivalen... |
| elecres 8684 | Elementhood in the restric... |
| elecreseq 8685 | The restricted coset of ` ... |
| elecex 8686 | Condition for a coset to b... |
| ecss 8687 | An equivalence class is a ... |
| ecdmn0 8688 | A representative of a none... |
| ereldm 8689 | Equality of equivalence cl... |
| erth 8690 | Basic property of equivale... |
| erth2 8691 | Basic property of equivale... |
| erthi 8692 | Basic property of equivale... |
| erdisj 8693 | Equivalence classes do not... |
| ecidsn 8694 | An equivalence class modul... |
| qseq1 8695 | Equality theorem for quoti... |
| qseq2 8696 | Equality theorem for quoti... |
| qseq2i 8697 | Equality theorem for quoti... |
| qseq1d 8698 | Equality theorem for quoti... |
| qseq2d 8699 | Equality theorem for quoti... |
| qseq12 8700 | Equality theorem for quoti... |
| 0qs 8701 | Quotient set with the empt... |
| elqsg 8702 | Closed form of ~ elqs . (... |
| elqs 8703 | Membership in a quotient s... |
| elqsi 8704 | Membership in a quotient s... |
| elqsecl 8705 | Membership in a quotient s... |
| ecelqs 8706 | Membership of an equivalen... |
| ecelqsw 8707 | Membership of an equivalen... |
| ecelqsi 8708 | Membership of an equivalen... |
| ecopqsi 8709 | "Closure" law for equivale... |
| qsexg 8710 | A quotient set exists. (C... |
| qsex 8711 | A quotient set exists. (C... |
| uniqs 8712 | The union of a quotient se... |
| uniqsw 8713 | The union of a quotient se... |
| qsss 8714 | A quotient set is a set of... |
| uniqs2 8715 | The union of a quotient se... |
| snecg 8716 | The singleton of a coset i... |
| snec 8717 | The singleton of an equiva... |
| ecqs 8718 | Equivalence class in terms... |
| ecid 8719 | A set is equal to its cose... |
| qsid 8720 | A set is equal to its quot... |
| ectocld 8721 | Implicit substitution of c... |
| ectocl 8722 | Implicit substitution of c... |
| elqsn0 8723 | A quotient set does not co... |
| ecelqsdm 8724 | Membership of an equivalen... |
| ecelqsdmb 8725 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8726 | ` R ` -coset in its domain... |
| xpider 8727 | A Cartesian square is an e... |
| iiner 8728 | The intersection of a none... |
| riiner 8729 | The relative intersection ... |
| erinxp 8730 | A restricted equivalence r... |
| ecinxp 8731 | Restrict the relation in a... |
| qsinxp 8732 | Restrict the equivalence r... |
| qsdisj 8733 | Members of a quotient set ... |
| qsdisj2 8734 | A quotient set is a disjoi... |
| qsel 8735 | If an element of a quotien... |
| uniinqs 8736 | Class union distributes ov... |
| qliftlem 8737 | Lemma for theorems about a... |
| qliftrel 8738 | ` F ` , a function lift, i... |
| qliftel 8739 | Elementhood in the relatio... |
| qliftel1 8740 | Elementhood in the relatio... |
| qliftfun 8741 | The function ` F ` is the ... |
| qliftfund 8742 | The function ` F ` is the ... |
| qliftfuns 8743 | The function ` F ` is the ... |
| qliftf 8744 | The domain and codomain of... |
| qliftval 8745 | The value of the function ... |
| ecoptocl 8746 | Implicit substitution of c... |
| 2ecoptocl 8747 | Implicit substitution of c... |
| 3ecoptocl 8748 | Implicit substitution of c... |
| brecop 8749 | Binary relation on a quoti... |
| brecop2 8750 | Binary relation on a quoti... |
| eroveu 8751 | Lemma for ~ erov and ~ ero... |
| erovlem 8752 | Lemma for ~ erov and ~ ero... |
| erov 8753 | The value of an operation ... |
| eroprf 8754 | Functionality of an operat... |
| erov2 8755 | The value of an operation ... |
| eroprf2 8756 | Functionality of an operat... |
| ecopoveq 8757 | This is the first of sever... |
| ecopovsym 8758 | Assuming the operation ` F... |
| ecopovtrn 8759 | Assuming that operation ` ... |
| ecopover 8760 | Assuming that operation ` ... |
| eceqoveq 8761 | Equality of equivalence re... |
| ecovcom 8762 | Lemma used to transfer a c... |
| ecovass 8763 | Lemma used to transfer an ... |
| ecovdi 8764 | Lemma used to transfer a d... |
| mapprc 8769 | When ` A ` is a proper cla... |
| pmex 8770 | The class of all partial f... |
| mapexOLD 8771 | Obsolete version of ~ mape... |
| fnmap 8772 | Set exponentiation has a u... |
| fnpm 8773 | Partial function exponenti... |
| reldmmap 8774 | Set exponentiation is a we... |
| mapvalg 8775 | The value of set exponenti... |
| pmvalg 8776 | The value of the partial m... |
| mapval 8777 | The value of set exponenti... |
| elmapg 8778 | Membership relation for se... |
| elmapd 8779 | Deduction form of ~ elmapg... |
| elmapdd 8780 | Deduction associated with ... |
| mapdm0 8781 | The empty set is the only ... |
| elpmg 8782 | The predicate "is a partia... |
| elpm2g 8783 | The predicate "is a partia... |
| elpm2r 8784 | Sufficient condition for b... |
| elpmi 8785 | A partial function is a fu... |
| pmfun 8786 | A partial function is a fu... |
| elmapex 8787 | Eliminate antecedent for m... |
| elmapi 8788 | A mapping is a function, f... |
| mapfset 8789 | If ` B ` is a set, the val... |
| mapssfset 8790 | The value of the set expon... |
| mapfoss 8791 | The value of the set expon... |
| fsetsspwxp 8792 | The class of all functions... |
| fset0 8793 | The set of functions from ... |
| fsetdmprc0 8794 | The set of functions with ... |
| fsetex 8795 | The set of functions betwe... |
| f1setex 8796 | The set of injections betw... |
| fosetex 8797 | The set of surjections bet... |
| f1osetex 8798 | The set of bijections betw... |
| fsetfcdm 8799 | The class of functions wit... |
| fsetfocdm 8800 | The class of functions wit... |
| fsetprcnex 8801 | The class of all functions... |
| fsetcdmex 8802 | The class of all functions... |
| fsetexb 8803 | The class of all functions... |
| elmapfn 8804 | A mapping is a function wi... |
| elmapfun 8805 | A mapping is always a func... |
| elmapssres 8806 | A restricted mapping is a ... |
| elmapssresd 8807 | A restricted mapping is a ... |
| fpmg 8808 | A total function is a part... |
| pmss12g 8809 | Subset relation for the se... |
| pmresg 8810 | Elementhood of a restricte... |
| elmap 8811 | Membership relation for se... |
| mapval2 8812 | Alternate expression for t... |
| elpm 8813 | The predicate "is a partia... |
| elpm2 8814 | The predicate "is a partia... |
| fpm 8815 | A total function is a part... |
| mapsspm 8816 | Set exponentiation is a su... |
| pmsspw 8817 | Partial maps are a subset ... |
| mapsspw 8818 | Set exponentiation is a su... |
| mapfvd 8819 | The value of a function th... |
| elmapresaun 8820 | ~ fresaun transposed to ma... |
| fvmptmap 8821 | Special case of ~ fvmpt fo... |
| map0e 8822 | Set exponentiation with an... |
| map0b 8823 | Set exponentiation with an... |
| map0g 8824 | Set exponentiation is empt... |
| 0map0sn0 8825 | The set of mappings of the... |
| mapsnd 8826 | The value of set exponenti... |
| map0 8827 | Set exponentiation is empt... |
| mapsn 8828 | The value of set exponenti... |
| mapss 8829 | Subset inheritance for set... |
| fdiagfn 8830 | Functionality of the diago... |
| fvdiagfn 8831 | Functionality of the diago... |
| mapsnconst 8832 | Every singleton map is a c... |
| mapsncnv 8833 | Expression for the inverse... |
| mapsnf1o2 8834 | Explicit bijection between... |
| mapsnf1o3 8835 | Explicit bijection in the ... |
| ralxpmap 8836 | Quantification over functi... |
| dfixp 8839 | Eliminate the expression `... |
| ixpsnval 8840 | The value of an infinite C... |
| elixp2 8841 | Membership in an infinite ... |
| fvixp 8842 | Projection of a factor of ... |
| ixpfn 8843 | A nuple is a function. (C... |
| elixp 8844 | Membership in an infinite ... |
| elixpconst 8845 | Membership in an infinite ... |
| ixpconstg 8846 | Infinite Cartesian product... |
| ixpconst 8847 | Infinite Cartesian product... |
| ixpeq1 8848 | Equality theorem for infin... |
| ixpeq1d 8849 | Equality theorem for infin... |
| ss2ixp 8850 | Subclass theorem for infin... |
| ixpeq2 8851 | Equality theorem for infin... |
| ixpeq2dva 8852 | Equality theorem for infin... |
| ixpeq2dv 8853 | Equality theorem for infin... |
| cbvixp 8854 | Change bound variable in a... |
| cbvixpv 8855 | Change bound variable in a... |
| nfixpw 8856 | Bound-variable hypothesis ... |
| nfixp 8857 | Bound-variable hypothesis ... |
| nfixp1 8858 | The index variable in an i... |
| ixpprc 8859 | A cartesian product of pro... |
| ixpf 8860 | A member of an infinite Ca... |
| uniixp 8861 | The union of an infinite C... |
| ixpexg 8862 | The existence of an infini... |
| ixpin 8863 | The intersection of two in... |
| ixpiin 8864 | The indexed intersection o... |
| ixpint 8865 | The intersection of a coll... |
| ixp0x 8866 | An infinite Cartesian prod... |
| ixpssmap2g 8867 | An infinite Cartesian prod... |
| ixpssmapg 8868 | An infinite Cartesian prod... |
| 0elixp 8869 | Membership of the empty se... |
| ixpn0 8870 | The infinite Cartesian pro... |
| ixp0 8871 | The infinite Cartesian pro... |
| ixpssmap 8872 | An infinite Cartesian prod... |
| resixp 8873 | Restriction of an element ... |
| undifixp 8874 | Union of two projections o... |
| mptelixpg 8875 | Condition for an explicit ... |
| resixpfo 8876 | Restriction of elements of... |
| elixpsn 8877 | Membership in a class of s... |
| ixpsnf1o 8878 | A bijection between a clas... |
| mapsnf1o 8879 | A bijection between a set ... |
| boxriin 8880 | A rectangular subset of a ... |
| boxcutc 8881 | The relative complement of... |
| relen 8890 | Equinumerosity is a relati... |
| reldom 8891 | Dominance is a relation. ... |
| relsdom 8892 | Strict dominance is a rela... |
| encv 8893 | If two classes are equinum... |
| breng 8894 | Equinumerosity relation. ... |
| bren 8895 | Equinumerosity relation. ... |
| brdom2g 8896 | Dominance relation. This ... |
| brdomg 8897 | Dominance relation. (Cont... |
| brdomi 8898 | Dominance relation. (Cont... |
| brdom 8899 | Dominance relation. (Cont... |
| domen 8900 | Dominance in terms of equi... |
| domeng 8901 | Dominance in terms of equi... |
| ctex 8902 | A countable set is a set. ... |
| f1oen4g 8903 | The domain and range of a ... |
| f1dom4g 8904 | The domain of a one-to-one... |
| f1oen3g 8905 | The domain and range of a ... |
| f1dom3g 8906 | The domain of a one-to-one... |
| f1oen2g 8907 | The domain and range of a ... |
| f1dom2g 8908 | The domain of a one-to-one... |
| f1oeng 8909 | The domain and range of a ... |
| f1domg 8910 | The domain of a one-to-one... |
| f1oen 8911 | The domain and range of a ... |
| f1dom 8912 | The domain of a one-to-one... |
| brsdom 8913 | Strict dominance relation,... |
| isfi 8914 | Express " ` A ` is finite"... |
| enssdom 8915 | Equinumerosity implies dom... |
| enssdomOLD 8916 | Obsolete version of ~ enss... |
| dfdom2 8917 | Alternate definition of do... |
| endom 8918 | Equinumerosity implies dom... |
| sdomdom 8919 | Strict dominance implies d... |
| sdomnen 8920 | Strict dominance implies n... |
| brdom2 8921 | Dominance in terms of stri... |
| bren2 8922 | Equinumerosity expressed i... |
| enrefg 8923 | Equinumerosity is reflexiv... |
| enref 8924 | Equinumerosity is reflexiv... |
| eqeng 8925 | Equality implies equinumer... |
| domrefg 8926 | Dominance is reflexive. (... |
| en2d 8927 | Equinumerosity inference f... |
| en3d 8928 | Equinumerosity inference f... |
| en2i 8929 | Equinumerosity inference f... |
| en3i 8930 | Equinumerosity inference f... |
| dom2lem 8931 | A mapping (first hypothesi... |
| dom2d 8932 | A mapping (first hypothesi... |
| dom3d 8933 | A mapping (first hypothesi... |
| dom2 8934 | A mapping (first hypothesi... |
| dom3 8935 | A mapping (first hypothesi... |
| idssen 8936 | Equality implies equinumer... |
| domssl 8937 | If ` A ` is a subset of ` ... |
| domssr 8938 | If ` C ` is a superset of ... |
| ssdomg 8939 | A set dominates its subset... |
| ener 8940 | Equinumerosity is an equiv... |
| ensymb 8941 | Symmetry of equinumerosity... |
| ensym 8942 | Symmetry of equinumerosity... |
| ensymi 8943 | Symmetry of equinumerosity... |
| ensymd 8944 | Symmetry of equinumerosity... |
| entr 8945 | Transitivity of equinumero... |
| domtr 8946 | Transitivity of dominance ... |
| entri 8947 | A chained equinumerosity i... |
| entr2i 8948 | A chained equinumerosity i... |
| entr3i 8949 | A chained equinumerosity i... |
| entr4i 8950 | A chained equinumerosity i... |
| endomtr 8951 | Transitivity of equinumero... |
| domentr 8952 | Transitivity of dominance ... |
| f1imaeng 8953 | If a function is one-to-on... |
| f1imaen2g 8954 | If a function is one-to-on... |
| f1imaen3g 8955 | If a set function is one-t... |
| f1imaen 8956 | If a function is one-to-on... |
| en0 8957 | The empty set is equinumer... |
| en0ALT 8958 | Shorter proof of ~ en0 , d... |
| en0r 8959 | The empty set is equinumer... |
| ensn1 8960 | A singleton is equinumerou... |
| ensn1g 8961 | A singleton is equinumerou... |
| enpr1g 8962 | ` { A , A } ` has only one... |
| en1 8963 | A set is equinumerous to o... |
| en1b 8964 | A set is equinumerous to o... |
| reuen1 8965 | Two ways to express "exact... |
| euen1 8966 | Two ways to express "exact... |
| euen1b 8967 | Two ways to express " ` A ... |
| en1uniel 8968 | A singleton contains its s... |
| 2dom 8969 | A set that dominates ordin... |
| fundmen 8970 | A function is equinumerous... |
| fundmeng 8971 | A function is equinumerous... |
| cnven 8972 | A relational set is equinu... |
| cnvct 8973 | If a set is countable, so ... |
| fndmeng 8974 | A function is equinumerate... |
| mapsnend 8975 | Set exponentiation to a si... |
| mapsnen 8976 | Set exponentiation to a si... |
| snmapen 8977 | Set exponentiation: a sing... |
| snmapen1 8978 | Set exponentiation: a sing... |
| map1 8979 | Set exponentiation: ordina... |
| en2sn 8980 | Two singletons are equinum... |
| 0fi 8981 | The empty set is finite. ... |
| snfi 8982 | A singleton is finite. (C... |
| fiprc 8983 | The class of finite sets i... |
| unen 8984 | Equinumerosity of union of... |
| enrefnn 8985 | Equinumerosity is reflexiv... |
| en2prd 8986 | Two proper unordered pairs... |
| enpr2d 8987 | A pair with distinct eleme... |
| ssct 8988 | Any subset of a countable ... |
| difsnen 8989 | All decrements of a set ar... |
| domdifsn 8990 | Dominance over a set with ... |
| xpsnen 8991 | A set is equinumerous to i... |
| xpsneng 8992 | A set is equinumerous to i... |
| xp1en 8993 | One times a cardinal numbe... |
| endisj 8994 | Any two sets are equinumer... |
| undom 8995 | Dominance law for union. ... |
| xpcomf1o 8996 | The canonical bijection fr... |
| xpcomco 8997 | Composition with the bijec... |
| xpcomen 8998 | Commutative law for equinu... |
| xpcomeng 8999 | Commutative law for equinu... |
| xpsnen2g 9000 | A set is equinumerous to i... |
| xpassen 9001 | Associative law for equinu... |
| xpdom2 9002 | Dominance law for Cartesia... |
| xpdom2g 9003 | Dominance law for Cartesia... |
| xpdom1g 9004 | Dominance law for Cartesia... |
| xpdom3 9005 | A set is dominated by its ... |
| xpdom1 9006 | Dominance law for Cartesia... |
| domunsncan 9007 | A singleton cancellation l... |
| omxpenlem 9008 | Lemma for ~ omxpen . (Con... |
| omxpen 9009 | The cardinal and ordinal p... |
| omf1o 9010 | Construct an explicit bije... |
| pw2f1olem 9011 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9012 | The power set of a set is ... |
| pw2eng 9013 | The power set of a set is ... |
| pw2en 9014 | The power set of a set is ... |
| fopwdom 9015 | Covering implies injection... |
| enfixsn 9016 | Given two equipollent sets... |
| sbthlem1 9017 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9018 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9019 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9020 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9021 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9022 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9023 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9024 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9025 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9026 | Lemma for ~ sbth . (Contr... |
| sbth 9027 | Schroeder-Bernstein Theore... |
| sbthb 9028 | Schroeder-Bernstein Theore... |
| sbthcl 9029 | Schroeder-Bernstein Theore... |
| dfsdom2 9030 | Alternate definition of st... |
| brsdom2 9031 | Alternate definition of st... |
| sdomnsym 9032 | Strict dominance is asymme... |
| domnsym 9033 | Theorem 22(i) of [Suppes] ... |
| 0domg 9034 | Any set dominates the empt... |
| dom0 9035 | A set dominated by the emp... |
| 0sdomg 9036 | A set strictly dominates t... |
| 0dom 9037 | Any set dominates the empt... |
| 0sdom 9038 | A set strictly dominates t... |
| sdom0 9039 | The empty set does not str... |
| sdomdomtr 9040 | Transitivity of strict dom... |
| sdomentr 9041 | Transitivity of strict dom... |
| domsdomtr 9042 | Transitivity of dominance ... |
| ensdomtr 9043 | Transitivity of equinumero... |
| sdomirr 9044 | Strict dominance is irrefl... |
| sdomtr 9045 | Strict dominance is transi... |
| sdomn2lp 9046 | Strict dominance has no 2-... |
| enen1 9047 | Equality-like theorem for ... |
| enen2 9048 | Equality-like theorem for ... |
| domen1 9049 | Equality-like theorem for ... |
| domen2 9050 | Equality-like theorem for ... |
| sdomen1 9051 | Equality-like theorem for ... |
| sdomen2 9052 | Equality-like theorem for ... |
| domtriord 9053 | Dominance is trichotomous ... |
| sdomel 9054 | For ordinals, strict domin... |
| sdomdif 9055 | The difference of a set fr... |
| onsdominel 9056 | An ordinal with more eleme... |
| domunsn 9057 | Dominance over a set with ... |
| fodomr 9058 | There exists a mapping fro... |
| pwdom 9059 | Injection of sets implies ... |
| canth2 9060 | Cantor's Theorem. No set ... |
| canth2g 9061 | Cantor's theorem with the ... |
| 2pwuninel 9062 | The power set of the power... |
| 2pwne 9063 | No set equals the power se... |
| disjen 9064 | A stronger form of ~ pwuni... |
| disjenex 9065 | Existence version of ~ dis... |
| domss2 9066 | A corollary of ~ disjenex ... |
| domssex2 9067 | A corollary of ~ disjenex ... |
| domssex 9068 | Weakening of ~ domssex2 to... |
| xpf1o 9069 | Construct a bijection on a... |
| xpen 9070 | Equinumerosity law for Car... |
| mapen 9071 | Two set exponentiations ar... |
| mapdom1 9072 | Order-preserving property ... |
| mapxpen 9073 | Equinumerosity law for dou... |
| xpmapenlem 9074 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9075 | Equinumerosity law for set... |
| mapunen 9076 | Equinumerosity law for set... |
| map2xp 9077 | A cardinal power with expo... |
| mapdom2 9078 | Order-preserving property ... |
| mapdom3 9079 | Set exponentiation dominat... |
| pwen 9080 | If two sets are equinumero... |
| ssenen 9081 | Equinumerosity of equinume... |
| limenpsi 9082 | A limit ordinal is equinum... |
| limensuci 9083 | A limit ordinal is equinum... |
| limensuc 9084 | A limit ordinal is equinum... |
| infensuc 9085 | Any infinite ordinal is eq... |
| dif1enlem 9086 | Lemma for ~ rexdif1en and ... |
| rexdif1en 9087 | If a set is equinumerous t... |
| dif1en 9088 | If a set ` A ` is equinume... |
| dif1ennn 9089 | If a set ` A ` is equinume... |
| findcard 9090 | Schema for induction on th... |
| findcard2 9091 | Schema for induction on th... |
| findcard2s 9092 | Variation of ~ findcard2 r... |
| findcard2d 9093 | Deduction version of ~ fin... |
| nnfi 9094 | Natural numbers are finite... |
| pssnn 9095 | A proper subset of a natur... |
| ssnnfi 9096 | A subset of a natural numb... |
| unfi 9097 | The union of two finite se... |
| unfid 9098 | The union of two finite se... |
| ssfi 9099 | A subset of a finite set i... |
| ssfiALT 9100 | Shorter proof of ~ ssfi us... |
| diffi 9101 | If ` A ` is finite, ` ( A ... |
| cnvfi 9102 | If a set is finite, its co... |
| pwssfi 9103 | Every element of the power... |
| fnfi 9104 | A version of ~ fnex for fi... |
| f1oenfi 9105 | If the domain of a one-to-... |
| f1oenfirn 9106 | If the range of a one-to-o... |
| f1domfi 9107 | If the codomain of a one-t... |
| f1domfi2 9108 | If the domain of a one-to-... |
| enreffi 9109 | Equinumerosity is reflexiv... |
| ensymfib 9110 | Symmetry of equinumerosity... |
| entrfil 9111 | Transitivity of equinumero... |
| enfii 9112 | A set equinumerous to a fi... |
| enfi 9113 | Equinumerous sets have the... |
| enfiALT 9114 | Shorter proof of ~ enfi us... |
| domfi 9115 | A set dominated by a finit... |
| entrfi 9116 | Transitivity of equinumero... |
| entrfir 9117 | Transitivity of equinumero... |
| domtrfil 9118 | Transitivity of dominance ... |
| domtrfi 9119 | Transitivity of dominance ... |
| domtrfir 9120 | Transitivity of dominance ... |
| f1imaenfi 9121 | If a function is one-to-on... |
| ssdomfi 9122 | A finite set dominates its... |
| ssdomfi2 9123 | A set dominates its finite... |
| sbthfilem 9124 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9125 | Schroeder-Bernstein Theore... |
| domnsymfi 9126 | If a set dominates a finit... |
| sdomdomtrfi 9127 | Transitivity of strict dom... |
| domsdomtrfi 9128 | Transitivity of dominance ... |
| sucdom2 9129 | Strict dominance of a set ... |
| phplem1 9130 | Lemma for Pigeonhole Princ... |
| phplem2 9131 | Lemma for Pigeonhole Princ... |
| nneneq 9132 | Two equinumerous natural n... |
| php 9133 | Pigeonhole Principle. A n... |
| php2 9134 | Corollary of Pigeonhole Pr... |
| php3 9135 | Corollary of Pigeonhole Pr... |
| php4 9136 | Corollary of the Pigeonhol... |
| php5 9137 | Corollary of the Pigeonhol... |
| phpeqd 9138 | Corollary of the Pigeonhol... |
| nndomog 9139 | Cardinal ordering agrees w... |
| onomeneq 9140 | An ordinal number equinume... |
| onfin 9141 | An ordinal number is finit... |
| ordfin 9142 | A generalization of ~ onfi... |
| onfin2 9143 | A set is a natural number ... |
| nndomo 9144 | Cardinal ordering agrees w... |
| nnsdomo 9145 | Cardinal ordering agrees w... |
| sucdom 9146 | Strict dominance of a set ... |
| snnen2o 9147 | A singleton ` { A } ` is n... |
| 0sdom1dom 9148 | Strict dominance over 0 is... |
| 0sdom1domALT 9149 | Alternate proof of ~ 0sdom... |
| 1sdom2 9150 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9151 | Alternate proof of ~ 1sdom... |
| sdom1 9152 | A set has less than one me... |
| modom 9153 | Two ways to express "at mo... |
| modom2 9154 | Two ways to express "at mo... |
| rex2dom 9155 | A set that has at least 2 ... |
| 1sdom2dom 9156 | Strict dominance over 1 is... |
| 1sdom 9157 | A set that strictly domina... |
| unxpdomlem1 9158 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9159 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9160 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9161 | Cartesian product dominate... |
| unxpdom2 9162 | Corollary of ~ unxpdom . ... |
| sucxpdom 9163 | Cartesian product dominate... |
| pssinf 9164 | A set equinumerous to a pr... |
| fisseneq 9165 | A finite set is equal to i... |
| ominf 9166 | The set of natural numbers... |
| isinf 9167 | Any set that is not finite... |
| fineqvlem 9168 | Lemma for ~ fineqv . (Con... |
| fineqv 9169 | If the Axiom of Infinity i... |
| xpfir 9170 | The components of a nonemp... |
| ssfid 9171 | A subset of a finite set i... |
| infi 9172 | The intersection of two se... |
| rabfi 9173 | A restricted class built f... |
| finresfin 9174 | The restriction of a finit... |
| f1finf1o 9175 | Any injection from one fin... |
| nfielex 9176 | If a class is not finite, ... |
| en1eqsn 9177 | A set with one element is ... |
| en1eqsnbi 9178 | A set containing an elemen... |
| dif1ennnALT 9179 | Alternate proof of ~ dif1e... |
| enp1ilem 9180 | Lemma for uses of ~ enp1i ... |
| enp1i 9181 | Proof induction for ~ en2 ... |
| en2 9182 | A set equinumerous to ordi... |
| en3 9183 | A set equinumerous to ordi... |
| en4 9184 | A set equinumerous to ordi... |
| findcard3 9185 | Schema for strong inductio... |
| ac6sfi 9186 | A version of ~ ac6s for fi... |
| frfi 9187 | A partial order is well-fo... |
| fimax2g 9188 | A finite set has a maximum... |
| fimaxg 9189 | A finite set has a maximum... |
| fisupg 9190 | Lemma showing existence an... |
| wofi 9191 | A total order on a finite ... |
| ordunifi 9192 | The maximum of a finite co... |
| nnunifi 9193 | The union (supremum) of a ... |
| unblem1 9194 | Lemma for ~ unbnn . After... |
| unblem2 9195 | Lemma for ~ unbnn . The v... |
| unblem3 9196 | Lemma for ~ unbnn . The v... |
| unblem4 9197 | Lemma for ~ unbnn . The f... |
| unbnn 9198 | Any unbounded subset of na... |
| unbnn2 9199 | Version of ~ unbnn that do... |
| isfinite2 9200 | Any set strictly dominated... |
| nnsdomg 9201 | Omega strictly dominates a... |
| isfiniteg 9202 | A set is finite iff it is ... |
| infsdomnn 9203 | An infinite set strictly d... |
| infn0 9204 | An infinite set is not emp... |
| infn0ALT 9205 | Shorter proof of ~ infn0 u... |
| fin2inf 9206 | This (useless) theorem, wh... |
| unfilem1 9207 | Lemma for proving that the... |
| unfilem2 9208 | Lemma for proving that the... |
| unfilem3 9209 | Lemma for proving that the... |
| unfir 9210 | If a union is finite, the ... |
| unfib 9211 | A union is finite if and o... |
| unfi2 9212 | The union of two finite se... |
| difinf 9213 | An infinite set ` A ` minu... |
| fodomfi 9214 | An onto function implies d... |
| fofi 9215 | If an onto function has a ... |
| f1fi 9216 | If a 1-to-1 function has a... |
| imafi 9217 | Images of finite sets are ... |
| imafiOLD 9218 | Obsolete version of ~ imaf... |
| pwfir 9219 | If the power set of a set ... |
| pwfilem 9220 | Lemma for ~ pwfi . (Contr... |
| pwfi 9221 | The power set of a finite ... |
| xpfi 9222 | The Cartesian product of t... |
| 3xpfi 9223 | The Cartesian product of t... |
| domunfican 9224 | A finite set union cancell... |
| infcntss 9225 | Every infinite set has a d... |
| prfi 9226 | An unordered pair is finit... |
| prfiALT 9227 | Shorter proof of ~ prfi us... |
| tpfi 9228 | An unordered triple is fin... |
| fiint 9229 | Equivalent ways of stating... |
| fodomfir 9230 | There exists a mapping fro... |
| fodomfib 9231 | Equivalence of an onto map... |
| fodomfiOLD 9232 | Obsolete version of ~ fodo... |
| fodomfibOLD 9233 | Obsolete version of ~ fodo... |
| fofinf1o 9234 | Any surjection from one fi... |
| rneqdmfinf1o 9235 | Any function from a finite... |
| fidomdm 9236 | Any finite set dominates i... |
| dmfi 9237 | The domain of a finite set... |
| fundmfibi 9238 | A function is finite if an... |
| resfnfinfin 9239 | The restriction of a funct... |
| residfi 9240 | A restricted identity func... |
| cnvfiALT 9241 | Shorter proof of ~ cnvfi u... |
| rnfi 9242 | The range of a finite set ... |
| f1dmvrnfibi 9243 | A one-to-one function whos... |
| f1vrnfibi 9244 | A one-to-one function whic... |
| iunfi 9245 | The finite union of finite... |
| unifi 9246 | The finite union of finite... |
| unifi2 9247 | The finite union of finite... |
| infssuni 9248 | If an infinite set ` A ` i... |
| unirnffid 9249 | The union of the range of ... |
| mapfi 9250 | Set exponentiation of fini... |
| ixpfi 9251 | A Cartesian product of fin... |
| ixpfi2 9252 | A Cartesian product of fin... |
| mptfi 9253 | A finite mapping set is fi... |
| abrexfi 9254 | An image set from a finite... |
| cnvimamptfin 9255 | A preimage of a mapping wi... |
| elfpw 9256 | Membership in a class of f... |
| unifpw 9257 | A set is the union of its ... |
| f1opwfi 9258 | A one-to-one mapping induc... |
| fissuni 9259 | A finite subset of a union... |
| fipreima 9260 | Given a finite subset ` A ... |
| finsschain 9261 | A finite subset of the uni... |
| indexfi 9262 | If for every element of a ... |
| imafi2 9263 | The image by a finite set ... |
| unifi3 9264 | If a union is finite, then... |
| tfsnfin2 9265 | A transfinite sequence is ... |
| relfsupp 9268 | The property of a function... |
| relprcnfsupp 9269 | A proper class is never fi... |
| isfsupp 9270 | The property of a class to... |
| isfsuppd 9271 | Deduction form of ~ isfsup... |
| funisfsupp 9272 | The property of a function... |
| fsuppimp 9273 | Implications of a class be... |
| fsuppimpd 9274 | A finitely supported funct... |
| fsuppfund 9275 | A finitely supported funct... |
| fisuppfi 9276 | A function on a finite set... |
| fidmfisupp 9277 | A function with a finite d... |
| finnzfsuppd 9278 | If a function is zero outs... |
| fdmfisuppfi 9279 | The support of a function ... |
| fdmfifsupp 9280 | A function with a finite d... |
| fsuppmptdm 9281 | A mapping with a finite do... |
| fndmfisuppfi 9282 | The support of a function ... |
| fndmfifsupp 9283 | A function with a finite d... |
| suppeqfsuppbi 9284 | If two functions have the ... |
| suppssfifsupp 9285 | If the support of a functi... |
| fsuppsssupp 9286 | If the support of a functi... |
| fsuppsssuppgd 9287 | If the support of a functi... |
| fsuppss 9288 | A subset of a finitely sup... |
| fsuppssov1 9289 | Formula building theorem f... |
| fsuppxpfi 9290 | The cartesian product of t... |
| fczfsuppd 9291 | A constant function with v... |
| fsuppun 9292 | The union of two finitely ... |
| fsuppunfi 9293 | The union of the support o... |
| fsuppunbi 9294 | If the union of two classe... |
| 0fsupp 9295 | The empty set is a finitel... |
| snopfsupp 9296 | A singleton containing an ... |
| funsnfsupp 9297 | Finite support for a funct... |
| fsuppres 9298 | The restriction of a finit... |
| fmptssfisupp 9299 | The restriction of a mappi... |
| ressuppfi 9300 | If the support of the rest... |
| resfsupp 9301 | If the restriction of a fu... |
| resfifsupp 9302 | The restriction of a funct... |
| ffsuppbi 9303 | Two ways of saying that a ... |
| fsuppmptif 9304 | A function mapping an argu... |
| sniffsupp 9305 | A function mapping all but... |
| fsuppcolem 9306 | Lemma for ~ fsuppco . For... |
| fsuppco 9307 | The composition of a 1-1 f... |
| fsuppco2 9308 | The composition of a funct... |
| fsuppcor 9309 | The composition of a funct... |
| mapfienlem1 9310 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9311 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9312 | Lemma 3 for ~ mapfien . (... |
| mapfien 9313 | A bijection of the base se... |
| mapfien2 9314 | Equinumerousity relation f... |
| fival 9317 | The set of all the finite ... |
| elfi 9318 | Specific properties of an ... |
| elfi2 9319 | The empty intersection nee... |
| elfir 9320 | Sufficient condition for a... |
| intrnfi 9321 | Sufficient condition for t... |
| iinfi 9322 | An indexed intersection of... |
| inelfi 9323 | The intersection of two se... |
| ssfii 9324 | Any element of a set ` A `... |
| fi0 9325 | The set of finite intersec... |
| fieq0 9326 | A set is empty iff the cla... |
| fiin 9327 | The elements of ` ( fi `` ... |
| dffi2 9328 | The set of finite intersec... |
| fiss 9329 | Subset relationship for fu... |
| inficl 9330 | A set which is closed unde... |
| fipwuni 9331 | The set of finite intersec... |
| fisn 9332 | A singleton is closed unde... |
| fiuni 9333 | The union of the finite in... |
| fipwss 9334 | If a set is a family of su... |
| elfiun 9335 | A finite intersection of e... |
| dffi3 9336 | The set of finite intersec... |
| fifo 9337 | Describe a surjection from... |
| marypha1lem 9338 | Core induction for Philip ... |
| marypha1 9339 | (Philip) Hall's marriage t... |
| marypha2lem1 9340 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9341 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9342 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9343 | Lemma for ~ marypha2 . Pr... |
| marypha2 9344 | Version of ~ marypha1 usin... |
| dfsup2 9349 | Quantifier-free definition... |
| supeq1 9350 | Equality theorem for supre... |
| supeq1d 9351 | Equality deduction for sup... |
| supeq1i 9352 | Equality inference for sup... |
| supeq2 9353 | Equality theorem for supre... |
| supeq3 9354 | Equality theorem for supre... |
| supeq123d 9355 | Equality deduction for sup... |
| nfsup 9356 | Hypothesis builder for sup... |
| supmo 9357 | Any class ` B ` has at mos... |
| supexd 9358 | A supremum is a set. (Con... |
| supeu 9359 | A supremum is unique. Sim... |
| supval2 9360 | Alternate expression for t... |
| eqsup 9361 | Sufficient condition for a... |
| eqsupd 9362 | Sufficient condition for a... |
| supcl 9363 | A supremum belongs to its ... |
| supub 9364 | A supremum is an upper bou... |
| suplub 9365 | A supremum is the least up... |
| suplub2 9366 | Bidirectional form of ~ su... |
| supnub 9367 | An upper bound is not less... |
| supssd 9368 | Inequality deduction for s... |
| supex 9369 | A supremum is a set. (Con... |
| sup00 9370 | The supremum under an empt... |
| sup0riota 9371 | The supremum of an empty s... |
| sup0 9372 | The supremum of an empty s... |
| supmax 9373 | The greatest element of a ... |
| fisup2g 9374 | A finite set satisfies the... |
| fisupcl 9375 | A nonempty finite set cont... |
| supgtoreq 9376 | The supremum of a finite s... |
| suppr 9377 | The supremum of a pair. (... |
| supsn 9378 | The supremum of a singleto... |
| supisolem 9379 | Lemma for ~ supiso . (Con... |
| supisoex 9380 | Lemma for ~ supiso . (Con... |
| supiso 9381 | Image of a supremum under ... |
| infeq1 9382 | Equality theorem for infim... |
| infeq1d 9383 | Equality deduction for inf... |
| infeq1i 9384 | Equality inference for inf... |
| infeq2 9385 | Equality theorem for infim... |
| infeq3 9386 | Equality theorem for infim... |
| infeq123d 9387 | Equality deduction for inf... |
| nfinf 9388 | Hypothesis builder for inf... |
| infexd 9389 | An infimum is a set. (Con... |
| eqinf 9390 | Sufficient condition for a... |
| eqinfd 9391 | Sufficient condition for a... |
| infval 9392 | Alternate expression for t... |
| infcllem 9393 | Lemma for ~ infcl , ~ infl... |
| infcl 9394 | An infimum belongs to its ... |
| inflb 9395 | An infimum is a lower boun... |
| infglb 9396 | An infimum is the greatest... |
| infglbb 9397 | Bidirectional form of ~ in... |
| infnlb 9398 | A lower bound is not great... |
| infssd 9399 | Inequality deduction for i... |
| infex 9400 | An infimum is a set. (Con... |
| infmin 9401 | The smallest element of a ... |
| infmo 9402 | Any class ` B ` has at mos... |
| infeu 9403 | An infimum is unique. (Co... |
| fimin2g 9404 | A finite set has a minimum... |
| fiming 9405 | A finite set has a minimum... |
| fiinfg 9406 | Lemma showing existence an... |
| fiinf2g 9407 | A finite set satisfies the... |
| fiinfcl 9408 | A nonempty finite set cont... |
| infltoreq 9409 | The infimum of a finite se... |
| infpr 9410 | The infimum of a pair. (C... |
| infsupprpr 9411 | The infimum of a proper pa... |
| infsn 9412 | The infimum of a singleton... |
| inf00 9413 | The infimum regarding an e... |
| infempty 9414 | The infimum of an empty se... |
| infiso 9415 | Image of an infimum under ... |
| dfoi 9418 | Rewrite ~ df-oi with abbre... |
| oieq1 9419 | Equality theorem for ordin... |
| oieq2 9420 | Equality theorem for ordin... |
| nfoi 9421 | Hypothesis builder for ord... |
| ordiso2 9422 | Generalize ~ ordiso to pro... |
| ordiso 9423 | Order-isomorphic ordinal n... |
| ordtypecbv 9424 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9425 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9426 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9427 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9428 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9429 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9430 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9431 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9432 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9433 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9434 | Lemma for ~ ordtype . Usi... |
| oi0 9435 | Definition of the ordinal ... |
| oicl 9436 | The order type of the well... |
| oif 9437 | The order isomorphism of t... |
| oiiso2 9438 | The order isomorphism of t... |
| ordtype 9439 | For any set-like well-orde... |
| oiiniseg 9440 | ` ran F ` is an initial se... |
| ordtype2 9441 | For any set-like well-orde... |
| oiexg 9442 | The order isomorphism on a... |
| oion 9443 | The order type of the well... |
| oiiso 9444 | The order isomorphism of t... |
| oien 9445 | The order type of a well-o... |
| oieu 9446 | Uniqueness of the unique o... |
| oismo 9447 | When ` A ` is a subclass o... |
| oiid 9448 | The order type of an ordin... |
| hartogslem1 9449 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9450 | Lemma for ~ hartogs . (Co... |
| hartogs 9451 | The class of ordinals domi... |
| wofib 9452 | The only sets which are we... |
| wemaplem1 9453 | Value of the lexicographic... |
| wemaplem2 9454 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9455 | Lemma for ~ wemapso . Tra... |
| wemappo 9456 | Construct lexicographic or... |
| wemapsolem 9457 | Lemma for ~ wemapso . (Co... |
| wemapso 9458 | Construct lexicographic or... |
| wemapso2lem 9459 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9460 | An alternative to having a... |
| card2on 9461 | The alternate definition o... |
| card2inf 9462 | The alternate definition o... |
| harf 9465 | Functionality of the Harto... |
| harcl 9466 | Values of the Hartogs func... |
| harval 9467 | Function value of the Hart... |
| elharval 9468 | The Hartogs number of a se... |
| harndom 9469 | The Hartogs number of a se... |
| harword 9470 | Weak ordering property of ... |
| relwdom 9473 | Weak dominance is a relati... |
| brwdom 9474 | Property of weak dominance... |
| brwdomi 9475 | Property of weak dominance... |
| brwdomn0 9476 | Weak dominance over nonemp... |
| 0wdom 9477 | Any set weakly dominates t... |
| fowdom 9478 | An onto function implies w... |
| wdomref 9479 | Reflexivity of weak domina... |
| brwdom2 9480 | Alternate characterization... |
| domwdom 9481 | Weak dominance is implied ... |
| wdomtr 9482 | Transitivity of weak domin... |
| wdomen1 9483 | Equality-like theorem for ... |
| wdomen2 9484 | Equality-like theorem for ... |
| wdompwdom 9485 | Weak dominance strengthens... |
| canthwdom 9486 | Cantor's Theorem, stated u... |
| wdom2d 9487 | Deduce weak dominance from... |
| wdomd 9488 | Deduce weak dominance from... |
| brwdom3 9489 | Condition for weak dominan... |
| brwdom3i 9490 | Weak dominance implies exi... |
| unwdomg 9491 | Weak dominance of a (disjo... |
| xpwdomg 9492 | Weak dominance of a Cartes... |
| wdomima2g 9493 | A set is weakly dominant o... |
| wdomimag 9494 | A set is weakly dominant o... |
| unxpwdom2 9495 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9496 | If a Cartesian product is ... |
| ixpiunwdom 9497 | Describe an onto function ... |
| harwdom 9498 | The value of the Hartogs f... |
| axreg2 9500 | Axiom of Regularity expres... |
| zfregcl 9501 | The Axiom of Regularity wi... |
| zfregclOLD 9502 | Obsolete version of ~ zfre... |
| zfreg 9503 | The Axiom of Regularity us... |
| elirrv 9504 | The membership relation is... |
| elirrvOLD 9505 | Obsolete version of ~ elir... |
| elirr 9506 | No class is a member of it... |
| elneq 9507 | A class is not equal to an... |
| nelaneq 9508 | A class is not an element ... |
| nelaneqOLD 9509 | Obsolete version of ~ nela... |
| epinid0 9510 | The membership relation an... |
| sucprcreg 9511 | A class is equal to its su... |
| ruv 9512 | The Russell class is equal... |
| ruALT 9513 | Alternate proof of ~ ru , ... |
| disjcsn 9514 | A class is disjoint from i... |
| zfregfr 9515 | The membership relation is... |
| elirrvALT 9516 | Alternate proof of ~ elirr... |
| en2lp 9517 | No class has 2-cycle membe... |
| elnanel 9518 | Two classes are not elemen... |
| cnvepnep 9519 | The membership (epsilon) r... |
| epnsym 9520 | The membership (epsilon) r... |
| elnotel 9521 | A class cannot be an eleme... |
| elnel 9522 | A class cannot be an eleme... |
| en3lplem1 9523 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9524 | Lemma for ~ en3lp . (Cont... |
| en3lp 9525 | No class has 3-cycle membe... |
| preleqg 9526 | Equality of two unordered ... |
| preleq 9527 | Equality of two unordered ... |
| preleqALT 9528 | Alternate proof of ~ prele... |
| opthreg 9529 | Theorem for alternate repr... |
| suc11reg 9530 | The successor operation be... |
| dford2 9531 | Assuming ~ ax-reg , an ord... |
| inf0 9532 | Existence of ` _om ` impli... |
| inf1 9533 | Variation of Axiom of Infi... |
| inf2 9534 | Variation of Axiom of Infi... |
| inf3lema 9535 | Lemma for our Axiom of Inf... |
| inf3lemb 9536 | Lemma for our Axiom of Inf... |
| inf3lemc 9537 | Lemma for our Axiom of Inf... |
| inf3lemd 9538 | Lemma for our Axiom of Inf... |
| inf3lem1 9539 | Lemma for our Axiom of Inf... |
| inf3lem2 9540 | Lemma for our Axiom of Inf... |
| inf3lem3 9541 | Lemma for our Axiom of Inf... |
| inf3lem4 9542 | Lemma for our Axiom of Inf... |
| inf3lem5 9543 | Lemma for our Axiom of Inf... |
| inf3lem6 9544 | Lemma for our Axiom of Inf... |
| inf3lem7 9545 | Lemma for our Axiom of Inf... |
| inf3 9546 | Our Axiom of Infinity ~ ax... |
| infeq5i 9547 | Half of ~ infeq5 . (Contr... |
| infeq5 9548 | The statement "there exist... |
| zfinf 9550 | Axiom of Infinity expresse... |
| axinf2 9551 | A standard version of Axio... |
| zfinf2 9553 | A standard version of the ... |
| omex 9554 | The existence of omega (th... |
| axinf 9555 | The first version of the A... |
| inf5 9556 | The statement "there exist... |
| omelon 9557 | Omega is an ordinal number... |
| dfom3 9558 | The class of natural numbe... |
| elom3 9559 | A simplification of ~ elom... |
| dfom4 9560 | A simplification of ~ df-o... |
| dfom5 9561 | ` _om ` is the smallest li... |
| oancom 9562 | Ordinal addition is not co... |
| isfinite 9563 | A set is finite iff it is ... |
| fict 9564 | A finite set is countable ... |
| nnsdom 9565 | A natural number is strict... |
| omenps 9566 | Omega is equinumerous to a... |
| omensuc 9567 | The set of natural numbers... |
| infdifsn 9568 | Removing a singleton from ... |
| infdiffi 9569 | Removing a finite set from... |
| unbnn3 9570 | Any unbounded subset of na... |
| noinfep 9571 | Using the Axiom of Regular... |
| cantnffval 9574 | The value of the Cantor no... |
| cantnfdm 9575 | The domain of the Cantor n... |
| cantnfvalf 9576 | Lemma for ~ cantnf . The ... |
| cantnfs 9577 | Elementhood in the set of ... |
| cantnfcl 9578 | Basic properties of the or... |
| cantnfval 9579 | The value of the Cantor no... |
| cantnfval2 9580 | Alternate expression for t... |
| cantnfsuc 9581 | The value of the recursive... |
| cantnfle 9582 | A lower bound on the ` CNF... |
| cantnflt 9583 | An upper bound on the part... |
| cantnflt2 9584 | An upper bound on the ` CN... |
| cantnff 9585 | The ` CNF ` function is a ... |
| cantnf0 9586 | The value of the zero func... |
| cantnfrescl 9587 | A function is finitely sup... |
| cantnfres 9588 | The ` CNF ` function respe... |
| cantnfp1lem1 9589 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9590 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9591 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9592 | If ` F ` is created by add... |
| oemapso 9593 | The relation ` T ` is a st... |
| oemapval 9594 | Value of the relation ` T ... |
| oemapvali 9595 | If ` F < G ` , then there ... |
| cantnflem1a 9596 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9597 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9598 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9599 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9600 | Lemma for ~ cantnf . This... |
| cantnflem2 9601 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9602 | Lemma for ~ cantnf . Here... |
| cantnflem4 9603 | Lemma for ~ cantnf . Comp... |
| cantnf 9604 | The Cantor Normal Form the... |
| oemapwe 9605 | The lexicographic order on... |
| cantnffval2 9606 | An alternate definition of... |
| cantnff1o 9607 | Simplify the isomorphism o... |
| wemapwe 9608 | Construct lexicographic or... |
| oef1o 9609 | A bijection of the base se... |
| cnfcomlem 9610 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9611 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9612 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9613 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9614 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9615 | Any infinite ordinal ` B `... |
| cnfcom3clem 9616 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9617 | Wrap the construction of ~... |
| ttrcleq 9620 | Equality theorem for trans... |
| nfttrcld 9621 | Bound variable hypothesis ... |
| nfttrcl 9622 | Bound variable hypothesis ... |
| relttrcl 9623 | The transitive closure of ... |
| brttrcl 9624 | Characterization of elemen... |
| brttrcl2 9625 | Characterization of elemen... |
| ssttrcl 9626 | If ` R ` is a relation, th... |
| ttrcltr 9627 | The transitive closure of ... |
| ttrclresv 9628 | The transitive closure of ... |
| ttrclco 9629 | Composition law for the tr... |
| cottrcl 9630 | Composition law for the tr... |
| ttrclss 9631 | If ` R ` is a subclass of ... |
| dmttrcl 9632 | The domain of a transitive... |
| rnttrcl 9633 | The range of a transitive ... |
| ttrclexg 9634 | If ` R ` is a set, then so... |
| dfttrcl2 9635 | When ` R ` is a set and a ... |
| ttrclselem1 9636 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9637 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9638 | If ` R ` is set-like over ... |
| trcl 9639 | For any set ` A ` , show t... |
| tz9.1 9640 | Every set has a transitive... |
| tz9.1c 9641 | Alternate expression for t... |
| epfrs 9642 | The strong form of the Axi... |
| zfregs 9643 | The strong form of the Axi... |
| zfregs2 9644 | Alternate strong form of t... |
| tcvalg 9647 | Value of the transitive cl... |
| tcid 9648 | Defining property of the t... |
| tctr 9649 | Defining property of the t... |
| tcmin 9650 | Defining property of the t... |
| tc2 9651 | A variant of the definitio... |
| tcsni 9652 | The transitive closure of ... |
| tcss 9653 | The transitive closure fun... |
| tcel 9654 | The transitive closure fun... |
| tcidm 9655 | The transitive closure fun... |
| tc0 9656 | The transitive closure of ... |
| tc00 9657 | The transitive closure is ... |
| setind 9658 | Set (epsilon) induction. ... |
| setind2 9659 | Set (epsilon) induction, s... |
| setinds 9660 | Principle of set induction... |
| setinds2f 9661 | ` _E ` induction schema, u... |
| setinds2 9662 | ` _E ` induction schema, u... |
| frmin 9663 | Every (possibly proper) su... |
| frind 9664 | A subclass of a well-found... |
| frinsg 9665 | Well-Founded Induction Sch... |
| frins 9666 | Well-Founded Induction Sch... |
| frins2f 9667 | Well-Founded Induction sch... |
| frins2 9668 | Well-Founded Induction sch... |
| frins3 9669 | Well-Founded Induction sch... |
| frr3g 9670 | Functions defined by well-... |
| frrlem15 9671 | Lemma for general well-fou... |
| frrlem16 9672 | Lemma for general well-fou... |
| frr1 9673 | Law of general well-founde... |
| frr2 9674 | Law of general well-founde... |
| frr3 9675 | Law of general well-founde... |
| r1funlim 9680 | The cumulative hierarchy o... |
| r1fnon 9681 | The cumulative hierarchy o... |
| r10 9682 | Value of the cumulative hi... |
| r1sucg 9683 | Value of the cumulative hi... |
| r1suc 9684 | Value of the cumulative hi... |
| r1limg 9685 | Value of the cumulative hi... |
| r1lim 9686 | Value of the cumulative hi... |
| r1fin 9687 | The first ` _om ` levels o... |
| r1sdom 9688 | Each stage in the cumulati... |
| r111 9689 | The cumulative hierarchy i... |
| r1tr 9690 | The cumulative hierarchy o... |
| r1tr2 9691 | The union of a cumulative ... |
| r1ordg 9692 | Ordering relation for the ... |
| r1ord3g 9693 | Ordering relation for the ... |
| r1ord 9694 | Ordering relation for the ... |
| r1ord2 9695 | Ordering relation for the ... |
| r1ord3 9696 | Ordering relation for the ... |
| r1sssuc 9697 | The value of the cumulativ... |
| r1pwss 9698 | Each set of the cumulative... |
| r1sscl 9699 | Each set of the cumulative... |
| r1val1 9700 | The value of the cumulativ... |
| tz9.12lem1 9701 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9702 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9703 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9704 | A set is well-founded if a... |
| tz9.13 9705 | Every set is well-founded,... |
| tz9.13g 9706 | Every set is well-founded,... |
| rankwflemb 9707 | Two ways of saying a set i... |
| rankf 9708 | The domain and codomain of... |
| rankon 9709 | The rank of a set is an or... |
| r1elwf 9710 | Any member of the cumulati... |
| rankvalb 9711 | Value of the rank function... |
| rankr1ai 9712 | One direction of ~ rankr1a... |
| rankvaln 9713 | Value of the rank function... |
| rankidb 9714 | Identity law for the rank ... |
| rankdmr1 9715 | A rank is a member of the ... |
| rankr1ag 9716 | A version of ~ rankr1a tha... |
| rankr1bg 9717 | A relationship between ran... |
| r1rankidb 9718 | Any set is a subset of the... |
| r1elssi 9719 | The range of the ` R1 ` fu... |
| r1elss 9720 | The range of the ` R1 ` fu... |
| pwwf 9721 | A power set is well-founde... |
| sswf 9722 | A subset of a well-founded... |
| snwf 9723 | A singleton is well-founde... |
| unwf 9724 | A binary union is well-fou... |
| prwf 9725 | An unordered pair is well-... |
| opwf 9726 | An ordered pair is well-fo... |
| unir1 9727 | The cumulative hierarchy o... |
| jech9.3 9728 | Every set belongs to some ... |
| rankwflem 9729 | Every set is well-founded,... |
| rankval 9730 | Value of the rank function... |
| rankvalg 9731 | Value of the rank function... |
| rankval2 9732 | Value of an alternate defi... |
| uniwf 9733 | A union is well-founded if... |
| rankr1clem 9734 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9735 | A relationship between the... |
| rankidn 9736 | A relationship between the... |
| rankpwi 9737 | The rank of a power set. ... |
| rankelb 9738 | The membership relation is... |
| wfelirr 9739 | A well-founded set is not ... |
| rankval3b 9740 | The value of the rank func... |
| ranksnb 9741 | The rank of a singleton. ... |
| rankonidlem 9742 | Lemma for ~ rankonid . (C... |
| rankonid 9743 | The rank of an ordinal num... |
| onwf 9744 | The ordinals are all well-... |
| onssr1 9745 | Initial segments of the or... |
| rankr1g 9746 | A relationship between the... |
| rankid 9747 | Identity law for the rank ... |
| rankr1 9748 | A relationship between the... |
| ssrankr1 9749 | A relationship between an ... |
| rankr1a 9750 | A relationship between ran... |
| r1val2 9751 | The value of the cumulativ... |
| r1val3 9752 | The value of the cumulativ... |
| rankel 9753 | The membership relation is... |
| rankval3 9754 | The value of the rank func... |
| bndrank 9755 | Any class whose elements h... |
| unbndrank 9756 | The elements of a proper c... |
| rankpw 9757 | The rank of a power set. ... |
| ranklim 9758 | The rank of a set belongs ... |
| r1pw 9759 | A stronger property of ` R... |
| r1pwALT 9760 | Alternate shorter proof of... |
| r1pwcl 9761 | The cumulative hierarchy o... |
| rankssb 9762 | The subset relation is inh... |
| rankss 9763 | The subset relation is inh... |
| rankunb 9764 | The rank of the union of t... |
| rankprb 9765 | The rank of an unordered p... |
| rankopb 9766 | The rank of an ordered pai... |
| rankuni2b 9767 | The value of the rank func... |
| ranksn 9768 | The rank of a singleton. ... |
| rankuni2 9769 | The rank of a union. Part... |
| rankun 9770 | The rank of the union of t... |
| rankpr 9771 | The rank of an unordered p... |
| rankop 9772 | The rank of an ordered pai... |
| r1rankid 9773 | Any set is a subset of the... |
| rankeq0b 9774 | A set is empty iff its ran... |
| rankeq0 9775 | A set is empty iff its ran... |
| rankr1id 9776 | The rank of the hierarchy ... |
| rankuni 9777 | The rank of a union. Part... |
| rankr1b 9778 | A relationship between ran... |
| ranksuc 9779 | The rank of a successor. ... |
| rankuniss 9780 | Upper bound of the rank of... |
| rankval4 9781 | The rank of a set is the s... |
| rankbnd 9782 | The rank of a set is bound... |
| rankbnd2 9783 | The rank of a set is bound... |
| rankc1 9784 | A relationship that can be... |
| rankc2 9785 | A relationship that can be... |
| rankelun 9786 | Rank membership is inherit... |
| rankelpr 9787 | Rank membership is inherit... |
| rankelop 9788 | Rank membership is inherit... |
| rankxpl 9789 | A lower bound on the rank ... |
| rankxpu 9790 | An upper bound on the rank... |
| rankfu 9791 | An upper bound on the rank... |
| rankmapu 9792 | An upper bound on the rank... |
| rankxplim 9793 | The rank of a Cartesian pr... |
| rankxplim2 9794 | If the rank of a Cartesian... |
| rankxplim3 9795 | The rank of a Cartesian pr... |
| rankxpsuc 9796 | The rank of a Cartesian pr... |
| tcwf 9797 | The transitive closure fun... |
| tcrank 9798 | This theorem expresses two... |
| scottex 9799 | Scott's trick collects all... |
| scott0 9800 | Scott's trick collects all... |
| scottexs 9801 | Theorem scheme version of ... |
| scott0s 9802 | Theorem scheme version of ... |
| cplem1 9803 | Lemma for the Collection P... |
| cplem2 9804 | Lemma for the Collection P... |
| cp 9805 | Collection Principle. Thi... |
| bnd 9806 | A very strong generalizati... |
| bnd2 9807 | A variant of the Boundedne... |
| kardex 9808 | The collection of all sets... |
| karden 9809 | If we allow the Axiom of R... |
| htalem 9810 | Lemma for defining an emul... |
| hta 9811 | A ZFC emulation of Hilbert... |
| djueq12 9818 | Equality theorem for disjo... |
| djueq1 9819 | Equality theorem for disjo... |
| djueq2 9820 | Equality theorem for disjo... |
| nfdju 9821 | Bound-variable hypothesis ... |
| djuex 9822 | The disjoint union of sets... |
| djuexb 9823 | The disjoint union of two ... |
| djulcl 9824 | Left closure of disjoint u... |
| djurcl 9825 | Right closure of disjoint ... |
| djulf1o 9826 | The left injection functio... |
| djurf1o 9827 | The right injection functi... |
| inlresf 9828 | The left injection restric... |
| inlresf1 9829 | The left injection restric... |
| inrresf 9830 | The right injection restri... |
| inrresf1 9831 | The right injection restri... |
| djuin 9832 | The images of any classes ... |
| djur 9833 | A member of a disjoint uni... |
| djuss 9834 | A disjoint union is a subc... |
| djuunxp 9835 | The union of a disjoint un... |
| djuexALT 9836 | Alternate proof of ~ djuex... |
| eldju1st 9837 | The first component of an ... |
| eldju2ndl 9838 | The second component of an... |
| eldju2ndr 9839 | The second component of an... |
| djuun 9840 | The disjoint union of two ... |
| 1stinl 9841 | The first component of the... |
| 2ndinl 9842 | The second component of th... |
| 1stinr 9843 | The first component of the... |
| 2ndinr 9844 | The second component of th... |
| updjudhf 9845 | The mapping of an element ... |
| updjudhcoinlf 9846 | The composition of the map... |
| updjudhcoinrg 9847 | The composition of the map... |
| updjud 9848 | Universal property of the ... |
| cardf2 9857 | The cardinality function i... |
| cardon 9858 | The cardinal number of a s... |
| isnum2 9859 | A way to express well-orde... |
| isnumi 9860 | A set equinumerous to an o... |
| ennum 9861 | Equinumerous sets are equi... |
| finnum 9862 | Every finite set is numera... |
| onenon 9863 | Every ordinal number is nu... |
| tskwe 9864 | A Tarski set is well-order... |
| xpnum 9865 | The cartesian product of n... |
| cardval3 9866 | An alternate definition of... |
| cardid2 9867 | Any numerable set is equin... |
| isnum3 9868 | A set is numerable iff it ... |
| oncardval 9869 | The value of the cardinal ... |
| oncardid 9870 | Any ordinal number is equi... |
| cardonle 9871 | The cardinal of an ordinal... |
| card0 9872 | The cardinality of the emp... |
| cardidm 9873 | The cardinality function i... |
| oncard 9874 | A set is a cardinal number... |
| ficardom 9875 | The cardinal number of a f... |
| ficardid 9876 | A finite set is equinumero... |
| cardnn 9877 | The cardinality of a natur... |
| cardnueq0 9878 | The empty set is the only ... |
| cardne 9879 | No member of a cardinal nu... |
| carden2a 9880 | If two sets have equal non... |
| carden2b 9881 | If two sets are equinumero... |
| card1 9882 | A set has cardinality one ... |
| cardsn 9883 | A singleton has cardinalit... |
| carddomi2 9884 | Two sets have the dominanc... |
| sdomsdomcardi 9885 | A set strictly dominates i... |
| cardlim 9886 | An infinite cardinal is a ... |
| cardsdomelir 9887 | A cardinal strictly domina... |
| cardsdomel 9888 | A cardinal strictly domina... |
| iscard 9889 | Two ways to express the pr... |
| iscard2 9890 | Two ways to express the pr... |
| carddom2 9891 | Two numerable sets have th... |
| harcard 9892 | The class of ordinal numbe... |
| cardprclem 9893 | Lemma for ~ cardprc . (Co... |
| cardprc 9894 | The class of all cardinal ... |
| carduni 9895 | The union of a set of card... |
| cardiun 9896 | The indexed union of a set... |
| cardennn 9897 | If ` A ` is equinumerous t... |
| cardsucinf 9898 | The cardinality of the suc... |
| cardsucnn 9899 | The cardinality of the suc... |
| cardom 9900 | The set of natural numbers... |
| carden2 9901 | Two numerable sets are equ... |
| cardsdom2 9902 | A numerable set is strictl... |
| domtri2 9903 | Trichotomy of dominance fo... |
| nnsdomel 9904 | Strict dominance and eleme... |
| cardval2 9905 | An alternate version of th... |
| isinffi 9906 | An infinite set contains s... |
| fidomtri 9907 | Trichotomy of dominance wi... |
| fidomtri2 9908 | Trichotomy of dominance wi... |
| harsdom 9909 | The Hartogs number of a we... |
| onsdom 9910 | Any well-orderable set is ... |
| harval2 9911 | An alternate expression fo... |
| harsucnn 9912 | The next cardinal after a ... |
| cardmin2 9913 | The smallest ordinal that ... |
| pm54.43lem 9914 | In Theorem *54.43 of [Whit... |
| pm54.43 9915 | Theorem *54.43 of [Whitehe... |
| enpr2 9916 | An unordered pair with dis... |
| pr2ne 9917 | If an unordered pair has t... |
| prdom2 9918 | An unordered pair has at m... |
| en2eqpr 9919 | Building a set with two el... |
| en2eleq 9920 | Express a set of pair card... |
| en2other2 9921 | Taking the other element t... |
| dif1card 9922 | The cardinality of a nonem... |
| leweon 9923 | Lexicographical order is a... |
| r0weon 9924 | A set-like well-ordering o... |
| infxpenlem 9925 | Lemma for ~ infxpen . (Co... |
| infxpen 9926 | Every infinite ordinal is ... |
| xpomen 9927 | The Cartesian product of o... |
| xpct 9928 | The cartesian product of t... |
| infxpidm2 9929 | Every infinite well-ordera... |
| infxpenc 9930 | A canonical version of ~ i... |
| infxpenc2lem1 9931 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9932 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9933 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9934 | Existence form of ~ infxpe... |
| iunmapdisj 9935 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9936 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9937 | Lemma for ~ fseqen . (Con... |
| fseqdom 9938 | One half of ~ fseqen . (C... |
| fseqen 9939 | A set that is equinumerous... |
| infpwfidom 9940 | The collection of finite s... |
| dfac8alem 9941 | Lemma for ~ dfac8a . If t... |
| dfac8a 9942 | Numeration theorem: every ... |
| dfac8b 9943 | The well-ordering theorem:... |
| dfac8clem 9944 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9945 | If the union of a set is w... |
| ac10ct 9946 | A proof of the well-orderi... |
| ween 9947 | A set is numerable iff it ... |
| ac5num 9948 | A version of ~ ac5b with t... |
| ondomen 9949 | If a set is dominated by a... |
| numdom 9950 | A set dominated by a numer... |
| ssnum 9951 | A subset of a numerable se... |
| onssnum 9952 | All subsets of the ordinal... |
| indcardi 9953 | Indirect strong induction ... |
| acnrcl 9954 | Reverse closure for the ch... |
| acneq 9955 | Equality theorem for the c... |
| isacn 9956 | The property of being a ch... |
| acni 9957 | The property of being a ch... |
| acni2 9958 | The property of being a ch... |
| acni3 9959 | The property of being a ch... |
| acnlem 9960 | Construct a mapping satisf... |
| numacn 9961 | A well-orderable set has c... |
| finacn 9962 | Every set has finite choic... |
| acndom 9963 | A set with long choice seq... |
| acnnum 9964 | A set ` X ` which has choi... |
| acnen 9965 | The class of choice sets o... |
| acndom2 9966 | A set smaller than one wit... |
| acnen2 9967 | The class of sets with cho... |
| fodomacn 9968 | A version of ~ fodom that ... |
| fodomnum 9969 | A version of ~ fodom that ... |
| fonum 9970 | A surjection maps numerabl... |
| numwdom 9971 | A surjection maps numerabl... |
| fodomfi2 9972 | Onto functions define domi... |
| wdomfil 9973 | Weak dominance agrees with... |
| infpwfien 9974 | Any infinite well-orderabl... |
| inffien 9975 | The set of finite intersec... |
| wdomnumr 9976 | Weak dominance agrees with... |
| alephfnon 9977 | The aleph function is a fu... |
| aleph0 9978 | The first infinite cardina... |
| alephlim 9979 | Value of the aleph functio... |
| alephsuc 9980 | Value of the aleph functio... |
| alephon 9981 | An aleph is an ordinal num... |
| alephcard 9982 | Every aleph is a cardinal ... |
| alephnbtwn 9983 | No cardinal can be sandwic... |
| alephnbtwn2 9984 | No set has equinumerosity ... |
| alephordilem1 9985 | Lemma for ~ alephordi . (... |
| alephordi 9986 | Strict ordering property o... |
| alephord 9987 | Ordering property of the a... |
| alephord2 9988 | Ordering property of the a... |
| alephord2i 9989 | Ordering property of the a... |
| alephord3 9990 | Ordering property of the a... |
| alephsucdom 9991 | A set dominated by an alep... |
| alephsuc2 9992 | An alternate representatio... |
| alephdom 9993 | Relationship between inclu... |
| alephgeom 9994 | Every aleph is greater tha... |
| alephislim 9995 | Every aleph is a limit ord... |
| aleph11 9996 | The aleph function is one-... |
| alephf1 9997 | The aleph function is a on... |
| alephsdom 9998 | If an ordinal is smaller t... |
| alephdom2 9999 | A dominated initial ordina... |
| alephle 10000 | The argument of the aleph ... |
| cardaleph 10001 | Given any transfinite card... |
| cardalephex 10002 | Every transfinite cardinal... |
| infenaleph 10003 | An infinite numerable set ... |
| isinfcard 10004 | Two ways to express the pr... |
| iscard3 10005 | Two ways to express the pr... |
| cardnum 10006 | Two ways to express the cl... |
| alephinit 10007 | An infinite initial ordina... |
| carduniima 10008 | The union of the image of ... |
| cardinfima 10009 | If a mapping to cardinals ... |
| alephiso 10010 | Aleph is an order isomorph... |
| alephprc 10011 | The class of all transfini... |
| alephsson 10012 | The class of transfinite c... |
| unialeph 10013 | The union of the class of ... |
| alephsmo 10014 | The aleph function is stri... |
| alephf1ALT 10015 | Alternate proof of ~ aleph... |
| alephfplem1 10016 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10017 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10018 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10019 | Lemma for ~ alephfp . (Co... |
| alephfp 10020 | The aleph function has a f... |
| alephfp2 10021 | The aleph function has at ... |
| alephval3 10022 | An alternate way to expres... |
| alephsucpw2 10023 | The power set of an aleph ... |
| mappwen 10024 | Power rule for cardinal ar... |
| finnisoeu 10025 | A finite totally ordered s... |
| iunfictbso 10026 | Countability of a countabl... |
| aceq1 10029 | Equivalence of two version... |
| aceq0 10030 | Equivalence of two version... |
| aceq2 10031 | Equivalence of two version... |
| aceq3lem 10032 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10033 | Equivalence of two version... |
| dfac4 10034 | Equivalence of two version... |
| dfac5lem1 10035 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10036 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10037 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10038 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10039 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4OLD 10040 | Obsolete version of ~ dfac... |
| dfac5 10041 | Equivalence of two version... |
| dfac2a 10042 | Our Axiom of Choice (in th... |
| dfac2b 10043 | Axiom of Choice (first for... |
| dfac2 10044 | Axiom of Choice (first for... |
| dfac7 10045 | Equivalence of the Axiom o... |
| dfac0 10046 | Equivalence of two version... |
| dfac1 10047 | Equivalence of two version... |
| dfac8 10048 | A proof of the equivalency... |
| dfac9 10049 | Equivalence of the axiom o... |
| dfac10 10050 | Axiom of Choice equivalent... |
| dfac10c 10051 | Axiom of Choice equivalent... |
| dfac10b 10052 | Axiom of Choice equivalent... |
| acacni 10053 | A choice equivalent: every... |
| dfacacn 10054 | A choice equivalent: every... |
| dfac13 10055 | The axiom of choice holds ... |
| dfac12lem1 10056 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10057 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10058 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10059 | The axiom of choice holds ... |
| dfac12k 10060 | Equivalence of ~ dfac12 an... |
| dfac12a 10061 | The axiom of choice holds ... |
| dfac12 10062 | The axiom of choice holds ... |
| kmlem1 10063 | Lemma for 5-quantifier AC ... |
| kmlem2 10064 | Lemma for 5-quantifier AC ... |
| kmlem3 10065 | Lemma for 5-quantifier AC ... |
| kmlem4 10066 | Lemma for 5-quantifier AC ... |
| kmlem5 10067 | Lemma for 5-quantifier AC ... |
| kmlem6 10068 | Lemma for 5-quantifier AC ... |
| kmlem7 10069 | Lemma for 5-quantifier AC ... |
| kmlem8 10070 | Lemma for 5-quantifier AC ... |
| kmlem9 10071 | Lemma for 5-quantifier AC ... |
| kmlem10 10072 | Lemma for 5-quantifier AC ... |
| kmlem11 10073 | Lemma for 5-quantifier AC ... |
| kmlem12 10074 | Lemma for 5-quantifier AC ... |
| kmlem13 10075 | Lemma for 5-quantifier AC ... |
| kmlem14 10076 | Lemma for 5-quantifier AC ... |
| kmlem15 10077 | Lemma for 5-quantifier AC ... |
| kmlem16 10078 | Lemma for 5-quantifier AC ... |
| dfackm 10079 | Equivalence of the Axiom o... |
| undjudom 10080 | Cardinal addition dominate... |
| endjudisj 10081 | Equinumerosity of a disjoi... |
| djuen 10082 | Disjoint unions of equinum... |
| djuenun 10083 | Disjoint union is equinume... |
| dju1en 10084 | Cardinal addition with car... |
| dju1dif 10085 | Adding and subtracting one... |
| dju1p1e2 10086 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10087 | Alternate proof of ~ dju1p... |
| dju0en 10088 | Cardinal addition with car... |
| xp2dju 10089 | Two times a cardinal numbe... |
| djucomen 10090 | Commutative law for cardin... |
| djuassen 10091 | Associative law for cardin... |
| xpdjuen 10092 | Cardinal multiplication di... |
| mapdjuen 10093 | Sum of exponents law for c... |
| pwdjuen 10094 | Sum of exponents law for c... |
| djudom1 10095 | Ordering law for cardinal ... |
| djudom2 10096 | Ordering law for cardinal ... |
| djudoml 10097 | A set is dominated by its ... |
| djuxpdom 10098 | Cartesian product dominate... |
| djufi 10099 | The disjoint union of two ... |
| cdainflem 10100 | Any partition of omega int... |
| djuinf 10101 | A set is infinite iff the ... |
| infdju1 10102 | An infinite set is equinum... |
| pwdju1 10103 | The sum of a powerset with... |
| pwdjuidm 10104 | If the natural numbers inj... |
| djulepw 10105 | If ` A ` is idempotent und... |
| onadju 10106 | The cardinal and ordinal s... |
| cardadju 10107 | The cardinal sum is equinu... |
| djunum 10108 | The disjoint union of two ... |
| unnum 10109 | The union of two numerable... |
| nnadju 10110 | The cardinal and ordinal s... |
| nnadjuALT 10111 | Shorter proof of ~ nnadju ... |
| ficardadju 10112 | The disjoint union of fini... |
| ficardun 10113 | The cardinality of the uni... |
| ficardun2 10114 | The cardinality of the uni... |
| pwsdompw 10115 | Lemma for ~ domtriom . Th... |
| unctb 10116 | The union of two countable... |
| infdjuabs 10117 | Absorption law for additio... |
| infunabs 10118 | An infinite set is equinum... |
| infdju 10119 | The sum of two cardinal nu... |
| infdif 10120 | The cardinality of an infi... |
| infdif2 10121 | Cardinality ordering for a... |
| infxpdom 10122 | Dominance law for multipli... |
| infxpabs 10123 | Absorption law for multipl... |
| infunsdom1 10124 | The union of two sets that... |
| infunsdom 10125 | The union of two sets that... |
| infxp 10126 | Absorption law for multipl... |
| pwdjudom 10127 | A property of dominance ov... |
| infpss 10128 | Every infinite set has an ... |
| infmap2 10129 | An exponentiation law for ... |
| ackbij2lem1 10130 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10131 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10132 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10133 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10134 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10135 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10136 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10137 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10138 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10139 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10140 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10141 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10142 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10143 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10144 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10145 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10146 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10147 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10148 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10149 | The Ackermann bijection, p... |
| ackbij1b 10150 | The Ackermann bijection, p... |
| ackbij2lem2 10151 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10152 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10153 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10154 | The Ackermann bijection, p... |
| r1om 10155 | The set of hereditarily fi... |
| fictb 10156 | A set is countable iff its... |
| cflem 10157 | A lemma used to simplify c... |
| cflemOLD 10158 | Obsolete version of ~ cfle... |
| cfval 10159 | Value of the cofinality fu... |
| cff 10160 | Cofinality is a function o... |
| cfub 10161 | An upper bound on cofinali... |
| cflm 10162 | Value of the cofinality fu... |
| cf0 10163 | Value of the cofinality fu... |
| cardcf 10164 | Cofinality is a cardinal n... |
| cflecard 10165 | Cofinality is bounded by t... |
| cfle 10166 | Cofinality is bounded by i... |
| cfon 10167 | The cofinality of any set ... |
| cfeq0 10168 | Only the ordinal zero has ... |
| cfsuc 10169 | Value of the cofinality fu... |
| cff1 10170 | There is always a map from... |
| cfflb 10171 | If there is a cofinal map ... |
| cfval2 10172 | Another expression for the... |
| coflim 10173 | A simpler expression for t... |
| cflim3 10174 | Another expression for the... |
| cflim2 10175 | The cofinality function is... |
| cfom 10176 | Value of the cofinality fu... |
| cfss 10177 | There is a cofinal subset ... |
| cfslb 10178 | Any cofinal subset of ` A ... |
| cfslbn 10179 | Any subset of ` A ` smalle... |
| cfslb2n 10180 | Any small collection of sm... |
| cofsmo 10181 | Any cofinal map implies th... |
| cfsmolem 10182 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10183 | The map in ~ cff1 can be a... |
| cfcoflem 10184 | Lemma for ~ cfcof , showin... |
| coftr 10185 | If there is a cofinal map ... |
| cfcof 10186 | If there is a cofinal map ... |
| cfidm 10187 | The cofinality function is... |
| alephsing 10188 | The cofinality of a limit ... |
| sornom 10189 | The range of a single-step... |
| isfin1a 10204 | Definition of a Ia-finite ... |
| fin1ai 10205 | Property of a Ia-finite se... |
| isfin2 10206 | Definition of a II-finite ... |
| fin2i 10207 | Property of a II-finite se... |
| isfin3 10208 | Definition of a III-finite... |
| isfin4 10209 | Definition of a IV-finite ... |
| fin4i 10210 | Infer that a set is IV-inf... |
| isfin5 10211 | Definition of a V-finite s... |
| isfin6 10212 | Definition of a VI-finite ... |
| isfin7 10213 | Definition of a VII-finite... |
| sdom2en01 10214 | A set with less than two e... |
| infpssrlem1 10215 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10216 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10217 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10218 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10219 | Lemma for ~ infpssr . (Co... |
| infpssr 10220 | Dedekind infinity implies ... |
| fin4en1 10221 | Dedekind finite is a cardi... |
| ssfin4 10222 | Dedekind finite sets have ... |
| domfin4 10223 | A set dominated by a Dedek... |
| ominf4 10224 | ` _om ` is Dedekind infini... |
| infpssALT 10225 | Alternate proof of ~ infps... |
| isfin4-2 10226 | Alternate definition of IV... |
| isfin4p1 10227 | Alternate definition of IV... |
| fin23lem7 10228 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10229 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10230 | A II-finite set contains m... |
| isfin2-2 10231 | ` Fin2 ` expressed in term... |
| ssfin2 10232 | A subset of a II-finite se... |
| enfin2i 10233 | II-finiteness is a cardina... |
| fin23lem24 10234 | Lemma for ~ fin23 . In a ... |
| fincssdom 10235 | In a chain of finite sets,... |
| fin23lem25 10236 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10237 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10238 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10239 | Lemma for ~ fin23 but coul... |
| fin23lem27 10240 | The mapping constructed in... |
| isfin3ds 10241 | Property of a III-finite s... |
| ssfin3ds 10242 | A subset of a III-finite s... |
| fin23lem12 10243 | The beginning of the proof... |
| fin23lem13 10244 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10245 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10246 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10247 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10248 | Lemma for ~ fin23 . The f... |
| fin23lem20 10249 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10250 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10251 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10252 | Lemma for ~ fin23 . The r... |
| fin23lem29 10253 | Lemma for ~ fin23 . The r... |
| fin23lem30 10254 | Lemma for ~ fin23 . The r... |
| fin23lem31 10255 | Lemma for ~ fin23 . The r... |
| fin23lem32 10256 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10257 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10258 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10259 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10260 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10261 | Lemma for ~ fin23 . The c... |
| fin23lem39 10262 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10263 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10264 | Lemma for ~ fin23 . A set... |
| isf32lem1 10265 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10266 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10267 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10268 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10269 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10270 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10271 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10272 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10273 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10274 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10275 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10276 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10277 | One half of ~ isfin3-2 . ... |
| isf33lem 10278 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10279 | Weakly Dedekind-infinite s... |
| isfin3-3 10280 | Weakly Dedekind-infinite s... |
| fin33i 10281 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10282 | Lemma for ~ compsscnv . (... |
| compsscnv 10283 | Complementation on a power... |
| isf34lem1 10284 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10285 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10286 | Complementation is an anti... |
| isf34lem3 10287 | Lemma for ~ isfin3-4 . (C... |
| compss 10288 | Express image under of the... |
| isf34lem4 10289 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10290 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10291 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10292 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10293 | Inference from ~ isfin3-4 ... |
| isfin3-4 10294 | Weakly Dedekind-infinite s... |
| fin11a 10295 | Every I-finite set is Ia-f... |
| enfin1ai 10296 | Ia-finiteness is a cardina... |
| isfin1-2 10297 | A set is finite in the usu... |
| isfin1-3 10298 | A set is I-finite iff ever... |
| isfin1-4 10299 | A set is I-finite iff ever... |
| dffin1-5 10300 | Compact quantifier-free ve... |
| fin23 10301 | Every II-finite set (every... |
| fin34 10302 | Every III-finite set is IV... |
| isfin5-2 10303 | Alternate definition of V-... |
| fin45 10304 | Every IV-finite set is V-f... |
| fin56 10305 | Every V-finite set is VI-f... |
| fin17 10306 | Every I-finite set is VII-... |
| fin67 10307 | Every VI-finite set is VII... |
| isfin7-2 10308 | A set is VII-finite iff it... |
| fin71num 10309 | A well-orderable set is VI... |
| dffin7-2 10310 | Class form of ~ isfin7-2 .... |
| dfacfin7 10311 | Axiom of Choice equivalent... |
| fin1a2lem1 10312 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10313 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10314 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10315 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10316 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10317 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10318 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10319 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10320 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10321 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10322 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10323 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10324 | Lemma for ~ fin1a2 . (Con... |
| fin12 10325 | Weak theorem which skips I... |
| fin1a2s 10326 | An II-infinite set can hav... |
| fin1a2 10327 | Every Ia-finite set is II-... |
| itunifval 10328 | Function value of iterated... |
| itunifn 10329 | Functionality of the itera... |
| ituni0 10330 | A zero-fold iterated union... |
| itunisuc 10331 | Successor iterated union. ... |
| itunitc1 10332 | Each union iterate is a me... |
| itunitc 10333 | The union of all union ite... |
| ituniiun 10334 | Unwrap an iterated union f... |
| hsmexlem7 10335 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10336 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10337 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10338 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10339 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10340 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10341 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10342 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10343 | Lemma for ~ hsmex . (Cont... |
| hsmex 10344 | The collection of heredita... |
| hsmex2 10345 | The set of hereditary size... |
| hsmex3 10346 | The set of hereditary size... |
| axcc2lem 10348 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10349 | A possibly more useful ver... |
| axcc3 10350 | A possibly more useful ver... |
| axcc4 10351 | A version of ~ axcc3 that ... |
| acncc 10352 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10353 | Relax the constraint on ~ ... |
| domtriomlem 10354 | Lemma for ~ domtriom . (C... |
| domtriom 10355 | Trichotomy of equinumerosi... |
| fin41 10356 | Under countable choice, th... |
| dominf 10357 | A nonempty set that is a s... |
| dcomex 10359 | The Axiom of Dependent Cho... |
| axdc2lem 10360 | Lemma for ~ axdc2 . We co... |
| axdc2 10361 | An apparent strengthening ... |
| axdc3lem 10362 | The class ` S ` of finite ... |
| axdc3lem2 10363 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10364 | Simple substitution lemma ... |
| axdc3lem4 10365 | Lemma for ~ axdc3 . We ha... |
| axdc3 10366 | Dependent Choice. Axiom D... |
| axdc4lem 10367 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10368 | A more general version of ... |
| axcclem 10369 | Lemma for ~ axcc . (Contr... |
| axcc 10370 | Although CC can be proven ... |
| zfac 10372 | Axiom of Choice expressed ... |
| ac2 10373 | Axiom of Choice equivalent... |
| ac3 10374 | Axiom of Choice using abbr... |
| axac3 10376 | This theorem asserts that ... |
| ackm 10377 | A remarkable equivalent to... |
| axac2 10378 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10379 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10380 | Apply a choice equivalent.... |
| cardeqv 10381 | All sets are well-orderabl... |
| numth3 10382 | All sets are well-orderabl... |
| numth2 10383 | Numeration theorem: any se... |
| numth 10384 | Numeration theorem: every ... |
| ac7 10385 | An Axiom of Choice equival... |
| ac7g 10386 | An Axiom of Choice equival... |
| ac4 10387 | Equivalent of Axiom of Cho... |
| ac4c 10388 | Equivalent of Axiom of Cho... |
| ac5 10389 | An Axiom of Choice equival... |
| ac5b 10390 | Equivalent of Axiom of Cho... |
| ac6num 10391 | A version of ~ ac6 which t... |
| ac6 10392 | Equivalent of Axiom of Cho... |
| ac6c4 10393 | Equivalent of Axiom of Cho... |
| ac6c5 10394 | Equivalent of Axiom of Cho... |
| ac9 10395 | An Axiom of Choice equival... |
| ac6s 10396 | Equivalent of Axiom of Cho... |
| ac6n 10397 | Equivalent of Axiom of Cho... |
| ac6s2 10398 | Generalization of the Axio... |
| ac6s3 10399 | Generalization of the Axio... |
| ac6sg 10400 | ~ ac6s with sethood as ant... |
| ac6sf 10401 | Version of ~ ac6 with boun... |
| ac6s4 10402 | Generalization of the Axio... |
| ac6s5 10403 | Generalization of the Axio... |
| ac8 10404 | An Axiom of Choice equival... |
| ac9s 10405 | An Axiom of Choice equival... |
| numthcor 10406 | Any set is strictly domina... |
| weth 10407 | Well-ordering theorem: any... |
| zorn2lem1 10408 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10409 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10410 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10411 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10412 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10413 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10414 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10415 | Zorn's Lemma of [Monk1] p.... |
| zorng 10416 | Zorn's Lemma. If the unio... |
| zornn0g 10417 | Variant of Zorn's lemma ~ ... |
| zorn2 10418 | Zorn's Lemma of [Monk1] p.... |
| zorn 10419 | Zorn's Lemma. If the unio... |
| zornn0 10420 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10421 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10422 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10423 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10424 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10425 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10426 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10427 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10428 | The Teichmüller-Tukey... |
| ttukeyg 10429 | The Teichmüller-Tukey... |
| ttukey 10430 | The Teichmüller-Tukey... |
| axdclem 10431 | Lemma for ~ axdc . (Contr... |
| axdclem2 10432 | Lemma for ~ axdc . Using ... |
| axdc 10433 | This theorem derives ~ ax-... |
| fodomg 10434 | An onto function implies d... |
| fodom 10435 | An onto function implies d... |
| dmct 10436 | The domain of a countable ... |
| rnct 10437 | The range of a countable s... |
| fodomb 10438 | Equivalence of an onto map... |
| wdomac 10439 | When assuming AC, weak and... |
| brdom3 10440 | Equivalence to a dominance... |
| brdom5 10441 | An equivalence to a domina... |
| brdom4 10442 | An equivalence to a domina... |
| brdom7disj 10443 | An equivalence to a domina... |
| brdom6disj 10444 | An equivalence to a domina... |
| fin71ac 10445 | Once we allow AC, the "str... |
| imadomg 10446 | An image of a function und... |
| fimact 10447 | The image by a function of... |
| fnrndomg 10448 | The range of a function is... |
| fnct 10449 | If the domain of a functio... |
| mptct 10450 | A countable mapping set is... |
| iunfo 10451 | Existence of an onto funct... |
| iundom2g 10452 | An upper bound for the car... |
| iundomg 10453 | An upper bound for the car... |
| iundom 10454 | An upper bound for the car... |
| unidom 10455 | An upper bound for the car... |
| uniimadom 10456 | An upper bound for the car... |
| uniimadomf 10457 | An upper bound for the car... |
| cardval 10458 | The value of the cardinal ... |
| cardid 10459 | Any set is equinumerous to... |
| cardidg 10460 | Any set is equinumerous to... |
| cardidd 10461 | Any set is equinumerous to... |
| cardf 10462 | The cardinality function i... |
| carden 10463 | Two sets are equinumerous ... |
| cardeq0 10464 | Only the empty set has car... |
| unsnen 10465 | Equinumerosity of a set wi... |
| carddom 10466 | Two sets have the dominanc... |
| cardsdom 10467 | Two sets have the strict d... |
| domtri 10468 | Trichotomy law for dominan... |
| entric 10469 | Trichotomy of equinumerosi... |
| entri2 10470 | Trichotomy of dominance an... |
| entri3 10471 | Trichotomy of dominance. ... |
| sdomsdomcard 10472 | A set strictly dominates i... |
| canth3 10473 | Cantor's theorem in terms ... |
| infxpidm 10474 | Every infinite class is eq... |
| ondomon 10475 | The class of ordinals domi... |
| cardmin 10476 | The smallest ordinal that ... |
| ficard 10477 | A set is finite iff its ca... |
| infinfg 10478 | Equivalence between two in... |
| infinf 10479 | Equivalence between two in... |
| unirnfdomd 10480 | The union of the range of ... |
| konigthlem 10481 | Lemma for ~ konigth . (Co... |
| konigth 10482 | Konig's Theorem. If ` m (... |
| alephsucpw 10483 | The power set of an aleph ... |
| aleph1 10484 | The set exponentiation of ... |
| alephval2 10485 | An alternate way to expres... |
| dominfac 10486 | A nonempty set that is a s... |
| iunctb 10487 | The countable union of cou... |
| unictb 10488 | The countable union of cou... |
| infmap 10489 | An exponentiation law for ... |
| alephadd 10490 | The sum of two alephs is t... |
| alephmul 10491 | The product of two alephs ... |
| alephexp1 10492 | An exponentiation law for ... |
| alephsuc3 10493 | An alternate representatio... |
| alephexp2 10494 | An expression equinumerous... |
| alephreg 10495 | A successor aleph is regul... |
| pwcfsdom 10496 | A corollary of Konig's The... |
| cfpwsdom 10497 | A corollary of Konig's The... |
| alephom 10498 | From ~ canth2 , we know th... |
| smobeth 10499 | The beth function is stric... |
| nd1 10500 | A lemma for proving condit... |
| nd2 10501 | A lemma for proving condit... |
| nd3 10502 | A lemma for proving condit... |
| nd4 10503 | A lemma for proving condit... |
| axextnd 10504 | A version of the Axiom of ... |
| axrepndlem1 10505 | Lemma for the Axiom of Rep... |
| axrepndlem2 10506 | Lemma for the Axiom of Rep... |
| axrepnd 10507 | A version of the Axiom of ... |
| axunndlem1 10508 | Lemma for the Axiom of Uni... |
| axunnd 10509 | A version of the Axiom of ... |
| axpowndlem1 10510 | Lemma for the Axiom of Pow... |
| axpowndlem2 10511 | Lemma for the Axiom of Pow... |
| axpowndlem3 10512 | Lemma for the Axiom of Pow... |
| axpowndlem4 10513 | Lemma for the Axiom of Pow... |
| axpownd 10514 | A version of the Axiom of ... |
| axregndlem1 10515 | Lemma for the Axiom of Reg... |
| axregndlem2 10516 | Lemma for the Axiom of Reg... |
| axregnd 10517 | A version of the Axiom of ... |
| axinfndlem1 10518 | Lemma for the Axiom of Inf... |
| axinfnd 10519 | A version of the Axiom of ... |
| axacndlem1 10520 | Lemma for the Axiom of Cho... |
| axacndlem2 10521 | Lemma for the Axiom of Cho... |
| axacndlem3 10522 | Lemma for the Axiom of Cho... |
| axacndlem4 10523 | Lemma for the Axiom of Cho... |
| axacndlem5 10524 | Lemma for the Axiom of Cho... |
| axacnd 10525 | A version of the Axiom of ... |
| zfcndext 10526 | Axiom of Extensionality ~ ... |
| zfcndrep 10527 | Axiom of Replacement ~ ax-... |
| zfcndun 10528 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10529 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10530 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10531 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10532 | Axiom of Choice ~ ax-ac , ... |
| elgch 10535 | Elementhood in the collect... |
| fingch 10536 | A finite set is a GCH-set.... |
| gchi 10537 | The only GCH-sets which ha... |
| gchen1 10538 | If ` A <_ B < ~P A ` , and... |
| gchen2 10539 | If ` A < B <_ ~P A ` , and... |
| gchor 10540 | If ` A <_ B <_ ~P A ` , an... |
| engch 10541 | The property of being a GC... |
| gchdomtri 10542 | Under certain conditions, ... |
| fpwwe2cbv 10543 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10544 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10545 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10546 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10547 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10548 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10549 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10550 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10551 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10552 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10553 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10554 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10555 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10556 | Given any function ` F ` f... |
| fpwwecbv 10557 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10558 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10559 | Given any function ` F ` f... |
| canth4 10560 | An "effective" form of Can... |
| canthnumlem 10561 | Lemma for ~ canthnum . (C... |
| canthnum 10562 | The set of well-orderable ... |
| canthwelem 10563 | Lemma for ~ canthwe . (Co... |
| canthwe 10564 | The set of well-orders of ... |
| canthp1lem1 10565 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10566 | Lemma for ~ canthp1 . (Co... |
| canthp1 10567 | A slightly stronger form o... |
| finngch 10568 | The exclusion of finite se... |
| gchdju1 10569 | An infinite GCH-set is ide... |
| gchinf 10570 | An infinite GCH-set is Ded... |
| pwfseqlem1 10571 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10572 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10573 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10574 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10575 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10576 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10577 | The powerset of a Dedekind... |
| pwxpndom2 10578 | The powerset of a Dedekind... |
| pwxpndom 10579 | The powerset of a Dedekind... |
| pwdjundom 10580 | The powerset of a Dedekind... |
| gchdjuidm 10581 | An infinite GCH-set is ide... |
| gchxpidm 10582 | An infinite GCH-set is ide... |
| gchpwdom 10583 | A relationship between dom... |
| gchaleph 10584 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10585 | If ` ( aleph `` A ) ` and ... |
| hargch 10586 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10587 | If ` ( aleph `` suc A ) ` ... |
| gch2 10588 | It is sufficient to requir... |
| gch3 10589 | An equivalent formulation ... |
| gch-kn 10590 | The equivalence of two ver... |
| gchaclem 10591 | Lemma for ~ gchac (obsolet... |
| gchhar 10592 | A "local" form of ~ gchac ... |
| gchacg 10593 | A "local" form of ~ gchac ... |
| gchac 10594 | The Generalized Continuum ... |
| elwina 10599 | Conditions of weak inacces... |
| elina 10600 | Conditions of strong inacc... |
| winaon 10601 | A weakly inaccessible card... |
| inawinalem 10602 | Lemma for ~ inawina . (Co... |
| inawina 10603 | Every strongly inaccessibl... |
| omina 10604 | ` _om ` is a strongly inac... |
| winacard 10605 | A weakly inaccessible card... |
| winainflem 10606 | A weakly inaccessible card... |
| winainf 10607 | A weakly inaccessible card... |
| winalim 10608 | A weakly inaccessible card... |
| winalim2 10609 | A nontrivial weakly inacce... |
| winafp 10610 | A nontrivial weakly inacce... |
| winafpi 10611 | This theorem, which states... |
| gchina 10612 | Assuming the GCH, weakly a... |
| iswun 10617 | Properties of a weak unive... |
| wuntr 10618 | A weak universe is transit... |
| wununi 10619 | A weak universe is closed ... |
| wunpw 10620 | A weak universe is closed ... |
| wunelss 10621 | The elements of a weak uni... |
| wunpr 10622 | A weak universe is closed ... |
| wunun 10623 | A weak universe is closed ... |
| wuntp 10624 | A weak universe is closed ... |
| wunss 10625 | A weak universe is closed ... |
| wunin 10626 | A weak universe is closed ... |
| wundif 10627 | A weak universe is closed ... |
| wunint 10628 | A weak universe is closed ... |
| wunsn 10629 | A weak universe is closed ... |
| wunsuc 10630 | A weak universe is closed ... |
| wun0 10631 | A weak universe contains t... |
| wunr1om 10632 | A weak universe is infinit... |
| wunom 10633 | A weak universe contains a... |
| wunfi 10634 | A weak universe contains a... |
| wunop 10635 | A weak universe is closed ... |
| wunot 10636 | A weak universe is closed ... |
| wunxp 10637 | A weak universe is closed ... |
| wunpm 10638 | A weak universe is closed ... |
| wunmap 10639 | A weak universe is closed ... |
| wunf 10640 | A weak universe is closed ... |
| wundm 10641 | A weak universe is closed ... |
| wunrn 10642 | A weak universe is closed ... |
| wuncnv 10643 | A weak universe is closed ... |
| wunres 10644 | A weak universe is closed ... |
| wunfv 10645 | A weak universe is closed ... |
| wunco 10646 | A weak universe is closed ... |
| wuntpos 10647 | A weak universe is closed ... |
| intwun 10648 | The intersection of a coll... |
| r1limwun 10649 | Each limit stage in the cu... |
| r1wunlim 10650 | The weak universes in the ... |
| wunex2 10651 | Construct a weak universe ... |
| wunex 10652 | Construct a weak universe ... |
| uniwun 10653 | Every set is contained in ... |
| wunex3 10654 | Construct a weak universe ... |
| wuncval 10655 | Value of the weak universe... |
| wuncid 10656 | The weak universe closure ... |
| wunccl 10657 | The weak universe closure ... |
| wuncss 10658 | The weak universe closure ... |
| wuncidm 10659 | The weak universe closure ... |
| wuncval2 10660 | Our earlier expression for... |
| eltskg 10663 | Properties of a Tarski cla... |
| eltsk2g 10664 | Properties of a Tarski cla... |
| tskpwss 10665 | First axiom of a Tarski cl... |
| tskpw 10666 | Second axiom of a Tarski c... |
| tsken 10667 | Third axiom of a Tarski cl... |
| 0tsk 10668 | The empty set is a (transi... |
| tsksdom 10669 | An element of a Tarski cla... |
| tskssel 10670 | A part of a Tarski class s... |
| tskss 10671 | The subsets of an element ... |
| tskin 10672 | The intersection of two el... |
| tsksn 10673 | A singleton of an element ... |
| tsktrss 10674 | A transitive element of a ... |
| tsksuc 10675 | If an element of a Tarski ... |
| tsk0 10676 | A nonempty Tarski class co... |
| tsk1 10677 | One is an element of a non... |
| tsk2 10678 | Two is an element of a non... |
| 2domtsk 10679 | If a Tarski class is not e... |
| tskr1om 10680 | A nonempty Tarski class is... |
| tskr1om2 10681 | A nonempty Tarski class co... |
| tskinf 10682 | A nonempty Tarski class is... |
| tskpr 10683 | If ` A ` and ` B ` are mem... |
| tskop 10684 | If ` A ` and ` B ` are mem... |
| tskxpss 10685 | A Cartesian product of two... |
| tskwe2 10686 | A Tarski class is well-ord... |
| inttsk 10687 | The intersection of a coll... |
| inar1 10688 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10689 | Alternate proof of ~ r1om ... |
| rankcf 10690 | Any set must be at least a... |
| inatsk 10691 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10692 | The set of hereditarily fi... |
| tskord 10693 | A Tarski class contains al... |
| tskcard 10694 | An even more direct relati... |
| r1tskina 10695 | There is a direct relation... |
| tskuni 10696 | The union of an element of... |
| tskwun 10697 | A nonempty transitive Tars... |
| tskint 10698 | The intersection of an ele... |
| tskun 10699 | The union of two elements ... |
| tskxp 10700 | The Cartesian product of t... |
| tskmap 10701 | Set exponentiation is an e... |
| tskurn 10702 | A transitive Tarski class ... |
| elgrug 10705 | Properties of a Grothendie... |
| grutr 10706 | A Grothendieck universe is... |
| gruelss 10707 | A Grothendieck universe is... |
| grupw 10708 | A Grothendieck universe co... |
| gruss 10709 | Any subset of an element o... |
| grupr 10710 | A Grothendieck universe co... |
| gruurn 10711 | A Grothendieck universe co... |
| gruiun 10712 | If ` B ( x ) ` is a family... |
| gruuni 10713 | A Grothendieck universe co... |
| grurn 10714 | A Grothendieck universe co... |
| gruima 10715 | A Grothendieck universe co... |
| gruel 10716 | Any element of an element ... |
| grusn 10717 | A Grothendieck universe co... |
| gruop 10718 | A Grothendieck universe co... |
| gruun 10719 | A Grothendieck universe co... |
| gruxp 10720 | A Grothendieck universe co... |
| grumap 10721 | A Grothendieck universe co... |
| gruixp 10722 | A Grothendieck universe co... |
| gruiin 10723 | A Grothendieck universe co... |
| gruf 10724 | A Grothendieck universe co... |
| gruen 10725 | A Grothendieck universe co... |
| gruwun 10726 | A nonempty Grothendieck un... |
| intgru 10727 | The intersection of a fami... |
| ingru 10728 | The intersection of a univ... |
| wfgru 10729 | The wellfounded part of a ... |
| grudomon 10730 | Each ordinal that is compa... |
| gruina 10731 | If a Grothendieck universe... |
| grur1a 10732 | A characterization of Grot... |
| grur1 10733 | A characterization of Grot... |
| grutsk1 10734 | Grothendieck universes are... |
| grutsk 10735 | Grothendieck universes are... |
| axgroth5 10737 | The Tarski-Grothendieck ax... |
| axgroth2 10738 | Alternate version of the T... |
| grothpw 10739 | Derive the Axiom of Power ... |
| grothpwex 10740 | Derive the Axiom of Power ... |
| axgroth6 10741 | The Tarski-Grothendieck ax... |
| grothomex 10742 | The Tarski-Grothendieck Ax... |
| grothac 10743 | The Tarski-Grothendieck Ax... |
| axgroth3 10744 | Alternate version of the T... |
| axgroth4 10745 | Alternate version of the T... |
| grothprimlem 10746 | Lemma for ~ grothprim . E... |
| grothprim 10747 | The Tarski-Grothendieck Ax... |
| grothtsk 10748 | The Tarski-Grothendieck Ax... |
| inaprc 10749 | An equivalent to the Tarsk... |
| tskmval 10752 | Value of our tarski map. ... |
| tskmid 10753 | The set ` A ` is an elemen... |
| tskmcl 10754 | A Tarski class that contai... |
| sstskm 10755 | Being a part of ` ( tarski... |
| eltskm 10756 | Belonging to ` ( tarskiMap... |
| elni 10789 | Membership in the class of... |
| elni2 10790 | Membership in the class of... |
| pinn 10791 | A positive integer is a na... |
| pion 10792 | A positive integer is an o... |
| piord 10793 | A positive integer is ordi... |
| niex 10794 | The class of positive inte... |
| 0npi 10795 | The empty set is not a pos... |
| 1pi 10796 | Ordinal 'one' is a positiv... |
| addpiord 10797 | Positive integer addition ... |
| mulpiord 10798 | Positive integer multiplic... |
| mulidpi 10799 | 1 is an identity element f... |
| ltpiord 10800 | Positive integer 'less tha... |
| ltsopi 10801 | Positive integer 'less tha... |
| ltrelpi 10802 | Positive integer 'less tha... |
| dmaddpi 10803 | Domain of addition on posi... |
| dmmulpi 10804 | Domain of multiplication o... |
| addclpi 10805 | Closure of addition of pos... |
| mulclpi 10806 | Closure of multiplication ... |
| addcompi 10807 | Addition of positive integ... |
| addasspi 10808 | Addition of positive integ... |
| mulcompi 10809 | Multiplication of positive... |
| mulasspi 10810 | Multiplication of positive... |
| distrpi 10811 | Multiplication of positive... |
| addcanpi 10812 | Addition cancellation law ... |
| mulcanpi 10813 | Multiplication cancellatio... |
| addnidpi 10814 | There is no identity eleme... |
| ltexpi 10815 | Ordering on positive integ... |
| ltapi 10816 | Ordering property of addit... |
| ltmpi 10817 | Ordering property of multi... |
| 1lt2pi 10818 | One is less than two (one ... |
| nlt1pi 10819 | No positive integer is les... |
| indpi 10820 | Principle of Finite Induct... |
| enqbreq 10832 | Equivalence relation for p... |
| enqbreq2 10833 | Equivalence relation for p... |
| enqer 10834 | The equivalence relation f... |
| enqex 10835 | The equivalence relation f... |
| nqex 10836 | The class of positive frac... |
| 0nnq 10837 | The empty set is not a pos... |
| elpqn 10838 | Each positive fraction is ... |
| ltrelnq 10839 | Positive fraction 'less th... |
| pinq 10840 | The representatives of pos... |
| 1nq 10841 | The positive fraction 'one... |
| nqereu 10842 | There is a unique element ... |
| nqerf 10843 | Corollary of ~ nqereu : th... |
| nqercl 10844 | Corollary of ~ nqereu : cl... |
| nqerrel 10845 | Any member of ` ( N. X. N.... |
| nqerid 10846 | Corollary of ~ nqereu : th... |
| enqeq 10847 | Corollary of ~ nqereu : if... |
| nqereq 10848 | The function ` /Q ` acts a... |
| addpipq2 10849 | Addition of positive fract... |
| addpipq 10850 | Addition of positive fract... |
| addpqnq 10851 | Addition of positive fract... |
| mulpipq2 10852 | Multiplication of positive... |
| mulpipq 10853 | Multiplication of positive... |
| mulpqnq 10854 | Multiplication of positive... |
| ordpipq 10855 | Ordering of positive fract... |
| ordpinq 10856 | Ordering of positive fract... |
| addpqf 10857 | Closure of addition on pos... |
| addclnq 10858 | Closure of addition on pos... |
| mulpqf 10859 | Closure of multiplication ... |
| mulclnq 10860 | Closure of multiplication ... |
| addnqf 10861 | Domain of addition on posi... |
| mulnqf 10862 | Domain of multiplication o... |
| addcompq 10863 | Addition of positive fract... |
| addcomnq 10864 | Addition of positive fract... |
| mulcompq 10865 | Multiplication of positive... |
| mulcomnq 10866 | Multiplication of positive... |
| adderpqlem 10867 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10868 | Lemma for ~ mulerpq . (Co... |
| adderpq 10869 | Addition is compatible wit... |
| mulerpq 10870 | Multiplication is compatib... |
| addassnq 10871 | Addition of positive fract... |
| mulassnq 10872 | Multiplication of positive... |
| mulcanenq 10873 | Lemma for distributive law... |
| distrnq 10874 | Multiplication of positive... |
| 1nqenq 10875 | The equivalence class of r... |
| mulidnq 10876 | Multiplication identity el... |
| recmulnq 10877 | Relationship between recip... |
| recidnq 10878 | A positive fraction times ... |
| recclnq 10879 | Closure law for positive f... |
| recrecnq 10880 | Reciprocal of reciprocal o... |
| dmrecnq 10881 | Domain of reciprocal on po... |
| ltsonq 10882 | 'Less than' is a strict or... |
| lterpq 10883 | Compatibility of ordering ... |
| ltanq 10884 | Ordering property of addit... |
| ltmnq 10885 | Ordering property of multi... |
| 1lt2nq 10886 | One is less than two (one ... |
| ltaddnq 10887 | The sum of two fractions i... |
| ltexnq 10888 | Ordering on positive fract... |
| halfnq 10889 | One-half of any positive f... |
| nsmallnq 10890 | The is no smallest positiv... |
| ltbtwnnq 10891 | There exists a number betw... |
| ltrnq 10892 | Ordering property of recip... |
| archnq 10893 | For any fraction, there is... |
| npex 10899 | The class of positive real... |
| elnp 10900 | Membership in positive rea... |
| elnpi 10901 | Membership in positive rea... |
| prn0 10902 | A positive real is not emp... |
| prpssnq 10903 | A positive real is a subse... |
| elprnq 10904 | A positive real is a set o... |
| 0npr 10905 | The empty set is not a pos... |
| prcdnq 10906 | A positive real is closed ... |
| prub 10907 | A positive fraction not in... |
| prnmax 10908 | A positive real has no lar... |
| npomex 10909 | A simplifying observation,... |
| prnmadd 10910 | A positive real has no lar... |
| ltrelpr 10911 | Positive real 'less than' ... |
| genpv 10912 | Value of general operation... |
| genpelv 10913 | Membership in value of gen... |
| genpprecl 10914 | Pre-closure law for genera... |
| genpdm 10915 | Domain of general operatio... |
| genpn0 10916 | The result of an operation... |
| genpss 10917 | The result of an operation... |
| genpnnp 10918 | The result of an operation... |
| genpcd 10919 | Downward closure of an ope... |
| genpnmax 10920 | An operation on positive r... |
| genpcl 10921 | Closure of an operation on... |
| genpass 10922 | Associativity of an operat... |
| plpv 10923 | Value of addition on posit... |
| mpv 10924 | Value of multiplication on... |
| dmplp 10925 | Domain of addition on posi... |
| dmmp 10926 | Domain of multiplication o... |
| nqpr 10927 | The canonical embedding of... |
| 1pr 10928 | The positive real number '... |
| addclprlem1 10929 | Lemma to prove downward cl... |
| addclprlem2 10930 | Lemma to prove downward cl... |
| addclpr 10931 | Closure of addition on pos... |
| mulclprlem 10932 | Lemma to prove downward cl... |
| mulclpr 10933 | Closure of multiplication ... |
| addcompr 10934 | Addition of positive reals... |
| addasspr 10935 | Addition of positive reals... |
| mulcompr 10936 | Multiplication of positive... |
| mulasspr 10937 | Multiplication of positive... |
| distrlem1pr 10938 | Lemma for distributive law... |
| distrlem4pr 10939 | Lemma for distributive law... |
| distrlem5pr 10940 | Lemma for distributive law... |
| distrpr 10941 | Multiplication of positive... |
| 1idpr 10942 | 1 is an identity element f... |
| ltprord 10943 | Positive real 'less than' ... |
| psslinpr 10944 | Proper subset is a linear ... |
| ltsopr 10945 | Positive real 'less than' ... |
| prlem934 10946 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10947 | The sum of two positive re... |
| ltaddpr2 10948 | The sum of two positive re... |
| ltexprlem1 10949 | Lemma for Proposition 9-3.... |
| ltexprlem2 10950 | Lemma for Proposition 9-3.... |
| ltexprlem3 10951 | Lemma for Proposition 9-3.... |
| ltexprlem4 10952 | Lemma for Proposition 9-3.... |
| ltexprlem5 10953 | Lemma for Proposition 9-3.... |
| ltexprlem6 10954 | Lemma for Proposition 9-3.... |
| ltexprlem7 10955 | Lemma for Proposition 9-3.... |
| ltexpri 10956 | Proposition 9-3.5(iv) of [... |
| ltaprlem 10957 | Lemma for Proposition 9-3.... |
| ltapr 10958 | Ordering property of addit... |
| addcanpr 10959 | Addition cancellation law ... |
| prlem936 10960 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 10961 | Lemma for Proposition 9-3.... |
| reclem3pr 10962 | Lemma for Proposition 9-3.... |
| reclem4pr 10963 | Lemma for Proposition 9-3.... |
| recexpr 10964 | The reciprocal of a positi... |
| suplem1pr 10965 | The union of a nonempty, b... |
| suplem2pr 10966 | The union of a set of posi... |
| supexpr 10967 | The union of a nonempty, b... |
| enrer 10976 | The equivalence relation f... |
| nrex1 10977 | The class of signed reals ... |
| enrbreq 10978 | Equivalence relation for s... |
| enreceq 10979 | Equivalence class equality... |
| enrex 10980 | The equivalence relation f... |
| ltrelsr 10981 | Signed real 'less than' is... |
| addcmpblnr 10982 | Lemma showing compatibilit... |
| mulcmpblnrlem 10983 | Lemma used in lemma showin... |
| mulcmpblnr 10984 | Lemma showing compatibilit... |
| prsrlem1 10985 | Decomposing signed reals i... |
| addsrmo 10986 | There is at most one resul... |
| mulsrmo 10987 | There is at most one resul... |
| addsrpr 10988 | Addition of signed reals i... |
| mulsrpr 10989 | Multiplication of signed r... |
| ltsrpr 10990 | Ordering of signed reals i... |
| gt0srpr 10991 | Greater than zero in terms... |
| 0nsr 10992 | The empty set is not a sig... |
| 0r 10993 | The constant ` 0R ` is a s... |
| 1sr 10994 | The constant ` 1R ` is a s... |
| m1r 10995 | The constant ` -1R ` is a ... |
| addclsr 10996 | Closure of addition on sig... |
| mulclsr 10997 | Closure of multiplication ... |
| dmaddsr 10998 | Domain of addition on sign... |
| dmmulsr 10999 | Domain of multiplication o... |
| addcomsr 11000 | Addition of signed reals i... |
| addasssr 11001 | Addition of signed reals i... |
| mulcomsr 11002 | Multiplication of signed r... |
| mulasssr 11003 | Multiplication of signed r... |
| distrsr 11004 | Multiplication of signed r... |
| m1p1sr 11005 | Minus one plus one is zero... |
| m1m1sr 11006 | Minus one times minus one ... |
| ltsosr 11007 | Signed real 'less than' is... |
| 0lt1sr 11008 | 0 is less than 1 for signe... |
| 1ne0sr 11009 | 1 and 0 are distinct for s... |
| 0idsr 11010 | The signed real number 0 i... |
| 1idsr 11011 | 1 is an identity element f... |
| 00sr 11012 | A signed real times 0 is 0... |
| ltasr 11013 | Ordering property of addit... |
| pn0sr 11014 | A signed real plus its neg... |
| negexsr 11015 | Existence of negative sign... |
| recexsrlem 11016 | The reciprocal of a positi... |
| addgt0sr 11017 | The sum of two positive si... |
| mulgt0sr 11018 | The product of two positiv... |
| sqgt0sr 11019 | The square of a nonzero si... |
| recexsr 11020 | The reciprocal of a nonzer... |
| mappsrpr 11021 | Mapping from positive sign... |
| ltpsrpr 11022 | Mapping of order from posi... |
| map2psrpr 11023 | Equivalence for positive s... |
| supsrlem 11024 | Lemma for supremum theorem... |
| supsr 11025 | A nonempty, bounded set of... |
| opelcn 11042 | Ordered pair membership in... |
| opelreal 11043 | Ordered pair membership in... |
| elreal 11044 | Membership in class of rea... |
| elreal2 11045 | Ordered pair membership in... |
| 0ncn 11046 | The empty set is not a com... |
| ltrelre 11047 | 'Less than' is a relation ... |
| addcnsr 11048 | Addition of complex number... |
| mulcnsr 11049 | Multiplication of complex ... |
| eqresr 11050 | Equality of real numbers i... |
| addresr 11051 | Addition of real numbers i... |
| mulresr 11052 | Multiplication of real num... |
| ltresr 11053 | Ordering of real subset of... |
| ltresr2 11054 | Ordering of real subset of... |
| dfcnqs 11055 | Technical trick to permit ... |
| addcnsrec 11056 | Technical trick to permit ... |
| mulcnsrec 11057 | Technical trick to permit ... |
| axaddf 11058 | Addition is an operation o... |
| axmulf 11059 | Multiplication is an opera... |
| axcnex 11060 | The complex numbers form a... |
| axresscn 11061 | The real numbers are a sub... |
| ax1cn 11062 | 1 is a complex number. Ax... |
| axicn 11063 | ` _i ` is a complex number... |
| axaddcl 11064 | Closure law for addition o... |
| axaddrcl 11065 | Closure law for addition i... |
| axmulcl 11066 | Closure law for multiplica... |
| axmulrcl 11067 | Closure law for multiplica... |
| axmulcom 11068 | Multiplication of complex ... |
| axaddass 11069 | Addition of complex number... |
| axmulass 11070 | Multiplication of complex ... |
| axdistr 11071 | Distributive law for compl... |
| axi2m1 11072 | i-squared equals -1 (expre... |
| ax1ne0 11073 | 1 and 0 are distinct. Axi... |
| ax1rid 11074 | ` 1 ` is an identity eleme... |
| axrnegex 11075 | Existence of negative of r... |
| axrrecex 11076 | Existence of reciprocal of... |
| axcnre 11077 | A complex number can be ex... |
| axpre-lttri 11078 | Ordering on reals satisfie... |
| axpre-lttrn 11079 | Ordering on reals is trans... |
| axpre-ltadd 11080 | Ordering property of addit... |
| axpre-mulgt0 11081 | The product of two positiv... |
| axpre-sup 11082 | A nonempty, bounded-above ... |
| wuncn 11083 | A weak universe containing... |
| cnex 11109 | Alias for ~ ax-cnex . See... |
| addcl 11110 | Alias for ~ ax-addcl , for... |
| readdcl 11111 | Alias for ~ ax-addrcl , fo... |
| mulcl 11112 | Alias for ~ ax-mulcl , for... |
| remulcl 11113 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11114 | Alias for ~ ax-mulcom , fo... |
| addass 11115 | Alias for ~ ax-addass , fo... |
| mulass 11116 | Alias for ~ ax-mulass , fo... |
| adddi 11117 | Alias for ~ ax-distr , for... |
| recn 11118 | A real number is a complex... |
| reex 11119 | The real numbers form a se... |
| reelprrecn 11120 | Reals are a subset of the ... |
| cnelprrecn 11121 | Complex numbers are a subs... |
| mpoaddf 11122 | Addition is an operation o... |
| mpomulf 11123 | Multiplication is an opera... |
| elimne0 11124 | Hypothesis for weak deduct... |
| adddir 11125 | Distributive law for compl... |
| 0cn 11126 | Zero is a complex number. ... |
| 0cnd 11127 | Zero is a complex number, ... |
| c0ex 11128 | Zero is a set. (Contribut... |
| 1cnd 11129 | One is a complex number, d... |
| 1ex 11130 | One is a set. (Contribute... |
| cnre 11131 | Alias for ~ ax-cnre , for ... |
| mulrid 11132 | The number 1 is an identit... |
| mullid 11133 | Identity law for multiplic... |
| 1re 11134 | The number 1 is real. Thi... |
| 1red 11135 | The number 1 is real, dedu... |
| 0re 11136 | The number 0 is real. Rem... |
| 0red 11137 | The number 0 is real, dedu... |
| mulridi 11138 | Identity law for multiplic... |
| mullidi 11139 | Identity law for multiplic... |
| addcli 11140 | Closure law for addition. ... |
| mulcli 11141 | Closure law for multiplica... |
| mulcomi 11142 | Commutative law for multip... |
| mulcomli 11143 | Commutative law for multip... |
| addassi 11144 | Associative law for additi... |
| mulassi 11145 | Associative law for multip... |
| adddii 11146 | Distributive law (left-dis... |
| adddiri 11147 | Distributive law (right-di... |
| recni 11148 | A real number is a complex... |
| readdcli 11149 | Closure law for addition o... |
| remulcli 11150 | Closure law for multiplica... |
| mulridd 11151 | Identity law for multiplic... |
| mullidd 11152 | Identity law for multiplic... |
| addcld 11153 | Closure law for addition. ... |
| mulcld 11154 | Closure law for multiplica... |
| mulcomd 11155 | Commutative law for multip... |
| addassd 11156 | Associative law for additi... |
| mulassd 11157 | Associative law for multip... |
| adddid 11158 | Distributive law (left-dis... |
| adddird 11159 | Distributive law (right-di... |
| adddirp1d 11160 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11161 | Join AB+CB into (A+C) on L... |
| recnd 11162 | Deduction from real number... |
| readdcld 11163 | Closure law for addition o... |
| remulcld 11164 | Closure law for multiplica... |
| pnfnre 11175 | Plus infinity is not a rea... |
| pnfnre2 11176 | Plus infinity is not a rea... |
| mnfnre 11177 | Minus infinity is not a re... |
| ressxr 11178 | The standard reals are a s... |
| rexpssxrxp 11179 | The Cartesian product of s... |
| rexr 11180 | A standard real is an exte... |
| 0xr 11181 | Zero is an extended real. ... |
| renepnf 11182 | No (finite) real equals pl... |
| renemnf 11183 | No real equals minus infin... |
| rexrd 11184 | A standard real is an exte... |
| renepnfd 11185 | No (finite) real equals pl... |
| renemnfd 11186 | No real equals minus infin... |
| pnfex 11187 | Plus infinity exists. (Co... |
| pnfxr 11188 | Plus infinity belongs to t... |
| pnfnemnf 11189 | Plus and minus infinity ar... |
| mnfnepnf 11190 | Minus and plus infinity ar... |
| mnfxr 11191 | Minus infinity belongs to ... |
| rexri 11192 | A standard real is an exte... |
| 1xr 11193 | ` 1 ` is an extended real ... |
| renfdisj 11194 | The reals and the infiniti... |
| ltrelxr 11195 | "Less than" is a relation ... |
| ltrel 11196 | "Less than" is a relation.... |
| lerelxr 11197 | "Less than or equal to" is... |
| lerel 11198 | "Less than or equal to" is... |
| xrlenlt 11199 | "Less than or equal to" ex... |
| xrlenltd 11200 | "Less than or equal to" ex... |
| xrltnle 11201 | "Less than" expressed in t... |
| xrltnled 11202 | 'Less than' in terms of 'l... |
| xrnltled 11203 | "Not less than" implies "l... |
| ssxr 11204 | The three (non-exclusive) ... |
| ltxrlt 11205 | The standard less-than ` <... |
| axlttri 11206 | Ordering on reals satisfie... |
| axlttrn 11207 | Ordering on reals is trans... |
| axltadd 11208 | Ordering property of addit... |
| axmulgt0 11209 | The product of two positiv... |
| axsup 11210 | A nonempty, bounded-above ... |
| lttr 11211 | Alias for ~ axlttrn , for ... |
| mulgt0 11212 | The product of two positiv... |
| lenlt 11213 | 'Less than or equal to' ex... |
| ltnle 11214 | 'Less than' expressed in t... |
| ltso 11215 | 'Less than' is a strict or... |
| gtso 11216 | 'Greater than' is a strict... |
| lttri2 11217 | Consequence of trichotomy.... |
| lttri3 11218 | Trichotomy law for 'less t... |
| lttri4 11219 | Trichotomy law for 'less t... |
| letri3 11220 | Trichotomy law. (Contribu... |
| leloe 11221 | 'Less than or equal to' ex... |
| eqlelt 11222 | Equality in terms of 'less... |
| ltle 11223 | 'Less than' implies 'less ... |
| leltne 11224 | 'Less than or equal to' im... |
| lelttr 11225 | Transitive law. (Contribu... |
| leltletr 11226 | Transitive law, weaker for... |
| ltletr 11227 | Transitive law. (Contribu... |
| ltleletr 11228 | Transitive law, weaker for... |
| letr 11229 | Transitive law. (Contribu... |
| ltnr 11230 | 'Less than' is irreflexive... |
| leid 11231 | 'Less than or equal to' is... |
| ltne 11232 | 'Less than' implies not eq... |
| ltnsym 11233 | 'Less than' is not symmetr... |
| ltnsym2 11234 | 'Less than' is antisymmetr... |
| letric 11235 | Trichotomy law. (Contribu... |
| ltlen 11236 | 'Less than' expressed in t... |
| eqle 11237 | Equality implies 'less tha... |
| eqled 11238 | Equality implies 'less tha... |
| ltadd2 11239 | Addition to both sides of ... |
| ne0gt0 11240 | A nonzero nonnegative numb... |
| lecasei 11241 | Ordering elimination by ca... |
| lelttric 11242 | Trichotomy law. (Contribu... |
| ltlecasei 11243 | Ordering elimination by ca... |
| ltnri 11244 | 'Less than' is irreflexive... |
| eqlei 11245 | Equality implies 'less tha... |
| eqlei2 11246 | Equality implies 'less tha... |
| gtneii 11247 | 'Less than' implies not eq... |
| ltneii 11248 | 'Greater than' implies not... |
| lttri2i 11249 | Consequence of trichotomy.... |
| lttri3i 11250 | Consequence of trichotomy.... |
| letri3i 11251 | Consequence of trichotomy.... |
| leloei 11252 | 'Less than or equal to' in... |
| ltleni 11253 | 'Less than' expressed in t... |
| ltnsymi 11254 | 'Less than' is not symmetr... |
| lenlti 11255 | 'Less than or equal to' in... |
| ltnlei 11256 | 'Less than' in terms of 'l... |
| ltlei 11257 | 'Less than' implies 'less ... |
| ltleii 11258 | 'Less than' implies 'less ... |
| ltnei 11259 | 'Less than' implies not eq... |
| letrii 11260 | Trichotomy law for 'less t... |
| lttri 11261 | 'Less than' is transitive.... |
| lelttri 11262 | 'Less than or equal to', '... |
| ltletri 11263 | 'Less than', 'less than or... |
| letri 11264 | 'Less than or equal to' is... |
| le2tri3i 11265 | Extended trichotomy law fo... |
| ltadd2i 11266 | Addition to both sides of ... |
| mulgt0i 11267 | The product of two positiv... |
| mulgt0ii 11268 | The product of two positiv... |
| ltnrd 11269 | 'Less than' is irreflexive... |
| gtned 11270 | 'Less than' implies not eq... |
| ltned 11271 | 'Greater than' implies not... |
| ne0gt0d 11272 | A nonzero nonnegative numb... |
| lttrid 11273 | Ordering on reals satisfie... |
| lttri2d 11274 | Consequence of trichotomy.... |
| lttri3d 11275 | Consequence of trichotomy.... |
| lttri4d 11276 | Trichotomy law for 'less t... |
| letri3d 11277 | Consequence of trichotomy.... |
| leloed 11278 | 'Less than or equal to' in... |
| eqleltd 11279 | Equality in terms of 'less... |
| ltlend 11280 | 'Less than' expressed in t... |
| lenltd 11281 | 'Less than or equal to' in... |
| ltnled 11282 | 'Less than' in terms of 'l... |
| ltled 11283 | 'Less than' implies 'less ... |
| ltnsymd 11284 | 'Less than' implies 'less ... |
| nltled 11285 | 'Not less than ' implies '... |
| lensymd 11286 | 'Less than or equal to' im... |
| letrid 11287 | Trichotomy law for 'less t... |
| leltned 11288 | 'Less than or equal to' im... |
| leneltd 11289 | 'Less than or equal to' an... |
| mulgt0d 11290 | The product of two positiv... |
| ltadd2d 11291 | Addition to both sides of ... |
| letrd 11292 | Transitive law deduction f... |
| lelttrd 11293 | Transitive law deduction f... |
| ltadd2dd 11294 | Addition to both sides of ... |
| ltletrd 11295 | Transitive law deduction f... |
| lttrd 11296 | Transitive law deduction f... |
| lelttrdi 11297 | If a number is less than a... |
| dedekind 11298 | The Dedekind cut theorem. ... |
| dedekindle 11299 | The Dedekind cut theorem, ... |
| mul12 11300 | Commutative/associative la... |
| mul32 11301 | Commutative/associative la... |
| mul31 11302 | Commutative/associative la... |
| mul4 11303 | Rearrangement of 4 factors... |
| mul4r 11304 | Rearrangement of 4 factors... |
| muladd11 11305 | A simple product of sums e... |
| 1p1times 11306 | Two times a number. (Cont... |
| peano2cn 11307 | A theorem for complex numb... |
| peano2re 11308 | A theorem for reals analog... |
| readdcan 11309 | Cancellation law for addit... |
| 00id 11310 | ` 0 ` is its own additive ... |
| mul02lem1 11311 | Lemma for ~ mul02 . If an... |
| mul02lem2 11312 | Lemma for ~ mul02 . Zero ... |
| mul02 11313 | Multiplication by ` 0 ` . ... |
| mul01 11314 | Multiplication by ` 0 ` . ... |
| addrid 11315 | ` 0 ` is an additive ident... |
| cnegex 11316 | Existence of the negative ... |
| cnegex2 11317 | Existence of a left invers... |
| addlid 11318 | ` 0 ` is a left identity f... |
| addcan 11319 | Cancellation law for addit... |
| addcan2 11320 | Cancellation law for addit... |
| addcom 11321 | Addition commutes. This u... |
| addridi 11322 | ` 0 ` is an additive ident... |
| addlidi 11323 | ` 0 ` is a left identity f... |
| mul02i 11324 | Multiplication by 0. Theo... |
| mul01i 11325 | Multiplication by ` 0 ` . ... |
| addcomi 11326 | Addition commutes. Based ... |
| addcomli 11327 | Addition commutes. (Contr... |
| addcani 11328 | Cancellation law for addit... |
| addcan2i 11329 | Cancellation law for addit... |
| mul12i 11330 | Commutative/associative la... |
| mul32i 11331 | Commutative/associative la... |
| mul4i 11332 | Rearrangement of 4 factors... |
| mul02d 11333 | Multiplication by 0. Theo... |
| mul01d 11334 | Multiplication by ` 0 ` . ... |
| addridd 11335 | ` 0 ` is an additive ident... |
| addlidd 11336 | ` 0 ` is a left identity f... |
| addcomd 11337 | Addition commutes. Based ... |
| addcand 11338 | Cancellation law for addit... |
| addcan2d 11339 | Cancellation law for addit... |
| addcanad 11340 | Cancelling a term on the l... |
| addcan2ad 11341 | Cancelling a term on the r... |
| addneintrd 11342 | Introducing a term on the ... |
| addneintr2d 11343 | Introducing a term on the ... |
| mul12d 11344 | Commutative/associative la... |
| mul32d 11345 | Commutative/associative la... |
| mul31d 11346 | Commutative/associative la... |
| mul4d 11347 | Rearrangement of 4 factors... |
| muladd11r 11348 | A simple product of sums e... |
| comraddd 11349 | Commute RHS addition, in d... |
| comraddi 11350 | Commute RHS addition. See... |
| ltaddneg 11351 | Adding a negative number t... |
| ltaddnegr 11352 | Adding a negative number t... |
| add12 11353 | Commutative/associative la... |
| add32 11354 | Commutative/associative la... |
| add32r 11355 | Commutative/associative la... |
| add4 11356 | Rearrangement of 4 terms i... |
| add42 11357 | Rearrangement of 4 terms i... |
| add12i 11358 | Commutative/associative la... |
| add32i 11359 | Commutative/associative la... |
| add4i 11360 | Rearrangement of 4 terms i... |
| add42i 11361 | Rearrangement of 4 terms i... |
| add12d 11362 | Commutative/associative la... |
| add32d 11363 | Commutative/associative la... |
| add4d 11364 | Rearrangement of 4 terms i... |
| add42d 11365 | Rearrangement of 4 terms i... |
| 0cnALT 11370 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11371 | Alternate proof of ~ 0cnAL... |
| negeu 11372 | Existential uniqueness of ... |
| subval 11373 | Value of subtraction, whic... |
| negeq 11374 | Equality theorem for negat... |
| negeqi 11375 | Equality inference for neg... |
| negeqd 11376 | Equality deduction for neg... |
| nfnegd 11377 | Deduction version of ~ nfn... |
| nfneg 11378 | Bound-variable hypothesis ... |
| csbnegg 11379 | Move class substitution in... |
| negex 11380 | A negative is a set. (Con... |
| subcl 11381 | Closure law for subtractio... |
| negcl 11382 | Closure law for negative. ... |
| negicn 11383 | ` -u _i ` is a complex num... |
| subf 11384 | Subtraction is an operatio... |
| subadd 11385 | Relationship between subtr... |
| subadd2 11386 | Relationship between subtr... |
| subsub23 11387 | Swap subtrahend and result... |
| pncan 11388 | Cancellation law for subtr... |
| pncan2 11389 | Cancellation law for subtr... |
| pncan3 11390 | Subtraction and addition o... |
| npcan 11391 | Cancellation law for subtr... |
| addsubass 11392 | Associative-type law for a... |
| addsub 11393 | Law for addition and subtr... |
| subadd23 11394 | Commutative/associative la... |
| addsub12 11395 | Commutative/associative la... |
| 2addsub 11396 | Law for subtraction and ad... |
| addsubeq4 11397 | Relation between sums and ... |
| pncan3oi 11398 | Subtraction and addition o... |
| mvrraddi 11399 | Move the right term in a s... |
| mvrladdi 11400 | Move the left term in a su... |
| mvlladdi 11401 | Move the left term in a su... |
| subid 11402 | Subtraction of a number fr... |
| subid1 11403 | Identity law for subtracti... |
| npncan 11404 | Cancellation law for subtr... |
| nppcan 11405 | Cancellation law for subtr... |
| nnpcan 11406 | Cancellation law for subtr... |
| nppcan3 11407 | Cancellation law for subtr... |
| subcan2 11408 | Cancellation law for subtr... |
| subeq0 11409 | If the difference between ... |
| npncan2 11410 | Cancellation law for subtr... |
| subsub2 11411 | Law for double subtraction... |
| nncan 11412 | Cancellation law for subtr... |
| subsub 11413 | Law for double subtraction... |
| nppcan2 11414 | Cancellation law for subtr... |
| subsub3 11415 | Law for double subtraction... |
| subsub4 11416 | Law for double subtraction... |
| sub32 11417 | Swap the second and third ... |
| nnncan 11418 | Cancellation law for subtr... |
| nnncan1 11419 | Cancellation law for subtr... |
| nnncan2 11420 | Cancellation law for subtr... |
| npncan3 11421 | Cancellation law for subtr... |
| pnpcan 11422 | Cancellation law for mixed... |
| pnpcan2 11423 | Cancellation law for mixed... |
| pnncan 11424 | Cancellation law for mixed... |
| ppncan 11425 | Cancellation law for mixed... |
| addsub4 11426 | Rearrangement of 4 terms i... |
| subadd4 11427 | Rearrangement of 4 terms i... |
| sub4 11428 | Rearrangement of 4 terms i... |
| neg0 11429 | Minus 0 equals 0. (Contri... |
| negid 11430 | Addition of a number and i... |
| negsub 11431 | Relationship between subtr... |
| subneg 11432 | Relationship between subtr... |
| negneg 11433 | A number is equal to the n... |
| neg11 11434 | Negative is one-to-one. (... |
| negcon1 11435 | Negative contraposition la... |
| negcon2 11436 | Negative contraposition la... |
| negeq0 11437 | A number is zero iff its n... |
| subcan 11438 | Cancellation law for subtr... |
| negsubdi 11439 | Distribution of negative o... |
| negdi 11440 | Distribution of negative o... |
| negdi2 11441 | Distribution of negative o... |
| negsubdi2 11442 | Distribution of negative o... |
| neg2sub 11443 | Relationship between subtr... |
| renegcli 11444 | Closure law for negative o... |
| resubcli 11445 | Closure law for subtractio... |
| renegcl 11446 | Closure law for negative o... |
| resubcl 11447 | Closure law for subtractio... |
| negreb 11448 | The negative of a real is ... |
| peano2cnm 11449 | "Reverse" second Peano pos... |
| peano2rem 11450 | "Reverse" second Peano pos... |
| negcli 11451 | Closure law for negative. ... |
| negidi 11452 | Addition of a number and i... |
| negnegi 11453 | A number is equal to the n... |
| subidi 11454 | Subtraction of a number fr... |
| subid1i 11455 | Identity law for subtracti... |
| negne0bi 11456 | A number is nonzero iff it... |
| negrebi 11457 | The negative of a real is ... |
| negne0i 11458 | The negative of a nonzero ... |
| subcli 11459 | Closure law for subtractio... |
| pncan3i 11460 | Subtraction and addition o... |
| negsubi 11461 | Relationship between subtr... |
| subnegi 11462 | Relationship between subtr... |
| subeq0i 11463 | If the difference between ... |
| neg11i 11464 | Negative is one-to-one. (... |
| negcon1i 11465 | Negative contraposition la... |
| negcon2i 11466 | Negative contraposition la... |
| negdii 11467 | Distribution of negative o... |
| negsubdii 11468 | Distribution of negative o... |
| negsubdi2i 11469 | Distribution of negative o... |
| subaddi 11470 | Relationship between subtr... |
| subadd2i 11471 | Relationship between subtr... |
| subaddrii 11472 | Relationship between subtr... |
| subsub23i 11473 | Swap subtrahend and result... |
| addsubassi 11474 | Associative-type law for s... |
| addsubi 11475 | Law for subtraction and ad... |
| subcani 11476 | Cancellation law for subtr... |
| subcan2i 11477 | Cancellation law for subtr... |
| pnncani 11478 | Cancellation law for mixed... |
| addsub4i 11479 | Rearrangement of 4 terms i... |
| 0reALT 11480 | Alternate proof of ~ 0re .... |
| negcld 11481 | Closure law for negative. ... |
| subidd 11482 | Subtraction of a number fr... |
| subid1d 11483 | Identity law for subtracti... |
| negidd 11484 | Addition of a number and i... |
| negnegd 11485 | A number is equal to the n... |
| negeq0d 11486 | A number is zero iff its n... |
| negne0bd 11487 | A number is nonzero iff it... |
| negcon1d 11488 | Contraposition law for una... |
| negcon1ad 11489 | Contraposition law for una... |
| neg11ad 11490 | The negatives of two compl... |
| negned 11491 | If two complex numbers are... |
| negne0d 11492 | The negative of a nonzero ... |
| negrebd 11493 | The negative of a real is ... |
| subcld 11494 | Closure law for subtractio... |
| pncand 11495 | Cancellation law for subtr... |
| pncan2d 11496 | Cancellation law for subtr... |
| pncan3d 11497 | Subtraction and addition o... |
| npcand 11498 | Cancellation law for subtr... |
| nncand 11499 | Cancellation law for subtr... |
| negsubd 11500 | Relationship between subtr... |
| subnegd 11501 | Relationship between subtr... |
| subeq0d 11502 | If the difference between ... |
| subne0d 11503 | Two unequal numbers have n... |
| subeq0ad 11504 | The difference of two comp... |
| subne0ad 11505 | If the difference of two c... |
| neg11d 11506 | If the difference between ... |
| negdid 11507 | Distribution of negative o... |
| negdi2d 11508 | Distribution of negative o... |
| negsubdid 11509 | Distribution of negative o... |
| negsubdi2d 11510 | Distribution of negative o... |
| neg2subd 11511 | Relationship between subtr... |
| subaddd 11512 | Relationship between subtr... |
| subadd2d 11513 | Relationship between subtr... |
| addsubassd 11514 | Associative-type law for s... |
| addsubd 11515 | Law for subtraction and ad... |
| subadd23d 11516 | Commutative/associative la... |
| addsub12d 11517 | Commutative/associative la... |
| npncand 11518 | Cancellation law for subtr... |
| nppcand 11519 | Cancellation law for subtr... |
| nppcan2d 11520 | Cancellation law for subtr... |
| nppcan3d 11521 | Cancellation law for subtr... |
| subsubd 11522 | Law for double subtraction... |
| subsub2d 11523 | Law for double subtraction... |
| subsub3d 11524 | Law for double subtraction... |
| subsub4d 11525 | Law for double subtraction... |
| sub32d 11526 | Swap the second and third ... |
| nnncand 11527 | Cancellation law for subtr... |
| nnncan1d 11528 | Cancellation law for subtr... |
| nnncan2d 11529 | Cancellation law for subtr... |
| npncan3d 11530 | Cancellation law for subtr... |
| pnpcand 11531 | Cancellation law for mixed... |
| pnpcan2d 11532 | Cancellation law for mixed... |
| pnncand 11533 | Cancellation law for mixed... |
| ppncand 11534 | Cancellation law for mixed... |
| subcand 11535 | Cancellation law for subtr... |
| subcan2d 11536 | Cancellation law for subtr... |
| subcanad 11537 | Cancellation law for subtr... |
| subneintrd 11538 | Introducing subtraction on... |
| subcan2ad 11539 | Cancellation law for subtr... |
| subneintr2d 11540 | Introducing subtraction on... |
| addsub4d 11541 | Rearrangement of 4 terms i... |
| subadd4d 11542 | Rearrangement of 4 terms i... |
| sub4d 11543 | Rearrangement of 4 terms i... |
| 2addsubd 11544 | Law for subtraction and ad... |
| addsubeq4d 11545 | Relation between sums and ... |
| subsubadd23 11546 | Swap the second and the th... |
| addsubsub23 11547 | Swap the second and the th... |
| subeqxfrd 11548 | Transfer two terms of a su... |
| mvlraddd 11549 | Move the right term in a s... |
| mvlladdd 11550 | Move the left term in a su... |
| mvrraddd 11551 | Move the right term in a s... |
| mvrladdd 11552 | Move the left term in a su... |
| assraddsubd 11553 | Associate RHS addition-sub... |
| subaddeqd 11554 | Transfer two terms of a su... |
| addlsub 11555 | Left-subtraction: Subtrac... |
| addrsub 11556 | Right-subtraction: Subtra... |
| subexsub 11557 | A subtraction law: Exchan... |
| addid0 11558 | If adding a number to a an... |
| addn0nid 11559 | Adding a nonzero number to... |
| pnpncand 11560 | Addition/subtraction cance... |
| subeqrev 11561 | Reverse the order of subtr... |
| addeq0 11562 | Two complex numbers add up... |
| pncan1 11563 | Cancellation law for addit... |
| npcan1 11564 | Cancellation law for subtr... |
| subeq0bd 11565 | If two complex numbers are... |
| renegcld 11566 | Closure law for negative o... |
| resubcld 11567 | Closure law for subtractio... |
| negn0 11568 | The image under negation o... |
| negf1o 11569 | Negation is an isomorphism... |
| kcnktkm1cn 11570 | k times k minus 1 is a com... |
| muladd 11571 | Product of two sums. (Con... |
| subdi 11572 | Distribution of multiplica... |
| subdir 11573 | Distribution of multiplica... |
| ine0 11574 | The imaginary unit ` _i ` ... |
| mulneg1 11575 | Product with negative is n... |
| mulneg2 11576 | The product with a negativ... |
| mulneg12 11577 | Swap the negative sign in ... |
| mul2neg 11578 | Product of two negatives. ... |
| submul2 11579 | Convert a subtraction to a... |
| mulm1 11580 | Product with minus one is ... |
| addneg1mul 11581 | Addition with product with... |
| mulsub 11582 | Product of two differences... |
| mulsub2 11583 | Swap the order of subtract... |
| mulm1i 11584 | Product with minus one is ... |
| mulneg1i 11585 | Product with negative is n... |
| mulneg2i 11586 | Product with negative is n... |
| mul2negi 11587 | Product of two negatives. ... |
| subdii 11588 | Distribution of multiplica... |
| subdiri 11589 | Distribution of multiplica... |
| muladdi 11590 | Product of two sums. (Con... |
| mulm1d 11591 | Product with minus one is ... |
| mulneg1d 11592 | Product with negative is n... |
| mulneg2d 11593 | Product with negative is n... |
| mul2negd 11594 | Product of two negatives. ... |
| subdid 11595 | Distribution of multiplica... |
| subdird 11596 | Distribution of multiplica... |
| muladdd 11597 | Product of two sums. (Con... |
| mulsubd 11598 | Product of two differences... |
| muls1d 11599 | Multiplication by one minu... |
| mulsubfacd 11600 | Multiplication followed by... |
| addmulsub 11601 | The product of a sum and a... |
| subaddmulsub 11602 | The difference with a prod... |
| mulsubaddmulsub 11603 | A special difference of a ... |
| gt0ne0 11604 | Positive implies nonzero. ... |
| lt0ne0 11605 | A number which is less tha... |
| ltadd1 11606 | Addition to both sides of ... |
| leadd1 11607 | Addition to both sides of ... |
| leadd2 11608 | Addition to both sides of ... |
| ltsubadd 11609 | 'Less than' relationship b... |
| ltsubadd2 11610 | 'Less than' relationship b... |
| lesubadd 11611 | 'Less than or equal to' re... |
| lesubadd2 11612 | 'Less than or equal to' re... |
| ltaddsub 11613 | 'Less than' relationship b... |
| ltaddsub2 11614 | 'Less than' relationship b... |
| leaddsub 11615 | 'Less than or equal to' re... |
| leaddsub2 11616 | 'Less than or equal to' re... |
| suble 11617 | Swap subtrahends in an ine... |
| lesub 11618 | Swap subtrahends in an ine... |
| ltsub23 11619 | 'Less than' relationship b... |
| ltsub13 11620 | 'Less than' relationship b... |
| le2add 11621 | Adding both sides of two '... |
| ltleadd 11622 | Adding both sides of two o... |
| leltadd 11623 | Adding both sides of two o... |
| lt2add 11624 | Adding both sides of two '... |
| addgt0 11625 | The sum of 2 positive numb... |
| addgegt0 11626 | The sum of nonnegative and... |
| addgtge0 11627 | The sum of nonnegative and... |
| addge0 11628 | The sum of 2 nonnegative n... |
| ltaddpos 11629 | Adding a positive number t... |
| ltaddpos2 11630 | Adding a positive number t... |
| ltsubpos 11631 | Subtracting a positive num... |
| posdif 11632 | Comparison of two numbers ... |
| lesub1 11633 | Subtraction from both side... |
| lesub2 11634 | Subtraction of both sides ... |
| ltsub1 11635 | Subtraction from both side... |
| ltsub2 11636 | Subtraction of both sides ... |
| lt2sub 11637 | Subtracting both sides of ... |
| le2sub 11638 | Subtracting both sides of ... |
| ltneg 11639 | Negative of both sides of ... |
| ltnegcon1 11640 | Contraposition of negative... |
| ltnegcon2 11641 | Contraposition of negative... |
| leneg 11642 | Negative of both sides of ... |
| lenegcon1 11643 | Contraposition of negative... |
| lenegcon2 11644 | Contraposition of negative... |
| lt0neg1 11645 | Comparison of a number and... |
| lt0neg2 11646 | Comparison of a number and... |
| le0neg1 11647 | Comparison of a number and... |
| le0neg2 11648 | Comparison of a number and... |
| addge01 11649 | A number is less than or e... |
| addge02 11650 | A number is less than or e... |
| add20 11651 | Two nonnegative numbers ar... |
| subge0 11652 | Nonnegative subtraction. ... |
| suble0 11653 | Nonpositive subtraction. ... |
| leaddle0 11654 | The sum of a real number a... |
| subge02 11655 | Nonnegative subtraction. ... |
| lesub0 11656 | Lemma to show a nonnegativ... |
| mulge0 11657 | The product of two nonnega... |
| mullt0 11658 | The product of two negativ... |
| msqgt0 11659 | A nonzero square is positi... |
| msqge0 11660 | A square is nonnegative. ... |
| 0lt1 11661 | 0 is less than 1. Theorem... |
| 0le1 11662 | 0 is less than or equal to... |
| relin01 11663 | An interval law for less t... |
| ltordlem 11664 | Lemma for ~ ltord1 . (Con... |
| ltord1 11665 | Infer an ordering relation... |
| leord1 11666 | Infer an ordering relation... |
| eqord1 11667 | A strictly increasing real... |
| ltord2 11668 | Infer an ordering relation... |
| leord2 11669 | Infer an ordering relation... |
| eqord2 11670 | A strictly decreasing real... |
| wloglei 11671 | Form of ~ wlogle where bot... |
| wlogle 11672 | If the predicate ` ch ( x ... |
| leidi 11673 | 'Less than or equal to' is... |
| gt0ne0i 11674 | Positive means nonzero (us... |
| gt0ne0ii 11675 | Positive implies nonzero. ... |
| msqgt0i 11676 | A nonzero square is positi... |
| msqge0i 11677 | A square is nonnegative. ... |
| addgt0i 11678 | Addition of 2 positive num... |
| addge0i 11679 | Addition of 2 nonnegative ... |
| addgegt0i 11680 | Addition of nonnegative an... |
| addgt0ii 11681 | Addition of 2 positive num... |
| add20i 11682 | Two nonnegative numbers ar... |
| ltnegi 11683 | Negative of both sides of ... |
| lenegi 11684 | Negative of both sides of ... |
| ltnegcon2i 11685 | Contraposition of negative... |
| mulge0i 11686 | The product of two nonnega... |
| lesub0i 11687 | Lemma to show a nonnegativ... |
| ltaddposi 11688 | Adding a positive number t... |
| posdifi 11689 | Comparison of two numbers ... |
| ltnegcon1i 11690 | Contraposition of negative... |
| lenegcon1i 11691 | Contraposition of negative... |
| subge0i 11692 | Nonnegative subtraction. ... |
| ltadd1i 11693 | Addition to both sides of ... |
| leadd1i 11694 | Addition to both sides of ... |
| leadd2i 11695 | Addition to both sides of ... |
| ltsubaddi 11696 | 'Less than' relationship b... |
| lesubaddi 11697 | 'Less than or equal to' re... |
| ltsubadd2i 11698 | 'Less than' relationship b... |
| lesubadd2i 11699 | 'Less than or equal to' re... |
| ltaddsubi 11700 | 'Less than' relationship b... |
| lt2addi 11701 | Adding both side of two in... |
| le2addi 11702 | Adding both side of two in... |
| gt0ne0d 11703 | Positive implies nonzero. ... |
| lt0ne0d 11704 | Something less than zero i... |
| leidd 11705 | 'Less than or equal to' is... |
| msqgt0d 11706 | A nonzero square is positi... |
| msqge0d 11707 | A square is nonnegative. ... |
| lt0neg1d 11708 | Comparison of a number and... |
| lt0neg2d 11709 | Comparison of a number and... |
| le0neg1d 11710 | Comparison of a number and... |
| le0neg2d 11711 | Comparison of a number and... |
| addgegt0d 11712 | Addition of nonnegative an... |
| addgtge0d 11713 | Addition of positive and n... |
| addgt0d 11714 | Addition of 2 positive num... |
| addge0d 11715 | Addition of 2 nonnegative ... |
| mulge0d 11716 | The product of two nonnega... |
| ltnegd 11717 | Negative of both sides of ... |
| lenegd 11718 | Negative of both sides of ... |
| ltnegcon1d 11719 | Contraposition of negative... |
| ltnegcon2d 11720 | Contraposition of negative... |
| lenegcon1d 11721 | Contraposition of negative... |
| lenegcon2d 11722 | Contraposition of negative... |
| ltaddposd 11723 | Adding a positive number t... |
| ltaddpos2d 11724 | Adding a positive number t... |
| ltsubposd 11725 | Subtracting a positive num... |
| posdifd 11726 | Comparison of two numbers ... |
| addge01d 11727 | A number is less than or e... |
| addge02d 11728 | A number is less than or e... |
| subge0d 11729 | Nonnegative subtraction. ... |
| suble0d 11730 | Nonpositive subtraction. ... |
| subge02d 11731 | Nonnegative subtraction. ... |
| ltadd1d 11732 | Addition to both sides of ... |
| leadd1d 11733 | Addition to both sides of ... |
| leadd2d 11734 | Addition to both sides of ... |
| ltsubaddd 11735 | 'Less than' relationship b... |
| lesubaddd 11736 | 'Less than or equal to' re... |
| ltsubadd2d 11737 | 'Less than' relationship b... |
| lesubadd2d 11738 | 'Less than or equal to' re... |
| ltaddsubd 11739 | 'Less than' relationship b... |
| ltaddsub2d 11740 | 'Less than' relationship b... |
| leaddsub2d 11741 | 'Less than or equal to' re... |
| subled 11742 | Swap subtrahends in an ine... |
| lesubd 11743 | Swap subtrahends in an ine... |
| ltsub23d 11744 | 'Less than' relationship b... |
| ltsub13d 11745 | 'Less than' relationship b... |
| lesub1d 11746 | Subtraction from both side... |
| lesub2d 11747 | Subtraction of both sides ... |
| ltsub1d 11748 | Subtraction from both side... |
| ltsub2d 11749 | Subtraction of both sides ... |
| ltadd1dd 11750 | Addition to both sides of ... |
| ltsub1dd 11751 | Subtraction from both side... |
| ltsub2dd 11752 | Subtraction of both sides ... |
| leadd1dd 11753 | Addition to both sides of ... |
| leadd2dd 11754 | Addition to both sides of ... |
| lesub1dd 11755 | Subtraction from both side... |
| lesub2dd 11756 | Subtraction of both sides ... |
| lesub3d 11757 | The result of subtracting ... |
| le2addd 11758 | Adding both side of two in... |
| le2subd 11759 | Subtracting both sides of ... |
| ltleaddd 11760 | Adding both sides of two o... |
| leltaddd 11761 | Adding both sides of two o... |
| lt2addd 11762 | Adding both side of two in... |
| lt2subd 11763 | Subtracting both sides of ... |
| possumd 11764 | Condition for a positive s... |
| sublt0d 11765 | When a subtraction gives a... |
| ltaddsublt 11766 | Addition and subtraction o... |
| 1le1 11767 | One is less than or equal ... |
| ixi 11768 | ` _i ` times itself is min... |
| recextlem1 11769 | Lemma for ~ recex . (Cont... |
| recextlem2 11770 | Lemma for ~ recex . (Cont... |
| recex 11771 | Existence of reciprocal of... |
| mulcand 11772 | Cancellation law for multi... |
| mulcan2d 11773 | Cancellation law for multi... |
| mulcanad 11774 | Cancellation of a nonzero ... |
| mulcan2ad 11775 | Cancellation of a nonzero ... |
| mulcan 11776 | Cancellation law for multi... |
| mulcan2 11777 | Cancellation law for multi... |
| mulcani 11778 | Cancellation law for multi... |
| mul0or 11779 | If a product is zero, one ... |
| mulne0b 11780 | The product of two nonzero... |
| mulne0 11781 | The product of two nonzero... |
| mulne0i 11782 | The product of two nonzero... |
| muleqadd 11783 | Property of numbers whose ... |
| receu 11784 | Existential uniqueness of ... |
| mulnzcnf 11785 | Multiplication maps nonzer... |
| mul0ori 11786 | If a product is zero, one ... |
| mul0ord 11787 | If a product is zero, one ... |
| msq0i 11788 | A number is zero iff its s... |
| msq0d 11789 | A number is zero iff its s... |
| mulne0bd 11790 | The product of two nonzero... |
| mulne0d 11791 | The product of two nonzero... |
| mulcan1g 11792 | A generalized form of the ... |
| mulcan2g 11793 | A generalized form of the ... |
| mulne0bad 11794 | A factor of a nonzero comp... |
| mulne0bbd 11795 | A factor of a nonzero comp... |
| 1div0 11798 | You can't divide by zero, ... |
| 1div0OLD 11799 | Obsolete version of ~ 1div... |
| divval 11800 | Value of division: if ` A ... |
| divmul 11801 | Relationship between divis... |
| divmul2 11802 | Relationship between divis... |
| divmul3 11803 | Relationship between divis... |
| divcl 11804 | Closure law for division. ... |
| reccl 11805 | Closure law for reciprocal... |
| divcan2 11806 | A cancellation law for div... |
| divcan1 11807 | A cancellation law for div... |
| diveq0 11808 | A ratio is zero iff the nu... |
| divne0b 11809 | The ratio of nonzero numbe... |
| divne0 11810 | The ratio of nonzero numbe... |
| recne0 11811 | The reciprocal of a nonzer... |
| recid 11812 | Multiplication of a number... |
| recid2 11813 | Multiplication of a number... |
| divrec 11814 | Relationship between divis... |
| divrec2 11815 | Relationship between divis... |
| divass 11816 | An associative law for div... |
| div23 11817 | A commutative/associative ... |
| div32 11818 | A commutative/associative ... |
| div13 11819 | A commutative/associative ... |
| div12 11820 | A commutative/associative ... |
| divmulass 11821 | An associative law for div... |
| divmulasscom 11822 | An associative/commutative... |
| divdir 11823 | Distribution of division o... |
| divcan3 11824 | A cancellation law for div... |
| divcan4 11825 | A cancellation law for div... |
| div11 11826 | One-to-one relationship fo... |
| div11OLD 11827 | Obsolete version of ~ div1... |
| diveq1 11828 | Equality in terms of unit ... |
| divid 11829 | A number divided by itself... |
| dividOLD 11830 | Obsolete version of ~ divi... |
| div0 11831 | Division into zero is zero... |
| div0OLD 11832 | Obsolete version of ~ div0... |
| div1 11833 | A number divided by 1 is i... |
| 1div1e1 11834 | 1 divided by 1 is 1. (Con... |
| divneg 11835 | Move negative sign inside ... |
| muldivdir 11836 | Distribution of division o... |
| divsubdir 11837 | Distribution of division o... |
| subdivcomb1 11838 | Bring a term in a subtract... |
| subdivcomb2 11839 | Bring a term in a subtract... |
| recrec 11840 | A number is equal to the r... |
| rec11 11841 | Reciprocal is one-to-one. ... |
| rec11r 11842 | Mutual reciprocals. (Cont... |
| divmuldiv 11843 | Multiplication of two rati... |
| divdivdiv 11844 | Division of two ratios. T... |
| divcan5 11845 | Cancellation of common fac... |
| divmul13 11846 | Swap the denominators in t... |
| divmul24 11847 | Swap the numerators in the... |
| divmuleq 11848 | Cross-multiply in an equal... |
| recdiv 11849 | The reciprocal of a ratio.... |
| divcan6 11850 | Cancellation of inverted f... |
| divdiv32 11851 | Swap denominators in a div... |
| divcan7 11852 | Cancel equal divisors in a... |
| dmdcan 11853 | Cancellation law for divis... |
| divdiv1 11854 | Division into a fraction. ... |
| divdiv2 11855 | Division by a fraction. (... |
| recdiv2 11856 | Division into a reciprocal... |
| ddcan 11857 | Cancellation in a double d... |
| divadddiv 11858 | Addition of two ratios. T... |
| divsubdiv 11859 | Subtraction of two ratios.... |
| conjmul 11860 | Two numbers whose reciproc... |
| rereccl 11861 | Closure law for reciprocal... |
| redivcl 11862 | Closure law for division o... |
| eqneg 11863 | A number equal to its nega... |
| eqnegd 11864 | A complex number equals it... |
| eqnegad 11865 | If a complex number equals... |
| div2neg 11866 | Quotient of two negatives.... |
| divneg2 11867 | Move negative sign inside ... |
| recclzi 11868 | Closure law for reciprocal... |
| recne0zi 11869 | The reciprocal of a nonzer... |
| recidzi 11870 | Multiplication of a number... |
| div1i 11871 | A number divided by 1 is i... |
| eqnegi 11872 | A number equal to its nega... |
| reccli 11873 | Closure law for reciprocal... |
| recidi 11874 | Multiplication of a number... |
| recreci 11875 | A number is equal to the r... |
| dividi 11876 | A number divided by itself... |
| div0i 11877 | Division into zero is zero... |
| divclzi 11878 | Closure law for division. ... |
| divcan1zi 11879 | A cancellation law for div... |
| divcan2zi 11880 | A cancellation law for div... |
| divreczi 11881 | Relationship between divis... |
| divcan3zi 11882 | A cancellation law for div... |
| divcan4zi 11883 | A cancellation law for div... |
| rec11i 11884 | Reciprocal is one-to-one. ... |
| divcli 11885 | Closure law for division. ... |
| divcan2i 11886 | A cancellation law for div... |
| divcan1i 11887 | A cancellation law for div... |
| divreci 11888 | Relationship between divis... |
| divcan3i 11889 | A cancellation law for div... |
| divcan4i 11890 | A cancellation law for div... |
| divne0i 11891 | The ratio of nonzero numbe... |
| rec11ii 11892 | Reciprocal is one-to-one. ... |
| divasszi 11893 | An associative law for div... |
| divmulzi 11894 | Relationship between divis... |
| divdirzi 11895 | Distribution of division o... |
| divdiv23zi 11896 | Swap denominators in a div... |
| divmuli 11897 | Relationship between divis... |
| divdiv32i 11898 | Swap denominators in a div... |
| divassi 11899 | An associative law for div... |
| divdiri 11900 | Distribution of division o... |
| div23i 11901 | A commutative/associative ... |
| div11i 11902 | One-to-one relationship fo... |
| divmuldivi 11903 | Multiplication of two rati... |
| divmul13i 11904 | Swap denominators of two r... |
| divadddivi 11905 | Addition of two ratios. T... |
| divdivdivi 11906 | Division of two ratios. T... |
| rerecclzi 11907 | Closure law for reciprocal... |
| rereccli 11908 | Closure law for reciprocal... |
| redivclzi 11909 | Closure law for division o... |
| redivcli 11910 | Closure law for division o... |
| div1d 11911 | A number divided by 1 is i... |
| reccld 11912 | Closure law for reciprocal... |
| recne0d 11913 | The reciprocal of a nonzer... |
| recidd 11914 | Multiplication of a number... |
| recid2d 11915 | Multiplication of a number... |
| recrecd 11916 | A number is equal to the r... |
| dividd 11917 | A number divided by itself... |
| div0d 11918 | Division into zero is zero... |
| divcld 11919 | Closure law for division. ... |
| divcan1d 11920 | A cancellation law for div... |
| divcan2d 11921 | A cancellation law for div... |
| divrecd 11922 | Relationship between divis... |
| divrec2d 11923 | Relationship between divis... |
| divcan3d 11924 | A cancellation law for div... |
| divcan4d 11925 | A cancellation law for div... |
| diveq0d 11926 | A ratio is zero iff the nu... |
| diveq1d 11927 | Equality in terms of unit ... |
| diveq1ad 11928 | The quotient of two comple... |
| diveq0ad 11929 | A fraction of complex numb... |
| divne1d 11930 | If two complex numbers are... |
| divne0bd 11931 | A ratio is zero iff the nu... |
| divnegd 11932 | Move negative sign inside ... |
| divneg2d 11933 | Move negative sign inside ... |
| div2negd 11934 | Quotient of two negatives.... |
| divne0d 11935 | The ratio of nonzero numbe... |
| recdivd 11936 | The reciprocal of a ratio.... |
| recdiv2d 11937 | Division into a reciprocal... |
| divcan6d 11938 | Cancellation of inverted f... |
| ddcand 11939 | Cancellation in a double d... |
| rec11d 11940 | Reciprocal is one-to-one. ... |
| divmuld 11941 | Relationship between divis... |
| div32d 11942 | A commutative/associative ... |
| div13d 11943 | A commutative/associative ... |
| divdiv32d 11944 | Swap denominators in a div... |
| divcan5d 11945 | Cancellation of common fac... |
| divcan5rd 11946 | Cancellation of common fac... |
| divcan7d 11947 | Cancel equal divisors in a... |
| dmdcand 11948 | Cancellation law for divis... |
| dmdcan2d 11949 | Cancellation law for divis... |
| divdiv1d 11950 | Division into a fraction. ... |
| divdiv2d 11951 | Division by a fraction. (... |
| divmul2d 11952 | Relationship between divis... |
| divmul3d 11953 | Relationship between divis... |
| divassd 11954 | An associative law for div... |
| div12d 11955 | A commutative/associative ... |
| div23d 11956 | A commutative/associative ... |
| divdird 11957 | Distribution of division o... |
| divsubdird 11958 | Distribution of division o... |
| div11d 11959 | One-to-one relationship fo... |
| divmuldivd 11960 | Multiplication of two rati... |
| divmul13d 11961 | Swap denominators of two r... |
| divmul24d 11962 | Swap the numerators in the... |
| divadddivd 11963 | Addition of two ratios. T... |
| divsubdivd 11964 | Subtraction of two ratios.... |
| divmuleqd 11965 | Cross-multiply in an equal... |
| divdivdivd 11966 | Division of two ratios. T... |
| diveq1bd 11967 | If two complex numbers are... |
| div2sub 11968 | Swap the order of subtract... |
| div2subd 11969 | Swap subtrahend and minuen... |
| rereccld 11970 | Closure law for reciprocal... |
| redivcld 11971 | Closure law for division o... |
| subrecd 11972 | Subtraction of reciprocals... |
| subrec 11973 | Subtraction of reciprocals... |
| subreci 11974 | Subtraction of reciprocals... |
| mvllmuld 11975 | Move the left term in a pr... |
| mvllmuli 11976 | Move the left term in a pr... |
| ldiv 11977 | Left-division. (Contribut... |
| rdiv 11978 | Right-division. (Contribu... |
| mdiv 11979 | A division law. (Contribu... |
| lineq 11980 | Solution of a (scalar) lin... |
| elimgt0 11981 | Hypothesis for weak deduct... |
| elimge0 11982 | Hypothesis for weak deduct... |
| ltp1 11983 | A number is less than itse... |
| lep1 11984 | A number is less than or e... |
| ltm1 11985 | A number minus 1 is less t... |
| lem1 11986 | A number minus 1 is less t... |
| letrp1 11987 | A transitive property of '... |
| p1le 11988 | A transitive property of p... |
| recgt0 11989 | The reciprocal of a positi... |
| prodgt0 11990 | Infer that a multiplicand ... |
| prodgt02 11991 | Infer that a multiplier is... |
| ltmul1a 11992 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 11993 | Multiplication of both sid... |
| ltmul2 11994 | Multiplication of both sid... |
| lemul1 11995 | Multiplication of both sid... |
| lemul2 11996 | Multiplication of both sid... |
| lemul1a 11997 | Multiplication of both sid... |
| lemul2a 11998 | Multiplication of both sid... |
| ltmul12a 11999 | Comparison of product of t... |
| lemul12b 12000 | Comparison of product of t... |
| lemul12a 12001 | Comparison of product of t... |
| mulgt1OLD 12002 | Obsolete version of ~ mulg... |
| ltmulgt11 12003 | Multiplication by a number... |
| ltmulgt12 12004 | Multiplication by a number... |
| mulgt1 12005 | The product of two numbers... |
| lemulge11 12006 | Multiplication by a number... |
| lemulge12 12007 | Multiplication by a number... |
| ltdiv1 12008 | Division of both sides of ... |
| lediv1 12009 | Division of both sides of ... |
| gt0div 12010 | Division of a positive num... |
| ge0div 12011 | Division of a nonnegative ... |
| divgt0 12012 | The ratio of two positive ... |
| divge0 12013 | The ratio of nonnegative a... |
| mulge0b 12014 | A condition for multiplica... |
| mulle0b 12015 | A condition for multiplica... |
| mulsuble0b 12016 | A condition for multiplica... |
| ltmuldiv 12017 | 'Less than' relationship b... |
| ltmuldiv2 12018 | 'Less than' relationship b... |
| ltdivmul 12019 | 'Less than' relationship b... |
| ledivmul 12020 | 'Less than or equal to' re... |
| ltdivmul2 12021 | 'Less than' relationship b... |
| lt2mul2div 12022 | 'Less than' relationship b... |
| ledivmul2 12023 | 'Less than or equal to' re... |
| lemuldiv 12024 | 'Less than or equal' relat... |
| lemuldiv2 12025 | 'Less than or equal' relat... |
| ltrec 12026 | The reciprocal of both sid... |
| lerec 12027 | The reciprocal of both sid... |
| lt2msq1 12028 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12029 | Two nonnegative numbers co... |
| ltdiv2 12030 | Division of a positive num... |
| ltrec1 12031 | Reciprocal swap in a 'less... |
| lerec2 12032 | Reciprocal swap in a 'less... |
| ledivdiv 12033 | Invert ratios of positive ... |
| lediv2 12034 | Division of a positive num... |
| ltdiv23 12035 | Swap denominator with othe... |
| lediv23 12036 | Swap denominator with othe... |
| lediv12a 12037 | Comparison of ratio of two... |
| lediv2a 12038 | Division of both sides of ... |
| reclt1 12039 | The reciprocal of a positi... |
| recgt1 12040 | The reciprocal of a positi... |
| recgt1i 12041 | The reciprocal of a number... |
| recp1lt1 12042 | Construct a number less th... |
| recreclt 12043 | Given a positive number ` ... |
| le2msq 12044 | The square function on non... |
| msq11 12045 | The square of a nonnegativ... |
| ledivp1 12046 | "Less than or equal to" an... |
| squeeze0 12047 | If a nonnegative number is... |
| ltp1i 12048 | A number is less than itse... |
| recgt0i 12049 | The reciprocal of a positi... |
| recgt0ii 12050 | The reciprocal of a positi... |
| prodgt0i 12051 | Infer that a multiplicand ... |
| divgt0i 12052 | The ratio of two positive ... |
| divge0i 12053 | The ratio of nonnegative a... |
| ltreci 12054 | The reciprocal of both sid... |
| lereci 12055 | The reciprocal of both sid... |
| lt2msqi 12056 | The square function on non... |
| le2msqi 12057 | The square function on non... |
| msq11i 12058 | The square of a nonnegativ... |
| divgt0i2i 12059 | The ratio of two positive ... |
| ltrecii 12060 | The reciprocal of both sid... |
| divgt0ii 12061 | The ratio of two positive ... |
| ltmul1i 12062 | Multiplication of both sid... |
| ltdiv1i 12063 | Division of both sides of ... |
| ltmuldivi 12064 | 'Less than' relationship b... |
| ltmul2i 12065 | Multiplication of both sid... |
| lemul1i 12066 | Multiplication of both sid... |
| lemul2i 12067 | Multiplication of both sid... |
| ltdiv23i 12068 | Swap denominator with othe... |
| ledivp1i 12069 | "Less than or equal to" an... |
| ltdivp1i 12070 | Less-than and division rel... |
| ltdiv23ii 12071 | Swap denominator with othe... |
| ltmul1ii 12072 | Multiplication of both sid... |
| ltdiv1ii 12073 | Division of both sides of ... |
| ltp1d 12074 | A number is less than itse... |
| lep1d 12075 | A number is less than or e... |
| ltm1d 12076 | A number minus 1 is less t... |
| lem1d 12077 | A number minus 1 is less t... |
| recgt0d 12078 | The reciprocal of a positi... |
| divgt0d 12079 | The ratio of two positive ... |
| mulgt1d 12080 | The product of two numbers... |
| lemulge11d 12081 | Multiplication by a number... |
| lemulge12d 12082 | Multiplication by a number... |
| lemul1ad 12083 | Multiplication of both sid... |
| lemul2ad 12084 | Multiplication of both sid... |
| ltmul12ad 12085 | Comparison of product of t... |
| lemul12ad 12086 | Comparison of product of t... |
| lemul12bd 12087 | Comparison of product of t... |
| fimaxre 12088 | A finite set of real numbe... |
| fimaxre2 12089 | A nonempty finite set of r... |
| fimaxre3 12090 | A nonempty finite set of r... |
| fiminre 12091 | A nonempty finite set of r... |
| fiminre2 12092 | A nonempty finite set of r... |
| negfi 12093 | The negation of a finite s... |
| lbreu 12094 | If a set of reals contains... |
| lbcl 12095 | If a set of reals contains... |
| lble 12096 | If a set of reals contains... |
| lbinf 12097 | If a set of reals contains... |
| lbinfcl 12098 | If a set of reals contains... |
| lbinfle 12099 | If a set of reals contains... |
| sup2 12100 | A nonempty, bounded-above ... |
| sup3 12101 | A version of the completen... |
| infm3lem 12102 | Lemma for ~ infm3 . (Cont... |
| infm3 12103 | The completeness axiom for... |
| suprcl 12104 | Closure of supremum of a n... |
| suprub 12105 | A member of a nonempty bou... |
| suprubd 12106 | Natural deduction form of ... |
| suprcld 12107 | Natural deduction form of ... |
| suprlub 12108 | The supremum of a nonempty... |
| suprnub 12109 | An upper bound is not less... |
| suprleub 12110 | The supremum of a nonempty... |
| supaddc 12111 | The supremum function dist... |
| supadd 12112 | The supremum function dist... |
| supmul1 12113 | The supremum function dist... |
| supmullem1 12114 | Lemma for ~ supmul . (Con... |
| supmullem2 12115 | Lemma for ~ supmul . (Con... |
| supmul 12116 | The supremum function dist... |
| sup3ii 12117 | A version of the completen... |
| suprclii 12118 | Closure of supremum of a n... |
| suprubii 12119 | A member of a nonempty bou... |
| suprlubii 12120 | The supremum of a nonempty... |
| suprnubii 12121 | An upper bound is not less... |
| suprleubii 12122 | The supremum of a nonempty... |
| riotaneg 12123 | The negative of the unique... |
| negiso 12124 | Negation is an order anti-... |
| dfinfre 12125 | The infimum of a set of re... |
| infrecl 12126 | Closure of infimum of a no... |
| infrenegsup 12127 | The infimum of a set of re... |
| infregelb 12128 | Any lower bound of a nonem... |
| infrelb 12129 | If a nonempty set of real ... |
| infrefilb 12130 | The infimum of a finite se... |
| supfirege 12131 | The supremum of a finite s... |
| neg1cn 12132 | -1 is a complex number. (... |
| neg1rr 12133 | -1 is a real number. (Con... |
| neg1ne0 12134 | -1 is nonzero. (Contribut... |
| neg1lt0 12135 | -1 is less than 0. (Contr... |
| negneg1e1 12136 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12137 | The imaginary unit ` _i ` ... |
| rimul 12138 | A real number times the im... |
| cru 12139 | The representation of comp... |
| crne0 12140 | The real representation of... |
| creur 12141 | The real part of a complex... |
| creui 12142 | The imaginary part of a co... |
| cju 12143 | The complex conjugate of a... |
| ofsubeq0 12144 | Function analogue of ~ sub... |
| ofnegsub 12145 | Function analogue of ~ neg... |
| ofsubge0 12146 | Function analogue of ~ sub... |
| nnexALT 12149 | Alternate proof of ~ nnex ... |
| peano5nni 12150 | Peano's inductive postulat... |
| nnssre 12151 | The positive integers are ... |
| nnsscn 12152 | The positive integers are ... |
| nnex 12153 | The set of positive intege... |
| nnre 12154 | A positive integer is a re... |
| nncn 12155 | A positive integer is a co... |
| nnrei 12156 | A positive integer is a re... |
| nncni 12157 | A positive integer is a co... |
| 1nn 12158 | Peano postulate: 1 is a po... |
| peano2nn 12159 | Peano postulate: a success... |
| dfnn2 12160 | Alternate definition of th... |
| dfnn3 12161 | Alternate definition of th... |
| nnred 12162 | A positive integer is a re... |
| nncnd 12163 | A positive integer is a co... |
| peano2nnd 12164 | Peano postulate: a success... |
| nnind 12165 | Principle of Mathematical ... |
| nnindALT 12166 | Principle of Mathematical ... |
| nnindd 12167 | Principle of Mathematical ... |
| nn1m1nn 12168 | Every positive integer is ... |
| nn1suc 12169 | If a statement holds for 1... |
| nnaddcl 12170 | Closure of addition of pos... |
| nnmulcl 12171 | Closure of multiplication ... |
| nnmulcli 12172 | Closure of multiplication ... |
| nnmtmip 12173 | "Minus times minus is plus... |
| nn2ge 12174 | There exists a positive in... |
| nnge1 12175 | A positive integer is one ... |
| nngt1ne1 12176 | A positive integer is grea... |
| nnle1eq1 12177 | A positive integer is less... |
| nngt0 12178 | A positive integer is posi... |
| nnnlt1 12179 | A positive integer is not ... |
| nnnle0 12180 | A positive integer is not ... |
| nnne0 12181 | A positive integer is nonz... |
| nnneneg 12182 | No positive integer is equ... |
| 0nnn 12183 | Zero is not a positive int... |
| 0nnnALT 12184 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12185 | Alternate version of ~ nnn... |
| nngt0i 12186 | A positive integer is posi... |
| nnne0i 12187 | A positive integer is nonz... |
| nndivre 12188 | The quotient of a real and... |
| nnrecre 12189 | The reciprocal of a positi... |
| nnrecgt0 12190 | The reciprocal of a positi... |
| nnsub 12191 | Subtraction of positive in... |
| nnsubi 12192 | Subtraction of positive in... |
| nndiv 12193 | Two ways to express " ` A ... |
| nndivtr 12194 | Transitive property of div... |
| nnge1d 12195 | A positive integer is one ... |
| nngt0d 12196 | A positive integer is posi... |
| nnne0d 12197 | A positive integer is nonz... |
| nnrecred 12198 | The reciprocal of a positi... |
| nnaddcld 12199 | Closure of addition of pos... |
| nnmulcld 12200 | Closure of multiplication ... |
| nndivred 12201 | A positive integer is one ... |
| 0ne1 12218 | Zero is different from one... |
| 1m1e0 12219 | One minus one equals zero.... |
| 2nn 12220 | 2 is a positive integer. ... |
| 2re 12221 | The number 2 is real. (Co... |
| 2cn 12222 | The number 2 is a complex ... |
| 2cnALT 12223 | Alternate proof of ~ 2cn .... |
| 2ex 12224 | The number 2 is a set. (C... |
| 2cnd 12225 | The number 2 is a complex ... |
| 3nn 12226 | 3 is a positive integer. ... |
| 3re 12227 | The number 3 is real. (Co... |
| 3cn 12228 | The number 3 is a complex ... |
| 3ex 12229 | The number 3 is a set. (C... |
| 4nn 12230 | 4 is a positive integer. ... |
| 4re 12231 | The number 4 is real. (Co... |
| 4cn 12232 | The number 4 is a complex ... |
| 5nn 12233 | 5 is a positive integer. ... |
| 5re 12234 | The number 5 is real. (Co... |
| 5cn 12235 | The number 5 is a complex ... |
| 6nn 12236 | 6 is a positive integer. ... |
| 6re 12237 | The number 6 is real. (Co... |
| 6cn 12238 | The number 6 is a complex ... |
| 7nn 12239 | 7 is a positive integer. ... |
| 7re 12240 | The number 7 is real. (Co... |
| 7cn 12241 | The number 7 is a complex ... |
| 8nn 12242 | 8 is a positive integer. ... |
| 8re 12243 | The number 8 is real. (Co... |
| 8cn 12244 | The number 8 is a complex ... |
| 9nn 12245 | 9 is a positive integer. ... |
| 9re 12246 | The number 9 is real. (Co... |
| 9cn 12247 | The number 9 is a complex ... |
| 0le0 12248 | Zero is nonnegative. (Con... |
| 0le2 12249 | The number 0 is less than ... |
| 2pos 12250 | The number 2 is positive. ... |
| 2ne0 12251 | The number 2 is nonzero. ... |
| 3pos 12252 | The number 3 is positive. ... |
| 3ne0 12253 | The number 3 is nonzero. ... |
| 4pos 12254 | The number 4 is positive. ... |
| 4ne0 12255 | The number 4 is nonzero. ... |
| 5pos 12256 | The number 5 is positive. ... |
| 6pos 12257 | The number 6 is positive. ... |
| 7pos 12258 | The number 7 is positive. ... |
| 8pos 12259 | The number 8 is positive. ... |
| 9pos 12260 | The number 9 is positive. ... |
| 1pneg1e0 12261 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12262 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12263 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12264 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12265 | Function value at ` N + 1 ... |
| 1p0e1 12266 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12267 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12268 | 2 - 1 = 1. The result is ... |
| 1e2m1 12269 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12270 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12271 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12272 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12273 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12274 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12275 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12276 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12277 | Two plus two equals four. ... |
| 2times 12278 | Two times a number. (Cont... |
| times2 12279 | A number times 2. (Contri... |
| 2timesi 12280 | Two times a number. (Cont... |
| times2i 12281 | A number times 2. (Contri... |
| 2txmxeqx 12282 | Two times a complex number... |
| 2div2e1 12283 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12284 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12285 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12286 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12287 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12288 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12289 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12290 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12291 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12292 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12293 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12294 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12295 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12296 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12297 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12298 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12299 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12300 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12301 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12302 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12303 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12304 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12305 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12306 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12307 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12308 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12309 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12310 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12311 | 2 times 0 equals 0. (Cont... |
| 4div2e2 12312 | One half of four is two. ... |
| 1lt2 12313 | 1 is less than 2. (Contri... |
| 2lt3 12314 | 2 is less than 3. (Contri... |
| 1lt3 12315 | 1 is less than 3. (Contri... |
| 3lt4 12316 | 3 is less than 4. (Contri... |
| 2lt4 12317 | 2 is less than 4. (Contri... |
| 1lt4 12318 | 1 is less than 4. (Contri... |
| 4lt5 12319 | 4 is less than 5. (Contri... |
| 3lt5 12320 | 3 is less than 5. (Contri... |
| 2lt5 12321 | 2 is less than 5. (Contri... |
| 1lt5 12322 | 1 is less than 5. (Contri... |
| 5lt6 12323 | 5 is less than 6. (Contri... |
| 4lt6 12324 | 4 is less than 6. (Contri... |
| 3lt6 12325 | 3 is less than 6. (Contri... |
| 2lt6 12326 | 2 is less than 6. (Contri... |
| 1lt6 12327 | 1 is less than 6. (Contri... |
| 6lt7 12328 | 6 is less than 7. (Contri... |
| 5lt7 12329 | 5 is less than 7. (Contri... |
| 4lt7 12330 | 4 is less than 7. (Contri... |
| 3lt7 12331 | 3 is less than 7. (Contri... |
| 2lt7 12332 | 2 is less than 7. (Contri... |
| 1lt7 12333 | 1 is less than 7. (Contri... |
| 7lt8 12334 | 7 is less than 8. (Contri... |
| 6lt8 12335 | 6 is less than 8. (Contri... |
| 5lt8 12336 | 5 is less than 8. (Contri... |
| 4lt8 12337 | 4 is less than 8. (Contri... |
| 3lt8 12338 | 3 is less than 8. (Contri... |
| 2lt8 12339 | 2 is less than 8. (Contri... |
| 1lt8 12340 | 1 is less than 8. (Contri... |
| 8lt9 12341 | 8 is less than 9. (Contri... |
| 7lt9 12342 | 7 is less than 9. (Contri... |
| 6lt9 12343 | 6 is less than 9. (Contri... |
| 5lt9 12344 | 5 is less than 9. (Contri... |
| 4lt9 12345 | 4 is less than 9. (Contri... |
| 3lt9 12346 | 3 is less than 9. (Contri... |
| 2lt9 12347 | 2 is less than 9. (Contri... |
| 1lt9 12348 | 1 is less than 9. (Contri... |
| 0ne2 12349 | 0 is not equal to 2. (Con... |
| 1ne2 12350 | 1 is not equal to 2. (Con... |
| 1le2 12351 | 1 is less than or equal to... |
| 2cnne0 12352 | 2 is a nonzero complex num... |
| 2rene0 12353 | 2 is a nonzero real number... |
| 1le3 12354 | 1 is less than or equal to... |
| neg1mulneg1e1 12355 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12356 | One-half is real. (Contri... |
| halfcn 12357 | One-half is a complex numb... |
| halfgt0 12358 | One-half is greater than z... |
| halfge0 12359 | One-half is not negative. ... |
| halflt1 12360 | One-half is less than one.... |
| 2halves 12361 | Two halves make a whole. ... |
| 1mhlfehlf 12362 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12363 | An eighth of four thirds i... |
| halfthird 12364 | Half minus a third. (Cont... |
| halfpm6th 12365 | One half plus or minus one... |
| it0e0 12366 | i times 0 equals 0. (Cont... |
| 2mulicn 12367 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12368 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12369 | Closure of half of a numbe... |
| rehalfcl 12370 | Real closure of half. (Co... |
| half0 12371 | Half of a number is zero i... |
| halfpos2 12372 | A number is positive iff i... |
| halfpos 12373 | A positive number is great... |
| halfnneg2 12374 | A number is nonnegative if... |
| halfaddsubcl 12375 | Closure of half-sum and ha... |
| halfaddsub 12376 | Sum and difference of half... |
| subhalfhalf 12377 | Subtracting the half of a ... |
| lt2halves 12378 | A sum is less than the who... |
| addltmul 12379 | Sum is less than product f... |
| nominpos 12380 | There is no smallest posit... |
| avglt1 12381 | Ordering property for aver... |
| avglt2 12382 | Ordering property for aver... |
| avgle1 12383 | Ordering property for aver... |
| avgle2 12384 | Ordering property for aver... |
| avgle 12385 | The average of two numbers... |
| 2timesd 12386 | Two times a number. (Cont... |
| times2d 12387 | A number times 2. (Contri... |
| halfcld 12388 | Closure of half of a numbe... |
| 2halvesd 12389 | Two halves make a whole. ... |
| rehalfcld 12390 | Real closure of half. (Co... |
| lt2halvesd 12391 | A sum is less than the who... |
| rehalfcli 12392 | Half a real number is real... |
| lt2addmuld 12393 | If two real numbers are le... |
| add1p1 12394 | Adding two times 1 to a nu... |
| sub1m1 12395 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12396 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12397 | A complex number increased... |
| div4p1lem1div2 12398 | An integer greater than 5,... |
| nnunb 12399 | The set of positive intege... |
| arch 12400 | Archimedean property of re... |
| nnrecl 12401 | There exists a positive in... |
| bndndx 12402 | A bounded real sequence ` ... |
| elnn0 12405 | Nonnegative integers expre... |
| nnssnn0 12406 | Positive naturals are a su... |
| nn0ssre 12407 | Nonnegative integers are a... |
| nn0sscn 12408 | Nonnegative integers are a... |
| nn0ex 12409 | The set of nonnegative int... |
| nnnn0 12410 | A positive integer is a no... |
| nnnn0i 12411 | A positive integer is a no... |
| nn0re 12412 | A nonnegative integer is a... |
| nn0cn 12413 | A nonnegative integer is a... |
| nn0rei 12414 | A nonnegative integer is a... |
| nn0cni 12415 | A nonnegative integer is a... |
| dfn2 12416 | The set of positive intege... |
| elnnne0 12417 | The positive integer prope... |
| 0nn0 12418 | 0 is a nonnegative integer... |
| 1nn0 12419 | 1 is a nonnegative integer... |
| 2nn0 12420 | 2 is a nonnegative integer... |
| 3nn0 12421 | 3 is a nonnegative integer... |
| 4nn0 12422 | 4 is a nonnegative integer... |
| 5nn0 12423 | 5 is a nonnegative integer... |
| 6nn0 12424 | 6 is a nonnegative integer... |
| 7nn0 12425 | 7 is a nonnegative integer... |
| 8nn0 12426 | 8 is a nonnegative integer... |
| 9nn0 12427 | 9 is a nonnegative integer... |
| nn0ge0 12428 | A nonnegative integer is g... |
| nn0nlt0 12429 | A nonnegative integer is n... |
| nn0ge0i 12430 | Nonnegative integers are n... |
| nn0le0eq0 12431 | A nonnegative integer is l... |
| nn0p1gt0 12432 | A nonnegative integer incr... |
| nnnn0addcl 12433 | A positive integer plus a ... |
| nn0nnaddcl 12434 | A nonnegative integer plus... |
| 0mnnnnn0 12435 | The result of subtracting ... |
| un0addcl 12436 | If ` S ` is closed under a... |
| un0mulcl 12437 | If ` S ` is closed under m... |
| nn0addcl 12438 | Closure of addition of non... |
| nn0mulcl 12439 | Closure of multiplication ... |
| nn0addcli 12440 | Closure of addition of non... |
| nn0mulcli 12441 | Closure of multiplication ... |
| nn0p1nn 12442 | A nonnegative integer plus... |
| peano2nn0 12443 | Second Peano postulate for... |
| nnm1nn0 12444 | A positive integer minus 1... |
| elnn0nn 12445 | The nonnegative integer pr... |
| elnnnn0 12446 | The positive integer prope... |
| elnnnn0b 12447 | The positive integer prope... |
| elnnnn0c 12448 | The positive integer prope... |
| nn0addge1 12449 | A number is less than or e... |
| nn0addge2 12450 | A number is less than or e... |
| nn0addge1i 12451 | A number is less than or e... |
| nn0addge2i 12452 | A number is less than or e... |
| nn0sub 12453 | Subtraction of nonnegative... |
| ltsubnn0 12454 | Subtracting a nonnegative ... |
| nn0negleid 12455 | A nonnegative integer is g... |
| difgtsumgt 12456 | If the difference of a rea... |
| nn0le2x 12457 | A nonnegative integer is l... |
| nn0le2xi 12458 | A nonnegative integer is l... |
| nn0lele2xi 12459 | 'Less than or equal to' im... |
| fcdmnn0supp 12460 | Two ways to write the supp... |
| fcdmnn0fsupp 12461 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12462 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12463 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12464 | A positive integer is a no... |
| nn0red 12465 | A nonnegative integer is a... |
| nn0cnd 12466 | A nonnegative integer is a... |
| nn0ge0d 12467 | A nonnegative integer is g... |
| nn0addcld 12468 | Closure of addition of non... |
| nn0mulcld 12469 | Closure of multiplication ... |
| nn0readdcl 12470 | Closure law for addition o... |
| nn0n0n1ge2 12471 | A nonnegative integer whic... |
| nn0n0n1ge2b 12472 | A nonnegative integer is n... |
| nn0ge2m1nn 12473 | If a nonnegative integer i... |
| nn0ge2m1nn0 12474 | If a nonnegative integer i... |
| nn0nndivcl 12475 | Closure law for dividing o... |
| elxnn0 12478 | An extended nonnegative in... |
| nn0ssxnn0 12479 | The standard nonnegative i... |
| nn0xnn0 12480 | A standard nonnegative int... |
| xnn0xr 12481 | An extended nonnegative in... |
| 0xnn0 12482 | Zero is an extended nonneg... |
| pnf0xnn0 12483 | Positive infinity is an ex... |
| nn0nepnf 12484 | No standard nonnegative in... |
| nn0xnn0d 12485 | A standard nonnegative int... |
| nn0nepnfd 12486 | No standard nonnegative in... |
| xnn0nemnf 12487 | No extended nonnegative in... |
| xnn0xrnemnf 12488 | The extended nonnegative i... |
| xnn0nnn0pnf 12489 | An extended nonnegative in... |
| elz 12492 | Membership in the set of i... |
| nnnegz 12493 | The negative of a positive... |
| zre 12494 | An integer is a real. (Co... |
| zcn 12495 | An integer is a complex nu... |
| zrei 12496 | An integer is a real numbe... |
| zssre 12497 | The integers are a subset ... |
| zsscn 12498 | The integers are a subset ... |
| zex 12499 | The set of integers exists... |
| elnnz 12500 | Positive integer property ... |
| 0z 12501 | Zero is an integer. (Cont... |
| 0zd 12502 | Zero is an integer, deduct... |
| elnn0z 12503 | Nonnegative integer proper... |
| elznn0nn 12504 | Integer property expressed... |
| elznn0 12505 | Integer property expressed... |
| elznn 12506 | Integer property expressed... |
| zle0orge1 12507 | There is no integer in the... |
| elz2 12508 | Membership in the set of i... |
| dfz2 12509 | Alternative definition of ... |
| zexALT 12510 | Alternate proof of ~ zex .... |
| nnz 12511 | A positive integer is an i... |
| nnssz 12512 | Positive integers are a su... |
| nn0ssz 12513 | Nonnegative integers are a... |
| nn0z 12514 | A nonnegative integer is a... |
| nn0zd 12515 | A nonnegative integer is a... |
| nnzd 12516 | A positive integer is an i... |
| nnzi 12517 | A positive integer is an i... |
| nn0zi 12518 | A nonnegative integer is a... |
| elnnz1 12519 | Positive integer property ... |
| znnnlt1 12520 | An integer is not a positi... |
| nnzrab 12521 | Positive integers expresse... |
| nn0zrab 12522 | Nonnegative integers expre... |
| 1z 12523 | One is an integer. (Contr... |
| 1zzd 12524 | One is an integer, deducti... |
| 2z 12525 | 2 is an integer. (Contrib... |
| 3z 12526 | 3 is an integer. (Contrib... |
| 4z 12527 | 4 is an integer. (Contrib... |
| znegcl 12528 | Closure law for negative i... |
| neg1z 12529 | -1 is an integer. (Contri... |
| znegclb 12530 | A complex number is an int... |
| nn0negz 12531 | The negative of a nonnegat... |
| nn0negzi 12532 | The negative of a nonnegat... |
| zaddcl 12533 | Closure of addition of int... |
| peano2z 12534 | Second Peano postulate gen... |
| zsubcl 12535 | Closure of subtraction of ... |
| peano2zm 12536 | "Reverse" second Peano pos... |
| zletr 12537 | Transitive law of ordering... |
| zrevaddcl 12538 | Reverse closure law for ad... |
| znnsub 12539 | The positive difference of... |
| znn0sub 12540 | The nonnegative difference... |
| nzadd 12541 | The sum of a real number n... |
| zmulcl 12542 | Closure of multiplication ... |
| zltp1le 12543 | Integer ordering relation.... |
| zleltp1 12544 | Integer ordering relation.... |
| zlem1lt 12545 | Integer ordering relation.... |
| zltlem1 12546 | Integer ordering relation.... |
| zltlem1d 12547 | Integer ordering relation,... |
| zgt0ge1 12548 | An integer greater than ` ... |
| nnleltp1 12549 | Positive integer ordering ... |
| nnltp1le 12550 | Positive integer ordering ... |
| nnaddm1cl 12551 | Closure of addition of pos... |
| nn0ltp1le 12552 | Nonnegative integer orderi... |
| nn0leltp1 12553 | Nonnegative integer orderi... |
| nn0ltlem1 12554 | Nonnegative integer orderi... |
| nn0sub2 12555 | Subtraction of nonnegative... |
| nn0lt10b 12556 | A nonnegative integer less... |
| nn0lt2 12557 | A nonnegative integer less... |
| nn0le2is012 12558 | A nonnegative integer whic... |
| nn0lem1lt 12559 | Nonnegative integer orderi... |
| nnlem1lt 12560 | Positive integer ordering ... |
| nnltlem1 12561 | Positive integer ordering ... |
| nnm1ge0 12562 | A positive integer decreas... |
| nn0ge0div 12563 | Division of a nonnegative ... |
| zdiv 12564 | Two ways to express " ` M ... |
| zdivadd 12565 | Property of divisibility: ... |
| zdivmul 12566 | Property of divisibility: ... |
| zextle 12567 | An extensionality-like pro... |
| zextlt 12568 | An extensionality-like pro... |
| recnz 12569 | The reciprocal of a number... |
| btwnnz 12570 | A number between an intege... |
| gtndiv 12571 | A larger number does not d... |
| halfnz 12572 | One-half is not an integer... |
| 3halfnz 12573 | Three halves is not an int... |
| suprzcl 12574 | The supremum of a bounded-... |
| prime 12575 | Two ways to express " ` A ... |
| msqznn 12576 | The square of a nonzero in... |
| zneo 12577 | No even integer equals an ... |
| nneo 12578 | A positive integer is even... |
| nneoi 12579 | A positive integer is even... |
| zeo 12580 | An integer is even or odd.... |
| zeo2 12581 | An integer is even or odd ... |
| peano2uz2 12582 | Second Peano postulate for... |
| peano5uzi 12583 | Peano's inductive postulat... |
| peano5uzti 12584 | Peano's inductive postulat... |
| dfuzi 12585 | An expression for the uppe... |
| uzind 12586 | Induction on the upper int... |
| uzind2 12587 | Induction on the upper int... |
| uzind3 12588 | Induction on the upper int... |
| nn0ind 12589 | Principle of Mathematical ... |
| nn0indALT 12590 | Principle of Mathematical ... |
| nn0indd 12591 | Principle of Mathematical ... |
| fzind 12592 | Induction on the integers ... |
| fnn0ind 12593 | Induction on the integers ... |
| nn0ind-raph 12594 | Principle of Mathematical ... |
| zindd 12595 | Principle of Mathematical ... |
| fzindd 12596 | Induction on the integers ... |
| btwnz 12597 | Any real number can be san... |
| zred 12598 | An integer is a real numbe... |
| zcnd 12599 | An integer is a complex nu... |
| znegcld 12600 | Closure law for negative i... |
| peano2zd 12601 | Deduction from second Pean... |
| zaddcld 12602 | Closure of addition of int... |
| zsubcld 12603 | Closure of subtraction of ... |
| zmulcld 12604 | Closure of multiplication ... |
| znnn0nn 12605 | The negative of a negative... |
| zadd2cl 12606 | Increasing an integer by 2... |
| zriotaneg 12607 | The negative of the unique... |
| suprfinzcl 12608 | The supremum of a nonempty... |
| 9p1e10 12611 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12612 | Version of the definition ... |
| decex 12613 | A decimal number is a set.... |
| deceq1 12614 | Equality theorem for the d... |
| deceq2 12615 | Equality theorem for the d... |
| deceq1i 12616 | Equality theorem for the d... |
| deceq2i 12617 | Equality theorem for the d... |
| deceq12i 12618 | Equality theorem for the d... |
| numnncl 12619 | Closure for a numeral (wit... |
| num0u 12620 | Add a zero in the units pl... |
| num0h 12621 | Add a zero in the higher p... |
| numcl 12622 | Closure for a decimal inte... |
| numsuc 12623 | The successor of a decimal... |
| deccl 12624 | Closure for a numeral. (C... |
| 10nn 12625 | 10 is a positive integer. ... |
| 10pos 12626 | The number 10 is positive.... |
| 10nn0 12627 | 10 is a nonnegative intege... |
| 10re 12628 | The number 10 is real. (C... |
| decnncl 12629 | Closure for a numeral. (C... |
| dec0u 12630 | Add a zero in the units pl... |
| dec0h 12631 | Add a zero in the higher p... |
| numnncl2 12632 | Closure for a decimal inte... |
| decnncl2 12633 | Closure for a decimal inte... |
| numlt 12634 | Comparing two decimal inte... |
| numltc 12635 | Comparing two decimal inte... |
| le9lt10 12636 | A "decimal digit" (i.e. a ... |
| declt 12637 | Comparing two decimal inte... |
| decltc 12638 | Comparing two decimal inte... |
| declth 12639 | Comparing two decimal inte... |
| decsuc 12640 | The successor of a decimal... |
| 3declth 12641 | Comparing two decimal inte... |
| 3decltc 12642 | Comparing two decimal inte... |
| decle 12643 | Comparing two decimal inte... |
| decleh 12644 | Comparing two decimal inte... |
| declei 12645 | Comparing a digit to a dec... |
| numlti 12646 | Comparing a digit to a dec... |
| declti 12647 | Comparing a digit to a dec... |
| decltdi 12648 | Comparing a digit to a dec... |
| numsucc 12649 | The successor of a decimal... |
| decsucc 12650 | The successor of a decimal... |
| 1e0p1 12651 | The successor of zero. (C... |
| dec10p 12652 | Ten plus an integer. (Con... |
| numma 12653 | Perform a multiply-add of ... |
| nummac 12654 | Perform a multiply-add of ... |
| numma2c 12655 | Perform a multiply-add of ... |
| numadd 12656 | Add two decimal integers `... |
| numaddc 12657 | Add two decimal integers `... |
| nummul1c 12658 | The product of a decimal i... |
| nummul2c 12659 | The product of a decimal i... |
| decma 12660 | Perform a multiply-add of ... |
| decmac 12661 | Perform a multiply-add of ... |
| decma2c 12662 | Perform a multiply-add of ... |
| decadd 12663 | Add two numerals ` M ` and... |
| decaddc 12664 | Add two numerals ` M ` and... |
| decaddc2 12665 | Add two numerals ` M ` and... |
| decrmanc 12666 | Perform a multiply-add of ... |
| decrmac 12667 | Perform a multiply-add of ... |
| decaddm10 12668 | The sum of two multiples o... |
| decaddi 12669 | Add two numerals ` M ` and... |
| decaddci 12670 | Add two numerals ` M ` and... |
| decaddci2 12671 | Add two numerals ` M ` and... |
| decsubi 12672 | Difference between a numer... |
| decmul1 12673 | The product of a numeral w... |
| decmul1c 12674 | The product of a numeral w... |
| decmul2c 12675 | The product of a numeral w... |
| decmulnc 12676 | The product of a numeral w... |
| 11multnc 12677 | The product of 11 (as nume... |
| decmul10add 12678 | A multiplication of a numb... |
| 6p5lem 12679 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12680 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12681 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12682 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12683 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12684 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12685 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12686 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12687 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12688 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12689 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12690 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12691 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12692 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12693 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12694 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12695 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12696 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12697 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12698 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12699 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12700 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12701 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12702 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12703 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12704 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12705 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12706 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12707 | 4 times 3 equals 12. (Con... |
| 4t4e16 12708 | 4 times 4 equals 16. (Con... |
| 5t2e10 12709 | 5 times 2 equals 10. (Con... |
| 5t3e15 12710 | 5 times 3 equals 15. (Con... |
| 5t4e20 12711 | 5 times 4 equals 20. (Con... |
| 5t5e25 12712 | 5 times 5 equals 25. (Con... |
| 6t2e12 12713 | 6 times 2 equals 12. (Con... |
| 6t3e18 12714 | 6 times 3 equals 18. (Con... |
| 6t4e24 12715 | 6 times 4 equals 24. (Con... |
| 6t5e30 12716 | 6 times 5 equals 30. (Con... |
| 6t6e36 12717 | 6 times 6 equals 36. (Con... |
| 7t2e14 12718 | 7 times 2 equals 14. (Con... |
| 7t3e21 12719 | 7 times 3 equals 21. (Con... |
| 7t4e28 12720 | 7 times 4 equals 28. (Con... |
| 7t5e35 12721 | 7 times 5 equals 35. (Con... |
| 7t6e42 12722 | 7 times 6 equals 42. (Con... |
| 7t7e49 12723 | 7 times 7 equals 49. (Con... |
| 8t2e16 12724 | 8 times 2 equals 16. (Con... |
| 8t3e24 12725 | 8 times 3 equals 24. (Con... |
| 8t4e32 12726 | 8 times 4 equals 32. (Con... |
| 8t5e40 12727 | 8 times 5 equals 40. (Con... |
| 8t6e48 12728 | 8 times 6 equals 48. (Con... |
| 8t7e56 12729 | 8 times 7 equals 56. (Con... |
| 8t8e64 12730 | 8 times 8 equals 64. (Con... |
| 9t2e18 12731 | 9 times 2 equals 18. (Con... |
| 9t3e27 12732 | 9 times 3 equals 27. (Con... |
| 9t4e36 12733 | 9 times 4 equals 36. (Con... |
| 9t5e45 12734 | 9 times 5 equals 45. (Con... |
| 9t6e54 12735 | 9 times 6 equals 54. (Con... |
| 9t7e63 12736 | 9 times 7 equals 63. (Con... |
| 9t8e72 12737 | 9 times 8 equals 72. (Con... |
| 9t9e81 12738 | 9 times 9 equals 81. (Con... |
| 9t11e99 12739 | 9 times 11 equals 99. (Co... |
| 9lt10 12740 | 9 is less than 10. (Contr... |
| 8lt10 12741 | 8 is less than 10. (Contr... |
| 7lt10 12742 | 7 is less than 10. (Contr... |
| 6lt10 12743 | 6 is less than 10. (Contr... |
| 5lt10 12744 | 5 is less than 10. (Contr... |
| 4lt10 12745 | 4 is less than 10. (Contr... |
| 3lt10 12746 | 3 is less than 10. (Contr... |
| 2lt10 12747 | 2 is less than 10. (Contr... |
| 1lt10 12748 | 1 is less than 10. (Contr... |
| decbin0 12749 | Decompose base 4 into base... |
| decbin2 12750 | Decompose base 4 into base... |
| decbin3 12751 | Decompose base 4 into base... |
| 5recm6rec 12752 | One fifth minus one sixth.... |
| uzval 12755 | The value of the upper int... |
| uzf 12756 | The domain and codomain of... |
| eluz1 12757 | Membership in the upper se... |
| eluzel2 12758 | Implication of membership ... |
| eluz2 12759 | Membership in an upper set... |
| eluzmn 12760 | Membership in an earlier u... |
| eluz1i 12761 | Membership in an upper set... |
| eluzuzle 12762 | An integer in an upper set... |
| eluzelz 12763 | A member of an upper set o... |
| eluzelre 12764 | A member of an upper set o... |
| eluzelcn 12765 | A member of an upper set o... |
| eluzle 12766 | Implication of membership ... |
| eluz 12767 | Membership in an upper set... |
| uzid 12768 | Membership of the least me... |
| uzidd 12769 | Membership of the least me... |
| uzn0 12770 | The upper integers are all... |
| uztrn 12771 | Transitive law for sets of... |
| uztrn2 12772 | Transitive law for sets of... |
| uzneg 12773 | Contraposition law for upp... |
| uzssz 12774 | An upper set of integers i... |
| uzssre 12775 | An upper set of integers i... |
| uzss 12776 | Subset relationship for tw... |
| uztric 12777 | Totality of the ordering r... |
| uz11 12778 | The upper integers functio... |
| eluzp1m1 12779 | Membership in the next upp... |
| eluzp1l 12780 | Strict ordering implied by... |
| eluzp1p1 12781 | Membership in the next upp... |
| eluzadd 12782 | Membership in a later uppe... |
| eluzsub 12783 | Membership in an earlier u... |
| eluzaddi 12784 | Membership in a later uppe... |
| eluzsubi 12785 | Membership in an earlier u... |
| subeluzsub 12786 | Membership of a difference... |
| uzm1 12787 | Choices for an element of ... |
| uznn0sub 12788 | The nonnegative difference... |
| uzin 12789 | Intersection of two upper ... |
| uzp1 12790 | Choices for an element of ... |
| nn0uz 12791 | Nonnegative integers expre... |
| nnuz 12792 | Positive integers expresse... |
| elnnuz 12793 | A positive integer express... |
| elnn0uz 12794 | A nonnegative integer expr... |
| 1eluzge0 12795 | 1 is an integer greater th... |
| 2eluzge0 12796 | 2 is an integer greater th... |
| 2eluzge1 12797 | 2 is an integer greater th... |
| 5eluz3 12798 | 5 is an integer greater th... |
| uzuzle23 12799 | An integer greater than or... |
| uzuzle24 12800 | An integer greater than or... |
| uzuzle34 12801 | An integer greater than or... |
| uzuzle35 12802 | An integer greater than or... |
| eluz2nn 12803 | An integer greater than or... |
| eluz3nn 12804 | An integer greater than or... |
| eluz4nn 12805 | An integer greater than or... |
| eluz5nn 12806 | An integer greater than or... |
| eluzge2nn0 12807 | If an integer is greater t... |
| eluz2n0 12808 | An integer greater than or... |
| uz3m2nn 12809 | An integer greater than or... |
| uznnssnn 12810 | The upper integers startin... |
| raluz 12811 | Restricted universal quant... |
| raluz2 12812 | Restricted universal quant... |
| rexuz 12813 | Restricted existential qua... |
| rexuz2 12814 | Restricted existential qua... |
| 2rexuz 12815 | Double existential quantif... |
| peano2uz 12816 | Second Peano postulate for... |
| peano2uzs 12817 | Second Peano postulate for... |
| peano2uzr 12818 | Reversed second Peano axio... |
| uzaddcl 12819 | Addition closure law for a... |
| nn0pzuz 12820 | The sum of a nonnegative i... |
| uzind4 12821 | Induction on the upper set... |
| uzind4ALT 12822 | Induction on the upper set... |
| uzind4s 12823 | Induction on the upper set... |
| uzind4s2 12824 | Induction on the upper set... |
| uzind4i 12825 | Induction on the upper int... |
| uzwo 12826 | Well-ordering principle: a... |
| uzwo2 12827 | Well-ordering principle: a... |
| nnwo 12828 | Well-ordering principle: a... |
| nnwof 12829 | Well-ordering principle: a... |
| nnwos 12830 | Well-ordering principle: a... |
| indstr 12831 | Strong Mathematical Induct... |
| eluznn0 12832 | Membership in a nonnegativ... |
| eluznn 12833 | Membership in a positive u... |
| eluz2b1 12834 | Two ways to say "an intege... |
| eluz2gt1 12835 | An integer greater than or... |
| eluz2b2 12836 | Two ways to say "an intege... |
| eluz2b3 12837 | Two ways to say "an intege... |
| uz2m1nn 12838 | One less than an integer g... |
| 1nuz2 12839 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12840 | A positive integer is eith... |
| uz2mulcl 12841 | Closure of multiplication ... |
| indstr2 12842 | Strong Mathematical Induct... |
| uzinfi 12843 | Extract the lower bound of... |
| nninf 12844 | The infimum of the set of ... |
| nn0inf 12845 | The infimum of the set of ... |
| infssuzle 12846 | The infimum of a subset of... |
| infssuzcl 12847 | The infimum of a subset of... |
| ublbneg 12848 | The image under negation o... |
| eqreznegel 12849 | Two ways to express the im... |
| supminf 12850 | The supremum of a bounded-... |
| lbzbi 12851 | If a set of reals is bound... |
| zsupss 12852 | Any nonempty bounded subse... |
| suprzcl2 12853 | The supremum of a bounded-... |
| suprzub 12854 | The supremum of a bounded-... |
| uzsupss 12855 | Any bounded subset of an u... |
| nn01to3 12856 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12857 | Alternate proof of ~ nn0ge... |
| uzwo3 12858 | Well-ordering principle: a... |
| zmin 12859 | There is a unique smallest... |
| zmax 12860 | There is a unique largest ... |
| zbtwnre 12861 | There is a unique integer ... |
| rebtwnz 12862 | There is a unique greatest... |
| elq 12865 | Membership in the set of r... |
| qmulz 12866 | If ` A ` is rational, then... |
| znq 12867 | The ratio of an integer an... |
| qre 12868 | A rational number is a rea... |
| zq 12869 | An integer is a rational n... |
| qred 12870 | A rational number is a rea... |
| zssq 12871 | The integers are a subset ... |
| nn0ssq 12872 | The nonnegative integers a... |
| nnssq 12873 | The positive integers are ... |
| qssre 12874 | The rationals are a subset... |
| qsscn 12875 | The rationals are a subset... |
| qex 12876 | The set of rational number... |
| nnq 12877 | A positive integer is rati... |
| qcn 12878 | A rational number is a com... |
| qexALT 12879 | Alternate proof of ~ qex .... |
| qaddcl 12880 | Closure of addition of rat... |
| qnegcl 12881 | Closure law for the negati... |
| qmulcl 12882 | Closure of multiplication ... |
| qsubcl 12883 | Closure of subtraction of ... |
| qreccl 12884 | Closure of reciprocal of r... |
| qdivcl 12885 | Closure of division of rat... |
| qrevaddcl 12886 | Reverse closure law for ad... |
| nnrecq 12887 | The reciprocal of a positi... |
| irradd 12888 | The sum of an irrational n... |
| irrmul 12889 | The product of an irration... |
| elpq 12890 | A positive rational is the... |
| elpqb 12891 | A class is a positive rati... |
| rpnnen1lem2 12892 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12893 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12894 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12895 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12896 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12897 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12898 | One half of ~ rpnnen , whe... |
| reexALT 12899 | Alternate proof of ~ reex ... |
| cnref1o 12900 | There is a natural one-to-... |
| cnexALT 12901 | The set of complex numbers... |
| xrex 12902 | The set of extended reals ... |
| mpoaddex 12903 | The addition operation is ... |
| addex 12904 | The addition operation is ... |
| mpomulex 12905 | The multiplication operati... |
| mulex 12906 | The multiplication operati... |
| elrp 12909 | Membership in the set of p... |
| elrpii 12910 | Membership in the set of p... |
| 1rp 12911 | 1 is a positive real. (Co... |
| 2rp 12912 | 2 is a positive real. (Co... |
| 3rp 12913 | 3 is a positive real. (Co... |
| 5rp 12914 | 5 is a positive real. (Co... |
| rpssre 12915 | The positive reals are a s... |
| rpre 12916 | A positive real is a real.... |
| rpxr 12917 | A positive real is an exte... |
| rpcn 12918 | A positive real is a compl... |
| nnrp 12919 | A positive integer is a po... |
| rpgt0 12920 | A positive real is greater... |
| rpge0 12921 | A positive real is greater... |
| rpregt0 12922 | A positive real is a posit... |
| rprege0 12923 | A positive real is a nonne... |
| rpne0 12924 | A positive real is nonzero... |
| rprene0 12925 | A positive real is a nonze... |
| rpcnne0 12926 | A positive real is a nonze... |
| neglt 12927 | The negative of a positive... |
| rpcndif0 12928 | A positive real number is ... |
| ralrp 12929 | Quantification over positi... |
| rexrp 12930 | Quantification over positi... |
| rpaddcl 12931 | Closure law for addition o... |
| rpmulcl 12932 | Closure law for multiplica... |
| rpmtmip 12933 | "Minus times minus is plus... |
| rpdivcl 12934 | Closure law for division o... |
| rpreccl 12935 | Closure law for reciprocat... |
| rphalfcl 12936 | Closure law for half of a ... |
| rpgecl 12937 | A number greater than or e... |
| rphalflt 12938 | Half of a positive real is... |
| rerpdivcl 12939 | Closure law for division o... |
| ge0p1rp 12940 | A nonnegative number plus ... |
| rpneg 12941 | Either a nonzero real or i... |
| negelrp 12942 | Elementhood of a negation ... |
| negelrpd 12943 | The negation of a negative... |
| 0nrp 12944 | Zero is not a positive rea... |
| ltsubrp 12945 | Subtracting a positive rea... |
| ltaddrp 12946 | Adding a positive number t... |
| difrp 12947 | Two ways to say one number... |
| elrpd 12948 | Membership in the set of p... |
| nnrpd 12949 | A positive integer is a po... |
| zgt1rpn0n1 12950 | An integer greater than 1 ... |
| rpred 12951 | A positive real is a real.... |
| rpxrd 12952 | A positive real is an exte... |
| rpcnd 12953 | A positive real is a compl... |
| rpgt0d 12954 | A positive real is greater... |
| rpge0d 12955 | A positive real is greater... |
| rpne0d 12956 | A positive real is nonzero... |
| rpregt0d 12957 | A positive real is real an... |
| rprege0d 12958 | A positive real is real an... |
| rprene0d 12959 | A positive real is a nonze... |
| rpcnne0d 12960 | A positive real is a nonze... |
| rpreccld 12961 | Closure law for reciprocat... |
| rprecred 12962 | Closure law for reciprocat... |
| rphalfcld 12963 | Closure law for half of a ... |
| reclt1d 12964 | The reciprocal of a positi... |
| recgt1d 12965 | The reciprocal of a positi... |
| rpaddcld 12966 | Closure law for addition o... |
| rpmulcld 12967 | Closure law for multiplica... |
| rpdivcld 12968 | Closure law for division o... |
| ltrecd 12969 | The reciprocal of both sid... |
| lerecd 12970 | The reciprocal of both sid... |
| ltrec1d 12971 | Reciprocal swap in a 'less... |
| lerec2d 12972 | Reciprocal swap in a 'less... |
| lediv2ad 12973 | Division of both sides of ... |
| ltdiv2d 12974 | Division of a positive num... |
| lediv2d 12975 | Division of a positive num... |
| ledivdivd 12976 | Invert ratios of positive ... |
| divge1 12977 | The ratio of a number over... |
| divlt1lt 12978 | A real number divided by a... |
| divle1le 12979 | A real number divided by a... |
| ledivge1le 12980 | If a number is less than o... |
| ge0p1rpd 12981 | A nonnegative number plus ... |
| rerpdivcld 12982 | Closure law for division o... |
| ltsubrpd 12983 | Subtracting a positive rea... |
| ltaddrpd 12984 | Adding a positive number t... |
| ltaddrp2d 12985 | Adding a positive number t... |
| ltmulgt11d 12986 | Multiplication by a number... |
| ltmulgt12d 12987 | Multiplication by a number... |
| gt0divd 12988 | Division of a positive num... |
| ge0divd 12989 | Division of a nonnegative ... |
| rpgecld 12990 | A number greater than or e... |
| divge0d 12991 | The ratio of nonnegative a... |
| ltmul1d 12992 | The ratio of nonnegative a... |
| ltmul2d 12993 | Multiplication of both sid... |
| lemul1d 12994 | Multiplication of both sid... |
| lemul2d 12995 | Multiplication of both sid... |
| ltdiv1d 12996 | Division of both sides of ... |
| lediv1d 12997 | Division of both sides of ... |
| ltmuldivd 12998 | 'Less than' relationship b... |
| ltmuldiv2d 12999 | 'Less than' relationship b... |
| lemuldivd 13000 | 'Less than or equal to' re... |
| lemuldiv2d 13001 | 'Less than or equal to' re... |
| ltdivmuld 13002 | 'Less than' relationship b... |
| ltdivmul2d 13003 | 'Less than' relationship b... |
| ledivmuld 13004 | 'Less than or equal to' re... |
| ledivmul2d 13005 | 'Less than or equal to' re... |
| ltmul1dd 13006 | The ratio of nonnegative a... |
| ltmul2dd 13007 | Multiplication of both sid... |
| ltdiv1dd 13008 | Division of both sides of ... |
| lediv1dd 13009 | Division of both sides of ... |
| lediv12ad 13010 | Comparison of ratio of two... |
| mul2lt0rlt0 13011 | If the result of a multipl... |
| mul2lt0rgt0 13012 | If the result of a multipl... |
| mul2lt0llt0 13013 | If the result of a multipl... |
| mul2lt0lgt0 13014 | If the result of a multipl... |
| mul2lt0bi 13015 | If the result of a multipl... |
| prodge0rd 13016 | Infer that a multiplicand ... |
| prodge0ld 13017 | Infer that a multiplier is... |
| ltdiv23d 13018 | Swap denominator with othe... |
| lediv23d 13019 | Swap denominator with othe... |
| lt2mul2divd 13020 | The ratio of nonnegative a... |
| nnledivrp 13021 | Division of a positive int... |
| nn0ledivnn 13022 | Division of a nonnegative ... |
| addlelt 13023 | If the sum of a real numbe... |
| ge2halflem1 13024 | Half of an integer greater... |
| ltxr 13031 | The 'less than' binary rel... |
| elxr 13032 | Membership in the set of e... |
| xrnemnf 13033 | An extended real other tha... |
| xrnepnf 13034 | An extended real other tha... |
| xrltnr 13035 | The extended real 'less th... |
| ltpnf 13036 | Any (finite) real is less ... |
| ltpnfd 13037 | Any (finite) real is less ... |
| 0ltpnf 13038 | Zero is less than plus inf... |
| mnflt 13039 | Minus infinity is less tha... |
| mnfltd 13040 | Minus infinity is less tha... |
| mnflt0 13041 | Minus infinity is less tha... |
| mnfltpnf 13042 | Minus infinity is less tha... |
| mnfltxr 13043 | Minus infinity is less tha... |
| pnfnlt 13044 | No extended real is greate... |
| nltmnf 13045 | No extended real is less t... |
| pnfge 13046 | Plus infinity is an upper ... |
| pnfged 13047 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13048 | An extended nonnegative in... |
| 0lepnf 13049 | 0 less than or equal to po... |
| xnn0ge0 13050 | An extended nonnegative in... |
| mnfle 13051 | Minus infinity is less tha... |
| mnfled 13052 | Minus infinity is less tha... |
| xrltnsym 13053 | Ordering on the extended r... |
| xrltnsym2 13054 | 'Less than' is antisymmetr... |
| xrlttri 13055 | Ordering on the extended r... |
| xrlttr 13056 | Ordering on the extended r... |
| xrltso 13057 | 'Less than' is a strict or... |
| xrlttri2 13058 | Trichotomy law for 'less t... |
| xrlttri3 13059 | Trichotomy law for 'less t... |
| xrleloe 13060 | 'Less than or equal' expre... |
| xrleltne 13061 | 'Less than or equal to' im... |
| xrltlen 13062 | 'Less than' expressed in t... |
| dfle2 13063 | Alternative definition of ... |
| dflt2 13064 | Alternative definition of ... |
| xrltle 13065 | 'Less than' implies 'less ... |
| xrltled 13066 | 'Less than' implies 'less ... |
| xrleid 13067 | 'Less than or equal to' is... |
| xrleidd 13068 | 'Less than or equal to' is... |
| xrletri 13069 | Trichotomy law for extende... |
| xrletri3 13070 | Trichotomy law for extende... |
| xrletrid 13071 | Trichotomy law for extende... |
| xrlelttr 13072 | Transitive law for orderin... |
| xrltletr 13073 | Transitive law for orderin... |
| xrletr 13074 | Transitive law for orderin... |
| xrlttrd 13075 | Transitive law for orderin... |
| xrlelttrd 13076 | Transitive law for orderin... |
| xrltletrd 13077 | Transitive law for orderin... |
| xrletrd 13078 | Transitive law for orderin... |
| xrltne 13079 | 'Less than' implies not eq... |
| xrgtned 13080 | 'Greater than' implies not... |
| nltpnft 13081 | An extended real is not le... |
| xgepnf 13082 | An extended real which is ... |
| ngtmnft 13083 | An extended real is not gr... |
| xlemnf 13084 | An extended real which is ... |
| xrrebnd 13085 | An extended real is real i... |
| xrre 13086 | A way of proving that an e... |
| xrre2 13087 | An extended real between t... |
| xrre3 13088 | A way of proving that an e... |
| ge0gtmnf 13089 | A nonnegative extended rea... |
| ge0nemnf 13090 | A nonnegative extended rea... |
| xrrege0 13091 | A nonnegative extended rea... |
| xrmax1 13092 | An extended real is less t... |
| xrmax2 13093 | An extended real is less t... |
| xrmin1 13094 | The minimum of two extende... |
| xrmin2 13095 | The minimum of two extende... |
| xrmaxeq 13096 | The maximum of two extende... |
| xrmineq 13097 | The minimum of two extende... |
| xrmaxlt 13098 | Two ways of saying the max... |
| xrltmin 13099 | Two ways of saying an exte... |
| xrmaxle 13100 | Two ways of saying the max... |
| xrlemin 13101 | Two ways of saying a numbe... |
| max1 13102 | A number is less than or e... |
| max1ALT 13103 | A number is less than or e... |
| max2 13104 | A number is less than or e... |
| 2resupmax 13105 | The supremum of two real n... |
| min1 13106 | The minimum of two numbers... |
| min2 13107 | The minimum of two numbers... |
| maxle 13108 | Two ways of saying the max... |
| lemin 13109 | Two ways of saying a numbe... |
| maxlt 13110 | Two ways of saying the max... |
| ltmin 13111 | Two ways of saying a numbe... |
| lemaxle 13112 | A real number which is les... |
| max0sub 13113 | Decompose a real number in... |
| ifle 13114 | An if statement transforms... |
| z2ge 13115 | There exists an integer gr... |
| qbtwnre 13116 | The rational numbers are d... |
| qbtwnxr 13117 | The rational numbers are d... |
| qsqueeze 13118 | If a nonnegative real is l... |
| qextltlem 13119 | Lemma for ~ qextlt and qex... |
| qextlt 13120 | An extensionality-like pro... |
| qextle 13121 | An extensionality-like pro... |
| xralrple 13122 | Show that ` A ` is less th... |
| alrple 13123 | Show that ` A ` is less th... |
| xnegeq 13124 | Equality of two extended n... |
| xnegex 13125 | A negative extended real e... |
| xnegpnf 13126 | Minus ` +oo ` . Remark of... |
| xnegmnf 13127 | Minus ` -oo ` . Remark of... |
| rexneg 13128 | Minus a real number. Rema... |
| xneg0 13129 | The negative of zero. (Co... |
| xnegcl 13130 | Closure of extended real n... |
| xnegneg 13131 | Extended real version of ~... |
| xneg11 13132 | Extended real version of ~... |
| xltnegi 13133 | Forward direction of ~ xlt... |
| xltneg 13134 | Extended real version of ~... |
| xleneg 13135 | Extended real version of ~... |
| xlt0neg1 13136 | Extended real version of ~... |
| xlt0neg2 13137 | Extended real version of ~... |
| xle0neg1 13138 | Extended real version of ~... |
| xle0neg2 13139 | Extended real version of ~... |
| xaddval 13140 | Value of the extended real... |
| xaddf 13141 | The extended real addition... |
| xmulval 13142 | Value of the extended real... |
| xaddpnf1 13143 | Addition of positive infin... |
| xaddpnf2 13144 | Addition of positive infin... |
| xaddmnf1 13145 | Addition of negative infin... |
| xaddmnf2 13146 | Addition of negative infin... |
| pnfaddmnf 13147 | Addition of positive and n... |
| mnfaddpnf 13148 | Addition of negative and p... |
| rexadd 13149 | The extended real addition... |
| rexsub 13150 | Extended real subtraction ... |
| rexaddd 13151 | The extended real addition... |
| xnn0xaddcl 13152 | The extended nonnegative i... |
| xaddnemnf 13153 | Closure of extended real a... |
| xaddnepnf 13154 | Closure of extended real a... |
| xnegid 13155 | Extended real version of ~... |
| xaddcl 13156 | The extended real addition... |
| xaddcom 13157 | The extended real addition... |
| xaddrid 13158 | Extended real version of ~... |
| xaddlid 13159 | Extended real version of ~... |
| xaddridd 13160 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13161 | Extended nonnegative integ... |
| xnn0lenn0nn0 13162 | An extended nonnegative in... |
| xnn0le2is012 13163 | An extended nonnegative in... |
| xnn0xadd0 13164 | The sum of two extended no... |
| xnegdi 13165 | Extended real version of ~... |
| xaddass 13166 | Associativity of extended ... |
| xaddass2 13167 | Associativity of extended ... |
| xpncan 13168 | Extended real version of ~... |
| xnpcan 13169 | Extended real version of ~... |
| xleadd1a 13170 | Extended real version of ~... |
| xleadd2a 13171 | Commuted form of ~ xleadd1... |
| xleadd1 13172 | Weakened version of ~ xlea... |
| xltadd1 13173 | Extended real version of ~... |
| xltadd2 13174 | Extended real version of ~... |
| xaddge0 13175 | The sum of nonnegative ext... |
| xle2add 13176 | Extended real version of ~... |
| xlt2add 13177 | Extended real version of ~... |
| xsubge0 13178 | Extended real version of ~... |
| xposdif 13179 | Extended real version of ~... |
| xlesubadd 13180 | Under certain conditions, ... |
| xmullem 13181 | Lemma for ~ rexmul . (Con... |
| xmullem2 13182 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13183 | Extended real multiplicati... |
| xmul01 13184 | Extended real version of ~... |
| xmul02 13185 | Extended real version of ~... |
| xmulneg1 13186 | Extended real version of ~... |
| xmulneg2 13187 | Extended real version of ~... |
| rexmul 13188 | The extended real multipli... |
| xmulf 13189 | The extended real multipli... |
| xmulcl 13190 | Closure of extended real m... |
| xmulpnf1 13191 | Multiplication by plus inf... |
| xmulpnf2 13192 | Multiplication by plus inf... |
| xmulmnf1 13193 | Multiplication by minus in... |
| xmulmnf2 13194 | Multiplication by minus in... |
| xmulpnf1n 13195 | Multiplication by plus inf... |
| xmulrid 13196 | Extended real version of ~... |
| xmullid 13197 | Extended real version of ~... |
| xmulm1 13198 | Extended real version of ~... |
| xmulasslem2 13199 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13200 | Extended real version of ~... |
| xmulge0 13201 | Extended real version of ~... |
| xmulasslem 13202 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13203 | Lemma for ~ xmulass . (Co... |
| xmulass 13204 | Associativity of the exten... |
| xlemul1a 13205 | Extended real version of ~... |
| xlemul2a 13206 | Extended real version of ~... |
| xlemul1 13207 | Extended real version of ~... |
| xlemul2 13208 | Extended real version of ~... |
| xltmul1 13209 | Extended real version of ~... |
| xltmul2 13210 | Extended real version of ~... |
| xadddilem 13211 | Lemma for ~ xadddi . (Con... |
| xadddi 13212 | Distributive property for ... |
| xadddir 13213 | Commuted version of ~ xadd... |
| xadddi2 13214 | The assumption that the mu... |
| xadddi2r 13215 | Commuted version of ~ xadd... |
| x2times 13216 | Extended real version of ~... |
| xnegcld 13217 | Closure of extended real n... |
| xaddcld 13218 | The extended real addition... |
| xmulcld 13219 | Closure of extended real m... |
| xadd4d 13220 | Rearrangement of 4 terms i... |
| xnn0add4d 13221 | Rearrangement of 4 terms i... |
| xrsupexmnf 13222 | Adding minus infinity to a... |
| xrinfmexpnf 13223 | Adding plus infinity to a ... |
| xrsupsslem 13224 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13225 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13226 | Any subset of extended rea... |
| xrinfmss 13227 | Any subset of extended rea... |
| xrinfmss2 13228 | Any subset of extended rea... |
| xrub 13229 | By quantifying only over r... |
| supxr 13230 | The supremum of a set of e... |
| supxr2 13231 | The supremum of a set of e... |
| supxrcl 13232 | The supremum of an arbitra... |
| supxrun 13233 | The supremum of the union ... |
| supxrmnf 13234 | Adding minus infinity to a... |
| supxrpnf 13235 | The supremum of a set of e... |
| supxrunb1 13236 | The supremum of an unbound... |
| supxrunb2 13237 | The supremum of an unbound... |
| supxrbnd1 13238 | The supremum of a bounded-... |
| supxrbnd2 13239 | The supremum of a bounded-... |
| xrsup0 13240 | The supremum of an empty s... |
| supxrub 13241 | A member of a set of exten... |
| supxrlub 13242 | The supremum of a set of e... |
| supxrleub 13243 | The supremum of a set of e... |
| supxrre 13244 | The real and extended real... |
| supxrbnd 13245 | The supremum of a bounded-... |
| supxrgtmnf 13246 | The supremum of a nonempty... |
| supxrre1 13247 | The supremum of a nonempty... |
| supxrre2 13248 | The supremum of a nonempty... |
| supxrss 13249 | Smaller sets of extended r... |
| xrsupssd 13250 | Inequality deduction for s... |
| infxrcl 13251 | The infimum of an arbitrar... |
| infxrlb 13252 | A member of a set of exten... |
| infxrgelb 13253 | The infimum of a set of ex... |
| infxrre 13254 | The real and extended real... |
| infxrmnf 13255 | The infinimum of a set of ... |
| xrinf0 13256 | The infimum of the empty s... |
| infxrss 13257 | Larger sets of extended re... |
| reltre 13258 | For all real numbers there... |
| rpltrp 13259 | For all positive real numb... |
| reltxrnmnf 13260 | For all extended real numb... |
| infmremnf 13261 | The infimum of the reals i... |
| infmrp1 13262 | The infimum of the positiv... |
| ixxval 13271 | Value of the interval func... |
| elixx1 13272 | Membership in an interval ... |
| ixxf 13273 | The set of intervals of ex... |
| ixxex 13274 | The set of intervals of ex... |
| ixxssxr 13275 | The set of intervals of ex... |
| elixx3g 13276 | Membership in a set of ope... |
| ixxssixx 13277 | An interval is a subset of... |
| ixxdisj 13278 | Split an interval into dis... |
| ixxun 13279 | Split an interval into two... |
| ixxin 13280 | Intersection of two interv... |
| ixxss1 13281 | Subset relationship for in... |
| ixxss2 13282 | Subset relationship for in... |
| ixxss12 13283 | Subset relationship for in... |
| ixxub 13284 | Extract the upper bound of... |
| ixxlb 13285 | Extract the lower bound of... |
| iooex 13286 | The set of open intervals ... |
| iooval 13287 | Value of the open interval... |
| ioo0 13288 | An empty open interval of ... |
| ioon0 13289 | An open interval of extend... |
| ndmioo 13290 | The open interval function... |
| iooid 13291 | An open interval with iden... |
| elioo3g 13292 | Membership in a set of ope... |
| elioore 13293 | A member of an open interv... |
| lbioo 13294 | An open interval does not ... |
| ubioo 13295 | An open interval does not ... |
| iooval2 13296 | Value of the open interval... |
| iooin 13297 | Intersection of two open i... |
| iooss1 13298 | Subset relationship for op... |
| iooss2 13299 | Subset relationship for op... |
| iocval 13300 | Value of the open-below, c... |
| icoval 13301 | Value of the closed-below,... |
| iccval 13302 | Value of the closed interv... |
| elioo1 13303 | Membership in an open inte... |
| elioo2 13304 | Membership in an open inte... |
| elioc1 13305 | Membership in an open-belo... |
| elico1 13306 | Membership in a closed-bel... |
| elicc1 13307 | Membership in a closed int... |
| iccid 13308 | A closed interval with ide... |
| ico0 13309 | An empty open interval of ... |
| ioc0 13310 | An empty open interval of ... |
| icc0 13311 | An empty closed interval o... |
| dfrp2 13312 | Alternate definition of th... |
| elicod 13313 | Membership in a left-close... |
| icogelb 13314 | An element of a left-close... |
| icogelbd 13315 | An element of a left-close... |
| elicore 13316 | A member of a left-closed ... |
| ubioc1 13317 | The upper bound belongs to... |
| lbico1 13318 | The lower bound belongs to... |
| iccleub 13319 | An element of a closed int... |
| iccgelb 13320 | An element of a closed int... |
| elioo5 13321 | Membership in an open inte... |
| eliooxr 13322 | A nonempty open interval s... |
| eliooord 13323 | Ordering implied by a memb... |
| elioo4g 13324 | Membership in an open inte... |
| ioossre 13325 | An open interval is a set ... |
| ioosscn 13326 | An open interval is a set ... |
| elioc2 13327 | Membership in an open-belo... |
| elico2 13328 | Membership in a closed-bel... |
| elicc2 13329 | Membership in a closed rea... |
| elicc2i 13330 | Inference for membership i... |
| elicc4 13331 | Membership in a closed rea... |
| iccss 13332 | Condition for a closed int... |
| iccssioo 13333 | Condition for a closed int... |
| icossico 13334 | Condition for a closed-bel... |
| iccss2 13335 | Condition for a closed int... |
| iccssico 13336 | Condition for a closed int... |
| iccssioo2 13337 | Condition for a closed int... |
| iccssico2 13338 | Condition for a closed int... |
| icossico2d 13339 | Condition for a closed-bel... |
| ioomax 13340 | The open interval from min... |
| iccmax 13341 | The closed interval from m... |
| ioopos 13342 | The set of positive reals ... |
| ioorp 13343 | The set of positive reals ... |
| iooshf 13344 | Shift the arguments of the... |
| iocssre 13345 | A closed-above interval wi... |
| icossre 13346 | A closed-below interval wi... |
| iccssre 13347 | A closed real interval is ... |
| iccssxr 13348 | A closed interval is a set... |
| iocssxr 13349 | An open-below, closed-abov... |
| icossxr 13350 | A closed-below, open-above... |
| ioossicc 13351 | An open interval is a subs... |
| iccssred 13352 | A closed real interval is ... |
| eliccxr 13353 | A member of a closed inter... |
| icossicc 13354 | A closed-below, open-above... |
| iocssicc 13355 | A closed-above, open-below... |
| ioossico 13356 | An open interval is a subs... |
| iocssioo 13357 | Condition for a closed int... |
| icossioo 13358 | Condition for a closed int... |
| ioossioo 13359 | Condition for an open inte... |
| iccsupr 13360 | A nonempty subset of a clo... |
| elioopnf 13361 | Membership in an unbounded... |
| elioomnf 13362 | Membership in an unbounded... |
| elicopnf 13363 | Membership in a closed unb... |
| repos 13364 | Two ways of saying that a ... |
| ioof 13365 | The set of open intervals ... |
| iccf 13366 | The set of closed interval... |
| unirnioo 13367 | The union of the range of ... |
| dfioo2 13368 | Alternate definition of th... |
| ioorebas 13369 | Open intervals are element... |
| xrge0neqmnf 13370 | A nonnegative extended rea... |
| xrge0nre 13371 | An extended real which is ... |
| elrege0 13372 | The predicate "is a nonneg... |
| nn0rp0 13373 | A nonnegative integer is a... |
| rge0ssre 13374 | Nonnegative real numbers a... |
| elxrge0 13375 | Elementhood in the set of ... |
| 0e0icopnf 13376 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13377 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13378 | The nonnegative reals are ... |
| ge0mulcl 13379 | The nonnegative reals are ... |
| ge0xaddcl 13380 | The nonnegative reals are ... |
| ge0xmulcl 13381 | The nonnegative extended r... |
| lbicc2 13382 | The lower bound of a close... |
| ubicc2 13383 | The upper bound of a close... |
| elicc01 13384 | Membership in the closed r... |
| elunitrn 13385 | The closed unit interval i... |
| elunitcn 13386 | The closed unit interval i... |
| 0elunit 13387 | Zero is an element of the ... |
| 1elunit 13388 | One is an element of the c... |
| iooneg 13389 | Membership in a negated op... |
| iccneg 13390 | Membership in a negated cl... |
| icoshft 13391 | A shifted real is a member... |
| icoshftf1o 13392 | Shifting a closed-below, o... |
| icoun 13393 | The union of two adjacent ... |
| icodisj 13394 | Adjacent left-closed right... |
| ioounsn 13395 | The union of an open inter... |
| snunioo 13396 | The closure of one end of ... |
| snunico 13397 | The closure of the open en... |
| snunioc 13398 | The closure of the open en... |
| prunioo 13399 | The closure of an open rea... |
| ioodisj 13400 | If the upper bound of one ... |
| ioojoin 13401 | Join two open intervals to... |
| difreicc 13402 | The class difference of ` ... |
| iccsplit 13403 | Split a closed interval in... |
| iccshftr 13404 | Membership in a shifted in... |
| iccshftri 13405 | Membership in a shifted in... |
| iccshftl 13406 | Membership in a shifted in... |
| iccshftli 13407 | Membership in a shifted in... |
| iccdil 13408 | Membership in a dilated in... |
| iccdili 13409 | Membership in a dilated in... |
| icccntr 13410 | Membership in a contracted... |
| icccntri 13411 | Membership in a contracted... |
| divelunit 13412 | A condition for a ratio to... |
| lincmb01cmp 13413 | A linear combination of tw... |
| iccf1o 13414 | Describe a bijection from ... |
| iccen 13415 | Any nontrivial closed inte... |
| xov1plusxeqvd 13416 | A complex number ` X ` is ... |
| unitssre 13417 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13418 | The closed unit interval i... |
| supicc 13419 | Supremum of a bounded set ... |
| supiccub 13420 | The supremum of a bounded ... |
| supicclub 13421 | The supremum of a bounded ... |
| supicclub2 13422 | The supremum of a bounded ... |
| zltaddlt1le 13423 | The sum of an integer and ... |
| xnn0xrge0 13424 | An extended nonnegative in... |
| fzval 13427 | The value of a finite set ... |
| fzval2 13428 | An alternative way of expr... |
| fzf 13429 | Establish the domain and c... |
| elfz1 13430 | Membership in a finite set... |
| elfz 13431 | Membership in a finite set... |
| elfz2 13432 | Membership in a finite set... |
| elfzd 13433 | Membership in a finite set... |
| elfz5 13434 | Membership in a finite set... |
| elfz4 13435 | Membership in a finite set... |
| elfzuzb 13436 | Membership in a finite set... |
| eluzfz 13437 | Membership in a finite set... |
| elfzuz 13438 | A member of a finite set o... |
| elfzuz3 13439 | Membership in a finite set... |
| elfzel2 13440 | Membership in a finite set... |
| elfzel1 13441 | Membership in a finite set... |
| elfzelz 13442 | A member of a finite set o... |
| elfzelzd 13443 | A member of a finite set o... |
| fzssz 13444 | A finite sequence of integ... |
| elfzle1 13445 | A member of a finite set o... |
| elfzle2 13446 | A member of a finite set o... |
| elfzuz2 13447 | Implication of membership ... |
| elfzle3 13448 | Membership in a finite set... |
| eluzfz1 13449 | Membership in a finite set... |
| eluzfz2 13450 | Membership in a finite set... |
| eluzfz2b 13451 | Membership in a finite set... |
| elfz3 13452 | Membership in a finite set... |
| elfz1eq 13453 | Membership in a finite set... |
| elfzubelfz 13454 | If there is a member in a ... |
| peano2fzr 13455 | A Peano-postulate-like the... |
| fzn0 13456 | Properties of a finite int... |
| fz0 13457 | A finite set of sequential... |
| fzn 13458 | A finite set of sequential... |
| fzen 13459 | A shifted finite set of se... |
| fz1n 13460 | A 1-based finite set of se... |
| 0nelfz1 13461 | 0 is not an element of a f... |
| 0fz1 13462 | Two ways to say a finite 1... |
| fz10 13463 | There are no integers betw... |
| uzsubsubfz 13464 | Membership of an integer g... |
| uzsubsubfz1 13465 | Membership of an integer g... |
| ige3m2fz 13466 | Membership of an integer g... |
| fzsplit2 13467 | Split a finite interval of... |
| fzsplit 13468 | Split a finite interval of... |
| fzdisj 13469 | Condition for two finite i... |
| fz01en 13470 | 0-based and 1-based finite... |
| elfznn 13471 | A member of a finite set o... |
| elfz1end 13472 | A nonempty finite range of... |
| fz1ssnn 13473 | A finite set of positive i... |
| fznn0sub 13474 | Subtraction closure for a ... |
| fzmmmeqm 13475 | Subtracting the difference... |
| fzaddel 13476 | Membership of a sum in a f... |
| fzadd2 13477 | Membership of a sum in a f... |
| fzsubel 13478 | Membership of a difference... |
| fzopth 13479 | A finite set of sequential... |
| fzass4 13480 | Two ways to express a nond... |
| fzss1 13481 | Subset relationship for fi... |
| fzss2 13482 | Subset relationship for fi... |
| fzssuz 13483 | A finite set of sequential... |
| fzsn 13484 | A finite interval of integ... |
| fzssp1 13485 | Subset relationship for fi... |
| fzssnn 13486 | Finite sets of sequential ... |
| ssfzunsnext 13487 | A subset of a finite seque... |
| ssfzunsn 13488 | A subset of a finite seque... |
| fzsuc 13489 | Join a successor to the en... |
| fzpred 13490 | Join a predecessor to the ... |
| fzpreddisj 13491 | A finite set of sequential... |
| elfzp1 13492 | Append an element to a fin... |
| fzp1ss 13493 | Subset relationship for fi... |
| fzelp1 13494 | Membership in a set of seq... |
| fzp1elp1 13495 | Add one to an element of a... |
| fznatpl1 13496 | Shift membership in a fini... |
| fzpr 13497 | A finite interval of integ... |
| fztp 13498 | A finite interval of integ... |
| fz12pr 13499 | An integer range between 1... |
| fzsuc2 13500 | Join a successor to the en... |
| fzp1disj 13501 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13502 | Remove a successor from th... |
| fzprval 13503 | Two ways of defining the f... |
| fztpval 13504 | Two ways of defining the f... |
| fzrev 13505 | Reversal of start and end ... |
| fzrev2 13506 | Reversal of start and end ... |
| fzrev2i 13507 | Reversal of start and end ... |
| fzrev3 13508 | The "complement" of a memb... |
| fzrev3i 13509 | The "complement" of a memb... |
| fznn 13510 | Finite set of sequential i... |
| elfz1b 13511 | Membership in a 1-based fi... |
| elfz1uz 13512 | Membership in a 1-based fi... |
| elfzm11 13513 | Membership in a finite set... |
| uzsplit 13514 | Express an upper integer s... |
| uzdisj 13515 | The first ` N ` elements o... |
| fseq1p1m1 13516 | Add/remove an item to/from... |
| fseq1m1p1 13517 | Add/remove an item to/from... |
| fz1sbc 13518 | Quantification over a one-... |
| elfzp1b 13519 | An integer is a member of ... |
| elfzm1b 13520 | An integer is a member of ... |
| elfzp12 13521 | Options for membership in ... |
| fzne1 13522 | Elementhood in a finite se... |
| fzdif1 13523 | Split the first element of... |
| fz0dif1 13524 | Split the first element of... |
| fzm1 13525 | Choices for an element of ... |
| fzneuz 13526 | No finite set of sequentia... |
| fznuz 13527 | Disjointness of the upper ... |
| uznfz 13528 | Disjointness of the upper ... |
| fzp1nel 13529 | One plus the upper bound o... |
| fzrevral 13530 | Reversal of scanning order... |
| fzrevral2 13531 | Reversal of scanning order... |
| fzrevral3 13532 | Reversal of scanning order... |
| fzshftral 13533 | Shift the scanning order i... |
| ige2m1fz1 13534 | Membership of an integer g... |
| ige2m1fz 13535 | Membership in a 0-based fi... |
| elfz2nn0 13536 | Membership in a finite set... |
| fznn0 13537 | Characterization of a fini... |
| elfznn0 13538 | A member of a finite set o... |
| elfz3nn0 13539 | The upper bound of a nonem... |
| fz0ssnn0 13540 | Finite sets of sequential ... |
| fz1ssfz0 13541 | Subset relationship for fi... |
| 0elfz 13542 | 0 is an element of a finit... |
| nn0fz0 13543 | A nonnegative integer is a... |
| elfz0add 13544 | An element of a finite set... |
| fz0sn 13545 | An integer range from 0 to... |
| fz0tp 13546 | An integer range from 0 to... |
| fz0to3un2pr 13547 | An integer range from 0 to... |
| fz0to4untppr 13548 | An integer range from 0 to... |
| fz0to5un2tp 13549 | An integer range from 0 to... |
| elfz0ubfz0 13550 | An element of a finite set... |
| elfz0fzfz0 13551 | A member of a finite set o... |
| fz0fzelfz0 13552 | If a member of a finite se... |
| fznn0sub2 13553 | Subtraction closure for a ... |
| uzsubfz0 13554 | Membership of an integer g... |
| fz0fzdiffz0 13555 | The difference of an integ... |
| elfzmlbm 13556 | Subtracting the lower boun... |
| elfzmlbp 13557 | Subtracting the lower boun... |
| fzctr 13558 | Lemma for theorems about t... |
| difelfzle 13559 | The difference of two inte... |
| difelfznle 13560 | The difference of two inte... |
| nn0split 13561 | Express the set of nonnega... |
| nn0disj 13562 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13563 | A finite set of sequential... |
| fvffz0 13564 | The function value of a fu... |
| 1fv 13565 | A function on a singleton.... |
| 4fvwrd4 13566 | The first four function va... |
| 2ffzeq 13567 | Two functions over 0-based... |
| preduz 13568 | The value of the predecess... |
| prednn 13569 | The value of the predecess... |
| prednn0 13570 | The value of the predecess... |
| predfz 13571 | Calculate the predecessor ... |
| fzof 13574 | Functionality of the half-... |
| elfzoel1 13575 | Reverse closure for half-o... |
| elfzoel2 13576 | Reverse closure for half-o... |
| elfzoelz 13577 | Reverse closure for half-o... |
| fzoval 13578 | Value of the half-open int... |
| elfzo 13579 | Membership in a half-open ... |
| elfzo2 13580 | Membership in a half-open ... |
| elfzouz 13581 | Membership in a half-open ... |
| nelfzo 13582 | An integer not being a mem... |
| fzolb 13583 | The left endpoint of a hal... |
| fzolb2 13584 | The left endpoint of a hal... |
| elfzole1 13585 | A member in a half-open in... |
| elfzolt2 13586 | A member in a half-open in... |
| elfzolt3 13587 | Membership in a half-open ... |
| elfzolt2b 13588 | A member in a half-open in... |
| elfzolt3b 13589 | Membership in a half-open ... |
| elfzop1le2 13590 | A member in a half-open in... |
| fzonel 13591 | A half-open range does not... |
| elfzouz2 13592 | The upper bound of a half-... |
| elfzofz 13593 | A half-open range is conta... |
| elfzo3 13594 | Express membership in a ha... |
| fzon0 13595 | A half-open integer interv... |
| fzossfz 13596 | A half-open range is conta... |
| fzossz 13597 | A half-open integer interv... |
| fzon 13598 | A half-open set of sequent... |
| fzo0n 13599 | A half-open range of nonne... |
| fzonlt0 13600 | A half-open integer range ... |
| fzo0 13601 | Half-open sets with equal ... |
| fzonnsub 13602 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13603 | If ` M < N ` then ` N - M ... |
| fzoss1 13604 | Subset relationship for ha... |
| fzoss2 13605 | Subset relationship for ha... |
| fzossrbm1 13606 | Subset of a half-open rang... |
| fzo0ss1 13607 | Subset relationship for ha... |
| fzossnn0 13608 | A half-open integer range ... |
| fzospliti 13609 | One direction of splitting... |
| fzosplit 13610 | Split a half-open integer ... |
| fzodisj 13611 | Abutting half-open integer... |
| fzouzsplit 13612 | Split an upper integer set... |
| fzouzdisj 13613 | A half-open integer range ... |
| fzoun 13614 | A half-open integer range ... |
| fzodisjsn 13615 | A half-open integer range ... |
| prinfzo0 13616 | The intersection of a half... |
| lbfzo0 13617 | An integer is strictly gre... |
| elfzo0 13618 | Membership in a half-open ... |
| elfzo0z 13619 | Membership in a half-open ... |
| nn0p1elfzo 13620 | A nonnegative integer incr... |
| elfzo0le 13621 | A member in a half-open ra... |
| elfzolem1 13622 | A member in a half-open in... |
| elfzo0subge1 13623 | The difference of the uppe... |
| elfzo0suble 13624 | The difference of the uppe... |
| elfzonn0 13625 | A member of a half-open ra... |
| fzonmapblen 13626 | The result of subtracting ... |
| fzofzim 13627 | If a nonnegative integer i... |
| fz1fzo0m1 13628 | Translation of one between... |
| fzossnn 13629 | Half-open integer ranges s... |
| elfzo1 13630 | Membership in a half-open ... |
| fzo1lb 13631 | 1 is the left endpoint of ... |
| 1elfzo1 13632 | 1 is in a half-open range ... |
| fzo1fzo0n0 13633 | An integer between 1 and a... |
| fzo0n0 13634 | A half-open integer range ... |
| fzoaddel 13635 | Translate membership in a ... |
| fzo0addel 13636 | Translate membership in a ... |
| fzo0addelr 13637 | Translate membership in a ... |
| fzoaddel2 13638 | Translate membership in a ... |
| elfzoextl 13639 | Membership of an integer i... |
| elfzoext 13640 | Membership of an integer i... |
| elincfzoext 13641 | Membership of an increased... |
| fzosubel 13642 | Translate membership in a ... |
| fzosubel2 13643 | Membership in a translated... |
| fzosubel3 13644 | Membership in a translated... |
| eluzgtdifelfzo 13645 | Membership of the differen... |
| ige2m2fzo 13646 | Membership of an integer g... |
| fzocatel 13647 | Translate membership in a ... |
| ubmelfzo 13648 | If an integer in a 1-based... |
| elfzodifsumelfzo 13649 | If an integer is in a half... |
| elfzom1elp1fzo 13650 | Membership of an integer i... |
| elfzom1elfzo 13651 | Membership in a half-open ... |
| fzval3 13652 | Expressing a closed intege... |
| fz0add1fz1 13653 | Translate membership in a ... |
| fzosn 13654 | Expressing a singleton as ... |
| elfzomin 13655 | Membership of an integer i... |
| zpnn0elfzo 13656 | Membership of an integer i... |
| zpnn0elfzo1 13657 | Membership of an integer i... |
| fzosplitsnm1 13658 | Removing a singleton from ... |
| elfzonlteqm1 13659 | If an element of a half-op... |
| fzonn0p1 13660 | A nonnegative integer is a... |
| fzossfzop1 13661 | A half-open range of nonne... |
| fzonn0p1p1 13662 | If a nonnegative integer i... |
| elfzom1p1elfzo 13663 | Increasing an element of a... |
| fzo0ssnn0 13664 | Half-open integer ranges s... |
| fzo01 13665 | Expressing the singleton o... |
| fzo12sn 13666 | A 1-based half-open intege... |
| fzo13pr 13667 | A 1-based half-open intege... |
| fzo0to2pr 13668 | A half-open integer range ... |
| fz01pr 13669 | An integer range between 0... |
| fzo0to3tp 13670 | A half-open integer range ... |
| fzo0to42pr 13671 | A half-open integer range ... |
| fzo1to4tp 13672 | A half-open integer range ... |
| fzo0sn0fzo1 13673 | A half-open range of nonne... |
| elfzo0l 13674 | A member of a half-open ra... |
| fzoend 13675 | The endpoint of a half-ope... |
| fzo0end 13676 | The endpoint of a zero-bas... |
| ssfzo12 13677 | Subset relationship for ha... |
| ssfzoulel 13678 | If a half-open integer ran... |
| ssfzo12bi 13679 | Subset relationship for ha... |
| fzoopth 13680 | A half-open integer range ... |
| ubmelm1fzo 13681 | The result of subtracting ... |
| fzofzp1 13682 | If a point is in a half-op... |
| fzofzp1b 13683 | If a point is in a half-op... |
| elfzom1b 13684 | An integer is a member of ... |
| elfzom1elp1fzo1 13685 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13686 | Membership of a positive i... |
| elfzonelfzo 13687 | If an element of a half-op... |
| elfzodif0 13688 | If an integer ` M ` is in ... |
| fzonfzoufzol 13689 | If an element of a half-op... |
| elfzomelpfzo 13690 | An integer increased by an... |
| elfznelfzo 13691 | A value in a finite set of... |
| elfznelfzob 13692 | A value in a finite set of... |
| peano2fzor 13693 | A Peano-postulate-like the... |
| fzosplitsn 13694 | Extending a half-open rang... |
| fzosplitpr 13695 | Extending a half-open inte... |
| fzosplitprm1 13696 | Extending a half-open inte... |
| fzosplitsni 13697 | Membership in a half-open ... |
| fzisfzounsn 13698 | A finite interval of integ... |
| elfzr 13699 | A member of a finite inter... |
| elfzlmr 13700 | A member of a finite inter... |
| elfz0lmr 13701 | A member of a finite inter... |
| fzone1 13702 | Elementhood in a half-open... |
| fzom1ne1 13703 | Elementhood in a half-open... |
| fzostep1 13704 | Two possibilities for a nu... |
| fzoshftral 13705 | Shift the scanning order i... |
| fzind2 13706 | Induction on the integers ... |
| fvinim0ffz 13707 | The function values for th... |
| injresinjlem 13708 | Lemma for ~ injresinj . (... |
| injresinj 13709 | A function whose restricti... |
| subfzo0 13710 | The difference between two... |
| fvf1tp 13711 | Values of a one-to-one fun... |
| flval 13716 | Value of the floor (greate... |
| flcl 13717 | The floor (greatest intege... |
| reflcl 13718 | The floor (greatest intege... |
| fllelt 13719 | A basic property of the fl... |
| flcld 13720 | The floor (greatest intege... |
| flle 13721 | A basic property of the fl... |
| flltp1 13722 | A basic property of the fl... |
| fllep1 13723 | A basic property of the fl... |
| fraclt1 13724 | The fractional part of a r... |
| fracle1 13725 | The fractional part of a r... |
| fracge0 13726 | The fractional part of a r... |
| flge 13727 | The floor function value i... |
| fllt 13728 | The floor function value i... |
| flflp1 13729 | Move floor function betwee... |
| flid 13730 | An integer is its own floo... |
| flidm 13731 | The floor function is idem... |
| flidz 13732 | A real number equals its f... |
| flltnz 13733 | The floor of a non-integer... |
| flwordi 13734 | Ordering relation for the ... |
| flword2 13735 | Ordering relation for the ... |
| flval2 13736 | An alternate way to define... |
| flval3 13737 | An alternate way to define... |
| flbi 13738 | A condition equivalent to ... |
| flbi2 13739 | A condition equivalent to ... |
| adddivflid 13740 | The floor of a sum of an i... |
| ico01fl0 13741 | The floor of a real number... |
| flge0nn0 13742 | The floor of a number grea... |
| flge1nn 13743 | The floor of a number grea... |
| fldivnn0 13744 | The floor function of a di... |
| refldivcl 13745 | The floor function of a di... |
| divfl0 13746 | The floor of a fraction is... |
| fladdz 13747 | An integer can be moved in... |
| flzadd 13748 | An integer can be moved in... |
| flmulnn0 13749 | Move a nonnegative integer... |
| btwnzge0 13750 | A real bounded between an ... |
| 2tnp1ge0ge0 13751 | Two times an integer plus ... |
| flhalf 13752 | Ordering relation for the ... |
| fldivle 13753 | The floor function of a di... |
| fldivnn0le 13754 | The floor function of a di... |
| flltdivnn0lt 13755 | The floor function of a di... |
| ltdifltdiv 13756 | If the dividend of a divis... |
| fldiv4p1lem1div2 13757 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13758 | The floor of an integer gr... |
| fldiv4lem1div2 13759 | The floor of a positive in... |
| ceilval 13760 | The value of the ceiling f... |
| dfceil2 13761 | Alternative definition of ... |
| ceilval2 13762 | The value of the ceiling f... |
| ceicl 13763 | The ceiling function retur... |
| ceilcl 13764 | Closure of the ceiling fun... |
| ceilcld 13765 | Closure of the ceiling fun... |
| ceige 13766 | The ceiling of a real numb... |
| ceilge 13767 | The ceiling of a real numb... |
| ceilged 13768 | The ceiling of a real numb... |
| ceim1l 13769 | One less than the ceiling ... |
| ceilm1lt 13770 | One less than the ceiling ... |
| ceile 13771 | The ceiling of a real numb... |
| ceille 13772 | The ceiling of a real numb... |
| ceilid 13773 | An integer is its own ceil... |
| ceilidz 13774 | A real number equals its c... |
| flleceil 13775 | The floor of a real number... |
| fleqceilz 13776 | A real number is an intege... |
| quoremz 13777 | Quotient and remainder of ... |
| quoremnn0 13778 | Quotient and remainder of ... |
| quoremnn0ALT 13779 | Alternate proof of ~ quore... |
| intfrac2 13780 | Decompose a real into inte... |
| intfracq 13781 | Decompose a rational numbe... |
| fldiv 13782 | Cancellation of the embedd... |
| fldiv2 13783 | Cancellation of an embedde... |
| fznnfl 13784 | Finite set of sequential i... |
| uzsup 13785 | An upper set of integers i... |
| ioopnfsup 13786 | An upper set of reals is u... |
| icopnfsup 13787 | An upper set of reals is u... |
| rpsup 13788 | The positive reals are unb... |
| resup 13789 | The real numbers are unbou... |
| xrsup 13790 | The extended real numbers ... |
| modval 13793 | The value of the modulo op... |
| modvalr 13794 | The value of the modulo op... |
| modcl 13795 | Closure law for the modulo... |
| flpmodeq 13796 | Partition of a division in... |
| modcld 13797 | Closure law for the modulo... |
| mod0 13798 | ` A mod B ` is zero iff ` ... |
| mulmod0 13799 | The product of an integer ... |
| negmod0 13800 | ` A ` is divisible by ` B ... |
| modge0 13801 | The modulo operation is no... |
| modlt 13802 | The modulo operation is le... |
| modelico 13803 | Modular reduction produces... |
| moddiffl 13804 | Value of the modulo operat... |
| moddifz 13805 | The modulo operation diffe... |
| modfrac 13806 | The fractional part of a n... |
| flmod 13807 | The floor function express... |
| intfrac 13808 | Break a number into its in... |
| zmod10 13809 | An integer modulo 1 is 0. ... |
| zmod1congr 13810 | Two arbitrary integers are... |
| modmulnn 13811 | Move a positive integer in... |
| modvalp1 13812 | The value of the modulo op... |
| zmodcl 13813 | Closure law for the modulo... |
| zmodcld 13814 | Closure law for the modulo... |
| zmodfz 13815 | An integer mod ` B ` lies ... |
| zmodfzo 13816 | An integer mod ` B ` lies ... |
| zmodfzp1 13817 | An integer mod ` B ` lies ... |
| modid 13818 | Identity law for modulo. ... |
| modid0 13819 | A positive real number mod... |
| modid2 13820 | Identity law for modulo. ... |
| zmodid2 13821 | Identity law for modulo re... |
| zmodidfzo 13822 | Identity law for modulo re... |
| zmodidfzoimp 13823 | Identity law for modulo re... |
| 0mod 13824 | Special case: 0 modulo a p... |
| 1mod 13825 | Special case: 1 modulo a r... |
| modabs 13826 | Absorption law for modulo.... |
| modabs2 13827 | Absorption law for modulo.... |
| modcyc 13828 | The modulo operation is pe... |
| modcyc2 13829 | The modulo operation is pe... |
| modadd1 13830 | Addition property of the m... |
| modaddb 13831 | Addition property of the m... |
| modaddid 13832 | The sums of two nonnegativ... |
| modaddabs 13833 | Absorption law for modulo.... |
| modaddmod 13834 | The sum of a real number m... |
| muladdmodid 13835 | The sum of a positive real... |
| mulp1mod1 13836 | The product of an integer ... |
| muladdmod 13837 | A real number is the sum o... |
| modmuladd 13838 | Decomposition of an intege... |
| modmuladdim 13839 | Implication of a decomposi... |
| modmuladdnn0 13840 | Implication of a decomposi... |
| negmod 13841 | The negation of a number m... |
| m1modnnsub1 13842 | Minus one modulo a positiv... |
| m1modge3gt1 13843 | Minus one modulo an intege... |
| addmodid 13844 | The sum of a positive inte... |
| addmodidr 13845 | The sum of a positive inte... |
| modadd2mod 13846 | The sum of a real number m... |
| modm1p1mod0 13847 | If a real number modulo a ... |
| modltm1p1mod 13848 | If a real number modulo a ... |
| modmul1 13849 | Multiplication property of... |
| modmul12d 13850 | Multiplication property of... |
| modnegd 13851 | Negation property of the m... |
| modadd12d 13852 | Additive property of the m... |
| modsub12d 13853 | Subtraction property of th... |
| modsubmod 13854 | The difference of a real n... |
| modsubmodmod 13855 | The difference of a real n... |
| 2txmodxeq0 13856 | Two times a positive real ... |
| 2submod 13857 | If a real number is betwee... |
| modifeq2int 13858 | If a nonnegative integer i... |
| modaddmodup 13859 | The sum of an integer modu... |
| modaddmodlo 13860 | The sum of an integer modu... |
| modmulmod 13861 | The product of a real numb... |
| modmulmodr 13862 | The product of an integer ... |
| modaddmulmod 13863 | The sum of a real number a... |
| moddi 13864 | Distribute multiplication ... |
| modsubdir 13865 | Distribute the modulo oper... |
| modeqmodmin 13866 | A real number equals the d... |
| modirr 13867 | A number modulo an irratio... |
| modfzo0difsn 13868 | For a number within a half... |
| modsumfzodifsn 13869 | The sum of a number within... |
| modlteq 13870 | Two nonnegative integers l... |
| addmodlteq 13871 | Two nonnegative integers l... |
| om2uz0i 13872 | The mapping ` G ` is a one... |
| om2uzsuci 13873 | The value of ` G ` (see ~ ... |
| om2uzuzi 13874 | The value ` G ` (see ~ om2... |
| om2uzlti 13875 | Less-than relation for ` G... |
| om2uzlt2i 13876 | The mapping ` G ` (see ~ o... |
| om2uzrani 13877 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13878 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13879 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13880 | An alternative definition ... |
| om2uzrdg 13881 | A helper lemma for the val... |
| uzrdglem 13882 | A helper lemma for the val... |
| uzrdgfni 13883 | The recursive definition g... |
| uzrdg0i 13884 | Initial value of a recursi... |
| uzrdgsuci 13885 | Successor value of a recur... |
| ltweuz 13886 | ` < ` is a well-founded re... |
| ltwenn 13887 | Less than well-orders the ... |
| ltwefz 13888 | Less than well-orders a se... |
| uzenom 13889 | An upper integer set is de... |
| uzinf 13890 | An upper integer set is in... |
| nnnfi 13891 | The set of positive intege... |
| uzrdgxfr 13892 | Transfer the value of the ... |
| fzennn 13893 | The cardinality of a finit... |
| fzen2 13894 | The cardinality of a finit... |
| cardfz 13895 | The cardinality of a finit... |
| hashgf1o 13896 | ` G ` maps ` _om ` one-to-... |
| fzfi 13897 | A finite interval of integ... |
| fzfid 13898 | Commonly used special case... |
| fzofi 13899 | Half-open integer sets are... |
| fsequb 13900 | The values of a finite rea... |
| fsequb2 13901 | The values of a finite rea... |
| fseqsupcl 13902 | The values of a finite rea... |
| fseqsupubi 13903 | The values of a finite rea... |
| nn0ennn 13904 | The nonnegative integers a... |
| nnenom 13905 | The set of positive intege... |
| nnct 13906 | ` NN ` is countable. (Con... |
| uzindi 13907 | Indirect strong induction ... |
| axdc4uzlem 13908 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13909 | A version of ~ axdc4 that ... |
| ssnn0fi 13910 | A subset of the nonnegativ... |
| rabssnn0fi 13911 | A subset of the nonnegativ... |
| uzsinds 13912 | Strong (or "total") induct... |
| nnsinds 13913 | Strong (or "total") induct... |
| nn0sinds 13914 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13915 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13916 | If all functions of a fini... |
| fsuppmapnn0fiubex 13917 | If all functions of a fini... |
| fsuppmapnn0fiub0 13918 | If all functions of a fini... |
| suppssfz 13919 | Condition for a function o... |
| fsuppmapnn0ub 13920 | If a function over the non... |
| fsuppmapnn0fz 13921 | If a function over the non... |
| mptnn0fsupp 13922 | A mapping from the nonnega... |
| mptnn0fsuppd 13923 | A mapping from the nonnega... |
| mptnn0fsuppr 13924 | A finitely supported mappi... |
| f13idfv 13925 | A one-to-one function with... |
| seqex 13928 | Existence of the sequence ... |
| seqeq1 13929 | Equality theorem for the s... |
| seqeq2 13930 | Equality theorem for the s... |
| seqeq3 13931 | Equality theorem for the s... |
| seqeq1d 13932 | Equality deduction for the... |
| seqeq2d 13933 | Equality deduction for the... |
| seqeq3d 13934 | Equality deduction for the... |
| seqeq123d 13935 | Equality deduction for the... |
| nfseq 13936 | Hypothesis builder for the... |
| seqval 13937 | Value of the sequence buil... |
| seqfn 13938 | The sequence builder funct... |
| seq1 13939 | Value of the sequence buil... |
| seq1i 13940 | Value of the sequence buil... |
| seqp1 13941 | Value of the sequence buil... |
| seqexw 13942 | Weak version of ~ seqex th... |
| seqp1d 13943 | Value of the sequence buil... |
| seqm1 13944 | Value of the sequence buil... |
| seqcl2 13945 | Closure properties of the ... |
| seqf2 13946 | Range of the recursive seq... |
| seqcl 13947 | Closure properties of the ... |
| seqf 13948 | Range of the recursive seq... |
| seqfveq2 13949 | Equality of sequences. (C... |
| seqfeq2 13950 | Equality of sequences. (C... |
| seqfveq 13951 | Equality of sequences. (C... |
| seqfeq 13952 | Equality of sequences. (C... |
| seqshft2 13953 | Shifting the index set of ... |
| seqres 13954 | Restricting its characteri... |
| serf 13955 | An infinite series of comp... |
| serfre 13956 | An infinite series of real... |
| monoord 13957 | Ordering relation for a mo... |
| monoord2 13958 | Ordering relation for a mo... |
| sermono 13959 | The partial sums in an inf... |
| seqsplit 13960 | Split a sequence into two ... |
| seq1p 13961 | Removing the first term fr... |
| seqcaopr3 13962 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 13963 | The sum of two infinite se... |
| seqcaopr 13964 | The sum of two infinite se... |
| seqf1olem2a 13965 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 13966 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 13967 | Lemma for ~ seqf1o . (Con... |
| seqf1o 13968 | Rearrange a sum via an arb... |
| seradd 13969 | The sum of two infinite se... |
| sersub 13970 | The difference of two infi... |
| seqid3 13971 | A sequence that consists e... |
| seqid 13972 | Discarding the first few t... |
| seqid2 13973 | The last few partial sums ... |
| seqhomo 13974 | Apply a homomorphism to a ... |
| seqz 13975 | If the operation ` .+ ` ha... |
| seqfeq4 13976 | Equality of series under d... |
| seqfeq3 13977 | Equality of series under d... |
| seqdistr 13978 | The distributive property ... |
| ser0 13979 | The value of the partial s... |
| ser0f 13980 | A zero-valued infinite ser... |
| serge0 13981 | A finite sum of nonnegativ... |
| serle 13982 | Comparison of partial sums... |
| ser1const 13983 | Value of the partial serie... |
| seqof 13984 | Distribute function operat... |
| seqof2 13985 | Distribute function operat... |
| expval 13988 | Value of exponentiation to... |
| expnnval 13989 | Value of exponentiation to... |
| exp0 13990 | Value of a complex number ... |
| 0exp0e1 13991 | The zeroth power of zero e... |
| exp1 13992 | Value of a complex number ... |
| expp1 13993 | Value of a complex number ... |
| expneg 13994 | Value of a complex number ... |
| expneg2 13995 | Value of a complex number ... |
| expn1 13996 | A complex number raised to... |
| expcllem 13997 | Lemma for proving nonnegat... |
| expcl2lem 13998 | Lemma for proving integer ... |
| nnexpcl 13999 | Closure of exponentiation ... |
| nn0expcl 14000 | Closure of exponentiation ... |
| zexpcl 14001 | Closure of exponentiation ... |
| qexpcl 14002 | Closure of exponentiation ... |
| reexpcl 14003 | Closure of exponentiation ... |
| expcl 14004 | Closure law for nonnegativ... |
| rpexpcl 14005 | Closure law for integer ex... |
| qexpclz 14006 | Closure of integer exponen... |
| reexpclz 14007 | Closure of integer exponen... |
| expclzlem 14008 | Lemma for ~ expclz . (Con... |
| expclz 14009 | Closure law for integer ex... |
| m1expcl2 14010 | Closure of integer exponen... |
| m1expcl 14011 | Closure of exponentiation ... |
| zexpcld 14012 | Closure of exponentiation ... |
| nn0expcli 14013 | Closure of exponentiation ... |
| nn0sqcl 14014 | The square of a nonnegativ... |
| expm1t 14015 | Exponentiation in terms of... |
| 1exp 14016 | Value of 1 raised to an in... |
| expeq0 14017 | A positive integer power i... |
| expne0 14018 | A positive integer power i... |
| expne0i 14019 | An integer power is nonzer... |
| expgt0 14020 | A positive real raised to ... |
| expnegz 14021 | Value of a nonzero complex... |
| 0exp 14022 | Value of zero raised to a ... |
| expge0 14023 | A nonnegative real raised ... |
| expge1 14024 | A real greater than or equ... |
| expgt1 14025 | A real greater than 1 rais... |
| mulexp 14026 | Nonnegative integer expone... |
| mulexpz 14027 | Integer exponentiation of ... |
| exprec 14028 | Integer exponentiation of ... |
| expadd 14029 | Sum of exponents law for n... |
| expaddzlem 14030 | Lemma for ~ expaddz . (Co... |
| expaddz 14031 | Sum of exponents law for i... |
| expmul 14032 | Product of exponents law f... |
| expmulz 14033 | Product of exponents law f... |
| m1expeven 14034 | Exponentiation of negative... |
| expsub 14035 | Exponent subtraction law f... |
| expp1z 14036 | Value of a nonzero complex... |
| expm1 14037 | Value of a nonzero complex... |
| expdiv 14038 | Nonnegative integer expone... |
| sqval 14039 | Value of the square of a c... |
| sqneg 14040 | The square of the negative... |
| sqnegd 14041 | The square of the negative... |
| sqsubswap 14042 | Swap the order of subtract... |
| sqcl 14043 | Closure of square. (Contr... |
| sqmul 14044 | Distribution of squaring o... |
| sqeq0 14045 | A complex number is zero i... |
| sqdiv 14046 | Distribution of squaring o... |
| sqdivid 14047 | The square of a nonzero co... |
| sqne0 14048 | A complex number is nonzer... |
| resqcl 14049 | Closure of squaring in rea... |
| resqcld 14050 | Closure of squaring in rea... |
| sqgt0 14051 | The square of a nonzero re... |
| sqn0rp 14052 | The square of a nonzero re... |
| nnsqcl 14053 | The positive naturals are ... |
| zsqcl 14054 | Integers are closed under ... |
| qsqcl 14055 | The square of a rational i... |
| sq11 14056 | The square function is one... |
| nn0sq11 14057 | The square function is one... |
| lt2sq 14058 | The square function is inc... |
| le2sq 14059 | The square function is non... |
| le2sq2 14060 | The square function is non... |
| sqge0 14061 | The square of a real is no... |
| sqge0d 14062 | The square of a real is no... |
| zsqcl2 14063 | The square of an integer i... |
| 0expd 14064 | Value of zero raised to a ... |
| exp0d 14065 | Value of a complex number ... |
| exp1d 14066 | Value of a complex number ... |
| expeq0d 14067 | If a positive integer powe... |
| sqvald 14068 | Value of square. Inferenc... |
| sqcld 14069 | Closure of square. (Contr... |
| sqeq0d 14070 | A number is zero iff its s... |
| expcld 14071 | Closure law for nonnegativ... |
| expp1d 14072 | Value of a complex number ... |
| expaddd 14073 | Sum of exponents law for n... |
| expmuld 14074 | Product of exponents law f... |
| sqrecd 14075 | Square of reciprocal is re... |
| expclzd 14076 | Closure law for integer ex... |
| expne0d 14077 | A nonnegative integer powe... |
| expnegd 14078 | Value of a nonzero complex... |
| exprecd 14079 | An integer power of a reci... |
| expp1zd 14080 | Value of a nonzero complex... |
| expm1d 14081 | Value of a nonzero complex... |
| expsubd 14082 | Exponent subtraction law f... |
| sqmuld 14083 | Distribution of squaring o... |
| sqdivd 14084 | Distribution of squaring o... |
| expdivd 14085 | Nonnegative integer expone... |
| mulexpd 14086 | Nonnegative integer expone... |
| znsqcld 14087 | The square of a nonzero in... |
| reexpcld 14088 | Closure of exponentiation ... |
| expge0d 14089 | A nonnegative real raised ... |
| expge1d 14090 | A real greater than or equ... |
| ltexp2a 14091 | Exponent ordering relation... |
| expmordi 14092 | Base ordering relationship... |
| rpexpmord 14093 | Base ordering relationship... |
| expcan 14094 | Cancellation law for integ... |
| ltexp2 14095 | Strict ordering law for ex... |
| leexp2 14096 | Ordering law for exponenti... |
| leexp2a 14097 | Weak ordering relationship... |
| ltexp2r 14098 | The integer powers of a fi... |
| leexp2r 14099 | Weak ordering relationship... |
| leexp1a 14100 | Weak base ordering relatio... |
| leexp1ad 14101 | Weak base ordering relatio... |
| exple1 14102 | A real between 0 and 1 inc... |
| expubnd 14103 | An upper bound on ` A ^ N ... |
| sumsqeq0 14104 | The sum of two squres of r... |
| sqvali 14105 | Value of square. Inferenc... |
| sqcli 14106 | Closure of square. (Contr... |
| sqeq0i 14107 | A complex number is zero i... |
| sqrecii 14108 | The square of a reciprocal... |
| sqmuli 14109 | Distribution of squaring o... |
| sqdivi 14110 | Distribution of squaring o... |
| resqcli 14111 | Closure of square in reals... |
| sqgt0i 14112 | The square of a nonzero re... |
| sqge0i 14113 | The square of a real is no... |
| lt2sqi 14114 | The square function on non... |
| le2sqi 14115 | The square function on non... |
| sq11i 14116 | The square function is one... |
| sq0 14117 | The square of 0 is 0. (Co... |
| sq0i 14118 | If a number is zero, then ... |
| sq0id 14119 | If a number is zero, then ... |
| sq1 14120 | The square of 1 is 1. (Co... |
| neg1sqe1 14121 | The square of ` -u 1 ` is ... |
| sq2 14122 | The square of 2 is 4. (Co... |
| sq3 14123 | The square of 3 is 9. (Co... |
| sq4e2t8 14124 | The square of 4 is 2 times... |
| cu2 14125 | The cube of 2 is 8. (Cont... |
| irec 14126 | The reciprocal of ` _i ` .... |
| i2 14127 | ` _i ` squared. (Contribu... |
| i3 14128 | ` _i ` cubed. (Contribute... |
| i4 14129 | ` _i ` to the fourth power... |
| nnlesq 14130 | A positive integer is less... |
| zzlesq 14131 | An integer is less than or... |
| iexpcyc 14132 | Taking ` _i ` to the ` K `... |
| expnass 14133 | A counterexample showing t... |
| sqlecan 14134 | Cancel one factor of a squ... |
| subsq 14135 | Factor the difference of t... |
| subsq2 14136 | Express the difference of ... |
| binom2i 14137 | The square of a binomial. ... |
| subsqi 14138 | Factor the difference of t... |
| sqeqori 14139 | The squares of two complex... |
| subsq0i 14140 | The two solutions to the d... |
| sqeqor 14141 | The squares of two complex... |
| binom2 14142 | The square of a binomial. ... |
| binom2d 14143 | Deduction form of ~ binom2... |
| binom21 14144 | Special case of ~ binom2 w... |
| binom2sub 14145 | Expand the square of a sub... |
| binom2sub1 14146 | Special case of ~ binom2su... |
| binom2subi 14147 | Expand the square of a sub... |
| mulbinom2 14148 | The square of a binomial w... |
| binom3 14149 | The cube of a binomial. (... |
| sq01 14150 | If a complex number equals... |
| zesq 14151 | An integer is even iff its... |
| nnesq 14152 | A positive integer is even... |
| crreczi 14153 | Reciprocal of a complex nu... |
| bernneq 14154 | Bernoulli's inequality, du... |
| bernneq2 14155 | Variation of Bernoulli's i... |
| bernneq3 14156 | A corollary of ~ bernneq .... |
| expnbnd 14157 | Exponentiation with a base... |
| expnlbnd 14158 | The reciprocal of exponent... |
| expnlbnd2 14159 | The reciprocal of exponent... |
| expmulnbnd 14160 | Exponentiation with a base... |
| digit2 14161 | Two ways to express the ` ... |
| digit1 14162 | Two ways to express the ` ... |
| modexp 14163 | Exponentiation property of... |
| discr1 14164 | A nonnegative quadratic fo... |
| discr 14165 | If a quadratic polynomial ... |
| expnngt1 14166 | If an integer power with a... |
| expnngt1b 14167 | An integer power with an i... |
| sqoddm1div8 14168 | A squared odd number minus... |
| nnsqcld 14169 | The naturals are closed un... |
| nnexpcld 14170 | Closure of exponentiation ... |
| nn0expcld 14171 | Closure of exponentiation ... |
| rpexpcld 14172 | Closure law for exponentia... |
| ltexp2rd 14173 | The power of a positive nu... |
| reexpclzd 14174 | Closure of exponentiation ... |
| sqgt0d 14175 | The square of a nonzero re... |
| ltexp2d 14176 | Ordering relationship for ... |
| leexp2d 14177 | Ordering law for exponenti... |
| expcand 14178 | Ordering relationship for ... |
| leexp2ad 14179 | Ordering relationship for ... |
| leexp2rd 14180 | Ordering relationship for ... |
| lt2sqd 14181 | The square function on non... |
| le2sqd 14182 | The square function on non... |
| sq11d 14183 | The square function is one... |
| ltexp1d 14184 | Elevating to a positive po... |
| ltexp1dd 14185 | Raising both sides of 'les... |
| exp11nnd 14186 | The function elevating non... |
| mulsubdivbinom2 14187 | The square of a binomial w... |
| muldivbinom2 14188 | The square of a binomial w... |
| sq10 14189 | The square of 10 is 100. ... |
| sq10e99m1 14190 | The square of 10 is 99 plu... |
| 3dec 14191 | A "decimal constructor" wh... |
| nn0le2msqi 14192 | The square function on non... |
| nn0opthlem1 14193 | A rather pretty lemma for ... |
| nn0opthlem2 14194 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14195 | An ordered pair theorem fo... |
| nn0opth2i 14196 | An ordered pair theorem fo... |
| nn0opth2 14197 | An ordered pair theorem fo... |
| facnn 14200 | Value of the factorial fun... |
| fac0 14201 | The factorial of 0. (Cont... |
| fac1 14202 | The factorial of 1. (Cont... |
| facp1 14203 | The factorial of a success... |
| fac2 14204 | The factorial of 2. (Cont... |
| fac3 14205 | The factorial of 3. (Cont... |
| fac4 14206 | The factorial of 4. (Cont... |
| facnn2 14207 | Value of the factorial fun... |
| faccl 14208 | Closure of the factorial f... |
| faccld 14209 | Closure of the factorial f... |
| facmapnn 14210 | The factorial function res... |
| facne0 14211 | The factorial function is ... |
| facdiv 14212 | A positive integer divides... |
| facndiv 14213 | No positive integer (great... |
| facwordi 14214 | Ordering property of facto... |
| faclbnd 14215 | A lower bound for the fact... |
| faclbnd2 14216 | A lower bound for the fact... |
| faclbnd3 14217 | A lower bound for the fact... |
| faclbnd4lem1 14218 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14219 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14220 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14221 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14222 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14223 | The factorial function gro... |
| faclbnd6 14224 | Geometric lower bound for ... |
| facubnd 14225 | An upper bound for the fac... |
| facavg 14226 | The product of two factori... |
| bcval 14229 | Value of the binomial coef... |
| bcval2 14230 | Value of the binomial coef... |
| bcval3 14231 | Value of the binomial coef... |
| bcval4 14232 | Value of the binomial coef... |
| bcrpcl 14233 | Closure of the binomial co... |
| bccmpl 14234 | "Complementing" its second... |
| bcn0 14235 | ` N ` choose 0 is 1. Rema... |
| bc0k 14236 | The binomial coefficient "... |
| bcnn 14237 | ` N ` choose ` N ` is 1. ... |
| bcn1 14238 | Binomial coefficient: ` N ... |
| bcnp1n 14239 | Binomial coefficient: ` N ... |
| bcm1k 14240 | The proportion of one bino... |
| bcp1n 14241 | The proportion of one bino... |
| bcp1nk 14242 | The proportion of one bino... |
| bcval5 14243 | Write out the top and bott... |
| bcn2 14244 | Binomial coefficient: ` N ... |
| bcp1m1 14245 | Compute the binomial coeff... |
| bcpasc 14246 | Pascal's rule for the bino... |
| bccl 14247 | A binomial coefficient, in... |
| bccl2 14248 | A binomial coefficient, in... |
| bcn2m1 14249 | Compute the binomial coeff... |
| bcn2p1 14250 | Compute the binomial coeff... |
| permnn 14251 | The number of permutations... |
| bcnm1 14252 | The binomial coefficient o... |
| 4bc3eq4 14253 | The value of four choose t... |
| 4bc2eq6 14254 | The value of four choose t... |
| hashkf 14257 | The finite part of the siz... |
| hashgval 14258 | The value of the ` # ` fun... |
| hashginv 14259 | The converse of ` G ` maps... |
| hashinf 14260 | The value of the ` # ` fun... |
| hashbnd 14261 | If ` A ` has size bounded ... |
| hashfxnn0 14262 | The size function is a fun... |
| hashf 14263 | The size function maps all... |
| hashxnn0 14264 | The value of the hash func... |
| hashresfn 14265 | Restriction of the domain ... |
| dmhashres 14266 | Restriction of the domain ... |
| hashnn0pnf 14267 | The value of the hash func... |
| hashnnn0genn0 14268 | If the size of a set is no... |
| hashnemnf 14269 | The size of a set is never... |
| hashv01gt1 14270 | The size of a set is eithe... |
| hashfz1 14271 | The set ` ( 1 ... N ) ` ha... |
| hashen 14272 | Two finite sets have the s... |
| hasheni 14273 | Equinumerous sets have the... |
| hasheqf1o 14274 | The size of two finite set... |
| fiinfnf1o 14275 | There is no bijection betw... |
| hasheqf1oi 14276 | The size of two sets is eq... |
| hashf1rn 14277 | The size of a finite set w... |
| hasheqf1od 14278 | The size of two sets is eq... |
| fz1eqb 14279 | Two possibly-empty 1-based... |
| hashcard 14280 | The size function of the c... |
| hashcl 14281 | Closure of the ` # ` funct... |
| hashxrcl 14282 | Extended real closure of t... |
| hashclb 14283 | Reverse closure of the ` #... |
| nfile 14284 | The size of any infinite s... |
| hashvnfin 14285 | A set of finite size is a ... |
| hashnfinnn0 14286 | The size of an infinite se... |
| isfinite4 14287 | A finite set is equinumero... |
| hasheq0 14288 | Two ways of saying a set i... |
| hashneq0 14289 | Two ways of saying a set i... |
| hashgt0n0 14290 | If the size of a set is gr... |
| hashnncl 14291 | Positive natural closure o... |
| hash0 14292 | The empty set has size zer... |
| hashelne0d 14293 | A set with an element has ... |
| hashsng 14294 | The size of a singleton. ... |
| hashen1 14295 | A set has size 1 if and on... |
| hash1elsn 14296 | A set of size 1 with a kno... |
| hashrabrsn 14297 | The size of a restricted c... |
| hashrabsn01 14298 | The size of a restricted c... |
| hashrabsn1 14299 | If the size of a restricte... |
| hashfn 14300 | A function is equinumerous... |
| fseq1hash 14301 | The value of the size func... |
| hashgadd 14302 | ` G ` maps ordinal additio... |
| hashgval2 14303 | A short expression for the... |
| hashdom 14304 | Dominance relation for the... |
| hashdomi 14305 | Non-strict order relation ... |
| hashsdom 14306 | Strict dominance relation ... |
| hashun 14307 | The size of the union of d... |
| hashun2 14308 | The size of the union of f... |
| hashun3 14309 | The size of the union of f... |
| hashinfxadd 14310 | The extended real addition... |
| hashunx 14311 | The size of the union of d... |
| hashge0 14312 | The cardinality of a set i... |
| hashgt0 14313 | The cardinality of a nonem... |
| hashge1 14314 | The cardinality of a nonem... |
| 1elfz0hash 14315 | 1 is an element of the fin... |
| hashnn0n0nn 14316 | If a nonnegative integer i... |
| hashunsng 14317 | The size of the union of a... |
| hashunsngx 14318 | The size of the union of a... |
| hashunsnggt 14319 | The size of a set is great... |
| hashprg 14320 | The size of an unordered p... |
| elprchashprn2 14321 | If one element of an unord... |
| hashprb 14322 | The size of an unordered p... |
| hashprdifel 14323 | The elements of an unorder... |
| prhash2ex 14324 | There is (at least) one se... |
| hashle00 14325 | If the size of a set is le... |
| hashgt0elex 14326 | If the size of a set is gr... |
| hashgt0elexb 14327 | The size of a set is great... |
| hashp1i 14328 | Size of a finite ordinal. ... |
| hash1 14329 | Size of a finite ordinal. ... |
| hash2 14330 | Size of a finite ordinal. ... |
| hash3 14331 | Size of a finite ordinal. ... |
| hash4 14332 | Size of a finite ordinal. ... |
| pr0hash2ex 14333 | There is (at least) one se... |
| hashss 14334 | The size of a subset is le... |
| prsshashgt1 14335 | The size of a superset of ... |
| hashin 14336 | The size of the intersecti... |
| hashssdif 14337 | The size of the difference... |
| hashdif 14338 | The size of the difference... |
| hashdifsn 14339 | The size of the difference... |
| hashdifpr 14340 | The size of the difference... |
| hashsn01 14341 | The size of a singleton is... |
| hashsnle1 14342 | The size of a singleton is... |
| hashsnlei 14343 | Get an upper bound on a co... |
| hash1snb 14344 | The size of a set is 1 if ... |
| euhash1 14345 | The size of a set is 1 in ... |
| hash1n0 14346 | If the size of a set is 1 ... |
| hashgt12el 14347 | In a set with more than on... |
| hashgt12el2 14348 | In a set with more than on... |
| hashgt23el 14349 | A set with more than two e... |
| hashunlei 14350 | Get an upper bound on a co... |
| hashsslei 14351 | Get an upper bound on a co... |
| hashfz 14352 | Value of the numeric cardi... |
| fzsdom2 14353 | Condition for finite range... |
| hashfzo 14354 | Cardinality of a half-open... |
| hashfzo0 14355 | Cardinality of a half-open... |
| hashfzp1 14356 | Value of the numeric cardi... |
| hashfz0 14357 | Value of the numeric cardi... |
| hashxplem 14358 | Lemma for ~ hashxp . (Con... |
| hashxp 14359 | The size of the Cartesian ... |
| hashmap 14360 | The size of the set expone... |
| hashpw 14361 | The size of the power set ... |
| hashfun 14362 | A finite set is a function... |
| hashres 14363 | The number of elements of ... |
| hashreshashfun 14364 | The number of elements of ... |
| hashimarn 14365 | The size of the image of a... |
| hashimarni 14366 | If the size of the image o... |
| hashfundm 14367 | The size of a set function... |
| hashf1dmrn 14368 | The size of the domain of ... |
| hashf1dmcdm 14369 | The size of the domain of ... |
| resunimafz0 14370 | TODO-AV: Revise using ` F... |
| fnfz0hash 14371 | The size of a function on ... |
| ffz0hash 14372 | The size of a function on ... |
| fnfz0hashnn0 14373 | The size of a function on ... |
| ffzo0hash 14374 | The size of a function on ... |
| fnfzo0hash 14375 | The size of a function on ... |
| fnfzo0hashnn0 14376 | The value of the size func... |
| hashbclem 14377 | Lemma for ~ hashbc : induc... |
| hashbc 14378 | The binomial coefficient c... |
| hashfacen 14379 | The number of bijections b... |
| hashf1lem1 14380 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14381 | Lemma for ~ hashf1 . (Con... |
| hashf1 14382 | The permutation number ` |... |
| hashfac 14383 | A factorial counts the num... |
| leiso 14384 | Two ways to write a strict... |
| leisorel 14385 | Version of ~ isorel for st... |
| fz1isolem 14386 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14387 | Any finite ordered set has... |
| ishashinf 14388 | Any set that is not finite... |
| seqcoll 14389 | The function ` F ` contain... |
| seqcoll2 14390 | The function ` F ` contain... |
| phphashd 14391 | Corollary of the Pigeonhol... |
| phphashrd 14392 | Corollary of the Pigeonhol... |
| hashprlei 14393 | An unordered pair has at m... |
| hash2pr 14394 | A set of size two is an un... |
| hash2prde 14395 | A set of size two is an un... |
| hash2exprb 14396 | A set of size two is an un... |
| hash2prb 14397 | A set of size two is a pro... |
| prprrab 14398 | The set of proper pairs of... |
| nehash2 14399 | The cardinality of a set w... |
| hash2prd 14400 | A set of size two is an un... |
| hash2pwpr 14401 | If the size of a subset of... |
| hashle2pr 14402 | A nonempty set of size les... |
| hashle2prv 14403 | A nonempty subset of a pow... |
| pr2pwpr 14404 | The set of subsets of a pa... |
| hashge2el2dif 14405 | A set with size at least 2... |
| hashge2el2difr 14406 | A set with at least 2 diff... |
| hashge2el2difb 14407 | A set has size at least 2 ... |
| hashdmpropge2 14408 | The size of the domain of ... |
| hashtplei 14409 | An unordered triple has at... |
| hashtpg 14410 | The size of an unordered t... |
| hash7g 14411 | The size of an unordered s... |
| hashge3el3dif 14412 | A set with size at least 3... |
| elss2prb 14413 | An element of the set of s... |
| hash2sspr 14414 | A subset of size two is an... |
| exprelprel 14415 | If there is an element of ... |
| hash3tr 14416 | A set of size three is an ... |
| hash1to3 14417 | If the size of a set is be... |
| hash3tpde 14418 | A set of size three is an ... |
| hash3tpexb 14419 | A set of size three is an ... |
| hash3tpb 14420 | A set of size three is a p... |
| tpf1ofv0 14421 | The value of a one-to-one ... |
| tpf1ofv1 14422 | The value of a one-to-one ... |
| tpf1ofv2 14423 | The value of a one-to-one ... |
| tpf 14424 | A function into a (proper)... |
| tpfo 14425 | A function onto a (proper)... |
| tpf1o 14426 | A bijection onto a (proper... |
| fundmge2nop0 14427 | A function with a domain c... |
| fundmge2nop 14428 | A function with a domain c... |
| fun2dmnop0 14429 | A function with a domain c... |
| fun2dmnop 14430 | A function with a domain c... |
| hashdifsnp1 14431 | If the size of a set is a ... |
| fi1uzind 14432 | Properties of an ordered p... |
| brfi1uzind 14433 | Properties of a binary rel... |
| brfi1ind 14434 | Properties of a binary rel... |
| brfi1indALT 14435 | Alternate proof of ~ brfi1... |
| opfi1uzind 14436 | Properties of an ordered p... |
| opfi1ind 14437 | Properties of an ordered p... |
| iswrd 14440 | Property of being a word o... |
| wrdval 14441 | Value of the set of words ... |
| iswrdi 14442 | A zero-based sequence is a... |
| wrdf 14443 | A word is a zero-based seq... |
| wrdfd 14444 | A word is a zero-based seq... |
| iswrdb 14445 | A word over an alphabet is... |
| wrddm 14446 | The indices of a word (i.e... |
| sswrd 14447 | The set of words respects ... |
| snopiswrd 14448 | A singleton of an ordered ... |
| wrdexg 14449 | The set of words over a se... |
| wrdexb 14450 | The set of words over a se... |
| wrdexi 14451 | The set of words over a se... |
| wrdsymbcl 14452 | A symbol within a word ove... |
| wrdfn 14453 | A word is a function with ... |
| wrdv 14454 | A word over an alphabet is... |
| wrdlndm 14455 | The length of a word is no... |
| iswrdsymb 14456 | An arbitrary word is a wor... |
| wrdfin 14457 | A word is a finite set. (... |
| lencl 14458 | The length of a word is a ... |
| lennncl 14459 | The length of a nonempty w... |
| wrdffz 14460 | A word is a function from ... |
| wrdeq 14461 | Equality theorem for the s... |
| wrdeqi 14462 | Equality theorem for the s... |
| iswrddm0 14463 | A function with empty doma... |
| wrd0 14464 | The empty set is a word (t... |
| 0wrd0 14465 | The empty word is the only... |
| ffz0iswrd 14466 | A sequence with zero-based... |
| wrdsymb 14467 | A word is a word over the ... |
| nfwrd 14468 | Hypothesis builder for ` W... |
| csbwrdg 14469 | Class substitution for the... |
| wrdnval 14470 | Words of a fixed length ar... |
| wrdmap 14471 | Words as a mapping. (Cont... |
| hashwrdn 14472 | If there is only a finite ... |
| wrdnfi 14473 | If there is only a finite ... |
| wrdsymb0 14474 | A symbol at a position "ou... |
| wrdlenge1n0 14475 | A word with length at leas... |
| len0nnbi 14476 | The length of a word is a ... |
| wrdlenge2n0 14477 | A word with length at leas... |
| wrdsymb1 14478 | The first symbol of a none... |
| wrdlen1 14479 | A word of length 1 starts ... |
| fstwrdne 14480 | The first symbol of a none... |
| fstwrdne0 14481 | The first symbol of a none... |
| eqwrd 14482 | Two words are equal iff th... |
| elovmpowrd 14483 | Implications for the value... |
| elovmptnn0wrd 14484 | Implications for the value... |
| wrdred1 14485 | A word truncated by a symb... |
| wrdred1hash 14486 | The length of a word trunc... |
| lsw 14489 | Extract the last symbol of... |
| lsw0 14490 | The last symbol of an empt... |
| lsw0g 14491 | The last symbol of an empt... |
| lsw1 14492 | The last symbol of a word ... |
| lswcl 14493 | Closure of the last symbol... |
| lswlgt0cl 14494 | The last symbol of a nonem... |
| ccatfn 14497 | The concatenation operator... |
| ccatfval 14498 | Value of the concatenation... |
| ccatcl 14499 | The concatenation of two w... |
| ccatlen 14500 | The length of a concatenat... |
| ccat0 14501 | The concatenation of two w... |
| ccatval1 14502 | Value of a symbol in the l... |
| ccatval2 14503 | Value of a symbol in the r... |
| ccatval3 14504 | Value of a symbol in the r... |
| elfzelfzccat 14505 | An element of a finite set... |
| ccatvalfn 14506 | The concatenation of two w... |
| ccatdmss 14507 | The domain of a concatenat... |
| ccatsymb 14508 | The symbol at a given posi... |
| ccatfv0 14509 | The first symbol of a conc... |
| ccatval1lsw 14510 | The last symbol of the lef... |
| ccatval21sw 14511 | The first symbol of the ri... |
| ccatlid 14512 | Concatenation of a word by... |
| ccatrid 14513 | Concatenation of a word by... |
| ccatass 14514 | Associative law for concat... |
| ccatrn 14515 | The range of a concatenate... |
| ccatidid 14516 | Concatenation of the empty... |
| lswccatn0lsw 14517 | The last symbol of a word ... |
| lswccat0lsw 14518 | The last symbol of a word ... |
| ccatalpha 14519 | A concatenation of two arb... |
| ccatrcl1 14520 | Reverse closure of a conca... |
| ids1 14523 | Identity function protecti... |
| s1val 14524 | Value of a singleton word.... |
| s1rn 14525 | The range of a singleton w... |
| s1eq 14526 | Equality theorem for a sin... |
| s1eqd 14527 | Equality theorem for a sin... |
| s1cl 14528 | A singleton word is a word... |
| s1cld 14529 | A singleton word is a word... |
| s1prc 14530 | Value of a singleton word ... |
| s1cli 14531 | A singleton word is a word... |
| s1len 14532 | Length of a singleton word... |
| s1nz 14533 | A singleton word is not th... |
| s1dm 14534 | The domain of a singleton ... |
| s1dmALT 14535 | Alternate version of ~ s1d... |
| s1fv 14536 | Sole symbol of a singleton... |
| lsws1 14537 | The last symbol of a singl... |
| eqs1 14538 | A word of length 1 is a si... |
| wrdl1exs1 14539 | A word of length 1 is a si... |
| wrdl1s1 14540 | A word of length 1 is a si... |
| s111 14541 | The singleton word functio... |
| ccatws1cl 14542 | The concatenation of a wor... |
| ccatws1clv 14543 | The concatenation of a wor... |
| ccat2s1cl 14544 | The concatenation of two s... |
| ccats1alpha 14545 | A concatenation of a word ... |
| ccatws1len 14546 | The length of the concaten... |
| ccatws1lenp1b 14547 | The length of a word is ` ... |
| wrdlenccats1lenm1 14548 | The length of a word is th... |
| ccat2s1len 14549 | The length of the concaten... |
| ccatw2s1cl 14550 | The concatenation of a wor... |
| ccatw2s1len 14551 | The length of the concaten... |
| ccats1val1 14552 | Value of a symbol in the l... |
| ccats1val2 14553 | Value of the symbol concat... |
| ccat1st1st 14554 | The first symbol of a word... |
| ccat2s1p1 14555 | Extract the first of two c... |
| ccat2s1p2 14556 | Extract the second of two ... |
| ccatw2s1ass 14557 | Associative law for a conc... |
| ccatws1n0 14558 | The concatenation of a wor... |
| ccatws1ls 14559 | The last symbol of the con... |
| lswccats1 14560 | The last symbol of a word ... |
| lswccats1fst 14561 | The last symbol of a nonem... |
| ccatw2s1p1 14562 | Extract the symbol of the ... |
| ccatw2s1p2 14563 | Extract the second of two ... |
| ccat2s1fvw 14564 | Extract a symbol of a word... |
| ccat2s1fst 14565 | The first symbol of the co... |
| swrdnznd 14568 | The value of a subword ope... |
| swrdval 14569 | Value of a subword. (Cont... |
| swrd00 14570 | A zero length substring. ... |
| swrdcl 14571 | Closure of the subword ext... |
| swrdval2 14572 | Value of the subword extra... |
| swrdlen 14573 | Length of an extracted sub... |
| swrdfv 14574 | A symbol in an extracted s... |
| swrdfv0 14575 | The first symbol in an ext... |
| swrdf 14576 | A subword of a word is a f... |
| swrdvalfn 14577 | Value of the subword extra... |
| swrdrn 14578 | The range of a subword of ... |
| swrdlend 14579 | The value of the subword e... |
| swrdnd 14580 | The value of the subword e... |
| swrdnd2 14581 | Value of the subword extra... |
| swrdnnn0nd 14582 | The value of a subword ope... |
| swrdnd0 14583 | The value of a subword ope... |
| swrd0 14584 | A subword of an empty set ... |
| swrdrlen 14585 | Length of a right-anchored... |
| swrdlen2 14586 | Length of an extracted sub... |
| swrdfv2 14587 | A symbol in an extracted s... |
| swrdwrdsymb 14588 | A subword is a word over t... |
| swrdsb0eq 14589 | Two subwords with the same... |
| swrdsbslen 14590 | Two subwords with the same... |
| swrdspsleq 14591 | Two words have a common su... |
| swrds1 14592 | Extract a single symbol fr... |
| swrdlsw 14593 | Extract the last single sy... |
| ccatswrd 14594 | Joining two adjacent subwo... |
| swrdccat2 14595 | Recover the right half of ... |
| pfxnndmnd 14598 | The value of a prefix oper... |
| pfxval 14599 | Value of a prefix operatio... |
| pfx00 14600 | The zero length prefix is ... |
| pfx0 14601 | A prefix of an empty set i... |
| pfxval0 14602 | Value of a prefix operatio... |
| pfxcl 14603 | Closure of the prefix extr... |
| pfxmpt 14604 | Value of the prefix extrac... |
| pfxres 14605 | Value of the prefix extrac... |
| pfxf 14606 | A prefix of a word is a fu... |
| pfxfn 14607 | Value of the prefix extrac... |
| pfxfv 14608 | A symbol in a prefix of a ... |
| pfxlen 14609 | Length of a prefix. (Cont... |
| pfxid 14610 | A word is a prefix of itse... |
| pfxrn 14611 | The range of a prefix of a... |
| pfxn0 14612 | A prefix consisting of at ... |
| pfxnd 14613 | The value of a prefix oper... |
| pfxnd0 14614 | The value of a prefix oper... |
| pfxwrdsymb 14615 | A prefix of a word is a wo... |
| addlenpfx 14616 | The sum of the lengths of ... |
| pfxfv0 14617 | The first symbol of a pref... |
| pfxtrcfv 14618 | A symbol in a word truncat... |
| pfxtrcfv0 14619 | The first symbol in a word... |
| pfxfvlsw 14620 | The last symbol in a nonem... |
| pfxeq 14621 | The prefixes of two words ... |
| pfxtrcfvl 14622 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14623 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14624 | Two (nonempty) words are e... |
| disjwrdpfx 14625 | Sets of words are disjoint... |
| ccatpfx 14626 | Concatenating a prefix wit... |
| pfxccat1 14627 | Recover the left half of a... |
| pfx1 14628 | The prefix of length one o... |
| swrdswrdlem 14629 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14630 | A subword of a subword is ... |
| pfxswrd 14631 | A prefix of a subword is a... |
| swrdpfx 14632 | A subword of a prefix is a... |
| pfxpfx 14633 | A prefix of a prefix is a ... |
| pfxpfxid 14634 | A prefix of a prefix with ... |
| pfxcctswrd 14635 | The concatenation of the p... |
| lenpfxcctswrd 14636 | The length of the concaten... |
| lenrevpfxcctswrd 14637 | The length of the concaten... |
| pfxlswccat 14638 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14639 | The last symbol of a word ... |
| ccats1pfxeqrex 14640 | There exists a symbol such... |
| ccatopth 14641 | An ~ opth -like theorem fo... |
| ccatopth2 14642 | An ~ opth -like theorem fo... |
| ccatlcan 14643 | Concatenation of words is ... |
| ccatrcan 14644 | Concatenation of words is ... |
| wrdeqs1cat 14645 | Decompose a nonempty word ... |
| cats1un 14646 | Express a word with an ext... |
| wrdind 14647 | Perform induction over the... |
| wrd2ind 14648 | Perform induction over the... |
| swrdccatfn 14649 | The subword of a concatena... |
| swrdccatin1 14650 | The subword of a concatena... |
| pfxccatin12lem4 14651 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14652 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14653 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14654 | The subword of a concatena... |
| pfxccatin12lem2c 14655 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14656 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14657 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14658 | The subword of a concatena... |
| pfxccat3 14659 | The subword of a concatena... |
| swrdccat 14660 | The subword of a concatena... |
| pfxccatpfx1 14661 | A prefix of a concatenatio... |
| pfxccatpfx2 14662 | A prefix of a concatenatio... |
| pfxccat3a 14663 | A prefix of a concatenatio... |
| swrdccat3blem 14664 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14665 | A suffix of a concatenatio... |
| pfxccatid 14666 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14667 | A word is a prefix of a wo... |
| swrdccatin1d 14668 | The subword of a concatena... |
| swrdccatin2d 14669 | The subword of a concatena... |
| pfxccatin12d 14670 | The subword of a concatena... |
| reuccatpfxs1lem 14671 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14672 | There is a unique word hav... |
| reuccatpfxs1v 14673 | There is a unique word hav... |
| splval 14676 | Value of the substring rep... |
| splcl 14677 | Closure of the substring r... |
| splid 14678 | Splicing a subword for the... |
| spllen 14679 | The length of a splice. (... |
| splfv1 14680 | Symbols to the left of a s... |
| splfv2a 14681 | Symbols within the replace... |
| splval2 14682 | Value of a splice, assumin... |
| revval 14685 | Value of the word reversin... |
| revcl 14686 | The reverse of a word is a... |
| revlen 14687 | The reverse of a word has ... |
| revfv 14688 | Reverse of a word at a poi... |
| rev0 14689 | The empty word is its own ... |
| revs1 14690 | Singleton words are their ... |
| revccat 14691 | Antiautomorphic property o... |
| revrev 14692 | Reversal is an involution ... |
| reps 14695 | Construct a function mappi... |
| repsundef 14696 | A function mapping a half-... |
| repsconst 14697 | Construct a function mappi... |
| repsf 14698 | The constructed function m... |
| repswsymb 14699 | The symbols of a "repeated... |
| repsw 14700 | A function mapping a half-... |
| repswlen 14701 | The length of a "repeated ... |
| repsw0 14702 | The "repeated symbol word"... |
| repsdf2 14703 | Alternative definition of ... |
| repswsymball 14704 | All the symbols of a "repe... |
| repswsymballbi 14705 | A word is a "repeated symb... |
| repswfsts 14706 | The first symbol of a none... |
| repswlsw 14707 | The last symbol of a nonem... |
| repsw1 14708 | The "repeated symbol word"... |
| repswswrd 14709 | A subword of a "repeated s... |
| repswpfx 14710 | A prefix of a repeated sym... |
| repswccat 14711 | The concatenation of two "... |
| repswrevw 14712 | The reverse of a "repeated... |
| cshfn 14715 | Perform a cyclical shift f... |
| cshword 14716 | Perform a cyclical shift f... |
| cshnz 14717 | A cyclical shift is the em... |
| 0csh0 14718 | Cyclically shifting an emp... |
| cshw0 14719 | A word cyclically shifted ... |
| cshwmodn 14720 | Cyclically shifting a word... |
| cshwsublen 14721 | Cyclically shifting a word... |
| cshwn 14722 | A word cyclically shifted ... |
| cshwcl 14723 | A cyclically shifted word ... |
| cshwlen 14724 | The length of a cyclically... |
| cshwf 14725 | A cyclically shifted word ... |
| cshwfn 14726 | A cyclically shifted word ... |
| cshwrn 14727 | The range of a cyclically ... |
| cshwidxmod 14728 | The symbol at a given inde... |
| cshwidxmodr 14729 | The symbol at a given inde... |
| cshwidx0mod 14730 | The symbol at index 0 of a... |
| cshwidx0 14731 | The symbol at index 0 of a... |
| cshwidxm1 14732 | The symbol at index ((n-N)... |
| cshwidxm 14733 | The symbol at index (n-N) ... |
| cshwidxn 14734 | The symbol at index (n-1) ... |
| cshf1 14735 | Cyclically shifting a word... |
| cshinj 14736 | If a word is injectiv (reg... |
| repswcshw 14737 | A cyclically shifted "repe... |
| 2cshw 14738 | Cyclically shifting a word... |
| 2cshwid 14739 | Cyclically shifting a word... |
| lswcshw 14740 | The last symbol of a word ... |
| 2cshwcom 14741 | Cyclically shifting a word... |
| cshwleneq 14742 | If the results of cyclical... |
| 3cshw 14743 | Cyclically shifting a word... |
| cshweqdif2 14744 | If cyclically shifting two... |
| cshweqdifid 14745 | If cyclically shifting a w... |
| cshweqrep 14746 | If cyclically shifting a w... |
| cshw1 14747 | If cyclically shifting a w... |
| cshw1repsw 14748 | If cyclically shifting a w... |
| cshwsexa 14749 | The class of (different!) ... |
| 2cshwcshw 14750 | If a word is a cyclically ... |
| scshwfzeqfzo 14751 | For a nonempty word the se... |
| cshwcshid 14752 | A cyclically shifted word ... |
| cshwcsh2id 14753 | A cyclically shifted word ... |
| cshimadifsn 14754 | The image of a cyclically ... |
| cshimadifsn0 14755 | The image of a cyclically ... |
| wrdco 14756 | Mapping a word by a functi... |
| lenco 14757 | Length of a mapped word is... |
| s1co 14758 | Mapping of a singleton wor... |
| revco 14759 | Mapping of words (i.e., a ... |
| ccatco 14760 | Mapping of words commutes ... |
| cshco 14761 | Mapping of words commutes ... |
| swrdco 14762 | Mapping of words commutes ... |
| pfxco 14763 | Mapping of words commutes ... |
| lswco 14764 | Mapping of (nonempty) word... |
| repsco 14765 | Mapping of words commutes ... |
| cats1cld 14780 | Closure of concatenation w... |
| cats1co 14781 | Closure of concatenation w... |
| cats1cli 14782 | Closure of concatenation w... |
| cats1fvn 14783 | The last symbol of a conca... |
| cats1fv 14784 | A symbol other than the la... |
| cats1len 14785 | The length of concatenatio... |
| cats1cat 14786 | Closure of concatenation w... |
| cats2cat 14787 | Closure of concatenation o... |
| s2eqd 14788 | Equality theorem for a dou... |
| s3eqd 14789 | Equality theorem for a len... |
| s4eqd 14790 | Equality theorem for a len... |
| s5eqd 14791 | Equality theorem for a len... |
| s6eqd 14792 | Equality theorem for a len... |
| s7eqd 14793 | Equality theorem for a len... |
| s8eqd 14794 | Equality theorem for a len... |
| s3eq2 14795 | Equality theorem for a len... |
| s2cld 14796 | A doubleton word is a word... |
| s3cld 14797 | A length 3 string is a wor... |
| s4cld 14798 | A length 4 string is a wor... |
| s5cld 14799 | A length 5 string is a wor... |
| s6cld 14800 | A length 6 string is a wor... |
| s7cld 14801 | A length 7 string is a wor... |
| s8cld 14802 | A length 8 string is a wor... |
| s2cl 14803 | A doubleton word is a word... |
| s3cl 14804 | A length 3 string is a wor... |
| s2cli 14805 | A doubleton word is a word... |
| s3cli 14806 | A length 3 string is a wor... |
| s4cli 14807 | A length 4 string is a wor... |
| s5cli 14808 | A length 5 string is a wor... |
| s6cli 14809 | A length 6 string is a wor... |
| s7cli 14810 | A length 7 string is a wor... |
| s8cli 14811 | A length 8 string is a wor... |
| s2fv0 14812 | Extract the first symbol f... |
| s2fv1 14813 | Extract the second symbol ... |
| s2len 14814 | The length of a doubleton ... |
| s2dm 14815 | The domain of a doubleton ... |
| s3fv0 14816 | Extract the first symbol f... |
| s3fv1 14817 | Extract the second symbol ... |
| s3fv2 14818 | Extract the third symbol f... |
| s3len 14819 | The length of a length 3 s... |
| s4fv0 14820 | Extract the first symbol f... |
| s4fv1 14821 | Extract the second symbol ... |
| s4fv2 14822 | Extract the third symbol f... |
| s4fv3 14823 | Extract the fourth symbol ... |
| s4len 14824 | The length of a length 4 s... |
| s5len 14825 | The length of a length 5 s... |
| s6len 14826 | The length of a length 6 s... |
| s7len 14827 | The length of a length 7 s... |
| s8len 14828 | The length of a length 8 s... |
| lsws2 14829 | The last symbol of a doubl... |
| lsws3 14830 | The last symbol of a 3 let... |
| lsws4 14831 | The last symbol of a 4 let... |
| s2prop 14832 | A length 2 word is an unor... |
| s2dmALT 14833 | Alternate version of ~ s2d... |
| s3tpop 14834 | A length 3 word is an unor... |
| s4prop 14835 | A length 4 word is a union... |
| s3fn 14836 | A length 3 word is a funct... |
| funcnvs1 14837 | The converse of a singleto... |
| funcnvs2 14838 | The converse of a length 2... |
| funcnvs3 14839 | The converse of a length 3... |
| funcnvs4 14840 | The converse of a length 4... |
| s2f1o 14841 | A length 2 word with mutua... |
| f1oun2prg 14842 | A union of unordered pairs... |
| s4f1o 14843 | A length 4 word with mutua... |
| s4dom 14844 | The domain of a length 4 w... |
| s2co 14845 | Mapping a doubleton word b... |
| s3co 14846 | Mapping a length 3 string ... |
| s0s1 14847 | Concatenation of fixed len... |
| s1s2 14848 | Concatenation of fixed len... |
| s1s3 14849 | Concatenation of fixed len... |
| s1s4 14850 | Concatenation of fixed len... |
| s1s5 14851 | Concatenation of fixed len... |
| s1s6 14852 | Concatenation of fixed len... |
| s1s7 14853 | Concatenation of fixed len... |
| s2s2 14854 | Concatenation of fixed len... |
| s4s2 14855 | Concatenation of fixed len... |
| s4s3 14856 | Concatenation of fixed len... |
| s4s4 14857 | Concatenation of fixed len... |
| s3s4 14858 | Concatenation of fixed len... |
| s2s5 14859 | Concatenation of fixed len... |
| s5s2 14860 | Concatenation of fixed len... |
| s2eq2s1eq 14861 | Two length 2 words are equ... |
| s2eq2seq 14862 | Two length 2 words are equ... |
| s3eqs2s1eq 14863 | Two length 3 words are equ... |
| s3eq3seq 14864 | Two length 3 words are equ... |
| swrds2 14865 | Extract two adjacent symbo... |
| swrds2m 14866 | Extract two adjacent symbo... |
| wrdlen2i 14867 | Implications of a word of ... |
| wrd2pr2op 14868 | A word of length two repre... |
| wrdlen2 14869 | A word of length two. (Co... |
| wrdlen2s2 14870 | A word of length two as do... |
| wrdl2exs2 14871 | A word of length two is a ... |
| pfx2 14872 | A prefix of length two. (... |
| wrd3tpop 14873 | A word of length three rep... |
| wrdlen3s3 14874 | A word of length three as ... |
| repsw2 14875 | The "repeated symbol word"... |
| repsw3 14876 | The "repeated symbol word"... |
| swrd2lsw 14877 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14878 | Two words of length at lea... |
| ccatw2s1ccatws2 14879 | The concatenation of a wor... |
| ccat2s1fvwALT 14880 | Alternate proof of ~ ccat2... |
| wwlktovf 14881 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14882 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14883 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14884 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14885 | There is a bijection betwe... |
| eqwrds3 14886 | A word is equal with a len... |
| wrdl3s3 14887 | A word of length 3 is a le... |
| s2rn 14888 | Range of a length 2 string... |
| s3rn 14889 | Range of a length 3 string... |
| s7rn 14890 | Range of a length 7 string... |
| s7f1o 14891 | A length 7 word with mutua... |
| s3sndisj 14892 | The singletons consisting ... |
| s3iunsndisj 14893 | The union of singletons co... |
| ofccat 14894 | Letterwise operations on w... |
| ofs1 14895 | Letterwise operations on a... |
| ofs2 14896 | Letterwise operations on a... |
| coss12d 14897 | Subset deduction for compo... |
| trrelssd 14898 | The composition of subclas... |
| xpcogend 14899 | The most interesting case ... |
| xpcoidgend 14900 | If two classes are not dis... |
| cotr2g 14901 | Two ways of saying that th... |
| cotr2 14902 | Two ways of saying a relat... |
| cotr3 14903 | Two ways of saying a relat... |
| coemptyd 14904 | Deduction about compositio... |
| xptrrel 14905 | The cross product is alway... |
| 0trrel 14906 | The empty class is a trans... |
| cleq1lem 14907 | Equality implies bijection... |
| cleq1 14908 | Equality of relations impl... |
| clsslem 14909 | The closure of a subclass ... |
| trcleq1 14914 | Equality of relations impl... |
| trclsslem 14915 | The transitive closure (as... |
| trcleq2lem 14916 | Equality implies bijection... |
| cvbtrcl 14917 | Change of bound variable i... |
| trcleq12lem 14918 | Equality implies bijection... |
| trclexlem 14919 | Existence of relation impl... |
| trclublem 14920 | If a relation exists then ... |
| trclubi 14921 | The Cartesian product of t... |
| trclubgi 14922 | The union with the Cartesi... |
| trclub 14923 | The Cartesian product of t... |
| trclubg 14924 | The union with the Cartesi... |
| trclfv 14925 | The transitive closure of ... |
| brintclab 14926 | Two ways to express a bina... |
| brtrclfv 14927 | Two ways of expressing the... |
| brcnvtrclfv 14928 | Two ways of expressing the... |
| brtrclfvcnv 14929 | Two ways of expressing the... |
| brcnvtrclfvcnv 14930 | Two ways of expressing the... |
| trclfvss 14931 | The transitive closure (as... |
| trclfvub 14932 | The transitive closure of ... |
| trclfvlb 14933 | The transitive closure of ... |
| trclfvcotr 14934 | The transitive closure of ... |
| trclfvlb2 14935 | The transitive closure of ... |
| trclfvlb3 14936 | The transitive closure of ... |
| cotrtrclfv 14937 | The transitive closure of ... |
| trclidm 14938 | The transitive closure of ... |
| trclun 14939 | Transitive closure of a un... |
| trclfvg 14940 | The value of the transitiv... |
| trclfvcotrg 14941 | The value of the transitiv... |
| reltrclfv 14942 | The transitive closure of ... |
| dmtrclfv 14943 | The domain of the transiti... |
| reldmrelexp 14946 | The domain of the repeated... |
| relexp0g 14947 | A relation composed zero t... |
| relexp0 14948 | A relation composed zero t... |
| relexp0d 14949 | A relation composed zero t... |
| relexpsucnnr 14950 | A reduction for relation e... |
| relexp1g 14951 | A relation composed once i... |
| dfid5 14952 | Identity relation is equal... |
| dfid6 14953 | Identity relation expresse... |
| relexp1d 14954 | A relation composed once i... |
| relexpsucnnl 14955 | A reduction for relation e... |
| relexpsucl 14956 | A reduction for relation e... |
| relexpsucr 14957 | A reduction for relation e... |
| relexpsucrd 14958 | A reduction for relation e... |
| relexpsucld 14959 | A reduction for relation e... |
| relexpcnv 14960 | Commutation of converse an... |
| relexpcnvd 14961 | Commutation of converse an... |
| relexp0rel 14962 | The exponentiation of a cl... |
| relexprelg 14963 | The exponentiation of a cl... |
| relexprel 14964 | The exponentiation of a re... |
| relexpreld 14965 | The exponentiation of a re... |
| relexpnndm 14966 | The domain of an exponenti... |
| relexpdmg 14967 | The domain of an exponenti... |
| relexpdm 14968 | The domain of an exponenti... |
| relexpdmd 14969 | The domain of an exponenti... |
| relexpnnrn 14970 | The range of an exponentia... |
| relexprng 14971 | The range of an exponentia... |
| relexprn 14972 | The range of an exponentia... |
| relexprnd 14973 | The range of an exponentia... |
| relexpfld 14974 | The field of an exponentia... |
| relexpfldd 14975 | The field of an exponentia... |
| relexpaddnn 14976 | Relation composition becom... |
| relexpuzrel 14977 | The exponentiation of a cl... |
| relexpaddg 14978 | Relation composition becom... |
| relexpaddd 14979 | Relation composition becom... |
| rtrclreclem1 14982 | The reflexive, transitive ... |
| dfrtrclrec2 14983 | If two elements are connec... |
| rtrclreclem2 14984 | The reflexive, transitive ... |
| rtrclreclem3 14985 | The reflexive, transitive ... |
| rtrclreclem4 14986 | The reflexive, transitive ... |
| dfrtrcl2 14987 | The two definitions ` t* `... |
| relexpindlem 14988 | Principle of transitive in... |
| relexpind 14989 | Principle of transitive in... |
| rtrclind 14990 | Principle of transitive in... |
| shftlem 14993 | Two ways to write a shifte... |
| shftuz 14994 | A shift of the upper integ... |
| shftfval 14995 | The value of the sequence ... |
| shftdm 14996 | Domain of a relation shift... |
| shftfib 14997 | Value of a fiber of the re... |
| shftfn 14998 | Functionality and domain o... |
| shftval 14999 | Value of a sequence shifte... |
| shftval2 15000 | Value of a sequence shifte... |
| shftval3 15001 | Value of a sequence shifte... |
| shftval4 15002 | Value of a sequence shifte... |
| shftval5 15003 | Value of a shifted sequenc... |
| shftf 15004 | Functionality of a shifted... |
| 2shfti 15005 | Composite shift operations... |
| shftidt2 15006 | Identity law for the shift... |
| shftidt 15007 | Identity law for the shift... |
| shftcan1 15008 | Cancellation law for the s... |
| shftcan2 15009 | Cancellation law for the s... |
| seqshft 15010 | Shifting the index set of ... |
| sgnval 15013 | Value of the signum functi... |
| sgn0 15014 | The signum of 0 is 0. (Co... |
| sgnp 15015 | The signum of a positive e... |
| sgnrrp 15016 | The signum of a positive r... |
| sgn1 15017 | The signum of 1 is 1. (Co... |
| sgnpnf 15018 | The signum of ` +oo ` is 1... |
| sgnn 15019 | The signum of a negative e... |
| sgnmnf 15020 | The signum of ` -oo ` is -... |
| cjval 15027 | The value of the conjugate... |
| cjth 15028 | The defining property of t... |
| cjf 15029 | Domain and codomain of the... |
| cjcl 15030 | The conjugate of a complex... |
| reval 15031 | The value of the real part... |
| imval 15032 | The value of the imaginary... |
| imre 15033 | The imaginary part of a co... |
| reim 15034 | The real part of a complex... |
| recl 15035 | The real part of a complex... |
| imcl 15036 | The imaginary part of a co... |
| ref 15037 | Domain and codomain of the... |
| imf 15038 | Domain and codomain of the... |
| crre 15039 | The real part of a complex... |
| crim 15040 | The real part of a complex... |
| replim 15041 | Reconstruct a complex numb... |
| remim 15042 | Value of the conjugate of ... |
| reim0 15043 | The imaginary part of a re... |
| reim0b 15044 | A number is real iff its i... |
| rereb 15045 | A number is real iff it eq... |
| mulre 15046 | A product with a nonzero r... |
| rere 15047 | A real number equals its r... |
| cjreb 15048 | A number is real iff it eq... |
| recj 15049 | Real part of a complex con... |
| reneg 15050 | Real part of negative. (C... |
| readd 15051 | Real part distributes over... |
| resub 15052 | Real part distributes over... |
| remullem 15053 | Lemma for ~ remul , ~ immu... |
| remul 15054 | Real part of a product. (... |
| remul2 15055 | Real part of a product. (... |
| rediv 15056 | Real part of a division. ... |
| imcj 15057 | Imaginary part of a comple... |
| imneg 15058 | The imaginary part of a ne... |
| imadd 15059 | Imaginary part distributes... |
| imsub 15060 | Imaginary part distributes... |
| immul 15061 | Imaginary part of a produc... |
| immul2 15062 | Imaginary part of a produc... |
| imdiv 15063 | Imaginary part of a divisi... |
| cjre 15064 | A real number equals its c... |
| cjcj 15065 | The conjugate of the conju... |
| cjadd 15066 | Complex conjugate distribu... |
| cjmul 15067 | Complex conjugate distribu... |
| ipcnval 15068 | Standard inner product on ... |
| cjmulrcl 15069 | A complex number times its... |
| cjmulval 15070 | A complex number times its... |
| cjmulge0 15071 | A complex number times its... |
| cjneg 15072 | Complex conjugate of negat... |
| addcj 15073 | A number plus its conjugat... |
| cjsub 15074 | Complex conjugate distribu... |
| cjexp 15075 | Complex conjugate of posit... |
| imval2 15076 | The imaginary part of a nu... |
| re0 15077 | The real part of zero. (C... |
| im0 15078 | The imaginary part of zero... |
| re1 15079 | The real part of one. (Co... |
| im1 15080 | The imaginary part of one.... |
| rei 15081 | The real part of ` _i ` . ... |
| imi 15082 | The imaginary part of ` _i... |
| cj0 15083 | The conjugate of zero. (C... |
| cji 15084 | The complex conjugate of t... |
| cjreim 15085 | The conjugate of a represe... |
| cjreim2 15086 | The conjugate of the repre... |
| cj11 15087 | Complex conjugate is a one... |
| cjne0 15088 | A number is nonzero iff it... |
| cjdiv 15089 | Complex conjugate distribu... |
| cnrecnv 15090 | The inverse to the canonic... |
| sqeqd 15091 | A deduction for showing tw... |
| recli 15092 | The real part of a complex... |
| imcli 15093 | The imaginary part of a co... |
| cjcli 15094 | Closure law for complex co... |
| replimi 15095 | Construct a complex number... |
| cjcji 15096 | The conjugate of the conju... |
| reim0bi 15097 | A number is real iff its i... |
| rerebi 15098 | A real number equals its r... |
| cjrebi 15099 | A number is real iff it eq... |
| recji 15100 | Real part of a complex con... |
| imcji 15101 | Imaginary part of a comple... |
| cjmulrcli 15102 | A complex number times its... |
| cjmulvali 15103 | A complex number times its... |
| cjmulge0i 15104 | A complex number times its... |
| renegi 15105 | Real part of negative. (C... |
| imnegi 15106 | Imaginary part of negative... |
| cjnegi 15107 | Complex conjugate of negat... |
| addcji 15108 | A number plus its conjugat... |
| readdi 15109 | Real part distributes over... |
| imaddi 15110 | Imaginary part distributes... |
| remuli 15111 | Real part of a product. (... |
| immuli 15112 | Imaginary part of a produc... |
| cjaddi 15113 | Complex conjugate distribu... |
| cjmuli 15114 | Complex conjugate distribu... |
| ipcni 15115 | Standard inner product on ... |
| cjdivi 15116 | Complex conjugate distribu... |
| crrei 15117 | The real part of a complex... |
| crimi 15118 | The imaginary part of a co... |
| recld 15119 | The real part of a complex... |
| imcld 15120 | The imaginary part of a co... |
| cjcld 15121 | Closure law for complex co... |
| replimd 15122 | Construct a complex number... |
| remimd 15123 | Value of the conjugate of ... |
| cjcjd 15124 | The conjugate of the conju... |
| reim0bd 15125 | A number is real iff its i... |
| rerebd 15126 | A real number equals its r... |
| cjrebd 15127 | A number is real iff it eq... |
| cjne0d 15128 | A number is nonzero iff it... |
| recjd 15129 | Real part of a complex con... |
| imcjd 15130 | Imaginary part of a comple... |
| cjmulrcld 15131 | A complex number times its... |
| cjmulvald 15132 | A complex number times its... |
| cjmulge0d 15133 | A complex number times its... |
| renegd 15134 | Real part of negative. (C... |
| imnegd 15135 | Imaginary part of negative... |
| cjnegd 15136 | Complex conjugate of negat... |
| addcjd 15137 | A number plus its conjugat... |
| cjexpd 15138 | Complex conjugate of posit... |
| readdd 15139 | Real part distributes over... |
| imaddd 15140 | Imaginary part distributes... |
| resubd 15141 | Real part distributes over... |
| imsubd 15142 | Imaginary part distributes... |
| remuld 15143 | Real part of a product. (... |
| immuld 15144 | Imaginary part of a produc... |
| cjaddd 15145 | Complex conjugate distribu... |
| cjmuld 15146 | Complex conjugate distribu... |
| ipcnd 15147 | Standard inner product on ... |
| cjdivd 15148 | Complex conjugate distribu... |
| rered 15149 | A real number equals its r... |
| reim0d 15150 | The imaginary part of a re... |
| cjred 15151 | A real number equals its c... |
| remul2d 15152 | Real part of a product. (... |
| immul2d 15153 | Imaginary part of a produc... |
| redivd 15154 | Real part of a division. ... |
| imdivd 15155 | Imaginary part of a divisi... |
| crred 15156 | The real part of a complex... |
| crimd 15157 | The imaginary part of a co... |
| sqrtval 15162 | Value of square root funct... |
| absval 15163 | The absolute value (modulu... |
| rennim 15164 | A real number does not lie... |
| cnpart 15165 | The specification of restr... |
| sqrt0 15166 | The square root of zero is... |
| 01sqrexlem1 15167 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15168 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15169 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15170 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15171 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15172 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15173 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15174 | Existence of a square root... |
| resqrex 15175 | Existence of a square root... |
| sqrmo 15176 | Uniqueness for the square ... |
| resqreu 15177 | Existence and uniqueness f... |
| resqrtcl 15178 | Closure of the square root... |
| resqrtthlem 15179 | Lemma for ~ resqrtth . (C... |
| resqrtth 15180 | Square root theorem over t... |
| remsqsqrt 15181 | Square of square root. (C... |
| sqrtge0 15182 | The square root function i... |
| sqrtgt0 15183 | The square root function i... |
| sqrtmul 15184 | Square root distributes ov... |
| sqrtle 15185 | Square root is monotonic. ... |
| sqrtlt 15186 | Square root is strictly mo... |
| sqrt11 15187 | The square root function i... |
| sqrt00 15188 | A square root is zero iff ... |
| rpsqrtcl 15189 | The square root of a posit... |
| sqrtdiv 15190 | Square root distributes ov... |
| sqrtneglem 15191 | The square root of a negat... |
| sqrtneg 15192 | The square root of a negat... |
| sqrtsq2 15193 | Relationship between squar... |
| sqrtsq 15194 | Square root of square. (C... |
| sqrtmsq 15195 | Square root of square. (C... |
| sqrt1 15196 | The square root of 1 is 1.... |
| sqrt4 15197 | The square root of 4 is 2.... |
| sqrt9 15198 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15199 | The square root of 2 is bo... |
| sqrtm1 15200 | The imaginary unit is the ... |
| nn0sqeq1 15201 | A natural number with squa... |
| absneg 15202 | Absolute value of the nega... |
| abscl 15203 | Real closure of absolute v... |
| abscj 15204 | The absolute value of a nu... |
| absvalsq 15205 | Square of value of absolut... |
| absvalsq2 15206 | Square of value of absolut... |
| sqabsadd 15207 | Square of absolute value o... |
| sqabssub 15208 | Square of absolute value o... |
| absval2 15209 | Value of absolute value fu... |
| abs0 15210 | The absolute value of 0. ... |
| absi 15211 | The absolute value of the ... |
| absge0 15212 | Absolute value is nonnegat... |
| absrpcl 15213 | The absolute value of a no... |
| abs00 15214 | The absolute value of a nu... |
| abs00ad 15215 | A complex number is zero i... |
| abs00bd 15216 | If a complex number is zer... |
| absreimsq 15217 | Square of the absolute val... |
| absreim 15218 | Absolute value of a number... |
| absmul 15219 | Absolute value distributes... |
| absdiv 15220 | Absolute value distributes... |
| absid 15221 | A nonnegative number is it... |
| abs1 15222 | The absolute value of one ... |
| absnid 15223 | For a negative number, its... |
| leabs 15224 | A real number is less than... |
| absor 15225 | The absolute value of a re... |
| absre 15226 | Absolute value of a real n... |
| absresq 15227 | Square of the absolute val... |
| absmod0 15228 | ` A ` is divisible by ` B ... |
| absexp 15229 | Absolute value of positive... |
| absexpz 15230 | Absolute value of integer ... |
| abssq 15231 | Square can be moved in and... |
| sqabs 15232 | The squares of two reals a... |
| absrele 15233 | The absolute value of a co... |
| absimle 15234 | The absolute value of a co... |
| max0add 15235 | The sum of the positive an... |
| absz 15236 | A real number is an intege... |
| nn0abscl 15237 | The absolute value of an i... |
| zabscl 15238 | The absolute value of an i... |
| zabs0b 15239 | An integer has an absolute... |
| abslt 15240 | Absolute value and 'less t... |
| absle 15241 | Absolute value and 'less t... |
| abssubne0 15242 | If the absolute value of a... |
| absdiflt 15243 | The absolute value of a di... |
| absdifle 15244 | The absolute value of a di... |
| elicc4abs 15245 | Membership in a symmetric ... |
| lenegsq 15246 | Comparison to a nonnegativ... |
| releabs 15247 | The real part of a number ... |
| recval 15248 | Reciprocal expressed with ... |
| absidm 15249 | The absolute value functio... |
| absgt0 15250 | The absolute value of a no... |
| nnabscl 15251 | The absolute value of a no... |
| abssub 15252 | Swapping order of subtract... |
| abssubge0 15253 | Absolute value of a nonneg... |
| abssuble0 15254 | Absolute value of a nonpos... |
| absmax 15255 | The maximum of two numbers... |
| abstri 15256 | Triangle inequality for ab... |
| abs3dif 15257 | Absolute value of differen... |
| abs2dif 15258 | Difference of absolute val... |
| abs2dif2 15259 | Difference of absolute val... |
| abs2difabs 15260 | Absolute value of differen... |
| abs1m 15261 | For any complex number, th... |
| recan 15262 | Cancellation law involving... |
| absf 15263 | Mapping domain and codomai... |
| abs3lem 15264 | Lemma involving absolute v... |
| abslem2 15265 | Lemma involving absolute v... |
| rddif 15266 | The difference between a r... |
| absrdbnd 15267 | Bound on the absolute valu... |
| fzomaxdiflem 15268 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15269 | A bound on the separation ... |
| uzin2 15270 | The upper integers are clo... |
| rexanuz 15271 | Combine two different uppe... |
| rexanre 15272 | Combine two different uppe... |
| rexfiuz 15273 | Combine finitely many diff... |
| rexuz3 15274 | Restrict the base of the u... |
| rexanuz2 15275 | Combine two different uppe... |
| r19.29uz 15276 | A version of ~ 19.29 for u... |
| r19.2uz 15277 | A version of ~ r19.2z for ... |
| rexuzre 15278 | Convert an upper real quan... |
| rexico 15279 | Restrict the base of an up... |
| cau3lem 15280 | Lemma for ~ cau3 . (Contr... |
| cau3 15281 | Convert between three-quan... |
| cau4 15282 | Change the base of a Cauch... |
| caubnd2 15283 | A Cauchy sequence of compl... |
| caubnd 15284 | A Cauchy sequence of compl... |
| sqreulem 15285 | Lemma for ~ sqreu : write ... |
| sqreu 15286 | Existence and uniqueness f... |
| sqrtcl 15287 | Closure of the square root... |
| sqrtthlem 15288 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15289 | Mapping domain and codomai... |
| sqrtth 15290 | Square root theorem over t... |
| sqrtrege0 15291 | The square root function m... |
| eqsqrtor 15292 | Solve an equation containi... |
| eqsqrtd 15293 | A deduction for showing th... |
| eqsqrt2d 15294 | A deduction for showing th... |
| amgm2 15295 | Arithmetic-geometric mean ... |
| sqrtthi 15296 | Square root theorem. Theo... |
| sqrtcli 15297 | The square root of a nonne... |
| sqrtgt0i 15298 | The square root of a posit... |
| sqrtmsqi 15299 | Square root of square. (C... |
| sqrtsqi 15300 | Square root of square. (C... |
| sqsqrti 15301 | Square of square root. (C... |
| sqrtge0i 15302 | The square root of a nonne... |
| absidi 15303 | A nonnegative number is it... |
| absnidi 15304 | A negative number is the n... |
| leabsi 15305 | A real number is less than... |
| absori 15306 | The absolute value of a re... |
| absrei 15307 | Absolute value of a real n... |
| sqrtpclii 15308 | The square root of a posit... |
| sqrtgt0ii 15309 | The square root of a posit... |
| sqrt11i 15310 | The square root function i... |
| sqrtmuli 15311 | Square root distributes ov... |
| sqrtmulii 15312 | Square root distributes ov... |
| sqrtmsq2i 15313 | Relationship between squar... |
| sqrtlei 15314 | Square root is monotonic. ... |
| sqrtlti 15315 | Square root is strictly mo... |
| abslti 15316 | Absolute value and 'less t... |
| abslei 15317 | Absolute value and 'less t... |
| cnsqrt00 15318 | A square root of a complex... |
| absvalsqi 15319 | Square of value of absolut... |
| absvalsq2i 15320 | Square of value of absolut... |
| abscli 15321 | Real closure of absolute v... |
| absge0i 15322 | Absolute value is nonnegat... |
| absval2i 15323 | Value of absolute value fu... |
| abs00i 15324 | The absolute value of a nu... |
| absgt0i 15325 | The absolute value of a no... |
| absnegi 15326 | Absolute value of negative... |
| abscji 15327 | The absolute value of a nu... |
| releabsi 15328 | The real part of a number ... |
| abssubi 15329 | Swapping order of subtract... |
| absmuli 15330 | Absolute value distributes... |
| sqabsaddi 15331 | Square of absolute value o... |
| sqabssubi 15332 | Square of absolute value o... |
| absdivzi 15333 | Absolute value distributes... |
| abstrii 15334 | Triangle inequality for ab... |
| abs3difi 15335 | Absolute value of differen... |
| abs3lemi 15336 | Lemma involving absolute v... |
| rpsqrtcld 15337 | The square root of a posit... |
| sqrtgt0d 15338 | The square root of a posit... |
| absnidd 15339 | A negative number is the n... |
| leabsd 15340 | A real number is less than... |
| absord 15341 | The absolute value of a re... |
| absred 15342 | Absolute value of a real n... |
| resqrtcld 15343 | The square root of a nonne... |
| sqrtmsqd 15344 | Square root of square. (C... |
| sqrtsqd 15345 | Square root of square. (C... |
| sqrtge0d 15346 | The square root of a nonne... |
| sqrtnegd 15347 | The square root of a negat... |
| absidd 15348 | A nonnegative number is it... |
| sqrtdivd 15349 | Square root distributes ov... |
| sqrtmuld 15350 | Square root distributes ov... |
| sqrtsq2d 15351 | Relationship between squar... |
| sqrtled 15352 | Square root is monotonic. ... |
| sqrtltd 15353 | Square root is strictly mo... |
| sqr11d 15354 | The square root function i... |
| nn0absid 15355 | A nonnegative integer is i... |
| nn0absidi 15356 | A nonnegative integer is i... |
| absltd 15357 | Absolute value and 'less t... |
| absled 15358 | Absolute value and 'less t... |
| abssubge0d 15359 | Absolute value of a nonneg... |
| abssuble0d 15360 | Absolute value of a nonpos... |
| absdifltd 15361 | The absolute value of a di... |
| absdifled 15362 | The absolute value of a di... |
| icodiamlt 15363 | Two elements in a half-ope... |
| abscld 15364 | Real closure of absolute v... |
| sqrtcld 15365 | Closure of the square root... |
| sqrtrege0d 15366 | The real part of the squar... |
| sqsqrtd 15367 | Square root theorem. Theo... |
| msqsqrtd 15368 | Square root theorem. Theo... |
| sqr00d 15369 | A square root is zero iff ... |
| absvalsqd 15370 | Square of value of absolut... |
| absvalsq2d 15371 | Square of value of absolut... |
| absge0d 15372 | Absolute value is nonnegat... |
| absval2d 15373 | Value of absolute value fu... |
| abs00d 15374 | The absolute value of a nu... |
| absne0d 15375 | The absolute value of a nu... |
| absrpcld 15376 | The absolute value of a no... |
| absnegd 15377 | Absolute value of negative... |
| abscjd 15378 | The absolute value of a nu... |
| releabsd 15379 | The real part of a number ... |
| absexpd 15380 | Absolute value of positive... |
| abssubd 15381 | Swapping order of subtract... |
| absmuld 15382 | Absolute value distributes... |
| absdivd 15383 | Absolute value distributes... |
| abstrid 15384 | Triangle inequality for ab... |
| abs2difd 15385 | Difference of absolute val... |
| abs2dif2d 15386 | Difference of absolute val... |
| abs2difabsd 15387 | Absolute value of differen... |
| abs3difd 15388 | Absolute value of differen... |
| abs3lemd 15389 | Lemma involving absolute v... |
| reusq0 15390 | A complex number is the sq... |
| bhmafibid1cn 15391 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15392 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15393 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15394 | The Brahmagupta-Fibonacci ... |
| limsupgord 15397 | Ordering property of the s... |
| limsupcl 15398 | Closure of the superior li... |
| limsupval 15399 | The superior limit of an i... |
| limsupgf 15400 | Closure of the superior li... |
| limsupgval 15401 | Value of the superior limi... |
| limsupgle 15402 | The defining property of t... |
| limsuple 15403 | The defining property of t... |
| limsuplt 15404 | The defining property of t... |
| limsupval2 15405 | The superior limit, relati... |
| limsupgre 15406 | If a sequence of real numb... |
| limsupbnd1 15407 | If a sequence is eventuall... |
| limsupbnd2 15408 | If a sequence is eventuall... |
| climrel 15417 | The limit relation is a re... |
| rlimrel 15418 | The limit relation is a re... |
| clim 15419 | Express the predicate: Th... |
| rlim 15420 | Express the predicate: Th... |
| rlim2 15421 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15422 | Use strictly less-than in ... |
| rlim3 15423 | Restrict the range of the ... |
| climcl 15424 | Closure of the limit of a ... |
| rlimpm 15425 | Closure of a function with... |
| rlimf 15426 | Closure of a function with... |
| rlimss 15427 | Domain closure of a functi... |
| rlimcl 15428 | Closure of the limit of a ... |
| clim2 15429 | Express the predicate: Th... |
| clim2c 15430 | Express the predicate ` F ... |
| clim0 15431 | Express the predicate ` F ... |
| clim0c 15432 | Express the predicate ` F ... |
| rlim0 15433 | Express the predicate ` B ... |
| rlim0lt 15434 | Use strictly less-than in ... |
| climi 15435 | Convergence of a sequence ... |
| climi2 15436 | Convergence of a sequence ... |
| climi0 15437 | Convergence of a sequence ... |
| rlimi 15438 | Convergence at infinity of... |
| rlimi2 15439 | Convergence at infinity of... |
| ello1 15440 | Elementhood in the set of ... |
| ello12 15441 | Elementhood in the set of ... |
| ello12r 15442 | Sufficient condition for e... |
| lo1f 15443 | An eventually upper bounde... |
| lo1dm 15444 | An eventually upper bounde... |
| lo1bdd 15445 | The defining property of a... |
| ello1mpt 15446 | Elementhood in the set of ... |
| ello1mpt2 15447 | Elementhood in the set of ... |
| ello1d 15448 | Sufficient condition for e... |
| lo1bdd2 15449 | If an eventually bounded f... |
| lo1bddrp 15450 | Refine ~ o1bdd2 to give a ... |
| elo1 15451 | Elementhood in the set of ... |
| elo12 15452 | Elementhood in the set of ... |
| elo12r 15453 | Sufficient condition for e... |
| o1f 15454 | An eventually bounded func... |
| o1dm 15455 | An eventually bounded func... |
| o1bdd 15456 | The defining property of a... |
| lo1o1 15457 | A function is eventually b... |
| lo1o12 15458 | A function is eventually b... |
| elo1mpt 15459 | Elementhood in the set of ... |
| elo1mpt2 15460 | Elementhood in the set of ... |
| elo1d 15461 | Sufficient condition for e... |
| o1lo1 15462 | A real function is eventua... |
| o1lo12 15463 | A lower bounded real funct... |
| o1lo1d 15464 | A real eventually bounded ... |
| icco1 15465 | Derive eventual boundednes... |
| o1bdd2 15466 | If an eventually bounded f... |
| o1bddrp 15467 | Refine ~ o1bdd2 to give a ... |
| climconst 15468 | An (eventually) constant s... |
| rlimconst 15469 | A constant sequence conver... |
| rlimclim1 15470 | Forward direction of ~ rli... |
| rlimclim 15471 | A sequence on an upper int... |
| climrlim2 15472 | Produce a real limit from ... |
| climconst2 15473 | A constant sequence conver... |
| climz 15474 | The zero sequence converge... |
| rlimuni 15475 | A real function whose doma... |
| rlimdm 15476 | Two ways to express that a... |
| climuni 15477 | An infinite sequence of co... |
| fclim 15478 | The limit relation is func... |
| climdm 15479 | Two ways to express that a... |
| climeu 15480 | An infinite sequence of co... |
| climreu 15481 | An infinite sequence of co... |
| climmo 15482 | An infinite sequence of co... |
| rlimres 15483 | The restriction of a funct... |
| lo1res 15484 | The restriction of an even... |
| o1res 15485 | The restriction of an even... |
| rlimres2 15486 | The restriction of a funct... |
| lo1res2 15487 | The restriction of a funct... |
| o1res2 15488 | The restriction of a funct... |
| lo1resb 15489 | The restriction of a funct... |
| rlimresb 15490 | The restriction of a funct... |
| o1resb 15491 | The restriction of a funct... |
| climeq 15492 | Two functions that are eve... |
| lo1eq 15493 | Two functions that are eve... |
| rlimeq 15494 | Two functions that are eve... |
| o1eq 15495 | Two functions that are eve... |
| climmpt 15496 | Exhibit a function ` G ` w... |
| 2clim 15497 | If two sequences converge ... |
| climmpt2 15498 | Relate an integer limit on... |
| climshftlem 15499 | A shifted function converg... |
| climres 15500 | A function restricted to u... |
| climshft 15501 | A shifted function converg... |
| serclim0 15502 | The zero series converges ... |
| rlimcld2 15503 | If ` D ` is a closed set i... |
| rlimrege0 15504 | The limit of a sequence of... |
| rlimrecl 15505 | The limit of a real sequen... |
| rlimge0 15506 | The limit of a sequence of... |
| climshft2 15507 | A shifted function converg... |
| climrecl 15508 | The limit of a convergent ... |
| climge0 15509 | A nonnegative sequence con... |
| climabs0 15510 | Convergence to zero of the... |
| o1co 15511 | Sufficient condition for t... |
| o1compt 15512 | Sufficient condition for t... |
| rlimcn1 15513 | Image of a limit under a c... |
| rlimcn1b 15514 | Image of a limit under a c... |
| rlimcn3 15515 | Image of a limit under a c... |
| rlimcn2 15516 | Image of a limit under a c... |
| climcn1 15517 | Image of a limit under a c... |
| climcn2 15518 | Image of a limit under a c... |
| addcn2 15519 | Complex number addition is... |
| subcn2 15520 | Complex number subtraction... |
| mulcn2 15521 | Complex number multiplicat... |
| reccn2 15522 | The reciprocal function is... |
| cn1lem 15523 | A sufficient condition for... |
| abscn2 15524 | The absolute value functio... |
| cjcn2 15525 | The complex conjugate func... |
| recn2 15526 | The real part function is ... |
| imcn2 15527 | The imaginary part functio... |
| climcn1lem 15528 | The limit of a continuous ... |
| climabs 15529 | Limit of the absolute valu... |
| climcj 15530 | Limit of the complex conju... |
| climre 15531 | Limit of the real part of ... |
| climim 15532 | Limit of the imaginary par... |
| rlimmptrcl 15533 | Reverse closure for a real... |
| rlimabs 15534 | Limit of the absolute valu... |
| rlimcj 15535 | Limit of the complex conju... |
| rlimre 15536 | Limit of the real part of ... |
| rlimim 15537 | Limit of the imaginary par... |
| o1of2 15538 | Show that a binary operati... |
| o1add 15539 | The sum of two eventually ... |
| o1mul 15540 | The product of two eventua... |
| o1sub 15541 | The difference of two even... |
| rlimo1 15542 | Any function with a finite... |
| rlimdmo1 15543 | A convergent function is e... |
| o1rlimmul 15544 | The product of an eventual... |
| o1const 15545 | A constant function is eve... |
| lo1const 15546 | A constant function is eve... |
| lo1mptrcl 15547 | Reverse closure for an eve... |
| o1mptrcl 15548 | Reverse closure for an eve... |
| o1add2 15549 | The sum of two eventually ... |
| o1mul2 15550 | The product of two eventua... |
| o1sub2 15551 | The product of two eventua... |
| lo1add 15552 | The sum of two eventually ... |
| lo1mul 15553 | The product of an eventual... |
| lo1mul2 15554 | The product of an eventual... |
| o1dif 15555 | If the difference of two f... |
| lo1sub 15556 | The difference of an event... |
| climadd 15557 | Limit of the sum of two co... |
| climmul 15558 | Limit of the product of tw... |
| climsub 15559 | Limit of the difference of... |
| climaddc1 15560 | Limit of a constant ` C ` ... |
| climaddc2 15561 | Limit of a constant ` C ` ... |
| climmulc2 15562 | Limit of a sequence multip... |
| climsubc1 15563 | Limit of a constant ` C ` ... |
| climsubc2 15564 | Limit of a constant ` C ` ... |
| climle 15565 | Comparison of the limits o... |
| climsqz 15566 | Convergence of a sequence ... |
| climsqz2 15567 | Convergence of a sequence ... |
| rlimadd 15568 | Limit of the sum of two co... |
| rlimsub 15569 | Limit of the difference of... |
| rlimmul 15570 | Limit of the product of tw... |
| rlimdiv 15571 | Limit of the quotient of t... |
| rlimneg 15572 | Limit of the negative of a... |
| rlimle 15573 | Comparison of the limits o... |
| rlimsqzlem 15574 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15575 | Convergence of a sequence ... |
| rlimsqz2 15576 | Convergence of a sequence ... |
| lo1le 15577 | Transfer eventual upper bo... |
| o1le 15578 | Transfer eventual boundedn... |
| rlimno1 15579 | A function whose inverse c... |
| clim2ser 15580 | The limit of an infinite s... |
| clim2ser2 15581 | The limit of an infinite s... |
| iserex 15582 | An infinite series converg... |
| isermulc2 15583 | Multiplication of an infin... |
| climlec2 15584 | Comparison of a constant t... |
| iserle 15585 | Comparison of the limits o... |
| iserge0 15586 | The limit of an infinite s... |
| climub 15587 | The limit of a monotonic s... |
| climserle 15588 | The partial sums of a conv... |
| isershft 15589 | Index shift of the limit o... |
| isercolllem1 15590 | Lemma for ~ isercoll . (C... |
| isercolllem2 15591 | Lemma for ~ isercoll . (C... |
| isercolllem3 15592 | Lemma for ~ isercoll . (C... |
| isercoll 15593 | Rearrange an infinite seri... |
| isercoll2 15594 | Generalize ~ isercoll so t... |
| climsup 15595 | A bounded monotonic sequen... |
| climcau 15596 | A converging sequence of c... |
| climbdd 15597 | A converging sequence of c... |
| caucvgrlem 15598 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15599 | A Cauchy sequence of real ... |
| caucvgrlem2 15600 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15601 | A Cauchy sequence of compl... |
| caurcvg 15602 | A Cauchy sequence of real ... |
| caurcvg2 15603 | A Cauchy sequence of real ... |
| caucvg 15604 | A Cauchy sequence of compl... |
| caucvgb 15605 | A function is convergent i... |
| serf0 15606 | If an infinite series conv... |
| iseraltlem1 15607 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15608 | Lemma for ~ iseralt . The... |
| iseraltlem3 15609 | Lemma for ~ iseralt . Fro... |
| iseralt 15610 | The alternating series tes... |
| sumex 15613 | A sum is a set. (Contribu... |
| sumeq1 15614 | Equality theorem for a sum... |
| nfsum1 15615 | Bound-variable hypothesis ... |
| nfsum 15616 | Bound-variable hypothesis ... |
| sumeq2w 15617 | Equality theorem for sum, ... |
| sumeq2ii 15618 | Equality theorem for sum, ... |
| sumeq2 15619 | Equality theorem for sum. ... |
| cbvsum 15620 | Change bound variable in a... |
| cbvsumv 15621 | Change bound variable in a... |
| sumeq1i 15622 | Equality inference for sum... |
| sumeq2i 15623 | Equality inference for sum... |
| sumeq12i 15624 | Equality inference for sum... |
| sumeq1d 15625 | Equality deduction for sum... |
| sumeq2d 15626 | Equality deduction for sum... |
| sumeq2dv 15627 | Equality deduction for sum... |
| sumeq2sdv 15628 | Equality deduction for sum... |
| sumeq2sdvOLD 15629 | Obsolete version of ~ sume... |
| 2sumeq2dv 15630 | Equality deduction for dou... |
| sumeq12dv 15631 | Equality deduction for sum... |
| sumeq12rdv 15632 | Equality deduction for sum... |
| sum2id 15633 | The second class argument ... |
| sumfc 15634 | A lemma to facilitate conv... |
| fz1f1o 15635 | A lemma for working with f... |
| sumrblem 15636 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15637 | The sequence of partial su... |
| sumrb 15638 | Rebase the starting point ... |
| summolem3 15639 | Lemma for ~ summo . (Cont... |
| summolem2a 15640 | Lemma for ~ summo . (Cont... |
| summolem2 15641 | Lemma for ~ summo . (Cont... |
| summo 15642 | A sum has at most one limi... |
| zsum 15643 | Series sum with index set ... |
| isum 15644 | Series sum with an upper i... |
| fsum 15645 | The value of a sum over a ... |
| sum0 15646 | Any sum over the empty set... |
| sumz 15647 | Any sum of zero over a sum... |
| fsumf1o 15648 | Re-index a finite sum usin... |
| sumss 15649 | Change the index set to a ... |
| fsumss 15650 | Change the index set to a ... |
| sumss2 15651 | Change the index set of a ... |
| fsumcvg2 15652 | The sequence of partial su... |
| fsumsers 15653 | Special case of series sum... |
| fsumcvg3 15654 | A finite sum is convergent... |
| fsumser 15655 | A finite sum expressed in ... |
| fsumcl2lem 15656 | - Lemma for finite sum clo... |
| fsumcllem 15657 | - Lemma for finite sum clo... |
| fsumcl 15658 | Closure of a finite sum of... |
| fsumrecl 15659 | Closure of a finite sum of... |
| fsumzcl 15660 | Closure of a finite sum of... |
| fsumnn0cl 15661 | Closure of a finite sum of... |
| fsumrpcl 15662 | Closure of a finite sum of... |
| fsumclf 15663 | Closure of a finite sum of... |
| fsumzcl2 15664 | A finite sum with integer ... |
| fsumadd 15665 | The sum of two finite sums... |
| fsumsplit 15666 | Split a sum into two parts... |
| fsumsplitf 15667 | Split a sum into two parts... |
| sumsnf 15668 | A sum of a singleton is th... |
| fsumsplitsn 15669 | Separate out a term in a f... |
| fsumsplit1 15670 | Separate out a term in a f... |
| sumsn 15671 | A sum of a singleton is th... |
| fsum1 15672 | The finite sum of ` A ( k ... |
| sumpr 15673 | A sum over a pair is the s... |
| sumtp 15674 | A sum over a triple is the... |
| sumsns 15675 | A sum of a singleton is th... |
| fsumm1 15676 | Separate out the last term... |
| fzosump1 15677 | Separate out the last term... |
| fsum1p 15678 | Separate out the first ter... |
| fsummsnunz 15679 | A finite sum all of whose ... |
| fsumsplitsnun 15680 | Separate out a term in a f... |
| fsump1 15681 | The addition of the next t... |
| isumclim 15682 | An infinite sum equals the... |
| isumclim2 15683 | A converging series conver... |
| isumclim3 15684 | The sequence of partial fi... |
| sumnul 15685 | The sum of a non-convergen... |
| isumcl 15686 | The sum of a converging in... |
| isummulc2 15687 | An infinite sum multiplied... |
| isummulc1 15688 | An infinite sum multiplied... |
| isumdivc 15689 | An infinite sum divided by... |
| isumrecl 15690 | The sum of a converging in... |
| isumge0 15691 | An infinite sum of nonnega... |
| isumadd 15692 | Addition of infinite sums.... |
| sumsplit 15693 | Split a sum into two parts... |
| fsump1i 15694 | Optimized version of ~ fsu... |
| fsum2dlem 15695 | Lemma for ~ fsum2d - induc... |
| fsum2d 15696 | Write a double sum as a su... |
| fsumxp 15697 | Combine two sums into a si... |
| fsumcnv 15698 | Transform a region of summ... |
| fsumcom2 15699 | Interchange order of summa... |
| fsumcom 15700 | Interchange order of summa... |
| fsum0diaglem 15701 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15702 | Two ways to express "the s... |
| mptfzshft 15703 | 1-1 onto function in maps-... |
| fsumrev 15704 | Reversal of a finite sum. ... |
| fsumshft 15705 | Index shift of a finite su... |
| fsumshftm 15706 | Negative index shift of a ... |
| fsumrev2 15707 | Reversal of a finite sum. ... |
| fsum0diag2 15708 | Two ways to express "the s... |
| fsummulc2 15709 | A finite sum multiplied by... |
| fsummulc1 15710 | A finite sum multiplied by... |
| fsumdivc 15711 | A finite sum divided by a ... |
| fsumneg 15712 | Negation of a finite sum. ... |
| fsumsub 15713 | Split a finite sum over a ... |
| fsum2mul 15714 | Separate the nested sum of... |
| fsumconst 15715 | The sum of constant terms ... |
| fsumdifsnconst 15716 | The sum of constant terms ... |
| modfsummodslem1 15717 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15718 | Induction step for ~ modfs... |
| modfsummod 15719 | A finite sum modulo a posi... |
| fsumge0 15720 | If all of the terms of a f... |
| fsumless 15721 | A shorter sum of nonnegati... |
| fsumge1 15722 | A sum of nonnegative numbe... |
| fsum00 15723 | A sum of nonnegative numbe... |
| fsumle 15724 | If all of the terms of fin... |
| fsumlt 15725 | If every term in one finit... |
| fsumabs 15726 | Generalized triangle inequ... |
| telfsumo 15727 | Sum of a telescoping serie... |
| telfsumo2 15728 | Sum of a telescoping serie... |
| telfsum 15729 | Sum of a telescoping serie... |
| telfsum2 15730 | Sum of a telescoping serie... |
| fsumparts 15731 | Summation by parts. (Cont... |
| fsumrelem 15732 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15733 | The real part of a sum. (... |
| fsumim 15734 | The imaginary part of a su... |
| fsumcj 15735 | The complex conjugate of a... |
| fsumrlim 15736 | Limit of a finite sum of c... |
| fsumo1 15737 | The finite sum of eventual... |
| o1fsum 15738 | If ` A ( k ) ` is O(1), th... |
| seqabs 15739 | Generalized triangle inequ... |
| iserabs 15740 | Generalized triangle inequ... |
| cvgcmp 15741 | A comparison test for conv... |
| cvgcmpub 15742 | An upper bound for the lim... |
| cvgcmpce 15743 | A comparison test for conv... |
| abscvgcvg 15744 | An absolutely convergent s... |
| climfsum 15745 | Limit of a finite sum of c... |
| fsumiun 15746 | Sum over a disjoint indexe... |
| hashiun 15747 | The cardinality of a disjo... |
| hash2iun 15748 | The cardinality of a neste... |
| hash2iun1dif1 15749 | The cardinality of a neste... |
| hashrabrex 15750 | The number of elements in ... |
| hashuni 15751 | The cardinality of a disjo... |
| qshash 15752 | The cardinality of a set w... |
| ackbijnn 15753 | Translate the Ackermann bi... |
| binomlem 15754 | Lemma for ~ binom (binomia... |
| binom 15755 | The binomial theorem: ` ( ... |
| binom1p 15756 | Special case of the binomi... |
| binom11 15757 | Special case of the binomi... |
| binom1dif 15758 | A summation for the differ... |
| bcxmaslem1 15759 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15760 | Parallel summation (Christ... |
| incexclem 15761 | Lemma for ~ incexc . (Con... |
| incexc 15762 | The inclusion/exclusion pr... |
| incexc2 15763 | The inclusion/exclusion pr... |
| isumshft 15764 | Index shift of an infinite... |
| isumsplit 15765 | Split off the first ` N ` ... |
| isum1p 15766 | The infinite sum of a conv... |
| isumnn0nn 15767 | Sum from 0 to infinity in ... |
| isumrpcl 15768 | The infinite sum of positi... |
| isumle 15769 | Comparison of two infinite... |
| isumless 15770 | A finite sum of nonnegativ... |
| isumsup2 15771 | An infinite sum of nonnega... |
| isumsup 15772 | An infinite sum of nonnega... |
| isumltss 15773 | A partial sum of a series ... |
| climcndslem1 15774 | Lemma for ~ climcnds : bou... |
| climcndslem2 15775 | Lemma for ~ climcnds : bou... |
| climcnds 15776 | The Cauchy condensation te... |
| divrcnv 15777 | The sequence of reciprocal... |
| divcnv 15778 | The sequence of reciprocal... |
| flo1 15779 | The floor function satisfi... |
| divcnvshft 15780 | Limit of a ratio function.... |
| supcvg 15781 | Extract a sequence ` f ` i... |
| infcvgaux1i 15782 | Auxiliary theorem for appl... |
| infcvgaux2i 15783 | Auxiliary theorem for appl... |
| harmonic 15784 | The harmonic series ` H ` ... |
| arisum 15785 | Arithmetic series sum of t... |
| arisum2 15786 | Arithmetic series sum of t... |
| trireciplem 15787 | Lemma for ~ trirecip . Sh... |
| trirecip 15788 | The sum of the reciprocals... |
| expcnv 15789 | A sequence of powers of a ... |
| explecnv 15790 | A sequence of terms conver... |
| geoserg 15791 | The value of the finite ge... |
| geoser 15792 | The value of the finite ge... |
| pwdif 15793 | The difference of two numb... |
| pwm1geoser 15794 | The n-th power of a number... |
| geolim 15795 | The partial sums in the in... |
| geolim2 15796 | The partial sums in the ge... |
| georeclim 15797 | The limit of a geometric s... |
| geo2sum 15798 | The value of the finite ge... |
| geo2sum2 15799 | The value of the finite ge... |
| geo2lim 15800 | The value of the infinite ... |
| geomulcvg 15801 | The geometric series conve... |
| geoisum 15802 | The infinite sum of ` 1 + ... |
| geoisumr 15803 | The infinite sum of recipr... |
| geoisum1 15804 | The infinite sum of ` A ^ ... |
| geoisum1c 15805 | The infinite sum of ` A x.... |
| 0.999... 15806 | The recurring decimal 0.99... |
| geoihalfsum 15807 | Prove that the infinite ge... |
| cvgrat 15808 | Ratio test for convergence... |
| mertenslem1 15809 | Lemma for ~ mertens . (Co... |
| mertenslem2 15810 | Lemma for ~ mertens . (Co... |
| mertens 15811 | Mertens' theorem. If ` A ... |
| prodf 15812 | An infinite product of com... |
| clim2prod 15813 | The limit of an infinite p... |
| clim2div 15814 | The limit of an infinite p... |
| prodfmul 15815 | The product of two infinit... |
| prodf1 15816 | The value of the partial p... |
| prodf1f 15817 | A one-valued infinite prod... |
| prodfclim1 15818 | The constant one product c... |
| prodfn0 15819 | No term of a nonzero infin... |
| prodfrec 15820 | The reciprocal of an infin... |
| prodfdiv 15821 | The quotient of two infini... |
| ntrivcvg 15822 | A non-trivially converging... |
| ntrivcvgn0 15823 | A product that converges t... |
| ntrivcvgfvn0 15824 | Any value of a product seq... |
| ntrivcvgtail 15825 | A tail of a non-trivially ... |
| ntrivcvgmullem 15826 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15827 | The product of two non-tri... |
| prodex 15830 | A product is a set. (Cont... |
| prodeq1f 15831 | Equality theorem for a pro... |
| prodeq1 15832 | Equality theorem for a pro... |
| nfcprod1 15833 | Bound-variable hypothesis ... |
| nfcprod 15834 | Bound-variable hypothesis ... |
| prodeq2w 15835 | Equality theorem for produ... |
| prodeq2ii 15836 | Equality theorem for produ... |
| prodeq2 15837 | Equality theorem for produ... |
| cbvprod 15838 | Change bound variable in a... |
| cbvprodv 15839 | Change bound variable in a... |
| cbvprodi 15840 | Change bound variable in a... |
| prodeq1i 15841 | Equality inference for pro... |
| prodeq1iOLD 15842 | Obsolete version of ~ prod... |
| prodeq2i 15843 | Equality inference for pro... |
| prodeq12i 15844 | Equality inference for pro... |
| prodeq1d 15845 | Equality deduction for pro... |
| prodeq2d 15846 | Equality deduction for pro... |
| prodeq2dv 15847 | Equality deduction for pro... |
| prodeq2sdv 15848 | Equality deduction for pro... |
| prodeq2sdvOLD 15849 | Obsolete version of ~ prod... |
| 2cprodeq2dv 15850 | Equality deduction for dou... |
| prodeq12dv 15851 | Equality deduction for pro... |
| prodeq12rdv 15852 | Equality deduction for pro... |
| prod2id 15853 | The second class argument ... |
| prodrblem 15854 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15855 | The sequence of partial pr... |
| prodrblem2 15856 | Lemma for ~ prodrb . (Con... |
| prodrb 15857 | Rebase the starting point ... |
| prodmolem3 15858 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15859 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15860 | Lemma for ~ prodmo . (Con... |
| prodmo 15861 | A product has at most one ... |
| zprod 15862 | Series product with index ... |
| iprod 15863 | Series product with an upp... |
| zprodn0 15864 | Nonzero series product wit... |
| iprodn0 15865 | Nonzero series product wit... |
| fprod 15866 | The value of a product ove... |
| fprodntriv 15867 | A non-triviality lemma for... |
| prod0 15868 | A product over the empty s... |
| prod1 15869 | Any product of one over a ... |
| prodfc 15870 | A lemma to facilitate conv... |
| fprodf1o 15871 | Re-index a finite product ... |
| prodss 15872 | Change the index set to a ... |
| fprodss 15873 | Change the index set to a ... |
| fprodser 15874 | A finite product expressed... |
| fprodcl2lem 15875 | Finite product closure lem... |
| fprodcllem 15876 | Finite product closure lem... |
| fprodcl 15877 | Closure of a finite produc... |
| fprodrecl 15878 | Closure of a finite produc... |
| fprodzcl 15879 | Closure of a finite produc... |
| fprodnncl 15880 | Closure of a finite produc... |
| fprodrpcl 15881 | Closure of a finite produc... |
| fprodnn0cl 15882 | Closure of a finite produc... |
| fprodcllemf 15883 | Finite product closure lem... |
| fprodreclf 15884 | Closure of a finite produc... |
| fprodmul 15885 | The product of two finite ... |
| fproddiv 15886 | The quotient of two finite... |
| prodsn 15887 | A product of a singleton i... |
| fprod1 15888 | A finite product of only o... |
| prodsnf 15889 | A product of a singleton i... |
| climprod1 15890 | The limit of a product ove... |
| fprodsplit 15891 | Split a finite product int... |
| fprodm1 15892 | Separate out the last term... |
| fprod1p 15893 | Separate out the first ter... |
| fprodp1 15894 | Multiply in the last term ... |
| fprodm1s 15895 | Separate out the last term... |
| fprodp1s 15896 | Multiply in the last term ... |
| prodsns 15897 | A product of the singleton... |
| fprodfac 15898 | Factorial using product no... |
| fprodabs 15899 | The absolute value of a fi... |
| fprodeq0 15900 | Any finite product contain... |
| fprodshft 15901 | Shift the index of a finit... |
| fprodrev 15902 | Reversal of a finite produ... |
| fprodconst 15903 | The product of constant te... |
| fprodn0 15904 | A finite product of nonzer... |
| fprod2dlem 15905 | Lemma for ~ fprod2d - indu... |
| fprod2d 15906 | Write a double product as ... |
| fprodxp 15907 | Combine two products into ... |
| fprodcnv 15908 | Transform a product region... |
| fprodcom2 15909 | Interchange order of multi... |
| fprodcom 15910 | Interchange product order.... |
| fprod0diag 15911 | Two ways to express "the p... |
| fproddivf 15912 | The quotient of two finite... |
| fprodsplitf 15913 | Split a finite product int... |
| fprodsplitsn 15914 | Separate out a term in a f... |
| fprodsplit1f 15915 | Separate out a term in a f... |
| fprodn0f 15916 | A finite product of nonzer... |
| fprodclf 15917 | Closure of a finite produc... |
| fprodge0 15918 | If all the terms of a fini... |
| fprodeq0g 15919 | Any finite product contain... |
| fprodge1 15920 | If all of the terms of a f... |
| fprodle 15921 | If all the terms of two fi... |
| fprodmodd 15922 | If all factors of two fini... |
| iprodclim 15923 | An infinite product equals... |
| iprodclim2 15924 | A converging product conve... |
| iprodclim3 15925 | The sequence of partial fi... |
| iprodcl 15926 | The product of a non-trivi... |
| iprodrecl 15927 | The product of a non-trivi... |
| iprodmul 15928 | Multiplication of infinite... |
| risefacval 15933 | The value of the rising fa... |
| fallfacval 15934 | The value of the falling f... |
| risefacval2 15935 | One-based value of rising ... |
| fallfacval2 15936 | One-based value of falling... |
| fallfacval3 15937 | A product representation o... |
| risefaccllem 15938 | Lemma for rising factorial... |
| fallfaccllem 15939 | Lemma for falling factoria... |
| risefaccl 15940 | Closure law for rising fac... |
| fallfaccl 15941 | Closure law for falling fa... |
| rerisefaccl 15942 | Closure law for rising fac... |
| refallfaccl 15943 | Closure law for falling fa... |
| nnrisefaccl 15944 | Closure law for rising fac... |
| zrisefaccl 15945 | Closure law for rising fac... |
| zfallfaccl 15946 | Closure law for falling fa... |
| nn0risefaccl 15947 | Closure law for rising fac... |
| rprisefaccl 15948 | Closure law for rising fac... |
| risefallfac 15949 | A relationship between ris... |
| fallrisefac 15950 | A relationship between fal... |
| risefall0lem 15951 | Lemma for ~ risefac0 and ~... |
| risefac0 15952 | The value of the rising fa... |
| fallfac0 15953 | The value of the falling f... |
| risefacp1 15954 | The value of the rising fa... |
| fallfacp1 15955 | The value of the falling f... |
| risefacp1d 15956 | The value of the rising fa... |
| fallfacp1d 15957 | The value of the falling f... |
| risefac1 15958 | The value of rising factor... |
| fallfac1 15959 | The value of falling facto... |
| risefacfac 15960 | Relate rising factorial to... |
| fallfacfwd 15961 | The forward difference of ... |
| 0fallfac 15962 | The value of the zero fall... |
| 0risefac 15963 | The value of the zero risi... |
| binomfallfaclem1 15964 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 15965 | Lemma for ~ binomfallfac .... |
| binomfallfac 15966 | A version of the binomial ... |
| binomrisefac 15967 | A version of the binomial ... |
| fallfacval4 15968 | Represent the falling fact... |
| bcfallfac 15969 | Binomial coefficient in te... |
| fallfacfac 15970 | Relate falling factorial t... |
| bpolylem 15973 | Lemma for ~ bpolyval . (C... |
| bpolyval 15974 | The value of the Bernoulli... |
| bpoly0 15975 | The value of the Bernoulli... |
| bpoly1 15976 | The value of the Bernoulli... |
| bpolycl 15977 | Closure law for Bernoulli ... |
| bpolysum 15978 | A sum for Bernoulli polyno... |
| bpolydiflem 15979 | Lemma for ~ bpolydif . (C... |
| bpolydif 15980 | Calculate the difference b... |
| fsumkthpow 15981 | A closed-form expression f... |
| bpoly2 15982 | The Bernoulli polynomials ... |
| bpoly3 15983 | The Bernoulli polynomials ... |
| bpoly4 15984 | The Bernoulli polynomials ... |
| fsumcube 15985 | Express the sum of cubes i... |
| eftcl 15998 | Closure of a term in the s... |
| reeftcl 15999 | The terms of the series ex... |
| eftabs 16000 | The absolute value of a te... |
| eftval 16001 | The value of a term in the... |
| efcllem 16002 | Lemma for ~ efcl . The se... |
| ef0lem 16003 | The series defining the ex... |
| efval 16004 | Value of the exponential f... |
| esum 16005 | Value of Euler's constant ... |
| eff 16006 | Domain and codomain of the... |
| efcl 16007 | Closure law for the expone... |
| efcld 16008 | Closure law for the expone... |
| efval2 16009 | Value of the exponential f... |
| efcvg 16010 | The series that defines th... |
| efcvgfsum 16011 | Exponential function conve... |
| reefcl 16012 | The exponential function i... |
| reefcld 16013 | The exponential function i... |
| ere 16014 | Euler's constant ` _e ` = ... |
| ege2le3 16015 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16016 | Value of the exponential f... |
| efcj 16017 | The exponential of a compl... |
| efaddlem 16018 | Lemma for ~ efadd (exponen... |
| efadd 16019 | Sum of exponents law for e... |
| fprodefsum 16020 | Move the exponential funct... |
| efcan 16021 | Cancellation law for expon... |
| efne0d 16022 | The exponential of a compl... |
| efne0 16023 | The exponential of a compl... |
| efne0OLD 16024 | Obsolete version of ~ efne... |
| efneg 16025 | The exponential of the opp... |
| eff2 16026 | The exponential function m... |
| efsub 16027 | Difference of exponents la... |
| efexp 16028 | The exponential of an inte... |
| efzval 16029 | Value of the exponential f... |
| efgt0 16030 | The exponential of a real ... |
| rpefcl 16031 | The exponential of a real ... |
| rpefcld 16032 | The exponential of a real ... |
| eftlcvg 16033 | The tail series of the exp... |
| eftlcl 16034 | Closure of the sum of an i... |
| reeftlcl 16035 | Closure of the sum of an i... |
| eftlub 16036 | An upper bound on the abso... |
| efsep 16037 | Separate out the next term... |
| effsumlt 16038 | The partial sums of the se... |
| eft0val 16039 | The value of the first ter... |
| ef4p 16040 | Separate out the first fou... |
| efgt1p2 16041 | The exponential of a posit... |
| efgt1p 16042 | The exponential of a posit... |
| efgt1 16043 | The exponential of a posit... |
| eflt 16044 | The exponential function o... |
| efle 16045 | The exponential function o... |
| reef11 16046 | The exponential function o... |
| reeff1 16047 | The exponential function m... |
| eflegeo 16048 | The exponential function o... |
| sinval 16049 | Value of the sine function... |
| cosval 16050 | Value of the cosine functi... |
| sinf 16051 | Domain and codomain of the... |
| cosf 16052 | Domain and codomain of the... |
| sincl 16053 | Closure of the sine functi... |
| coscl 16054 | Closure of the cosine func... |
| tanval 16055 | Value of the tangent funct... |
| tancl 16056 | The closure of the tangent... |
| sincld 16057 | Closure of the sine functi... |
| coscld 16058 | Closure of the cosine func... |
| tancld 16059 | Closure of the tangent fun... |
| tanval2 16060 | Express the tangent functi... |
| tanval3 16061 | Express the tangent functi... |
| resinval 16062 | The sine of a real number ... |
| recosval 16063 | The cosine of a real numbe... |
| efi4p 16064 | Separate out the first fou... |
| resin4p 16065 | Separate out the first fou... |
| recos4p 16066 | Separate out the first fou... |
| resincl 16067 | The sine of a real number ... |
| recoscl 16068 | The cosine of a real numbe... |
| retancl 16069 | The closure of the tangent... |
| resincld 16070 | Closure of the sine functi... |
| recoscld 16071 | Closure of the cosine func... |
| retancld 16072 | Closure of the tangent fun... |
| sinneg 16073 | The sine of a negative is ... |
| cosneg 16074 | The cosines of a number an... |
| tanneg 16075 | The tangent of a negative ... |
| sin0 16076 | Value of the sine function... |
| cos0 16077 | Value of the cosine functi... |
| tan0 16078 | The value of the tangent f... |
| efival 16079 | The exponential function i... |
| efmival 16080 | The exponential function i... |
| sinhval 16081 | Value of the hyperbolic si... |
| coshval 16082 | Value of the hyperbolic co... |
| resinhcl 16083 | The hyperbolic sine of a r... |
| rpcoshcl 16084 | The hyperbolic cosine of a... |
| recoshcl 16085 | The hyperbolic cosine of a... |
| retanhcl 16086 | The hyperbolic tangent of ... |
| tanhlt1 16087 | The hyperbolic tangent of ... |
| tanhbnd 16088 | The hyperbolic tangent of ... |
| efeul 16089 | Eulerian representation of... |
| efieq 16090 | The exponentials of two im... |
| sinadd 16091 | Addition formula for sine.... |
| cosadd 16092 | Addition formula for cosin... |
| tanaddlem 16093 | A useful intermediate step... |
| tanadd 16094 | Addition formula for tange... |
| sinsub 16095 | Sine of difference. (Cont... |
| cossub 16096 | Cosine of difference. (Co... |
| addsin 16097 | Sum of sines. (Contribute... |
| subsin 16098 | Difference of sines. (Con... |
| sinmul 16099 | Product of sines can be re... |
| cosmul 16100 | Product of cosines can be ... |
| addcos 16101 | Sum of cosines. (Contribu... |
| subcos 16102 | Difference of cosines. (C... |
| sincossq 16103 | Sine squared plus cosine s... |
| sin2t 16104 | Double-angle formula for s... |
| cos2t 16105 | Double-angle formula for c... |
| cos2tsin 16106 | Double-angle formula for c... |
| sinbnd 16107 | The sine of a real number ... |
| cosbnd 16108 | The cosine of a real numbe... |
| sinbnd2 16109 | The sine of a real number ... |
| cosbnd2 16110 | The cosine of a real numbe... |
| ef01bndlem 16111 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16112 | Bounds on the sine of a po... |
| cos01bnd 16113 | Bounds on the cosine of a ... |
| cos1bnd 16114 | Bounds on the cosine of 1.... |
| cos2bnd 16115 | Bounds on the cosine of 2.... |
| sinltx 16116 | The sine of a positive rea... |
| sin01gt0 16117 | The sine of a positive rea... |
| cos01gt0 16118 | The cosine of a positive r... |
| sin02gt0 16119 | The sine of a positive rea... |
| sincos1sgn 16120 | The signs of the sine and ... |
| sincos2sgn 16121 | The signs of the sine and ... |
| sin4lt0 16122 | The sine of 4 is negative.... |
| absefi 16123 | The absolute value of the ... |
| absef 16124 | The absolute value of the ... |
| absefib 16125 | A complex number is real i... |
| efieq1re 16126 | A number whose imaginary e... |
| demoivre 16127 | De Moivre's Formula. Proo... |
| demoivreALT 16128 | Alternate proof of ~ demoi... |
| eirrlem 16131 | Lemma for ~ eirr . (Contr... |
| eirr 16132 | ` _e ` is irrational. (Co... |
| egt2lt3 16133 | Euler's constant ` _e ` = ... |
| epos 16134 | Euler's constant ` _e ` is... |
| epr 16135 | Euler's constant ` _e ` is... |
| ene0 16136 | ` _e ` is not 0. (Contrib... |
| ene1 16137 | ` _e ` is not 1. (Contrib... |
| xpnnen 16138 | The Cartesian product of t... |
| znnen 16139 | The set of integers and th... |
| qnnen 16140 | The rational numbers are c... |
| rpnnen2lem1 16141 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16142 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16143 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16144 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16145 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16146 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16147 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16148 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16149 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16150 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16151 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16152 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16153 | The other half of ~ rpnnen... |
| rpnnen 16154 | The cardinality of the con... |
| rexpen 16155 | The real numbers are equin... |
| cpnnen 16156 | The complex numbers are eq... |
| rucALT 16157 | Alternate proof of ~ ruc .... |
| ruclem1 16158 | Lemma for ~ ruc (the reals... |
| ruclem2 16159 | Lemma for ~ ruc . Orderin... |
| ruclem3 16160 | Lemma for ~ ruc . The con... |
| ruclem4 16161 | Lemma for ~ ruc . Initial... |
| ruclem6 16162 | Lemma for ~ ruc . Domain ... |
| ruclem7 16163 | Lemma for ~ ruc . Success... |
| ruclem8 16164 | Lemma for ~ ruc . The int... |
| ruclem9 16165 | Lemma for ~ ruc . The fir... |
| ruclem10 16166 | Lemma for ~ ruc . Every f... |
| ruclem11 16167 | Lemma for ~ ruc . Closure... |
| ruclem12 16168 | Lemma for ~ ruc . The sup... |
| ruclem13 16169 | Lemma for ~ ruc . There i... |
| ruc 16170 | The set of positive intege... |
| resdomq 16171 | The set of rationals is st... |
| aleph1re 16172 | There are at least aleph-o... |
| aleph1irr 16173 | There are at least aleph-o... |
| cnso 16174 | The complex numbers can be... |
| sqrt2irrlem 16175 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16176 | The square root of 2 is ir... |
| sqrt2re 16177 | The square root of 2 exist... |
| sqrt2irr0 16178 | The square root of 2 is an... |
| nthruc 16179 | The sequence ` NN ` , ` ZZ... |
| nthruz 16180 | The sequence ` NN ` , ` NN... |
| divides 16183 | Define the divides relatio... |
| dvdsval2 16184 | One nonzero integer divide... |
| dvdsval3 16185 | One nonzero integer divide... |
| dvdszrcl 16186 | Reverse closure for the di... |
| dvdsmod0 16187 | If a positive integer divi... |
| p1modz1 16188 | If a number greater than 1... |
| dvdsmodexp 16189 | If a positive integer divi... |
| nndivdvds 16190 | Strong form of ~ dvdsval2 ... |
| nndivides 16191 | Definition of the divides ... |
| moddvds 16192 | Two ways to say ` A == B `... |
| modm1div 16193 | An integer greater than on... |
| addmulmodb 16194 | An integer plus a product ... |
| dvds0lem 16195 | A lemma to assist theorems... |
| dvds1lem 16196 | A lemma to assist theorems... |
| dvds2lem 16197 | A lemma to assist theorems... |
| iddvds 16198 | An integer divides itself.... |
| 1dvds 16199 | 1 divides any integer. Th... |
| dvds0 16200 | Any integer divides 0. Th... |
| negdvdsb 16201 | An integer divides another... |
| dvdsnegb 16202 | An integer divides another... |
| absdvdsb 16203 | An integer divides another... |
| dvdsabsb 16204 | An integer divides another... |
| 0dvds 16205 | Only 0 is divisible by 0. ... |
| dvdsmul1 16206 | An integer divides a multi... |
| dvdsmul2 16207 | An integer divides a multi... |
| iddvdsexp 16208 | An integer divides a posit... |
| muldvds1 16209 | If a product divides an in... |
| muldvds2 16210 | If a product divides an in... |
| dvdscmul 16211 | Multiplication by a consta... |
| dvdsmulc 16212 | Multiplication by a consta... |
| dvdscmulr 16213 | Cancellation law for the d... |
| dvdsmulcr 16214 | Cancellation law for the d... |
| summodnegmod 16215 | The sum of two integers mo... |
| difmod0 16216 | The difference of two inte... |
| modmulconst 16217 | Constant multiplication in... |
| dvds2ln 16218 | If an integer divides each... |
| dvds2add 16219 | If an integer divides each... |
| dvds2sub 16220 | If an integer divides each... |
| dvds2addd 16221 | Deduction form of ~ dvds2a... |
| dvds2subd 16222 | Deduction form of ~ dvds2s... |
| dvdstr 16223 | The divides relation is tr... |
| dvdstrd 16224 | The divides relation is tr... |
| dvdsmultr1 16225 | If an integer divides anot... |
| dvdsmultr1d 16226 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16227 | If an integer divides anot... |
| dvdsmultr2d 16228 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16229 | If an integer divides eith... |
| dvdssub2 16230 | If an integer divides a di... |
| dvdsadd 16231 | An integer divides another... |
| dvdsaddr 16232 | An integer divides another... |
| dvdssub 16233 | An integer divides another... |
| dvdssubr 16234 | An integer divides another... |
| dvdsadd2b 16235 | Adding a multiple of the b... |
| dvdsaddre2b 16236 | Adding a multiple of the b... |
| fsumdvds 16237 | If every term in a sum is ... |
| dvdslelem 16238 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16239 | The divisors of a positive... |
| dvdsleabs 16240 | The divisors of a nonzero ... |
| dvdsleabs2 16241 | Transfer divisibility to a... |
| dvdsabseq 16242 | If two integers divide eac... |
| dvdseq 16243 | If two nonnegative integer... |
| divconjdvds 16244 | If a nonzero integer ` M `... |
| dvdsdivcl 16245 | The complement of a diviso... |
| dvdsflip 16246 | An involution of the divis... |
| dvdsssfz1 16247 | The set of divisors of a n... |
| dvds1 16248 | The only nonnegative integ... |
| alzdvds 16249 | Only 0 is divisible by all... |
| dvdsext 16250 | Poset extensionality for d... |
| fzm1ndvds 16251 | No number between ` 1 ` an... |
| fzo0dvdseq 16252 | Zero is the only one of th... |
| fzocongeq 16253 | Two different elements of ... |
| addmodlteqALT 16254 | Two nonnegative integers l... |
| dvdsfac 16255 | A positive integer divides... |
| dvdsexp2im 16256 | If an integer divides anot... |
| dvdsexp 16257 | A power divides a power wi... |
| dvdsmod 16258 | Any number ` K ` whose mod... |
| mulmoddvds 16259 | If an integer is divisible... |
| 3dvds 16260 | A rule for divisibility by... |
| 3dvdsdec 16261 | A decimal number is divisi... |
| 3dvds2dec 16262 | A decimal number is divisi... |
| fprodfvdvdsd 16263 | A finite product of intege... |
| fproddvdsd 16264 | A finite product of intege... |
| evenelz 16265 | An even number is an integ... |
| zeo3 16266 | An integer is even or odd.... |
| zeo4 16267 | An integer is even or odd ... |
| zeneo 16268 | No even integer equals an ... |
| odd2np1lem 16269 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16270 | An integer is odd iff it i... |
| even2n 16271 | An integer is even iff it ... |
| oddm1even 16272 | An integer is odd iff its ... |
| oddp1even 16273 | An integer is odd iff its ... |
| oexpneg 16274 | The exponential of the neg... |
| mod2eq0even 16275 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16276 | An integer is 1 modulo 2 i... |
| oddnn02np1 16277 | A nonnegative integer is o... |
| oddge22np1 16278 | An integer greater than on... |
| evennn02n 16279 | A nonnegative integer is e... |
| evennn2n 16280 | A positive integer is even... |
| 2tp1odd 16281 | A number which is twice an... |
| mulsucdiv2z 16282 | An integer multiplied with... |
| sqoddm1div8z 16283 | A squared odd number minus... |
| 2teven 16284 | A number which is twice an... |
| zeo5 16285 | An integer is either even ... |
| evend2 16286 | An integer is even iff its... |
| oddp1d2 16287 | An integer is odd iff its ... |
| zob 16288 | Alternate characterization... |
| oddm1d2 16289 | An integer is odd iff its ... |
| ltoddhalfle 16290 | An integer is less than ha... |
| halfleoddlt 16291 | An integer is greater than... |
| opoe 16292 | The sum of two odds is eve... |
| omoe 16293 | The difference of two odds... |
| opeo 16294 | The sum of an odd and an e... |
| omeo 16295 | The difference of an odd a... |
| z0even 16296 | 2 divides 0. That means 0... |
| n2dvds1 16297 | 2 does not divide 1. That... |
| n2dvdsm1 16298 | 2 does not divide -1. Tha... |
| z2even 16299 | 2 divides 2. That means 2... |
| n2dvds3 16300 | 2 does not divide 3. That... |
| z4even 16301 | 2 divides 4. That means 4... |
| 4dvdseven 16302 | An integer which is divisi... |
| m1expe 16303 | Exponentiation of -1 by an... |
| m1expo 16304 | Exponentiation of -1 by an... |
| m1exp1 16305 | Exponentiation of negative... |
| nn0enne 16306 | A positive integer is an e... |
| nn0ehalf 16307 | The half of an even nonneg... |
| nnehalf 16308 | The half of an even positi... |
| nn0onn 16309 | An odd nonnegative integer... |
| nn0o1gt2 16310 | An odd nonnegative integer... |
| nno 16311 | An alternate characterizat... |
| nn0o 16312 | An alternate characterizat... |
| nn0ob 16313 | Alternate characterization... |
| nn0oddm1d2 16314 | A positive integer is odd ... |
| nnoddm1d2 16315 | A positive integer is odd ... |
| sumeven 16316 | If every term in a sum is ... |
| sumodd 16317 | If every term in a sum is ... |
| evensumodd 16318 | If every term in a sum wit... |
| oddsumodd 16319 | If every term in a sum wit... |
| pwp1fsum 16320 | The n-th power of a number... |
| oddpwp1fsum 16321 | An odd power of a number i... |
| divalglem0 16322 | Lemma for ~ divalg . (Con... |
| divalglem1 16323 | Lemma for ~ divalg . (Con... |
| divalglem2 16324 | Lemma for ~ divalg . (Con... |
| divalglem4 16325 | Lemma for ~ divalg . (Con... |
| divalglem5 16326 | Lemma for ~ divalg . (Con... |
| divalglem6 16327 | Lemma for ~ divalg . (Con... |
| divalglem7 16328 | Lemma for ~ divalg . (Con... |
| divalglem8 16329 | Lemma for ~ divalg . (Con... |
| divalglem9 16330 | Lemma for ~ divalg . (Con... |
| divalglem10 16331 | Lemma for ~ divalg . (Con... |
| divalg 16332 | The division algorithm (th... |
| divalgb 16333 | Express the division algor... |
| divalg2 16334 | The division algorithm (th... |
| divalgmod 16335 | The result of the ` mod ` ... |
| divalgmodcl 16336 | The result of the ` mod ` ... |
| modremain 16337 | The result of the modulo o... |
| ndvdssub 16338 | Corollary of the division ... |
| ndvdsadd 16339 | Corollary of the division ... |
| ndvdsp1 16340 | Special case of ~ ndvdsadd... |
| ndvdsi 16341 | A quick test for non-divis... |
| 5ndvds3 16342 | 5 does not divide 3. (Con... |
| 5ndvds6 16343 | 5 does not divide 6. (Con... |
| flodddiv4 16344 | The floor of an odd intege... |
| fldivndvdslt 16345 | The floor of an integer di... |
| flodddiv4lt 16346 | The floor of an odd number... |
| flodddiv4t2lthalf 16347 | The floor of an odd number... |
| bitsfval 16352 | Expand the definition of t... |
| bitsval 16353 | Expand the definition of t... |
| bitsval2 16354 | Expand the definition of t... |
| bitsss 16355 | The set of bits of an inte... |
| bitsf 16356 | The ` bits ` function is a... |
| bits0 16357 | Value of the zeroth bit. ... |
| bits0e 16358 | The zeroth bit of an even ... |
| bits0o 16359 | The zeroth bit of an odd n... |
| bitsp1 16360 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16361 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16362 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16363 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16364 | The bits of a number are a... |
| bitsmod 16365 | Truncating the bit sequenc... |
| bitsfi 16366 | Every number is associated... |
| bitscmp 16367 | The bit complement of ` N ... |
| 0bits 16368 | The bits of zero. (Contri... |
| m1bits 16369 | The bits of negative one. ... |
| bitsinv1lem 16370 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16371 | There is an explicit inver... |
| bitsinv2 16372 | There is an explicit inver... |
| bitsf1ocnv 16373 | The ` bits ` function rest... |
| bitsf1o 16374 | The ` bits ` function rest... |
| bitsf1 16375 | The ` bits ` function is a... |
| 2ebits 16376 | The bits of a power of two... |
| bitsinv 16377 | The inverse of the ` bits ... |
| bitsinvp1 16378 | Recursive definition of th... |
| sadadd2lem2 16379 | The core of the proof of ~... |
| sadfval 16381 | Define the addition of two... |
| sadcf 16382 | The carry sequence is a se... |
| sadc0 16383 | The initial element of the... |
| sadcp1 16384 | The carry sequence (which ... |
| sadval 16385 | The full adder sequence is... |
| sadcaddlem 16386 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16387 | Non-recursive definition o... |
| sadadd2lem 16388 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16389 | Sum of initial segments of... |
| sadadd3 16390 | Sum of initial segments of... |
| sadcl 16391 | The sum of two sequences i... |
| sadcom 16392 | The adder sequence functio... |
| saddisjlem 16393 | Lemma for ~ sadadd . (Con... |
| saddisj 16394 | The sum of disjoint sequen... |
| sadaddlem 16395 | Lemma for ~ sadadd . (Con... |
| sadadd 16396 | For sequences that corresp... |
| sadid1 16397 | The adder sequence functio... |
| sadid2 16398 | The adder sequence functio... |
| sadasslem 16399 | Lemma for ~ sadass . (Con... |
| sadass 16400 | Sequence addition is assoc... |
| sadeq 16401 | Any element of a sequence ... |
| bitsres 16402 | Restrict the bits of a num... |
| bitsuz 16403 | The bits of a number are a... |
| bitsshft 16404 | Shifting a bit sequence to... |
| smufval 16406 | The multiplication of two ... |
| smupf 16407 | The sequence of partial su... |
| smup0 16408 | The initial element of the... |
| smupp1 16409 | The initial element of the... |
| smuval 16410 | Define the addition of two... |
| smuval2 16411 | The partial sum sequence s... |
| smupvallem 16412 | If ` A ` only has elements... |
| smucl 16413 | The product of two sequenc... |
| smu01lem 16414 | Lemma for ~ smu01 and ~ sm... |
| smu01 16415 | Multiplication of a sequen... |
| smu02 16416 | Multiplication of a sequen... |
| smupval 16417 | Rewrite the elements of th... |
| smup1 16418 | Rewrite ~ smupp1 using onl... |
| smueqlem 16419 | Any element of a sequence ... |
| smueq 16420 | Any element of a sequence ... |
| smumullem 16421 | Lemma for ~ smumul . (Con... |
| smumul 16422 | For sequences that corresp... |
| gcdval 16425 | The value of the ` gcd ` o... |
| gcd0val 16426 | The value, by convention, ... |
| gcdn0val 16427 | The value of the ` gcd ` o... |
| gcdcllem1 16428 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16429 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16430 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16431 | Closure of the ` gcd ` ope... |
| gcddvds 16432 | The gcd of two integers di... |
| dvdslegcd 16433 | An integer which divides b... |
| nndvdslegcd 16434 | A positive integer which d... |
| gcdcl 16435 | Closure of the ` gcd ` ope... |
| gcdnncl 16436 | Closure of the ` gcd ` ope... |
| gcdcld 16437 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16438 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16439 | An integer is the product ... |
| divgcdz 16440 | An integer divided by the ... |
| gcdf 16441 | Domain and codomain of the... |
| gcdcom 16442 | The ` gcd ` operator is co... |
| gcdcomd 16443 | The ` gcd ` operator is co... |
| divgcdnn 16444 | A positive integer divided... |
| divgcdnnr 16445 | A positive integer divided... |
| gcdeq0 16446 | The gcd of two integers is... |
| gcdn0gt0 16447 | The gcd of two integers is... |
| gcd0id 16448 | The gcd of 0 and an intege... |
| gcdid0 16449 | The gcd of an integer and ... |
| nn0gcdid0 16450 | The gcd of a nonnegative i... |
| gcdneg 16451 | Negating one operand of th... |
| neggcd 16452 | Negating one operand of th... |
| gcdaddmlem 16453 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16454 | Adding a multiple of one o... |
| gcdadd 16455 | The GCD of two numbers is ... |
| gcdid 16456 | The gcd of a number and it... |
| gcd1 16457 | The gcd of a number with 1... |
| gcdabs1 16458 | ` gcd ` of the absolute va... |
| gcdabs2 16459 | ` gcd ` of the absolute va... |
| gcdabs 16460 | The gcd of two integers is... |
| modgcd 16461 | The gcd remains unchanged ... |
| 1gcd 16462 | The GCD of one and an inte... |
| gcdmultipled 16463 | The greatest common diviso... |
| gcdmultiplez 16464 | The GCD of a multiple of a... |
| gcdmultiple 16465 | The GCD of a multiple of a... |
| dvdsgcdidd 16466 | The greatest common diviso... |
| 6gcd4e2 16467 | The greatest common diviso... |
| bezoutlem1 16468 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16469 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16470 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16471 | Lemma for ~ bezout . (Con... |
| bezout 16472 | Bézout's identity: ... |
| dvdsgcd 16473 | An integer which divides e... |
| dvdsgcdb 16474 | Biconditional form of ~ dv... |
| dfgcd2 16475 | Alternate definition of th... |
| gcdass 16476 | Associative law for ` gcd ... |
| mulgcd 16477 | Distribute multiplication ... |
| absmulgcd 16478 | Distribute absolute value ... |
| mulgcdr 16479 | Reverse distribution law f... |
| gcddiv 16480 | Division law for GCD. (Con... |
| gcdzeq 16481 | A positive integer ` A ` i... |
| gcdeq 16482 | ` A ` is equal to its gcd ... |
| dvdssqim 16483 | Unidirectional form of ~ d... |
| dvdsexpim 16484 | If two numbers are divisib... |
| dvdsmulgcd 16485 | A divisibility equivalent ... |
| rpmulgcd 16486 | If ` K ` and ` M ` are rel... |
| rplpwr 16487 | If ` A ` and ` B ` are rel... |
| rprpwr 16488 | If ` A ` and ` B ` are rel... |
| rppwr 16489 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16490 | If ` A ` and ` B ` are rel... |
| sqgcd 16491 | Square distributes over gc... |
| expgcd 16492 | Exponentiation distributes... |
| nn0expgcd 16493 | Exponentiation distributes... |
| zexpgcd 16494 | Exponentiation distributes... |
| dvdssqlem 16495 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16496 | Two numbers are divisible ... |
| bezoutr 16497 | Partial converse to ~ bezo... |
| bezoutr1 16498 | Converse of ~ bezout for w... |
| nn0seqcvgd 16499 | A strictly-decreasing nonn... |
| seq1st 16500 | A sequence whose iteration... |
| algr0 16501 | The value of the algorithm... |
| algrf 16502 | An algorithm is a step fun... |
| algrp1 16503 | The value of the algorithm... |
| alginv 16504 | If ` I ` is an invariant o... |
| algcvg 16505 | One way to prove that an a... |
| algcvgblem 16506 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16507 | Two ways of expressing tha... |
| algcvga 16508 | The countdown function ` C... |
| algfx 16509 | If ` F ` reaches a fixed p... |
| eucalgval2 16510 | The value of the step func... |
| eucalgval 16511 | Euclid's Algorithm ~ eucal... |
| eucalgf 16512 | Domain and codomain of the... |
| eucalginv 16513 | The invariant of the step ... |
| eucalglt 16514 | The second member of the s... |
| eucalgcvga 16515 | Once Euclid's Algorithm ha... |
| eucalg 16516 | Euclid's Algorithm compute... |
| lcmval 16521 | Value of the ` lcm ` opera... |
| lcmcom 16522 | The ` lcm ` operator is co... |
| lcm0val 16523 | The value, by convention, ... |
| lcmn0val 16524 | The value of the ` lcm ` o... |
| lcmcllem 16525 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16526 | Closure of the ` lcm ` ope... |
| dvdslcm 16527 | The lcm of two integers is... |
| lcmledvds 16528 | A positive integer which b... |
| lcmeq0 16529 | The lcm of two integers is... |
| lcmcl 16530 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16531 | The greatest common diviso... |
| lcmneg 16532 | Negating one operand of th... |
| neglcm 16533 | Negating one operand of th... |
| lcmabs 16534 | The lcm of two integers is... |
| lcmgcdlem 16535 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16536 | The product of two numbers... |
| lcmdvds 16537 | The lcm of two integers di... |
| lcmid 16538 | The lcm of an integer and ... |
| lcm1 16539 | The lcm of an integer and ... |
| lcmgcdnn 16540 | The product of two positiv... |
| lcmgcdeq 16541 | Two integers' absolute val... |
| lcmdvdsb 16542 | Biconditional form of ~ lc... |
| lcmass 16543 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16544 | The least common multiple ... |
| 6lcm4e12 16545 | The least common multiple ... |
| absproddvds 16546 | The absolute value of the ... |
| absprodnn 16547 | The absolute value of the ... |
| fissn0dvds 16548 | For each finite subset of ... |
| fissn0dvdsn0 16549 | For each finite subset of ... |
| lcmfval 16550 | Value of the ` _lcm ` func... |
| lcmf0val 16551 | The value, by convention, ... |
| lcmfn0val 16552 | The value of the ` _lcm ` ... |
| lcmfnnval 16553 | The value of the ` _lcm ` ... |
| lcmfcllem 16554 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16555 | Closure of the ` _lcm ` fu... |
| lcmfpr 16556 | The value of the ` _lcm ` ... |
| lcmfcl 16557 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16558 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16559 | The least common multiple ... |
| dvdslcmf 16560 | The least common multiple ... |
| lcmfledvds 16561 | A positive integer which i... |
| lcmf 16562 | Characterization of the le... |
| lcmf0 16563 | The least common multiple ... |
| lcmfsn 16564 | The least common multiple ... |
| lcmftp 16565 | The least common multiple ... |
| lcmfunsnlem1 16566 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16567 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16568 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16569 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16570 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16571 | The least common multiple ... |
| lcmfdvdsb 16572 | Biconditional form of ~ lc... |
| lcmfunsn 16573 | The ` _lcm ` function for ... |
| lcmfun 16574 | The ` _lcm ` function for ... |
| lcmfass 16575 | Associative law for the ` ... |
| lcmf2a3a4e12 16576 | The least common multiple ... |
| lcmflefac 16577 | The least common multiple ... |
| coprmgcdb 16578 | Two positive integers are ... |
| ncoprmgcdne1b 16579 | Two positive integers are ... |
| ncoprmgcdgt1b 16580 | Two positive integers are ... |
| coprmdvds1 16581 | If two positive integers a... |
| coprmdvds 16582 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16583 | If an integer is divisible... |
| mulgcddvds 16584 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16585 | If ` M ` is relatively pri... |
| qredeq 16586 | Two equal reduced fraction... |
| qredeu 16587 | Every rational number has ... |
| rpmul 16588 | If ` K ` is relatively pri... |
| rpdvds 16589 | If ` K ` is relatively pri... |
| coprmprod 16590 | The product of the element... |
| coprmproddvdslem 16591 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16592 | If a positive integer is d... |
| congr 16593 | Definition of congruence b... |
| divgcdcoprm0 16594 | Integers divided by gcd ar... |
| divgcdcoprmex 16595 | Integers divided by gcd ar... |
| cncongr1 16596 | One direction of the bicon... |
| cncongr2 16597 | The other direction of the... |
| cncongr 16598 | Cancellability of Congruen... |
| cncongrcoprm 16599 | Corollary 1 of Cancellabil... |
| isprm 16602 | The predicate "is a prime ... |
| prmnn 16603 | A prime number is a positi... |
| prmz 16604 | A prime number is an integ... |
| prmssnn 16605 | The prime numbers are a su... |
| prmex 16606 | The set of prime numbers e... |
| 0nprm 16607 | 0 is not a prime number. ... |
| 1nprm 16608 | 1 is not a prime number. ... |
| 1idssfct 16609 | The positive divisors of a... |
| isprm2lem 16610 | Lemma for ~ isprm2 . (Con... |
| isprm2 16611 | The predicate "is a prime ... |
| isprm3 16612 | The predicate "is a prime ... |
| isprm4 16613 | The predicate "is a prime ... |
| prmind2 16614 | A variation on ~ prmind as... |
| prmind 16615 | Perform induction over the... |
| dvdsprime 16616 | If ` M ` divides a prime, ... |
| nprm 16617 | A product of two integers ... |
| nprmi 16618 | An inference for composite... |
| dvdsnprmd 16619 | If a number is divisible b... |
| prm2orodd 16620 | A prime number is either 2... |
| 2prm 16621 | 2 is a prime number. (Con... |
| 2mulprm 16622 | A multiple of two is prime... |
| 3prm 16623 | 3 is a prime number. (Con... |
| 4nprm 16624 | 4 is not a prime number. ... |
| prmuz2 16625 | A prime number is an integ... |
| prmgt1 16626 | A prime number is an integ... |
| prmm2nn0 16627 | Subtracting 2 from a prime... |
| oddprmgt2 16628 | An odd prime is greater th... |
| oddprmge3 16629 | An odd prime is greater th... |
| ge2nprmge4 16630 | A composite integer greate... |
| sqnprm 16631 | A square is never prime. ... |
| dvdsprm 16632 | An integer greater than or... |
| exprmfct 16633 | Every integer greater than... |
| prmdvdsfz 16634 | Each integer greater than ... |
| nprmdvds1 16635 | No prime number divides 1.... |
| isprm5 16636 | One need only check prime ... |
| isprm7 16637 | One need only check prime ... |
| maxprmfct 16638 | The set of prime factors o... |
| divgcdodd 16639 | Either ` A / ( A gcd B ) `... |
| coprm 16640 | A prime number either divi... |
| prmrp 16641 | Unequal prime numbers are ... |
| euclemma 16642 | Euclid's lemma. A prime n... |
| isprm6 16643 | A number is prime iff it s... |
| prmdvdsexp 16644 | A prime divides a positive... |
| prmdvdsexpb 16645 | A prime divides a positive... |
| prmdvdsexpr 16646 | If a prime divides a nonne... |
| prmdvdssq 16647 | Condition for a prime divi... |
| prmexpb 16648 | Two positive prime powers ... |
| prmfac1 16649 | The factorial of a number ... |
| dvdszzq 16650 | Divisibility for an intege... |
| rpexp 16651 | If two numbers ` A ` and `... |
| rpexp1i 16652 | Relative primality passes ... |
| rpexp12i 16653 | Relative primality passes ... |
| prmndvdsfaclt 16654 | A prime number does not di... |
| prmdvdsbc 16655 | Condition for a prime numb... |
| prmdvdsncoprmbd 16656 | Two positive integers are ... |
| ncoprmlnprm 16657 | If two positive integers a... |
| cncongrprm 16658 | Corollary 2 of Cancellabil... |
| isevengcd2 16659 | The predicate "is an even ... |
| isoddgcd1 16660 | The predicate "is an odd n... |
| 3lcm2e6 16661 | The least common multiple ... |
| qnumval 16666 | Value of the canonical num... |
| qdenval 16667 | Value of the canonical den... |
| qnumdencl 16668 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16669 | The canonical numerator of... |
| qdencl 16670 | The canonical denominator ... |
| fnum 16671 | Canonical numerator define... |
| fden 16672 | Canonical denominator defi... |
| qnumdenbi 16673 | Two numbers are the canoni... |
| qnumdencoprm 16674 | The canonical representati... |
| qeqnumdivden 16675 | Recover a rational number ... |
| qmuldeneqnum 16676 | Multiplying a rational by ... |
| divnumden 16677 | Calculate the reduced form... |
| divdenle 16678 | Reducing a quotient never ... |
| qnumgt0 16679 | A rational is positive iff... |
| qgt0numnn 16680 | A rational is positive iff... |
| nn0gcdsq 16681 | Squaring commutes with GCD... |
| zgcdsq 16682 | ~ nn0gcdsq extended to int... |
| numdensq 16683 | Squaring a rational square... |
| numsq 16684 | Square commutes with canon... |
| densq 16685 | Square commutes with canon... |
| qden1elz 16686 | A rational is an integer i... |
| zsqrtelqelz 16687 | If an integer has a ration... |
| nonsq 16688 | Any integer strictly betwe... |
| numdenexp 16689 | Elevating a rational numbe... |
| numexp 16690 | Elevating to a nonnegative... |
| denexp 16691 | Elevating to a nonnegative... |
| phival 16696 | Value of the Euler ` phi `... |
| phicl2 16697 | Bounds and closure for the... |
| phicl 16698 | Closure for the value of t... |
| phibndlem 16699 | Lemma for ~ phibnd . (Con... |
| phibnd 16700 | A slightly tighter bound o... |
| phicld 16701 | Closure for the value of t... |
| phi1 16702 | Value of the Euler ` phi `... |
| dfphi2 16703 | Alternate definition of th... |
| hashdvds 16704 | The number of numbers in a... |
| phiprmpw 16705 | Value of the Euler ` phi `... |
| phiprm 16706 | Value of the Euler ` phi `... |
| crth 16707 | The Chinese Remainder Theo... |
| phimullem 16708 | Lemma for ~ phimul . (Con... |
| phimul 16709 | The Euler ` phi ` function... |
| eulerthlem1 16710 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16711 | Lemma for ~ eulerth . (Co... |
| eulerth 16712 | Euler's theorem, a general... |
| fermltl 16713 | Fermat's little theorem. ... |
| prmdiv 16714 | Show an explicit expressio... |
| prmdiveq 16715 | The modular inverse of ` A... |
| prmdivdiv 16716 | The (modular) inverse of t... |
| hashgcdlem 16717 | A correspondence between e... |
| dvdsfi 16718 | A natural number has finit... |
| hashgcdeq 16719 | Number of initial positive... |
| phisum 16720 | The divisor sum identity o... |
| odzval 16721 | Value of the order functio... |
| odzcllem 16722 | - Lemma for ~ odzcl , show... |
| odzcl 16723 | The order of a group eleme... |
| odzid 16724 | Any element raised to the ... |
| odzdvds 16725 | The only powers of ` A ` t... |
| odzphi 16726 | The order of any group ele... |
| modprm1div 16727 | A prime number divides an ... |
| m1dvdsndvds 16728 | If an integer minus 1 is d... |
| modprminv 16729 | Show an explicit expressio... |
| modprminveq 16730 | The modular inverse of ` A... |
| vfermltl 16731 | Variant of Fermat's little... |
| vfermltlALT 16732 | Alternate proof of ~ vferm... |
| powm2modprm 16733 | If an integer minus 1 is d... |
| reumodprminv 16734 | For any prime number and f... |
| modprm0 16735 | For two positive integers ... |
| nnnn0modprm0 16736 | For a positive integer and... |
| modprmn0modprm0 16737 | For an integer not being 0... |
| coprimeprodsq 16738 | If three numbers are copri... |
| coprimeprodsq2 16739 | If three numbers are copri... |
| oddprm 16740 | A prime not equal to ` 2 `... |
| nnoddn2prm 16741 | A prime not equal to ` 2 `... |
| oddn2prm 16742 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16743 | A number is a prime number... |
| prm23lt5 16744 | A prime less than 5 is eit... |
| prm23ge5 16745 | A prime is either 2 or 3 o... |
| pythagtriplem1 16746 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16747 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16748 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16749 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16750 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16751 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16752 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16753 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16754 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16755 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16756 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16757 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16758 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16759 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16760 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16761 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16762 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16763 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16764 | Parameterize the Pythagore... |
| iserodd 16765 | Collect the odd terms in a... |
| pclem 16768 | - Lemma for the prime powe... |
| pcprecl 16769 | Closure of the prime power... |
| pcprendvds 16770 | Non-divisibility property ... |
| pcprendvds2 16771 | Non-divisibility property ... |
| pcpre1 16772 | Value of the prime power p... |
| pcpremul 16773 | Multiplicative property of... |
| pcval 16774 | The value of the prime pow... |
| pceulem 16775 | Lemma for ~ pceu . (Contr... |
| pceu 16776 | Uniqueness for the prime p... |
| pczpre 16777 | Connect the prime count pr... |
| pczcl 16778 | Closure of the prime power... |
| pccl 16779 | Closure of the prime power... |
| pccld 16780 | Closure of the prime power... |
| pcmul 16781 | Multiplication property of... |
| pcdiv 16782 | Division property of the p... |
| pcqmul 16783 | Multiplication property of... |
| pc0 16784 | The value of the prime pow... |
| pc1 16785 | Value of the prime count f... |
| pcqcl 16786 | Closure of the general pri... |
| pcqdiv 16787 | Division property of the p... |
| pcrec 16788 | Prime power of a reciproca... |
| pcexp 16789 | Prime power of an exponent... |
| pcxnn0cl 16790 | Extended nonnegative integ... |
| pcxcl 16791 | Extended real closure of t... |
| pcge0 16792 | The prime count of an inte... |
| pczdvds 16793 | Defining property of the p... |
| pcdvds 16794 | Defining property of the p... |
| pczndvds 16795 | Defining property of the p... |
| pcndvds 16796 | Defining property of the p... |
| pczndvds2 16797 | The remainder after dividi... |
| pcndvds2 16798 | The remainder after dividi... |
| pcdvdsb 16799 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16800 | There are a positive numbe... |
| pceq0 16801 | There are zero powers of a... |
| pcidlem 16802 | The prime count of a prime... |
| pcid 16803 | The prime count of a prime... |
| pcneg 16804 | The prime count of a negat... |
| pcabs 16805 | The prime count of an abso... |
| pcdvdstr 16806 | The prime count increases ... |
| pcgcd1 16807 | The prime count of a GCD i... |
| pcgcd 16808 | The prime count of a GCD i... |
| pc2dvds 16809 | A characterization of divi... |
| pc11 16810 | The prime count function, ... |
| pcz 16811 | The prime count function c... |
| pcprmpw2 16812 | Self-referential expressio... |
| pcprmpw 16813 | Self-referential expressio... |
| dvdsprmpweq 16814 | If a positive integer divi... |
| dvdsprmpweqnn 16815 | If an integer greater than... |
| dvdsprmpweqle 16816 | If a positive integer divi... |
| difsqpwdvds 16817 | If the difference of two s... |
| pcaddlem 16818 | Lemma for ~ pcadd . The o... |
| pcadd 16819 | An inequality for the prim... |
| pcadd2 16820 | The inequality of ~ pcadd ... |
| pcmptcl 16821 | Closure for the prime powe... |
| pcmpt 16822 | Construct a function with ... |
| pcmpt2 16823 | Dividing two prime count m... |
| pcmptdvds 16824 | The partial products of th... |
| pcprod 16825 | The product of the primes ... |
| sumhash 16826 | The sum of 1 over a set is... |
| fldivp1 16827 | The difference between the... |
| pcfaclem 16828 | Lemma for ~ pcfac . (Cont... |
| pcfac 16829 | Calculate the prime count ... |
| pcbc 16830 | Calculate the prime count ... |
| qexpz 16831 | If a power of a rational n... |
| expnprm 16832 | A second or higher power o... |
| oddprmdvds 16833 | Every positive integer whi... |
| prmpwdvds 16834 | A relation involving divis... |
| pockthlem 16835 | Lemma for ~ pockthg . (Co... |
| pockthg 16836 | The generalized Pocklingto... |
| pockthi 16837 | Pocklington's theorem, whi... |
| unbenlem 16838 | Lemma for ~ unben . (Cont... |
| unben 16839 | An unbounded set of positi... |
| infpnlem1 16840 | Lemma for ~ infpn . The s... |
| infpnlem2 16841 | Lemma for ~ infpn . For a... |
| infpn 16842 | There exist infinitely man... |
| infpn2 16843 | There exist infinitely man... |
| prmunb 16844 | The primes are unbounded. ... |
| prminf 16845 | There are an infinite numb... |
| prmreclem1 16846 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16847 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16848 | Lemma for ~ prmrec . The ... |
| prmreclem4 16849 | Lemma for ~ prmrec . Show... |
| prmreclem5 16850 | Lemma for ~ prmrec . Here... |
| prmreclem6 16851 | Lemma for ~ prmrec . If t... |
| prmrec 16852 | The sum of the reciprocals... |
| 1arithlem1 16853 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16854 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16855 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16856 | Lemma for ~ 1arith . (Con... |
| 1arith 16857 | Fundamental theorem of ari... |
| 1arith2 16858 | Fundamental theorem of ari... |
| elgz 16861 | Elementhood in the gaussia... |
| gzcn 16862 | A gaussian integer is a co... |
| zgz 16863 | An integer is a gaussian i... |
| igz 16864 | ` _i ` is a gaussian integ... |
| gznegcl 16865 | The gaussian integers are ... |
| gzcjcl 16866 | The gaussian integers are ... |
| gzaddcl 16867 | The gaussian integers are ... |
| gzmulcl 16868 | The gaussian integers are ... |
| gzreim 16869 | Construct a gaussian integ... |
| gzsubcl 16870 | The gaussian integers are ... |
| gzabssqcl 16871 | The squared norm of a gaus... |
| 4sqlem5 16872 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16873 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16874 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16875 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16876 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16877 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16878 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16879 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16880 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16881 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16882 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16883 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16884 | Euler's four-square identi... |
| 4sqlem11 16885 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16886 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16887 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16888 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16889 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16890 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16891 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16892 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16893 | Lemma for ~ 4sq . The pro... |
| 4sq 16894 | Lagrange's four-square the... |
| vdwapfval 16901 | Define the arithmetic prog... |
| vdwapf 16902 | The arithmetic progression... |
| vdwapval 16903 | Value of the arithmetic pr... |
| vdwapun 16904 | Remove the first element o... |
| vdwapid1 16905 | The first element of an ar... |
| vdwap0 16906 | Value of a length-1 arithm... |
| vdwap1 16907 | Value of a length-1 arithm... |
| vdwmc 16908 | The predicate " The ` <. R... |
| vdwmc2 16909 | Expand out the definition ... |
| vdwpc 16910 | The predicate " The colori... |
| vdwlem1 16911 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16912 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16913 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16914 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16915 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16916 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16917 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16918 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16919 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16920 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16921 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16922 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16923 | Lemma for ~ vdw . Main in... |
| vdw 16924 | Van der Waerden's theorem.... |
| vdwnnlem1 16925 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16926 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16927 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16928 | Van der Waerden's theorem,... |
| ramtlecl 16930 | The set ` T ` of numbers w... |
| hashbcval 16932 | Value of the "binomial set... |
| hashbccl 16933 | The binomial set is a fini... |
| hashbcss 16934 | Subset relation for the bi... |
| hashbc0 16935 | The set of subsets of size... |
| hashbc2 16936 | The size of the binomial s... |
| 0hashbc 16937 | There are no subsets of th... |
| ramval 16938 | The value of the Ramsey nu... |
| ramcl2lem 16939 | Lemma for extended real cl... |
| ramtcl 16940 | The Ramsey number has the ... |
| ramtcl2 16941 | The Ramsey number is an in... |
| ramtub 16942 | The Ramsey number is a low... |
| ramub 16943 | The Ramsey number is a low... |
| ramub2 16944 | It is sufficient to check ... |
| rami 16945 | The defining property of a... |
| ramcl2 16946 | The Ramsey number is eithe... |
| ramxrcl 16947 | The Ramsey number is an ex... |
| ramubcl 16948 | If the Ramsey number is up... |
| ramlb 16949 | Establish a lower bound on... |
| 0ram 16950 | The Ramsey number when ` M... |
| 0ram2 16951 | The Ramsey number when ` M... |
| ram0 16952 | The Ramsey number when ` R... |
| 0ramcl 16953 | Lemma for ~ ramcl : Exist... |
| ramz2 16954 | The Ramsey number when ` F... |
| ramz 16955 | The Ramsey number when ` F... |
| ramub1lem1 16956 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16957 | Lemma for ~ ramub1 . (Con... |
| ramub1 16958 | Inductive step for Ramsey'... |
| ramcl 16959 | Ramsey's theorem: the Rams... |
| ramsey 16960 | Ramsey's theorem with the ... |
| prmoval 16963 | Value of the primorial fun... |
| prmocl 16964 | Closure of the primorial f... |
| prmone0 16965 | The primorial function is ... |
| prmo0 16966 | The primorial of 0. (Cont... |
| prmo1 16967 | The primorial of 1. (Cont... |
| prmop1 16968 | The primorial of a success... |
| prmonn2 16969 | Value of the primorial fun... |
| prmo2 16970 | The primorial of 2. (Cont... |
| prmo3 16971 | The primorial of 3. (Cont... |
| prmdvdsprmo 16972 | The primorial of a number ... |
| prmdvdsprmop 16973 | The primorial of a number ... |
| fvprmselelfz 16974 | The value of the prime sel... |
| fvprmselgcd1 16975 | The greatest common diviso... |
| prmolefac 16976 | The primorial of a positiv... |
| prmodvdslcmf 16977 | The primorial of a nonnega... |
| prmolelcmf 16978 | The primorial of a positiv... |
| prmgaplem1 16979 | Lemma for ~ prmgap : The ... |
| prmgaplem2 16980 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 16981 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 16982 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 16983 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 16984 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 16985 | Lemma for ~ prmgap : for e... |
| prmgaplem6 16986 | Lemma for ~ prmgap : for e... |
| prmgaplem7 16987 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 16988 | Lemma for ~ prmgap . (Con... |
| prmgap 16989 | The prime gap theorem: for... |
| prmgaplcm 16990 | Alternate proof of ~ prmga... |
| prmgapprmolem 16991 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 16992 | Alternate proof of ~ prmga... |
| dec2dvds 16993 | Divisibility by two is obv... |
| dec5dvds 16994 | Divisibility by five is ob... |
| dec5dvds2 16995 | Divisibility by five is ob... |
| dec5nprm 16996 | A decimal number greater t... |
| dec2nprm 16997 | A decimal number greater t... |
| modxai 16998 | Add exponents in a power m... |
| mod2xi 16999 | Double exponents in a powe... |
| modxp1i 17000 | Add one to an exponent in ... |
| mod2xnegi 17001 | Version of ~ mod2xi with a... |
| modsubi 17002 | Subtract from within a mod... |
| gcdi 17003 | Calculate a GCD via Euclid... |
| gcdmodi 17004 | Calculate a GCD via Euclid... |
| numexp0 17005 | Calculate an integer power... |
| numexp1 17006 | Calculate an integer power... |
| numexpp1 17007 | Calculate an integer power... |
| numexp2x 17008 | Double an integer power. ... |
| decsplit0b 17009 | Split a decimal number int... |
| decsplit0 17010 | Split a decimal number int... |
| decsplit1 17011 | Split a decimal number int... |
| decsplit 17012 | Split a decimal number int... |
| karatsuba 17013 | The Karatsuba multiplicati... |
| 2exp4 17014 | Two to the fourth power is... |
| 2exp5 17015 | Two to the fifth power is ... |
| 2exp6 17016 | Two to the sixth power is ... |
| 2exp7 17017 | Two to the seventh power i... |
| 2exp8 17018 | Two to the eighth power is... |
| 2exp11 17019 | Two to the eleventh power ... |
| 2exp16 17020 | Two to the sixteenth power... |
| 3exp3 17021 | Three to the third power i... |
| 2expltfac 17022 | The factorial grows faster... |
| cshwsidrepsw 17023 | If cyclically shifting a w... |
| cshwsidrepswmod0 17024 | If cyclically shifting a w... |
| cshwshashlem1 17025 | If cyclically shifting a w... |
| cshwshashlem2 17026 | If cyclically shifting a w... |
| cshwshashlem3 17027 | If cyclically shifting a w... |
| cshwsdisj 17028 | The singletons resulting b... |
| cshwsiun 17029 | The set of (different!) wo... |
| cshwsex 17030 | The class of (different!) ... |
| cshws0 17031 | The size of the set of (di... |
| cshwrepswhash1 17032 | The size of the set of (di... |
| cshwshashnsame 17033 | If a word (not consisting ... |
| cshwshash 17034 | If a word has a length bei... |
| prmlem0 17035 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17036 | A quick proof skeleton to ... |
| prmlem1 17037 | A quick proof skeleton to ... |
| 5prm 17038 | 5 is a prime number. (Con... |
| 6nprm 17039 | 6 is not a prime number. ... |
| 7prm 17040 | 7 is a prime number. (Con... |
| 8nprm 17041 | 8 is not a prime number. ... |
| 9nprm 17042 | 9 is not a prime number. ... |
| 10nprm 17043 | 10 is not a prime number. ... |
| 11prm 17044 | 11 is a prime number. (Co... |
| 13prm 17045 | 13 is a prime number. (Co... |
| 17prm 17046 | 17 is a prime number. (Co... |
| 19prm 17047 | 19 is a prime number. (Co... |
| 23prm 17048 | 23 is a prime number. (Co... |
| prmlem2 17049 | Our last proving session g... |
| 37prm 17050 | 37 is a prime number. (Co... |
| 43prm 17051 | 43 is a prime number. (Co... |
| 83prm 17052 | 83 is a prime number. (Co... |
| 139prm 17053 | 139 is a prime number. (C... |
| 163prm 17054 | 163 is a prime number. (C... |
| 317prm 17055 | 317 is a prime number. (C... |
| 631prm 17056 | 631 is a prime number. (C... |
| prmo4 17057 | The primorial of 4. (Cont... |
| prmo5 17058 | The primorial of 5. (Cont... |
| prmo6 17059 | The primorial of 6. (Cont... |
| 1259lem1 17060 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17061 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17062 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17063 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17064 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17065 | 1259 is a prime number. (... |
| 2503lem1 17066 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17067 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17068 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17069 | 2503 is a prime number. (... |
| 4001lem1 17070 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17071 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17072 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17073 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17074 | 4001 is a prime number. (... |
| brstruct 17077 | The structure relation is ... |
| isstruct2 17078 | The property of being a st... |
| structex 17079 | A structure is a set. (Co... |
| structn0fun 17080 | A structure without the em... |
| isstruct 17081 | The property of being a st... |
| structcnvcnv 17082 | Two ways to express the re... |
| structfung 17083 | The converse of the conver... |
| structfun 17084 | Convert between two kinds ... |
| structfn 17085 | Convert between two kinds ... |
| strleun 17086 | Combine two structures int... |
| strle1 17087 | Make a structure from a si... |
| strle2 17088 | Make a structure from a pa... |
| strle3 17089 | Make a structure from a tr... |
| sbcie2s 17090 | A special version of class... |
| sbcie3s 17091 | A special version of class... |
| reldmsets 17094 | The structure override ope... |
| setsvalg 17095 | Value of the structure rep... |
| setsval 17096 | Value of the structure rep... |
| fvsetsid 17097 | The value of the structure... |
| fsets 17098 | The structure replacement ... |
| setsdm 17099 | The domain of a structure ... |
| setsfun 17100 | A structure with replaceme... |
| setsfun0 17101 | A structure with replaceme... |
| setsn0fun 17102 | The value of the structure... |
| setsstruct2 17103 | An extensible structure wi... |
| setsexstruct2 17104 | An extensible structure wi... |
| setsstruct 17105 | An extensible structure wi... |
| wunsets 17106 | Closure of structure repla... |
| setsres 17107 | The structure replacement ... |
| setsabs 17108 | Replacing the same compone... |
| setscom 17109 | Different components can b... |
| sloteq 17112 | Equality theorem for the `... |
| slotfn 17113 | A slot is a function on se... |
| strfvnd 17114 | Deduction version of ~ str... |
| strfvn 17115 | Value of a structure compo... |
| strfvss 17116 | A structure component extr... |
| wunstr 17117 | Closure of a structure ind... |
| str0 17118 | All components of the empt... |
| strfvi 17119 | Structure slot extractors ... |
| fveqprc 17120 | Lemma for showing the equa... |
| oveqprc 17121 | Lemma for showing the equa... |
| wunndx 17124 | Closure of the index extra... |
| ndxarg 17125 | Get the numeric argument f... |
| ndxid 17126 | A structure component extr... |
| strndxid 17127 | The value of a structure c... |
| setsidvald 17128 | Value of the structure rep... |
| strfvd 17129 | Deduction version of ~ str... |
| strfv2d 17130 | Deduction version of ~ str... |
| strfv2 17131 | A variation on ~ strfv to ... |
| strfv 17132 | Extract a structure compon... |
| strfv3 17133 | Variant on ~ strfv for lar... |
| strssd 17134 | Deduction version of ~ str... |
| strss 17135 | Propagate component extrac... |
| setsid 17136 | Value of the structure rep... |
| setsnid 17137 | Value of the structure rep... |
| baseval 17140 | Value of the base set extr... |
| baseid 17141 | Utility theorem: index-ind... |
| basfn 17142 | The base set extractor is ... |
| base0 17143 | The base set of the empty ... |
| elbasfv 17144 | Utility theorem: reverse c... |
| elbasov 17145 | Utility theorem: reverse c... |
| strov2rcl 17146 | Partial reverse closure fo... |
| basendx 17147 | Index value of the base se... |
| basendxnn 17148 | The index value of the bas... |
| basndxelwund 17149 | The index of the base set ... |
| basprssdmsets 17150 | The pair of the base index... |
| opelstrbas 17151 | The base set of a structur... |
| 1strstr 17152 | A constructed one-slot str... |
| 1strbas 17153 | The base set of a construc... |
| 1strwunbndx 17154 | A constructed one-slot str... |
| 1strwun 17155 | A constructed one-slot str... |
| 2strstr 17156 | A constructed two-slot str... |
| 2strbas 17157 | The base set of a construc... |
| 2strop 17158 | The other slot of a constr... |
| reldmress 17161 | The structure restriction ... |
| ressval 17162 | Value of structure restric... |
| ressid2 17163 | General behavior of trivia... |
| ressval2 17164 | Value of nontrivial struct... |
| ressbas 17165 | Base set of a structure re... |
| ressbasssg 17166 | The base set of a restrict... |
| ressbas2 17167 | Base set of a structure re... |
| ressbasss 17168 | The base set of a restrict... |
| ressbasssOLD 17169 | Obsolete version of ~ ress... |
| ressbasss2 17170 | The base set of a restrict... |
| resseqnbas 17171 | The components of an exten... |
| ress0 17172 | All restrictions of the nu... |
| ressid 17173 | Behavior of trivial restri... |
| ressinbas 17174 | Restriction only cares abo... |
| ressval3d 17175 | Value of structure restric... |
| ressress 17176 | Restriction composition la... |
| ressabs 17177 | Restriction absorption law... |
| wunress 17178 | Closure of structure restr... |
| plusgndx 17205 | Index value of the ~ df-pl... |
| plusgid 17206 | Utility theorem: index-ind... |
| plusgndxnn 17207 | The index of the slot for ... |
| basendxltplusgndx 17208 | The index of the slot for ... |
| basendxnplusgndx 17209 | The slot for the base set ... |
| grpstr 17210 | A constructed group is a s... |
| grpbase 17211 | The base set of a construc... |
| grpplusg 17212 | The operation of a constru... |
| ressplusg 17213 | ` +g ` is unaffected by re... |
| grpbasex 17214 | The base of an explicitly ... |
| grpplusgx 17215 | The operation of an explic... |
| mulrndx 17216 | Index value of the ~ df-mu... |
| mulridx 17217 | Utility theorem: index-ind... |
| basendxnmulrndx 17218 | The slot for the base set ... |
| plusgndxnmulrndx 17219 | The slot for the group (ad... |
| rngstr 17220 | A constructed ring is a st... |
| rngbase 17221 | The base set of a construc... |
| rngplusg 17222 | The additive operation of ... |
| rngmulr 17223 | The multiplicative operati... |
| starvndx 17224 | Index value of the ~ df-st... |
| starvid 17225 | Utility theorem: index-ind... |
| starvndxnbasendx 17226 | The slot for the involutio... |
| starvndxnplusgndx 17227 | The slot for the involutio... |
| starvndxnmulrndx 17228 | The slot for the involutio... |
| ressmulr 17229 | ` .r ` is unaffected by re... |
| ressstarv 17230 | ` *r ` is unaffected by re... |
| srngstr 17231 | A constructed star ring is... |
| srngbase 17232 | The base set of a construc... |
| srngplusg 17233 | The addition operation of ... |
| srngmulr 17234 | The multiplication operati... |
| srnginvl 17235 | The involution function of... |
| scandx 17236 | Index value of the ~ df-sc... |
| scaid 17237 | Utility theorem: index-ind... |
| scandxnbasendx 17238 | The slot for the scalar is... |
| scandxnplusgndx 17239 | The slot for the scalar fi... |
| scandxnmulrndx 17240 | The slot for the scalar fi... |
| vscandx 17241 | Index value of the ~ df-vs... |
| vscaid 17242 | Utility theorem: index-ind... |
| vscandxnbasendx 17243 | The slot for the scalar pr... |
| vscandxnplusgndx 17244 | The slot for the scalar pr... |
| vscandxnmulrndx 17245 | The slot for the scalar pr... |
| vscandxnscandx 17246 | The slot for the scalar pr... |
| lmodstr 17247 | A constructed left module ... |
| lmodbase 17248 | The base set of a construc... |
| lmodplusg 17249 | The additive operation of ... |
| lmodsca 17250 | The set of scalars of a co... |
| lmodvsca 17251 | The scalar product operati... |
| ipndx 17252 | Index value of the ~ df-ip... |
| ipid 17253 | Utility theorem: index-ind... |
| ipndxnbasendx 17254 | The slot for the inner pro... |
| ipndxnplusgndx 17255 | The slot for the inner pro... |
| ipndxnmulrndx 17256 | The slot for the inner pro... |
| slotsdifipndx 17257 | The slot for the scalar is... |
| ipsstr 17258 | Lemma to shorten proofs of... |
| ipsbase 17259 | The base set of a construc... |
| ipsaddg 17260 | The additive operation of ... |
| ipsmulr 17261 | The multiplicative operati... |
| ipssca 17262 | The set of scalars of a co... |
| ipsvsca 17263 | The scalar product operati... |
| ipsip 17264 | The multiplicative operati... |
| resssca 17265 | ` Scalar ` is unaffected b... |
| ressvsca 17266 | ` .s ` is unaffected by re... |
| ressip 17267 | The inner product is unaff... |
| phlstr 17268 | A constructed pre-Hilbert ... |
| phlbase 17269 | The base set of a construc... |
| phlplusg 17270 | The additive operation of ... |
| phlsca 17271 | The ring of scalars of a c... |
| phlvsca 17272 | The scalar product operati... |
| phlip 17273 | The inner product (Hermiti... |
| tsetndx 17274 | Index value of the ~ df-ts... |
| tsetid 17275 | Utility theorem: index-ind... |
| tsetndxnn 17276 | The index of the slot for ... |
| basendxlttsetndx 17277 | The index of the slot for ... |
| tsetndxnbasendx 17278 | The slot for the topology ... |
| tsetndxnplusgndx 17279 | The slot for the topology ... |
| tsetndxnmulrndx 17280 | The slot for the topology ... |
| tsetndxnstarvndx 17281 | The slot for the topology ... |
| slotstnscsi 17282 | The slots ` Scalar ` , ` .... |
| topgrpstr 17283 | A constructed topological ... |
| topgrpbas 17284 | The base set of a construc... |
| topgrpplusg 17285 | The additive operation of ... |
| topgrptset 17286 | The topology of a construc... |
| resstset 17287 | ` TopSet ` is unaffected b... |
| plendx 17288 | Index value of the ~ df-pl... |
| pleid 17289 | Utility theorem: self-refe... |
| plendxnn 17290 | The index value of the ord... |
| basendxltplendx 17291 | The index value of the ` B... |
| plendxnbasendx 17292 | The slot for the order is ... |
| plendxnplusgndx 17293 | The slot for the "less tha... |
| plendxnmulrndx 17294 | The slot for the "less tha... |
| plendxnscandx 17295 | The slot for the "less tha... |
| plendxnvscandx 17296 | The slot for the "less tha... |
| slotsdifplendx 17297 | The index of the slot for ... |
| otpsstr 17298 | Functionality of a topolog... |
| otpsbas 17299 | The base set of a topologi... |
| otpstset 17300 | The open sets of a topolog... |
| otpsle 17301 | The order of a topological... |
| ressle 17302 | ` le ` is unaffected by re... |
| ocndx 17303 | Index value of the ~ df-oc... |
| ocid 17304 | Utility theorem: index-ind... |
| basendxnocndx 17305 | The slot for the orthocomp... |
| plendxnocndx 17306 | The slot for the orthocomp... |
| dsndx 17307 | Index value of the ~ df-ds... |
| dsid 17308 | Utility theorem: index-ind... |
| dsndxnn 17309 | The index of the slot for ... |
| basendxltdsndx 17310 | The index of the slot for ... |
| dsndxnbasendx 17311 | The slot for the distance ... |
| dsndxnplusgndx 17312 | The slot for the distance ... |
| dsndxnmulrndx 17313 | The slot for the distance ... |
| slotsdnscsi 17314 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17315 | The slot for the distance ... |
| slotsdifdsndx 17316 | The index of the slot for ... |
| unifndx 17317 | Index value of the ~ df-un... |
| unifid 17318 | Utility theorem: index-ind... |
| unifndxnn 17319 | The index of the slot for ... |
| basendxltunifndx 17320 | The index of the slot for ... |
| unifndxnbasendx 17321 | The slot for the uniform s... |
| unifndxntsetndx 17322 | The slot for the uniform s... |
| slotsdifunifndx 17323 | The index of the slot for ... |
| ressunif 17324 | ` UnifSet ` is unaffected ... |
| odrngstr 17325 | Functionality of an ordere... |
| odrngbas 17326 | The base set of an ordered... |
| odrngplusg 17327 | The addition operation of ... |
| odrngmulr 17328 | The multiplication operati... |
| odrngtset 17329 | The open sets of an ordere... |
| odrngle 17330 | The order of an ordered me... |
| odrngds 17331 | The metric of an ordered m... |
| ressds 17332 | ` dist ` is unaffected by ... |
| homndx 17333 | Index value of the ~ df-ho... |
| homid 17334 | Utility theorem: index-ind... |
| ccondx 17335 | Index value of the ~ df-cc... |
| ccoid 17336 | Utility theorem: index-ind... |
| slotsbhcdif 17337 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17338 | The index of the slot for ... |
| slotsdifocndx 17339 | The index of the slot for ... |
| resshom 17340 | ` Hom ` is unaffected by r... |
| ressco 17341 | ` comp ` is unaffected by ... |
| restfn 17346 | The subspace topology oper... |
| topnfn 17347 | The topology extractor fun... |
| restval 17348 | The subspace topology indu... |
| elrest 17349 | The predicate "is an open ... |
| elrestr 17350 | Sufficient condition for b... |
| 0rest 17351 | Value of the structure res... |
| restid2 17352 | The subspace topology over... |
| restsspw 17353 | The subspace topology is a... |
| firest 17354 | The finite intersections o... |
| restid 17355 | The subspace topology of t... |
| topnval 17356 | Value of the topology extr... |
| topnid 17357 | Value of the topology extr... |
| topnpropd 17358 | The topology extractor fun... |
| reldmprds 17370 | The structure product is a... |
| prdsbasex 17372 | Lemma for structure produc... |
| imasvalstr 17373 | An image structure value i... |
| prdsvalstr 17374 | Structure product value is... |
| prdsbaslem 17375 | Lemma for ~ prdsbas and si... |
| prdsvallem 17376 | Lemma for ~ prdsval . (Co... |
| prdsval 17377 | Value of the structure pro... |
| prdssca 17378 | Scalar ring of a structure... |
| prdsbas 17379 | Base set of a structure pr... |
| prdsplusg 17380 | Addition in a structure pr... |
| prdsmulr 17381 | Multiplication in a struct... |
| prdsvsca 17382 | Scalar multiplication in a... |
| prdsip 17383 | Inner product in a structu... |
| prdsle 17384 | Structure product weak ord... |
| prdsless 17385 | Closure of the order relat... |
| prdsds 17386 | Structure product distance... |
| prdsdsfn 17387 | Structure product distance... |
| prdstset 17388 | Structure product topology... |
| prdshom 17389 | Structure product hom-sets... |
| prdsco 17390 | Structure product composit... |
| prdsbas2 17391 | The base set of a structur... |
| prdsbasmpt 17392 | A constructed tuple is a p... |
| prdsbasfn 17393 | Points in the structure pr... |
| prdsbasprj 17394 | Each point in a structure ... |
| prdsplusgval 17395 | Value of a componentwise s... |
| prdsplusgfval 17396 | Value of a structure produ... |
| prdsmulrval 17397 | Value of a componentwise r... |
| prdsmulrfval 17398 | Value of a structure produ... |
| prdsleval 17399 | Value of the product order... |
| prdsdsval 17400 | Value of the metric in a s... |
| prdsvscaval 17401 | Scalar multiplication in a... |
| prdsvscafval 17402 | Scalar multiplication of a... |
| prdsbas3 17403 | The base set of an indexed... |
| prdsbasmpt2 17404 | A constructed tuple is a p... |
| prdsbascl 17405 | An element of the base has... |
| prdsdsval2 17406 | Value of the metric in a s... |
| prdsdsval3 17407 | Value of the metric in a s... |
| pwsval 17408 | Value of a structure power... |
| pwsbas 17409 | Base set of a structure po... |
| pwselbasb 17410 | Membership in the base set... |
| pwselbas 17411 | An element of a structure ... |
| pwselbasr 17412 | The reverse direction of ~... |
| pwsplusgval 17413 | Value of addition in a str... |
| pwsmulrval 17414 | Value of multiplication in... |
| pwsle 17415 | Ordering in a structure po... |
| pwsleval 17416 | Ordering in a structure po... |
| pwsvscafval 17417 | Scalar multiplication in a... |
| pwsvscaval 17418 | Scalar multiplication of a... |
| pwssca 17419 | The ring of scalars of a s... |
| pwsdiagel 17420 | Membership of diagonal ele... |
| pwssnf1o 17421 | Triviality of singleton po... |
| imasval 17434 | Value of an image structur... |
| imasbas 17435 | The base set of an image s... |
| imasds 17436 | The distance function of a... |
| imasdsfn 17437 | The distance function is a... |
| imasdsval 17438 | The distance function of a... |
| imasdsval2 17439 | The distance function of a... |
| imasplusg 17440 | The group operation in an ... |
| imasmulr 17441 | The ring multiplication in... |
| imassca 17442 | The scalar field of an ima... |
| imasvsca 17443 | The scalar multiplication ... |
| imasip 17444 | The inner product of an im... |
| imastset 17445 | The topology of an image s... |
| imasle 17446 | The ordering of an image s... |
| f1ocpbllem 17447 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17448 | An injection is compatible... |
| f1ovscpbl 17449 | An injection is compatible... |
| f1olecpbl 17450 | An injection is compatible... |
| imasaddfnlem 17451 | The image structure operat... |
| imasaddvallem 17452 | The operation of an image ... |
| imasaddflem 17453 | The image set operations a... |
| imasaddfn 17454 | The image structure's grou... |
| imasaddval 17455 | The value of an image stru... |
| imasaddf 17456 | The image structure's grou... |
| imasmulfn 17457 | The image structure's ring... |
| imasmulval 17458 | The value of an image stru... |
| imasmulf 17459 | The image structure's ring... |
| imasvscafn 17460 | The image structure's scal... |
| imasvscaval 17461 | The value of an image stru... |
| imasvscaf 17462 | The image structure's scal... |
| imasless 17463 | The order relation defined... |
| imasleval 17464 | The value of the image str... |
| qusval 17465 | Value of a quotient struct... |
| quslem 17466 | The function in ~ qusval i... |
| qusin 17467 | Restrict the equivalence r... |
| qusbas 17468 | Base set of a quotient str... |
| quss 17469 | The scalar field of a quot... |
| divsfval 17470 | Value of the function in ~... |
| ercpbllem 17471 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17472 | Translate the function com... |
| erlecpbl 17473 | Translate the relation com... |
| qusaddvallem 17474 | Value of an operation defi... |
| qusaddflem 17475 | The operation of a quotien... |
| qusaddval 17476 | The addition in a quotient... |
| qusaddf 17477 | The addition in a quotient... |
| qusmulval 17478 | The multiplication in a qu... |
| qusmulf 17479 | The multiplication in a qu... |
| fnpr2o 17480 | Function with a domain of ... |
| fnpr2ob 17481 | Biconditional version of ~... |
| fvpr0o 17482 | The value of a function wi... |
| fvpr1o 17483 | The value of a function wi... |
| fvprif 17484 | The value of the pair func... |
| xpsfrnel 17485 | Elementhood in the target ... |
| xpsfeq 17486 | A function on ` 2o ` is de... |
| xpsfrnel2 17487 | Elementhood in the target ... |
| xpscf 17488 | Equivalent condition for t... |
| xpsfval 17489 | The value of the function ... |
| xpsff1o 17490 | The function appearing in ... |
| xpsfrn 17491 | A short expression for the... |
| xpsff1o2 17492 | The function appearing in ... |
| xpsval 17493 | Value of the binary struct... |
| xpsrnbas 17494 | The indexed structure prod... |
| xpsbas 17495 | The base set of the binary... |
| xpsaddlem 17496 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17497 | Value of the addition oper... |
| xpsmul 17498 | Value of the multiplicatio... |
| xpssca 17499 | Value of the scalar field ... |
| xpsvsca 17500 | Value of the scalar multip... |
| xpsless 17501 | Closure of the ordering in... |
| xpsle 17502 | Value of the ordering in a... |
| ismre 17511 | Property of being a Moore ... |
| fnmre 17512 | The Moore collection gener... |
| mresspw 17513 | A Moore collection is a su... |
| mress 17514 | A Moore-closed subset is a... |
| mre1cl 17515 | In any Moore collection th... |
| mreintcl 17516 | A nonempty collection of c... |
| mreiincl 17517 | A nonempty indexed interse... |
| mrerintcl 17518 | The relative intersection ... |
| mreriincl 17519 | The relative intersection ... |
| mreincl 17520 | Two closed sets have a clo... |
| mreuni 17521 | Since the entire base set ... |
| mreunirn 17522 | Two ways to express the no... |
| ismred 17523 | Properties that determine ... |
| ismred2 17524 | Properties that determine ... |
| mremre 17525 | The Moore collections of s... |
| submre 17526 | The subcollection of a clo... |
| xrsle 17527 | The ordering of the extend... |
| xrge0le 17528 | The "less than or equal to... |
| xrsbas 17529 | The base set of the extend... |
| xrge0base 17530 | The base of the extended n... |
| mrcflem 17531 | The domain and codomain of... |
| fnmrc 17532 | Moore-closure is a well-be... |
| mrcfval 17533 | Value of the function expr... |
| mrcf 17534 | The Moore closure is a fun... |
| mrcval 17535 | Evaluation of the Moore cl... |
| mrccl 17536 | The Moore closure of a set... |
| mrcsncl 17537 | The Moore closure of a sin... |
| mrcid 17538 | The closure of a closed se... |
| mrcssv 17539 | The closure of a set is a ... |
| mrcidb 17540 | A set is closed iff it is ... |
| mrcss 17541 | Closure preserves subset o... |
| mrcssid 17542 | The closure of a set is a ... |
| mrcidb2 17543 | A set is closed iff it con... |
| mrcidm 17544 | The closure operation is i... |
| mrcsscl 17545 | The closure is the minimal... |
| mrcuni 17546 | Idempotence of closure und... |
| mrcun 17547 | Idempotence of closure und... |
| mrcssvd 17548 | The Moore closure of a set... |
| mrcssd 17549 | Moore closure preserves su... |
| mrcssidd 17550 | A set is contained in its ... |
| mrcidmd 17551 | Moore closure is idempoten... |
| mressmrcd 17552 | In a Moore system, if a se... |
| submrc 17553 | In a closure system which ... |
| mrieqvlemd 17554 | In a Moore system, if ` Y ... |
| mrisval 17555 | Value of the set of indepe... |
| ismri 17556 | Criterion for a set to be ... |
| ismri2 17557 | Criterion for a subset of ... |
| ismri2d 17558 | Criterion for a subset of ... |
| ismri2dd 17559 | Definition of independence... |
| mriss 17560 | An independent set of a Mo... |
| mrissd 17561 | An independent set of a Mo... |
| ismri2dad 17562 | Consequence of a set in a ... |
| mrieqvd 17563 | In a Moore system, a set i... |
| mrieqv2d 17564 | In a Moore system, a set i... |
| mrissmrcd 17565 | In a Moore system, if an i... |
| mrissmrid 17566 | In a Moore system, subsets... |
| mreexd 17567 | In a Moore system, the clo... |
| mreexmrid 17568 | In a Moore system whose cl... |
| mreexexlemd 17569 | This lemma is used to gene... |
| mreexexlem2d 17570 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17571 | Base case of the induction... |
| mreexexlem4d 17572 | Induction step of the indu... |
| mreexexd 17573 | Exchange-type theorem. In... |
| mreexdomd 17574 | In a Moore system whose cl... |
| mreexfidimd 17575 | In a Moore system whose cl... |
| isacs 17576 | A set is an algebraic clos... |
| acsmre 17577 | Algebraic closure systems ... |
| isacs2 17578 | In the definition of an al... |
| acsfiel 17579 | A set is closed in an alge... |
| acsfiel2 17580 | A set is closed in an alge... |
| acsmred 17581 | An algebraic closure syste... |
| isacs1i 17582 | A closure system determine... |
| mreacs 17583 | Algebraicity is a composab... |
| acsfn 17584 | Algebraicity of a conditio... |
| acsfn0 17585 | Algebraicity of a point cl... |
| acsfn1 17586 | Algebraicity of a one-argu... |
| acsfn1c 17587 | Algebraicity of a one-argu... |
| acsfn2 17588 | Algebraicity of a two-argu... |
| iscat 17597 | The predicate "is a catego... |
| iscatd 17598 | Properties that determine ... |
| catidex 17599 | Each object in a category ... |
| catideu 17600 | Each object in a category ... |
| cidfval 17601 | Each object in a category ... |
| cidval 17602 | Each object in a category ... |
| cidffn 17603 | The identity arrow constru... |
| cidfn 17604 | The identity arrow operato... |
| catidd 17605 | Deduce the identity arrow ... |
| iscatd2 17606 | Version of ~ iscatd with a... |
| catidcl 17607 | Each object in a category ... |
| catlid 17608 | Left identity property of ... |
| catrid 17609 | Right identity property of... |
| catcocl 17610 | Closure of a composition a... |
| catass 17611 | Associativity of compositi... |
| catcone0 17612 | Composition of non-empty h... |
| 0catg 17613 | Any structure with an empt... |
| 0cat 17614 | The empty set is a categor... |
| homffval 17615 | Value of the functionalize... |
| fnhomeqhomf 17616 | If the Hom-set operation i... |
| homfval 17617 | Value of the functionalize... |
| homffn 17618 | The functionalized Hom-set... |
| homfeq 17619 | Condition for two categori... |
| homfeqd 17620 | If two structures have the... |
| homfeqbas 17621 | Deduce equality of base se... |
| homfeqval 17622 | Value of the functionalize... |
| comfffval 17623 | Value of the functionalize... |
| comffval 17624 | Value of the functionalize... |
| comfval 17625 | Value of the functionalize... |
| comfffval2 17626 | Value of the functionalize... |
| comffval2 17627 | Value of the functionalize... |
| comfval2 17628 | Value of the functionalize... |
| comfffn 17629 | The functionalized composi... |
| comffn 17630 | The functionalized composi... |
| comfeq 17631 | Condition for two categori... |
| comfeqd 17632 | Condition for two categori... |
| comfeqval 17633 | Equality of two compositio... |
| catpropd 17634 | Two structures with the sa... |
| cidpropd 17635 | Two structures with the sa... |
| oppcval 17638 | Value of the opposite cate... |
| oppchomfval 17639 | Hom-sets of the opposite c... |
| oppchom 17640 | Hom-sets of the opposite c... |
| oppccofval 17641 | Composition in the opposit... |
| oppcco 17642 | Composition in the opposit... |
| oppcbas 17643 | Base set of an opposite ca... |
| oppccatid 17644 | Lemma for ~ oppccat . (Co... |
| oppchomf 17645 | Hom-sets of the opposite c... |
| oppcid 17646 | Identity function of an op... |
| oppccat 17647 | An opposite category is a ... |
| 2oppcbas 17648 | The double opposite catego... |
| 2oppchomf 17649 | The double opposite catego... |
| 2oppccomf 17650 | The double opposite catego... |
| oppchomfpropd 17651 | If two categories have the... |
| oppccomfpropd 17652 | If two categories have the... |
| oppccatf 17653 | ` oppCat ` restricted to `... |
| monfval 17658 | Definition of a monomorphi... |
| ismon 17659 | Definition of a monomorphi... |
| ismon2 17660 | Write out the monomorphism... |
| monhom 17661 | A monomorphism is a morphi... |
| moni 17662 | Property of a monomorphism... |
| monpropd 17663 | If two categories have the... |
| oppcmon 17664 | A monomorphism in the oppo... |
| oppcepi 17665 | An epimorphism in the oppo... |
| isepi 17666 | Definition of an epimorphi... |
| isepi2 17667 | Write out the epimorphism ... |
| epihom 17668 | An epimorphism is a morphi... |
| epii 17669 | Property of an epimorphism... |
| sectffval 17676 | Value of the section opera... |
| sectfval 17677 | Value of the section relat... |
| sectss 17678 | The section relation is a ... |
| issect 17679 | The property " ` F ` is a ... |
| issect2 17680 | Property of being a sectio... |
| sectcan 17681 | If ` G ` is a section of `... |
| sectco 17682 | Composition of two section... |
| isofval 17683 | Function value of the func... |
| invffval 17684 | Value of the inverse relat... |
| invfval 17685 | Value of the inverse relat... |
| isinv 17686 | Value of the inverse relat... |
| invss 17687 | The inverse relation is a ... |
| invsym 17688 | The inverse relation is sy... |
| invsym2 17689 | The inverse relation is sy... |
| invfun 17690 | The inverse relation is a ... |
| isoval 17691 | The isomorphisms are the d... |
| inviso1 17692 | If ` G ` is an inverse to ... |
| inviso2 17693 | If ` G ` is an inverse to ... |
| invf 17694 | The inverse relation is a ... |
| invf1o 17695 | The inverse relation is a ... |
| invinv 17696 | The inverse of the inverse... |
| invco 17697 | The composition of two iso... |
| dfiso2 17698 | Alternate definition of an... |
| dfiso3 17699 | Alternate definition of an... |
| inveq 17700 | If there are two inverses ... |
| isofn 17701 | The function value of the ... |
| isohom 17702 | An isomorphism is a homomo... |
| isoco 17703 | The composition of two iso... |
| oppcsect 17704 | A section in the opposite ... |
| oppcsect2 17705 | A section in the opposite ... |
| oppcinv 17706 | An inverse in the opposite... |
| oppciso 17707 | An isomorphism in the oppo... |
| sectmon 17708 | If ` F ` is a section of `... |
| monsect 17709 | If ` F ` is a monomorphism... |
| sectepi 17710 | If ` F ` is a section of `... |
| episect 17711 | If ` F ` is an epimorphism... |
| sectid 17712 | The identity is a section ... |
| invid 17713 | The inverse of the identit... |
| idiso 17714 | The identity is an isomorp... |
| idinv 17715 | The inverse of the identit... |
| invisoinvl 17716 | The inverse of an isomorph... |
| invisoinvr 17717 | The inverse of an isomorph... |
| invcoisoid 17718 | The inverse of an isomorph... |
| isocoinvid 17719 | The inverse of an isomorph... |
| rcaninv 17720 | Right cancellation of an i... |
| cicfval 17723 | The set of isomorphic obje... |
| brcic 17724 | The relation "is isomorphi... |
| cic 17725 | Objects ` X ` and ` Y ` in... |
| brcici 17726 | Prove that two objects are... |
| cicref 17727 | Isomorphism is reflexive. ... |
| ciclcl 17728 | Isomorphism implies the le... |
| cicrcl 17729 | Isomorphism implies the ri... |
| cicsym 17730 | Isomorphism is symmetric. ... |
| cictr 17731 | Isomorphism is transitive.... |
| cicer 17732 | Isomorphism is an equivale... |
| sscrel 17739 | The subcategory subset rel... |
| brssc 17740 | The subcategory subset rel... |
| sscpwex 17741 | An analogue of ~ pwex for ... |
| subcrcl 17742 | Reverse closure for the su... |
| sscfn1 17743 | The subcategory subset rel... |
| sscfn2 17744 | The subcategory subset rel... |
| ssclem 17745 | Lemma for ~ ssc1 and simil... |
| isssc 17746 | Value of the subcategory s... |
| ssc1 17747 | Infer subset relation on o... |
| ssc2 17748 | Infer subset relation on m... |
| sscres 17749 | Any function restricted to... |
| sscid 17750 | The subcategory subset rel... |
| ssctr 17751 | The subcategory subset rel... |
| ssceq 17752 | The subcategory subset rel... |
| rescval 17753 | Value of the category rest... |
| rescval2 17754 | Value of the category rest... |
| rescbas 17755 | Base set of the category r... |
| reschom 17756 | Hom-sets of the category r... |
| reschomf 17757 | Hom-sets of the category r... |
| rescco 17758 | Composition in the categor... |
| rescabs 17759 | Restriction absorption law... |
| rescabs2 17760 | Restriction absorption law... |
| issubc 17761 | Elementhood in the set of ... |
| issubc2 17762 | Elementhood in the set of ... |
| 0ssc 17763 | For any category ` C ` , t... |
| 0subcat 17764 | For any category ` C ` , t... |
| catsubcat 17765 | For any category ` C ` , `... |
| subcssc 17766 | An element in the set of s... |
| subcfn 17767 | An element in the set of s... |
| subcss1 17768 | The objects of a subcatego... |
| subcss2 17769 | The morphisms of a subcate... |
| subcidcl 17770 | The identity of the origin... |
| subccocl 17771 | A subcategory is closed un... |
| subccatid 17772 | A subcategory is a categor... |
| subcid 17773 | The identity in a subcateg... |
| subccat 17774 | A subcategory is a categor... |
| issubc3 17775 | Alternate definition of a ... |
| fullsubc 17776 | The full subcategory gener... |
| fullresc 17777 | The category formed by str... |
| resscat 17778 | A category restricted to a... |
| subsubc 17779 | A subcategory of a subcate... |
| relfunc 17788 | The set of functors is a r... |
| funcrcl 17789 | Reverse closure for a func... |
| isfunc 17790 | Value of the set of functo... |
| isfuncd 17791 | Deduce that an operation i... |
| funcf1 17792 | The object part of a funct... |
| funcixp 17793 | The morphism part of a fun... |
| funcf2 17794 | The morphism part of a fun... |
| funcfn2 17795 | The morphism part of a fun... |
| funcid 17796 | A functor maps each identi... |
| funcco 17797 | A functor maps composition... |
| funcsect 17798 | The image of a section und... |
| funcinv 17799 | The image of an inverse un... |
| funciso 17800 | The image of an isomorphis... |
| funcoppc 17801 | A functor on categories yi... |
| idfuval 17802 | Value of the identity func... |
| idfu2nd 17803 | Value of the morphism part... |
| idfu2 17804 | Value of the morphism part... |
| idfu1st 17805 | Value of the object part o... |
| idfu1 17806 | Value of the object part o... |
| idfucl 17807 | The identity functor is a ... |
| cofuval 17808 | Value of the composition o... |
| cofu1st 17809 | Value of the object part o... |
| cofu1 17810 | Value of the object part o... |
| cofu2nd 17811 | Value of the morphism part... |
| cofu2 17812 | Value of the morphism part... |
| cofuval2 17813 | Value of the composition o... |
| cofucl 17814 | The composition of two fun... |
| cofuass 17815 | Functor composition is ass... |
| cofulid 17816 | The identity functor is a ... |
| cofurid 17817 | The identity functor is a ... |
| resfval 17818 | Value of the functor restr... |
| resfval2 17819 | Value of the functor restr... |
| resf1st 17820 | Value of the functor restr... |
| resf2nd 17821 | Value of the functor restr... |
| funcres 17822 | A functor restricted to a ... |
| funcres2b 17823 | Condition for a functor to... |
| funcres2 17824 | A functor into a restricte... |
| idfusubc0 17825 | The identity functor for a... |
| idfusubc 17826 | The identity functor for a... |
| wunfunc 17827 | A weak universe is closed ... |
| funcpropd 17828 | If two categories have the... |
| funcres2c 17829 | Condition for a functor to... |
| fullfunc 17834 | A full functor is a functo... |
| fthfunc 17835 | A faithful functor is a fu... |
| relfull 17836 | The set of full functors i... |
| relfth 17837 | The set of faithful functo... |
| isfull 17838 | Value of the set of full f... |
| isfull2 17839 | Equivalent condition for a... |
| fullfo 17840 | The morphism map of a full... |
| fulli 17841 | The morphism map of a full... |
| isfth 17842 | Value of the set of faithf... |
| isfth2 17843 | Equivalent condition for a... |
| isffth2 17844 | A fully faithful functor i... |
| fthf1 17845 | The morphism map of a fait... |
| fthi 17846 | The morphism map of a fait... |
| ffthf1o 17847 | The morphism map of a full... |
| fullpropd 17848 | If two categories have the... |
| fthpropd 17849 | If two categories have the... |
| fulloppc 17850 | The opposite functor of a ... |
| fthoppc 17851 | The opposite functor of a ... |
| ffthoppc 17852 | The opposite functor of a ... |
| fthsect 17853 | A faithful functor reflect... |
| fthinv 17854 | A faithful functor reflect... |
| fthmon 17855 | A faithful functor reflect... |
| fthepi 17856 | A faithful functor reflect... |
| ffthiso 17857 | A fully faithful functor r... |
| fthres2b 17858 | Condition for a faithful f... |
| fthres2c 17859 | Condition for a faithful f... |
| fthres2 17860 | A faithful functor into a ... |
| idffth 17861 | The identity functor is a ... |
| cofull 17862 | The composition of two ful... |
| cofth 17863 | The composition of two fai... |
| coffth 17864 | The composition of two ful... |
| rescfth 17865 | The inclusion functor from... |
| ressffth 17866 | The inclusion functor from... |
| fullres2c 17867 | Condition for a full funct... |
| ffthres2c 17868 | Condition for a fully fait... |
| inclfusubc 17869 | The "inclusion functor" fr... |
| fnfuc 17874 | The ` FuncCat ` operation ... |
| natfval 17875 | Value of the function givi... |
| isnat 17876 | Property of being a natura... |
| isnat2 17877 | Property of being a natura... |
| natffn 17878 | The natural transformation... |
| natrcl 17879 | Reverse closure for a natu... |
| nat1st2nd 17880 | Rewrite the natural transf... |
| natixp 17881 | A natural transformation i... |
| natcl 17882 | A component of a natural t... |
| natfn 17883 | A natural transformation i... |
| nati 17884 | Naturality property of a n... |
| wunnat 17885 | A weak universe is closed ... |
| catstr 17886 | A category structure is a ... |
| fucval 17887 | Value of the functor categ... |
| fuccofval 17888 | Value of the functor categ... |
| fucbas 17889 | The objects of the functor... |
| fuchom 17890 | The morphisms in the funct... |
| fucco 17891 | Value of the composition o... |
| fuccoval 17892 | Value of the functor categ... |
| fuccocl 17893 | The composition of two nat... |
| fucidcl 17894 | The identity natural trans... |
| fuclid 17895 | Left identity of natural t... |
| fucrid 17896 | Right identity of natural ... |
| fucass 17897 | Associativity of natural t... |
| fuccatid 17898 | The functor category is a ... |
| fuccat 17899 | The functor category is a ... |
| fucid 17900 | The identity morphism in t... |
| fucsect 17901 | Two natural transformation... |
| fucinv 17902 | Two natural transformation... |
| invfuc 17903 | If ` V ( x ) ` is an inver... |
| fuciso 17904 | A natural transformation i... |
| natpropd 17905 | If two categories have the... |
| fucpropd 17906 | If two categories have the... |
| initofn 17913 | ` InitO ` is a function on... |
| termofn 17914 | ` TermO ` is a function on... |
| zeroofn 17915 | ` ZeroO ` is a function on... |
| initorcl 17916 | Reverse closure for an ini... |
| termorcl 17917 | Reverse closure for a term... |
| zeroorcl 17918 | Reverse closure for a zero... |
| initoval 17919 | The value of the initial o... |
| termoval 17920 | The value of the terminal ... |
| zerooval 17921 | The value of the zero obje... |
| isinito 17922 | The predicate "is an initi... |
| istermo 17923 | The predicate "is a termin... |
| iszeroo 17924 | The predicate "is a zero o... |
| isinitoi 17925 | Implication of a class bei... |
| istermoi 17926 | Implication of a class bei... |
| initoid 17927 | For an initial object, the... |
| termoid 17928 | For a terminal object, the... |
| dfinito2 17929 | An initial object is a ter... |
| dftermo2 17930 | A terminal object is an in... |
| dfinito3 17931 | An alternate definition of... |
| dftermo3 17932 | An alternate definition of... |
| initoo 17933 | An initial object is an ob... |
| termoo 17934 | A terminal object is an ob... |
| iszeroi 17935 | Implication of a class bei... |
| 2initoinv 17936 | Morphisms between two init... |
| initoeu1 17937 | Initial objects are essent... |
| initoeu1w 17938 | Initial objects are essent... |
| initoeu2lem0 17939 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17940 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17941 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17942 | Initial objects are essent... |
| 2termoinv 17943 | Morphisms between two term... |
| termoeu1 17944 | Terminal objects are essen... |
| termoeu1w 17945 | Terminal objects are essen... |
| homarcl 17954 | Reverse closure for an arr... |
| homafval 17955 | Value of the disjointified... |
| homaf 17956 | Functionality of the disjo... |
| homaval 17957 | Value of the disjointified... |
| elhoma 17958 | Value of the disjointified... |
| elhomai 17959 | Produce an arrow from a mo... |
| elhomai2 17960 | Produce an arrow from a mo... |
| homarcl2 17961 | Reverse closure for the do... |
| homarel 17962 | An arrow is an ordered pai... |
| homa1 17963 | The first component of an ... |
| homahom2 17964 | The second component of an... |
| homahom 17965 | The second component of an... |
| homadm 17966 | The domain of an arrow wit... |
| homacd 17967 | The codomain of an arrow w... |
| homadmcd 17968 | Decompose an arrow into do... |
| arwval 17969 | The set of arrows is the u... |
| arwrcl 17970 | The first component of an ... |
| arwhoma 17971 | An arrow is contained in t... |
| homarw 17972 | A hom-set is a subset of t... |
| arwdm 17973 | The domain of an arrow is ... |
| arwcd 17974 | The codomain of an arrow i... |
| dmaf 17975 | The domain function is a f... |
| cdaf 17976 | The codomain function is a... |
| arwhom 17977 | The second component of an... |
| arwdmcd 17978 | Decompose an arrow into do... |
| idafval 17983 | Value of the identity arro... |
| idaval 17984 | Value of the identity arro... |
| ida2 17985 | Morphism part of the ident... |
| idahom 17986 | Domain and codomain of the... |
| idadm 17987 | Domain of the identity arr... |
| idacd 17988 | Codomain of the identity a... |
| idaf 17989 | The identity arrow functio... |
| coafval 17990 | The value of the compositi... |
| eldmcoa 17991 | A pair ` <. G , F >. ` is ... |
| dmcoass 17992 | The domain of composition ... |
| homdmcoa 17993 | If ` F : X --> Y ` and ` G... |
| coaval 17994 | Value of composition for c... |
| coa2 17995 | The morphism part of arrow... |
| coahom 17996 | The composition of two com... |
| coapm 17997 | Composition of arrows is a... |
| arwlid 17998 | Left identity of a categor... |
| arwrid 17999 | Right identity of a catego... |
| arwass 18000 | Associativity of compositi... |
| setcval 18003 | Value of the category of s... |
| setcbas 18004 | Set of objects of the cate... |
| setchomfval 18005 | Set of arrows of the categ... |
| setchom 18006 | Set of arrows of the categ... |
| elsetchom 18007 | A morphism of sets is a fu... |
| setccofval 18008 | Composition in the categor... |
| setcco 18009 | Composition in the categor... |
| setccatid 18010 | Lemma for ~ setccat . (Co... |
| setccat 18011 | The category of sets is a ... |
| setcid 18012 | The identity arrow in the ... |
| setcmon 18013 | A monomorphism of sets is ... |
| setcepi 18014 | An epimorphism of sets is ... |
| setcsect 18015 | A section in the category ... |
| setcinv 18016 | An inverse in the category... |
| setciso 18017 | An isomorphism in the cate... |
| resssetc 18018 | The restriction of the cat... |
| funcsetcres2 18019 | A functor into a smaller c... |
| setc2obas 18020 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18021 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18022 | The category of sets in a ... |
| cat1 18023 | The definition of category... |
| catcval 18026 | Value of the category of c... |
| catcbas 18027 | Set of objects of the cate... |
| catchomfval 18028 | Set of arrows of the categ... |
| catchom 18029 | Set of arrows of the categ... |
| catccofval 18030 | Composition in the categor... |
| catcco 18031 | Composition in the categor... |
| catccatid 18032 | Lemma for ~ catccat . (Co... |
| catcid 18033 | The identity arrow in the ... |
| catccat 18034 | The category of categories... |
| resscatc 18035 | The restriction of the cat... |
| catcisolem 18036 | Lemma for ~ catciso . (Co... |
| catciso 18037 | A functor is an isomorphis... |
| catcbascl 18038 | An element of the base set... |
| catcslotelcl 18039 | A slot entry of an element... |
| catcbaselcl 18040 | The base set of an element... |
| catchomcl 18041 | The Hom-set of an element ... |
| catcccocl 18042 | The composition operation ... |
| catcoppccl 18043 | The category of categories... |
| catcfuccl 18044 | The category of categories... |
| fncnvimaeqv 18045 | The inverse images of the ... |
| bascnvimaeqv 18046 | The inverse image of the u... |
| estrcval 18049 | Value of the category of e... |
| estrcbas 18050 | Set of objects of the cate... |
| estrchomfval 18051 | Set of morphisms ("arrows"... |
| estrchom 18052 | The morphisms between exte... |
| elestrchom 18053 | A morphism between extensi... |
| estrccofval 18054 | Composition in the categor... |
| estrcco 18055 | Composition in the categor... |
| estrcbasbas 18056 | An element of the base set... |
| estrccatid 18057 | Lemma for ~ estrccat . (C... |
| estrccat 18058 | The category of extensible... |
| estrcid 18059 | The identity arrow in the ... |
| estrchomfn 18060 | The Hom-set operation in t... |
| estrchomfeqhom 18061 | The functionalized Hom-set... |
| estrreslem1 18062 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18063 | Lemma 2 for ~ estrres . (... |
| estrres 18064 | Any restriction of a categ... |
| funcestrcsetclem1 18065 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18066 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18067 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18068 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18069 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18070 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18071 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18072 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18073 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18074 | The "natural forgetful fun... |
| fthestrcsetc 18075 | The "natural forgetful fun... |
| fullestrcsetc 18076 | The "natural forgetful fun... |
| equivestrcsetc 18077 | The "natural forgetful fun... |
| setc1strwun 18078 | A constructed one-slot str... |
| funcsetcestrclem1 18079 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18080 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18081 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18082 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18083 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18084 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18085 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18086 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18087 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18088 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18089 | The "embedding functor" fr... |
| fthsetcestrc 18090 | The "embedding functor" fr... |
| fullsetcestrc 18091 | The "embedding functor" fr... |
| embedsetcestrc 18092 | The "embedding functor" fr... |
| fnxpc 18101 | The binary product of cate... |
| xpcval 18102 | Value of the binary produc... |
| xpcbas 18103 | Set of objects of the bina... |
| xpchomfval 18104 | Set of morphisms of the bi... |
| xpchom 18105 | Set of morphisms of the bi... |
| relxpchom 18106 | A hom-set in the binary pr... |
| xpccofval 18107 | Value of composition in th... |
| xpcco 18108 | Value of composition in th... |
| xpcco1st 18109 | Value of composition in th... |
| xpcco2nd 18110 | Value of composition in th... |
| xpchom2 18111 | Value of the set of morphi... |
| xpcco2 18112 | Value of composition in th... |
| xpccatid 18113 | The product of two categor... |
| xpcid 18114 | The identity morphism in t... |
| xpccat 18115 | The product of two categor... |
| 1stfval 18116 | Value of the first project... |
| 1stf1 18117 | Value of the first project... |
| 1stf2 18118 | Value of the first project... |
| 2ndfval 18119 | Value of the first project... |
| 2ndf1 18120 | Value of the first project... |
| 2ndf2 18121 | Value of the first project... |
| 1stfcl 18122 | The first projection funct... |
| 2ndfcl 18123 | The second projection func... |
| prfval 18124 | Value of the pairing funct... |
| prf1 18125 | Value of the pairing funct... |
| prf2fval 18126 | Value of the pairing funct... |
| prf2 18127 | Value of the pairing funct... |
| prfcl 18128 | The pairing of functors ` ... |
| prf1st 18129 | Cancellation of pairing wi... |
| prf2nd 18130 | Cancellation of pairing wi... |
| 1st2ndprf 18131 | Break a functor into a pro... |
| catcxpccl 18132 | The category of categories... |
| xpcpropd 18133 | If two categories have the... |
| evlfval 18142 | Value of the evaluation fu... |
| evlf2 18143 | Value of the evaluation fu... |
| evlf2val 18144 | Value of the evaluation na... |
| evlf1 18145 | Value of the evaluation fu... |
| evlfcllem 18146 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18147 | The evaluation functor is ... |
| curfval 18148 | Value of the curry functor... |
| curf1fval 18149 | Value of the object part o... |
| curf1 18150 | Value of the object part o... |
| curf11 18151 | Value of the double evalua... |
| curf12 18152 | The partially evaluated cu... |
| curf1cl 18153 | The partially evaluated cu... |
| curf2 18154 | Value of the curry functor... |
| curf2val 18155 | Value of a component of th... |
| curf2cl 18156 | The curry functor at a mor... |
| curfcl 18157 | The curry functor of a fun... |
| curfpropd 18158 | If two categories have the... |
| uncfval 18159 | Value of the uncurry funct... |
| uncfcl 18160 | The uncurry operation take... |
| uncf1 18161 | Value of the uncurry funct... |
| uncf2 18162 | Value of the uncurry funct... |
| curfuncf 18163 | Cancellation of curry with... |
| uncfcurf 18164 | Cancellation of uncurry wi... |
| diagval 18165 | Define the diagonal functo... |
| diagcl 18166 | The diagonal functor is a ... |
| diag1cl 18167 | The constant functor of ` ... |
| diag11 18168 | Value of the constant func... |
| diag12 18169 | Value of the constant func... |
| diag2 18170 | Value of the diagonal func... |
| diag2cl 18171 | The diagonal functor at a ... |
| curf2ndf 18172 | As shown in ~ diagval , th... |
| hofval 18177 | Value of the Hom functor, ... |
| hof1fval 18178 | The object part of the Hom... |
| hof1 18179 | The object part of the Hom... |
| hof2fval 18180 | The morphism part of the H... |
| hof2val 18181 | The morphism part of the H... |
| hof2 18182 | The morphism part of the H... |
| hofcllem 18183 | Lemma for ~ hofcl . (Cont... |
| hofcl 18184 | Closure of the Hom functor... |
| oppchofcl 18185 | Closure of the opposite Ho... |
| yonval 18186 | Value of the Yoneda embedd... |
| yoncl 18187 | The Yoneda embedding is a ... |
| yon1cl 18188 | The Yoneda embedding at an... |
| yon11 18189 | Value of the Yoneda embedd... |
| yon12 18190 | Value of the Yoneda embedd... |
| yon2 18191 | Value of the Yoneda embedd... |
| hofpropd 18192 | If two categories have the... |
| yonpropd 18193 | If two categories have the... |
| oppcyon 18194 | Value of the opposite Yone... |
| oyoncl 18195 | The opposite Yoneda embedd... |
| oyon1cl 18196 | The opposite Yoneda embedd... |
| yonedalem1 18197 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18198 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18199 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18200 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18201 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18202 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18203 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18204 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18205 | Lemma for ~ yoneda . (Con... |
| yonedainv 18206 | The Yoneda Lemma with expl... |
| yonffthlem 18207 | Lemma for ~ yonffth . (Co... |
| yoneda 18208 | The Yoneda Lemma. There i... |
| yonffth 18209 | The Yoneda Lemma. The Yon... |
| yoniso 18210 | If the codomain is recover... |
| oduval 18213 | Value of an order dual str... |
| oduleval 18214 | Value of the less-equal re... |
| oduleg 18215 | Truth of the less-equal re... |
| odubas 18216 | Base set of an order dual ... |
| isprs 18221 | Property of being a preord... |
| prslem 18222 | Lemma for ~ prsref and ~ p... |
| prsref 18223 | "Less than or equal to" is... |
| prstr 18224 | "Less than or equal to" is... |
| oduprs 18225 | Being a proset is a self-d... |
| isdrs 18226 | Property of being a direct... |
| drsdir 18227 | Direction of a directed se... |
| drsprs 18228 | A directed set is a proset... |
| drsbn0 18229 | The base of a directed set... |
| drsdirfi 18230 | Any _finite_ number of ele... |
| isdrs2 18231 | Directed sets may be defin... |
| ispos 18239 | The predicate "is a poset"... |
| ispos2 18240 | A poset is an antisymmetri... |
| posprs 18241 | A poset is a proset. (Con... |
| posi 18242 | Lemma for poset properties... |
| posref 18243 | A poset ordering is reflex... |
| posasymb 18244 | A poset ordering is asymme... |
| postr 18245 | A poset ordering is transi... |
| 0pos 18246 | Technical lemma to simplif... |
| isposd 18247 | Properties that determine ... |
| isposi 18248 | Properties that determine ... |
| isposix 18249 | Properties that determine ... |
| pospropd 18250 | Posethood is determined on... |
| odupos 18251 | Being a poset is a self-du... |
| oduposb 18252 | Being a poset is a self-du... |
| pltfval 18254 | Value of the less-than rel... |
| pltval 18255 | Less-than relation. ( ~ d... |
| pltle 18256 | "Less than" implies "less ... |
| pltne 18257 | The "less than" relation i... |
| pltirr 18258 | The "less than" relation i... |
| pleval2i 18259 | One direction of ~ pleval2... |
| pleval2 18260 | "Less than or equal to" in... |
| pltnle 18261 | "Less than" implies not co... |
| pltval3 18262 | Alternate expression for t... |
| pltnlt 18263 | The less-than relation imp... |
| pltn2lp 18264 | The less-than relation has... |
| plttr 18265 | The less-than relation is ... |
| pltletr 18266 | Transitive law for chained... |
| plelttr 18267 | Transitive law for chained... |
| pospo 18268 | Write a poset structure in... |
| lubfval 18273 | Value of the least upper b... |
| lubdm 18274 | Domain of the least upper ... |
| lubfun 18275 | The LUB is a function. (C... |
| lubeldm 18276 | Member of the domain of th... |
| lubelss 18277 | A member of the domain of ... |
| lubeu 18278 | Unique existence proper of... |
| lubval 18279 | Value of the least upper b... |
| lubcl 18280 | The least upper bound func... |
| lubprop 18281 | Properties of greatest low... |
| luble 18282 | The greatest lower bound i... |
| lublecllem 18283 | Lemma for ~ lublecl and ~ ... |
| lublecl 18284 | The set of all elements le... |
| lubid 18285 | The LUB of elements less t... |
| glbfval 18286 | Value of the greatest lowe... |
| glbdm 18287 | Domain of the greatest low... |
| glbfun 18288 | The GLB is a function. (C... |
| glbeldm 18289 | Member of the domain of th... |
| glbelss 18290 | A member of the domain of ... |
| glbeu 18291 | Unique existence proper of... |
| glbval 18292 | Value of the greatest lowe... |
| glbcl 18293 | The least upper bound func... |
| glbprop 18294 | Properties of greatest low... |
| glble 18295 | The greatest lower bound i... |
| joinfval 18296 | Value of join function for... |
| joinfval2 18297 | Value of join function for... |
| joindm 18298 | Domain of join function fo... |
| joindef 18299 | Two ways to say that a joi... |
| joinval 18300 | Join value. Since both si... |
| joincl 18301 | Closure of join of element... |
| joindmss 18302 | Subset property of domain ... |
| joinval2lem 18303 | Lemma for ~ joinval2 and ~... |
| joinval2 18304 | Value of join for a poset ... |
| joineu 18305 | Uniqueness of join of elem... |
| joinlem 18306 | Lemma for join properties.... |
| lejoin1 18307 | A join's first argument is... |
| lejoin2 18308 | A join's second argument i... |
| joinle 18309 | A join is less than or equ... |
| meetfval 18310 | Value of meet function for... |
| meetfval2 18311 | Value of meet function for... |
| meetdm 18312 | Domain of meet function fo... |
| meetdef 18313 | Two ways to say that a mee... |
| meetval 18314 | Meet value. Since both si... |
| meetcl 18315 | Closure of meet of element... |
| meetdmss 18316 | Subset property of domain ... |
| meetval2lem 18317 | Lemma for ~ meetval2 and ~... |
| meetval2 18318 | Value of meet for a poset ... |
| meeteu 18319 | Uniqueness of meet of elem... |
| meetlem 18320 | Lemma for meet properties.... |
| lemeet1 18321 | A meet's first argument is... |
| lemeet2 18322 | A meet's second argument i... |
| meetle 18323 | A meet is less than or equ... |
| joincomALT 18324 | The join of a poset is com... |
| joincom 18325 | The join of a poset is com... |
| meetcomALT 18326 | The meet of a poset is com... |
| meetcom 18327 | The meet of a poset is com... |
| join0 18328 | Lemma for ~ odumeet . (Co... |
| meet0 18329 | Lemma for ~ odujoin . (Co... |
| odulub 18330 | Least upper bounds in a du... |
| odujoin 18331 | Joins in a dual order are ... |
| oduglb 18332 | Greatest lower bounds in a... |
| odumeet 18333 | Meets in a dual order are ... |
| poslubmo 18334 | Least upper bounds in a po... |
| posglbmo 18335 | Greatest lower bounds in a... |
| poslubd 18336 | Properties which determine... |
| poslubdg 18337 | Properties which determine... |
| posglbdg 18338 | Properties which determine... |
| istos 18341 | The predicate "is a toset"... |
| tosso 18342 | Write the totally ordered ... |
| tospos 18343 | A Toset is a Poset. (Cont... |
| tleile 18344 | In a Toset, any two elemen... |
| tltnle 18345 | In a Toset, "less than" is... |
| p0val 18350 | Value of poset zero. (Con... |
| p1val 18351 | Value of poset zero. (Con... |
| p0le 18352 | Any element is less than o... |
| ple1 18353 | Any element is less than o... |
| resspos 18354 | The restriction of a Poset... |
| resstos 18355 | The restriction of a Toset... |
| islat 18358 | The predicate "is a lattic... |
| odulatb 18359 | Being a lattice is self-du... |
| odulat 18360 | Being a lattice is self-du... |
| latcl2 18361 | The join and meet of any t... |
| latlem 18362 | Lemma for lattice properti... |
| latpos 18363 | A lattice is a poset. (Co... |
| latjcl 18364 | Closure of join operation ... |
| latmcl 18365 | Closure of meet operation ... |
| latref 18366 | A lattice ordering is refl... |
| latasymb 18367 | A lattice ordering is asym... |
| latasym 18368 | A lattice ordering is asym... |
| lattr 18369 | A lattice ordering is tran... |
| latasymd 18370 | Deduce equality from latti... |
| lattrd 18371 | A lattice ordering is tran... |
| latjcom 18372 | The join of a lattice comm... |
| latlej1 18373 | A join's first argument is... |
| latlej2 18374 | A join's second argument i... |
| latjle12 18375 | A join is less than or equ... |
| latleeqj1 18376 | "Less than or equal to" in... |
| latleeqj2 18377 | "Less than or equal to" in... |
| latjlej1 18378 | Add join to both sides of ... |
| latjlej2 18379 | Add join to both sides of ... |
| latjlej12 18380 | Add join to both sides of ... |
| latnlej 18381 | An idiom to express that a... |
| latnlej1l 18382 | An idiom to express that a... |
| latnlej1r 18383 | An idiom to express that a... |
| latnlej2 18384 | An idiom to express that a... |
| latnlej2l 18385 | An idiom to express that a... |
| latnlej2r 18386 | An idiom to express that a... |
| latjidm 18387 | Lattice join is idempotent... |
| latmcom 18388 | The join of a lattice comm... |
| latmle1 18389 | A meet is less than or equ... |
| latmle2 18390 | A meet is less than or equ... |
| latlem12 18391 | An element is less than or... |
| latleeqm1 18392 | "Less than or equal to" in... |
| latleeqm2 18393 | "Less than or equal to" in... |
| latmlem1 18394 | Add meet to both sides of ... |
| latmlem2 18395 | Add meet to both sides of ... |
| latmlem12 18396 | Add join to both sides of ... |
| latnlemlt 18397 | Negation of "less than or ... |
| latnle 18398 | Equivalent expressions for... |
| latmidm 18399 | Lattice meet is idempotent... |
| latabs1 18400 | Lattice absorption law. F... |
| latabs2 18401 | Lattice absorption law. F... |
| latledi 18402 | An ortholattice is distrib... |
| latmlej11 18403 | Ordering of a meet and joi... |
| latmlej12 18404 | Ordering of a meet and joi... |
| latmlej21 18405 | Ordering of a meet and joi... |
| latmlej22 18406 | Ordering of a meet and joi... |
| lubsn 18407 | The least upper bound of a... |
| latjass 18408 | Lattice join is associativ... |
| latj12 18409 | Swap 1st and 2nd members o... |
| latj32 18410 | Swap 2nd and 3rd members o... |
| latj13 18411 | Swap 1st and 3rd members o... |
| latj31 18412 | Swap 2nd and 3rd members o... |
| latjrot 18413 | Rotate lattice join of 3 c... |
| latj4 18414 | Rearrangement of lattice j... |
| latj4rot 18415 | Rotate lattice join of 4 c... |
| latjjdi 18416 | Lattice join distributes o... |
| latjjdir 18417 | Lattice join distributes o... |
| mod1ile 18418 | The weak direction of the ... |
| mod2ile 18419 | The weak direction of the ... |
| latmass 18420 | Lattice meet is associativ... |
| latdisdlem 18421 | Lemma for ~ latdisd . (Co... |
| latdisd 18422 | In a lattice, joins distri... |
| isclat 18425 | The predicate "is a comple... |
| clatpos 18426 | A complete lattice is a po... |
| clatlem 18427 | Lemma for properties of a ... |
| clatlubcl 18428 | Any subset of the base set... |
| clatlubcl2 18429 | Any subset of the base set... |
| clatglbcl 18430 | Any subset of the base set... |
| clatglbcl2 18431 | Any subset of the base set... |
| oduclatb 18432 | Being a complete lattice i... |
| clatl 18433 | A complete lattice is a la... |
| isglbd 18434 | Properties that determine ... |
| lublem 18435 | Lemma for the least upper ... |
| lubub 18436 | The LUB of a complete latt... |
| lubl 18437 | The LUB of a complete latt... |
| lubss 18438 | Subset law for least upper... |
| lubel 18439 | An element of a set is les... |
| lubun 18440 | The LUB of a union. (Cont... |
| clatglb 18441 | Properties of greatest low... |
| clatglble 18442 | The greatest lower bound i... |
| clatleglb 18443 | Two ways of expressing "le... |
| clatglbss 18444 | Subset law for greatest lo... |
| isdlat 18447 | Property of being a distri... |
| dlatmjdi 18448 | In a distributive lattice,... |
| dlatl 18449 | A distributive lattice is ... |
| odudlatb 18450 | The dual of a distributive... |
| dlatjmdi 18451 | In a distributive lattice,... |
| ipostr 18454 | The structure of ~ df-ipo ... |
| ipoval 18455 | Value of the inclusion pos... |
| ipobas 18456 | Base set of the inclusion ... |
| ipolerval 18457 | Relation of the inclusion ... |
| ipotset 18458 | Topology of the inclusion ... |
| ipole 18459 | Weak order condition of th... |
| ipolt 18460 | Strict order condition of ... |
| ipopos 18461 | The inclusion poset on a f... |
| isipodrs 18462 | Condition for a family of ... |
| ipodrscl 18463 | Direction by inclusion as ... |
| ipodrsfi 18464 | Finite upper bound propert... |
| fpwipodrs 18465 | The finite subsets of any ... |
| ipodrsima 18466 | The monotone image of a di... |
| isacs3lem 18467 | An algebraic closure syste... |
| acsdrsel 18468 | An algebraic closure syste... |
| isacs4lem 18469 | In a closure system in whi... |
| isacs5lem 18470 | If closure commutes with d... |
| acsdrscl 18471 | In an algebraic closure sy... |
| acsficl 18472 | A closure in an algebraic ... |
| isacs5 18473 | A closure system is algebr... |
| isacs4 18474 | A closure system is algebr... |
| isacs3 18475 | A closure system is algebr... |
| acsficld 18476 | In an algebraic closure sy... |
| acsficl2d 18477 | In an algebraic closure sy... |
| acsfiindd 18478 | In an algebraic closure sy... |
| acsmapd 18479 | In an algebraic closure sy... |
| acsmap2d 18480 | In an algebraic closure sy... |
| acsinfd 18481 | In an algebraic closure sy... |
| acsdomd 18482 | In an algebraic closure sy... |
| acsinfdimd 18483 | In an algebraic closure sy... |
| acsexdimd 18484 | In an algebraic closure sy... |
| mrelatglb 18485 | Greatest lower bounds in a... |
| mrelatglb0 18486 | The empty intersection in ... |
| mrelatlub 18487 | Least upper bounds in a Mo... |
| mreclatBAD 18488 | A Moore space is a complet... |
| isps 18493 | The predicate "is a poset"... |
| psrel 18494 | A poset is a relation. (C... |
| psref2 18495 | A poset is antisymmetric a... |
| pstr2 18496 | A poset is transitive. (C... |
| pslem 18497 | Lemma for ~ psref and othe... |
| psdmrn 18498 | The domain and range of a ... |
| psref 18499 | A poset is reflexive. (Co... |
| psrn 18500 | The range of a poset equal... |
| psasym 18501 | A poset is antisymmetric. ... |
| pstr 18502 | A poset is transitive. (C... |
| cnvps 18503 | The converse of a poset is... |
| cnvpsb 18504 | The converse of a poset is... |
| psss 18505 | Any subset of a partially ... |
| psssdm2 18506 | Field of a subposet. (Con... |
| psssdm 18507 | Field of a subposet. (Con... |
| istsr 18508 | The predicate is a toset. ... |
| istsr2 18509 | The predicate is a toset. ... |
| tsrlin 18510 | A toset is a linear order.... |
| tsrlemax 18511 | Two ways of saying a numbe... |
| tsrps 18512 | A toset is a poset. (Cont... |
| cnvtsr 18513 | The converse of a toset is... |
| tsrss 18514 | Any subset of a totally or... |
| ledm 18515 | The domain of ` <_ ` is ` ... |
| lern 18516 | The range of ` <_ ` is ` R... |
| lefld 18517 | The field of the 'less or ... |
| letsr 18518 | The "less than or equal to... |
| isdir 18523 | A condition for a relation... |
| reldir 18524 | A direction is a relation.... |
| dirdm 18525 | A direction's domain is eq... |
| dirref 18526 | A direction is reflexive. ... |
| dirtr 18527 | A direction is transitive.... |
| dirge 18528 | For any two elements of a ... |
| tsrdir 18529 | A totally ordered set is a... |
| ischn 18532 | Property of being a chain.... |
| chnwrd 18533 | A chain is an ordered sequ... |
| chnltm1 18534 | Basic property of a chain.... |
| pfxchn 18535 | A prefix of a chain is sti... |
| nfchnd 18536 | Bound-variable hypothesis ... |
| chneq1 18537 | Equality theorem for chain... |
| chneq2 18538 | Equality theorem for chain... |
| chneq12 18539 | Equality theorem for chain... |
| chnrss 18540 | Chains under a relation ar... |
| chndss 18541 | Chains with an alphabet ar... |
| chnrdss 18542 | Subset theorem for chains.... |
| chnexg 18543 | Chains with a set given fo... |
| nulchn 18544 | Empty set is an increasing... |
| s1chn 18545 | A singleton word is always... |
| chnind 18546 | Induction over a chain. S... |
| chnub 18547 | In a chain, the last eleme... |
| chnlt 18548 | Compare any two elements i... |
| chnso 18549 | A chain induces a total or... |
| chnccats1 18550 | Extend a chain with a sing... |
| chnccat 18551 | Concatenate two chains. (... |
| chnrev 18552 | Reverse of a chain is chai... |
| chnflenfi 18553 | There is a finite number o... |
| chnf 18554 | A chain is a zero-based fi... |
| chnpof1 18555 | A chain under relation whi... |
| chnpoadomd 18556 | A chain under relation whi... |
| chnpolleha 18557 | A chain under relation whi... |
| chnpolfz 18558 | Provided that chain's rela... |
| chnfi 18559 | There is a finite number o... |
| chninf 18560 | There is an infinite numbe... |
| chnfibg 18561 | Given a partial order, the... |
| ex-chn1 18562 | Example: a doubleton of tw... |
| ex-chn2 18563 | Example: sequence <" ZZ NN... |
| ismgm 18568 | The predicate "is a magma"... |
| ismgmn0 18569 | The predicate "is a magma"... |
| mgmcl 18570 | Closure of the operation o... |
| isnmgm 18571 | A condition for a structur... |
| mgmsscl 18572 | If the base set of a magma... |
| plusffval 18573 | The group addition operati... |
| plusfval 18574 | The group addition operati... |
| plusfeq 18575 | If the addition operation ... |
| plusffn 18576 | The group addition operati... |
| mgmplusf 18577 | The group addition functio... |
| mgmpropd 18578 | If two structures have the... |
| ismgmd 18579 | Deduce a magma from its pr... |
| issstrmgm 18580 | Characterize a substructur... |
| intopsn 18581 | The internal operation for... |
| mgmb1mgm1 18582 | The only magma with a base... |
| mgm0 18583 | Any set with an empty base... |
| mgm0b 18584 | The structure with an empt... |
| mgm1 18585 | The structure with one ele... |
| opifismgm 18586 | A structure with a group a... |
| mgmidmo 18587 | A two-sided identity eleme... |
| grpidval 18588 | The value of the identity ... |
| grpidpropd 18589 | If two structures have the... |
| fn0g 18590 | The group zero extractor i... |
| 0g0 18591 | The identity element funct... |
| ismgmid 18592 | The identity element of a ... |
| mgmidcl 18593 | The identity element of a ... |
| mgmlrid 18594 | The identity element of a ... |
| ismgmid2 18595 | Show that a given element ... |
| lidrideqd 18596 | If there is a left and rig... |
| lidrididd 18597 | If there is a left and rig... |
| grpidd 18598 | Deduce the identity elemen... |
| mgmidsssn0 18599 | Property of the set of ide... |
| grpinvalem 18600 | Lemma for ~ grpinva . (Co... |
| grpinva 18601 | Deduce right inverse from ... |
| grprida 18602 | Deduce right identity from... |
| gsumvalx 18603 | Expand out the substitutio... |
| gsumval 18604 | Expand out the substitutio... |
| gsumpropd 18605 | The group sum depends only... |
| gsumpropd2lem 18606 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18607 | A stronger version of ~ gs... |
| gsummgmpropd 18608 | A stronger version of ~ gs... |
| gsumress 18609 | The group sum in a substru... |
| gsumval1 18610 | Value of the group sum ope... |
| gsum0 18611 | Value of the empty group s... |
| gsumval2a 18612 | Value of the group sum ope... |
| gsumval2 18613 | Value of the group sum ope... |
| gsumsplit1r 18614 | Splitting off the rightmos... |
| gsumprval 18615 | Value of the group sum ope... |
| gsumpr12val 18616 | Value of the group sum ope... |
| mgmhmrcl 18621 | Reverse closure of a magma... |
| submgmrcl 18622 | Reverse closure for submag... |
| ismgmhm 18623 | Property of a magma homomo... |
| mgmhmf 18624 | A magma homomorphism is a ... |
| mgmhmpropd 18625 | Magma homomorphism depends... |
| mgmhmlin 18626 | A magma homomorphism prese... |
| mgmhmf1o 18627 | A magma homomorphism is bi... |
| idmgmhm 18628 | The identity homomorphism ... |
| issubmgm 18629 | Expand definition of a sub... |
| issubmgm2 18630 | Submagmas are subsets that... |
| rabsubmgmd 18631 | Deduction for proving that... |
| submgmss 18632 | Submagmas are subsets of t... |
| submgmid 18633 | Every magma is trivially a... |
| submgmcl 18634 | Submagmas are closed under... |
| submgmmgm 18635 | Submagmas are themselves m... |
| submgmbas 18636 | The base set of a submagma... |
| subsubmgm 18637 | A submagma of a submagma i... |
| resmgmhm 18638 | Restriction of a magma hom... |
| resmgmhm2 18639 | One direction of ~ resmgmh... |
| resmgmhm2b 18640 | Restriction of the codomai... |
| mgmhmco 18641 | The composition of magma h... |
| mgmhmima 18642 | The homomorphic image of a... |
| mgmhmeql 18643 | The equalizer of two magma... |
| submgmacs 18644 | Submagmas are an algebraic... |
| issgrp 18647 | The predicate "is a semigr... |
| issgrpv 18648 | The predicate "is a semigr... |
| issgrpn0 18649 | The predicate "is a semigr... |
| isnsgrp 18650 | A condition for a structur... |
| sgrpmgm 18651 | A semigroup is a magma. (... |
| sgrpass 18652 | A semigroup operation is a... |
| sgrpcl 18653 | Closure of the operation o... |
| sgrp0 18654 | Any set with an empty base... |
| sgrp0b 18655 | The structure with an empt... |
| sgrp1 18656 | The structure with one ele... |
| issgrpd 18657 | Deduce a semigroup from it... |
| sgrppropd 18658 | If two structures are sets... |
| prdsplusgsgrpcl 18659 | Structure product pointwis... |
| prdssgrpd 18660 | The product of a family of... |
| ismnddef 18663 | The predicate "is a monoid... |
| ismnd 18664 | The predicate "is a monoid... |
| isnmnd 18665 | A condition for a structur... |
| sgrpidmnd 18666 | A semigroup with an identi... |
| mndsgrp 18667 | A monoid is a semigroup. ... |
| mndmgm 18668 | A monoid is a magma. (Con... |
| mndcl 18669 | Closure of the operation o... |
| mndass 18670 | A monoid operation is asso... |
| mndid 18671 | A monoid has a two-sided i... |
| mndideu 18672 | The two-sided identity ele... |
| mnd32g 18673 | Commutative/associative la... |
| mnd12g 18674 | Commutative/associative la... |
| mnd4g 18675 | Commutative/associative la... |
| mndidcl 18676 | The identity element of a ... |
| mndbn0 18677 | The base set of a monoid i... |
| hashfinmndnn 18678 | A finite monoid has positi... |
| mndplusf 18679 | The group addition operati... |
| mndlrid 18680 | A monoid's identity elemen... |
| mndlid 18681 | The identity element of a ... |
| mndrid 18682 | The identity element of a ... |
| ismndd 18683 | Deduce a monoid from its p... |
| mndpfo 18684 | The addition operation of ... |
| mndfo 18685 | The addition operation of ... |
| mndpropd 18686 | If two structures have the... |
| mndprop 18687 | If two structures have the... |
| issubmnd 18688 | Characterize a submonoid b... |
| ress0g 18689 | ` 0g ` is unaffected by re... |
| submnd0 18690 | The zero of a submonoid is... |
| mndinvmod 18691 | Uniqueness of an inverse e... |
| mndpsuppss 18692 | The support of a mapping o... |
| mndpsuppfi 18693 | The support of a mapping o... |
| mndpfsupp 18694 | A mapping of a scalar mult... |
| prdsplusgcl 18695 | Structure product pointwis... |
| prdsidlem 18696 | Characterization of identi... |
| prdsmndd 18697 | The product of a family of... |
| prds0g 18698 | The identity in a product ... |
| pwsmnd 18699 | The structure power of a m... |
| pws0g 18700 | The identity in a structur... |
| imasmnd2 18701 | The image structure of a m... |
| imasmnd 18702 | The image structure of a m... |
| imasmndf1 18703 | The image of a monoid unde... |
| xpsmnd 18704 | The binary product of mono... |
| xpsmnd0 18705 | The identity element of a ... |
| mnd1 18706 | The (smallest) structure r... |
| mnd1id 18707 | The singleton element of a... |
| ismhm 18712 | Property of a monoid homom... |
| ismhmd 18713 | Deduction version of ~ ism... |
| mhmrcl1 18714 | Reverse closure of a monoi... |
| mhmrcl2 18715 | Reverse closure of a monoi... |
| mhmf 18716 | A monoid homomorphism is a... |
| ismhm0 18717 | Property of a monoid homom... |
| mhmismgmhm 18718 | Each monoid homomorphism i... |
| mhmpropd 18719 | Monoid homomorphism depend... |
| mhmlin 18720 | A monoid homomorphism comm... |
| mhm0 18721 | A monoid homomorphism pres... |
| idmhm 18722 | The identity homomorphism ... |
| mhmf1o 18723 | A monoid homomorphism is b... |
| mndvcl 18724 | Tuple-wise additive closur... |
| mndvass 18725 | Tuple-wise associativity i... |
| mndvlid 18726 | Tuple-wise left identity i... |
| mndvrid 18727 | Tuple-wise right identity ... |
| mhmvlin 18728 | Tuple extension of monoid ... |
| submrcl 18729 | Reverse closure for submon... |
| issubm 18730 | Expand definition of a sub... |
| issubm2 18731 | Submonoids are subsets tha... |
| issubmndb 18732 | The submonoid predicate. ... |
| issubmd 18733 | Deduction for proving a su... |
| mndissubm 18734 | If the base set of a monoi... |
| resmndismnd 18735 | If the base set of a monoi... |
| submss 18736 | Submonoids are subsets of ... |
| submid 18737 | Every monoid is trivially ... |
| subm0cl 18738 | Submonoids contain zero. ... |
| submcl 18739 | Submonoids are closed unde... |
| submmnd 18740 | Submonoids are themselves ... |
| submbas 18741 | The base set of a submonoi... |
| subm0 18742 | Submonoids have the same i... |
| subsubm 18743 | A submonoid of a submonoid... |
| 0subm 18744 | The zero submonoid of an a... |
| insubm 18745 | The intersection of two su... |
| 0mhm 18746 | The constant zero linear f... |
| resmhm 18747 | Restriction of a monoid ho... |
| resmhm2 18748 | One direction of ~ resmhm2... |
| resmhm2b 18749 | Restriction of the codomai... |
| mhmco 18750 | The composition of monoid ... |
| mhmimalem 18751 | Lemma for ~ mhmima and sim... |
| mhmima 18752 | The homomorphic image of a... |
| mhmeql 18753 | The equalizer of two monoi... |
| submacs 18754 | Submonoids are an algebrai... |
| mndind 18755 | Induction in a monoid. In... |
| prdspjmhm 18756 | A projection from a produc... |
| pwspjmhm 18757 | A projection from a struct... |
| pwsdiagmhm 18758 | Diagonal monoid homomorphi... |
| pwsco1mhm 18759 | Right composition with a f... |
| pwsco2mhm 18760 | Left composition with a mo... |
| gsumvallem2 18761 | Lemma for properties of th... |
| gsumsubm 18762 | Evaluate a group sum in a ... |
| gsumz 18763 | Value of a group sum over ... |
| gsumwsubmcl 18764 | Closure of the composite i... |
| gsumws1 18765 | A singleton composite reco... |
| gsumwcl 18766 | Closure of the composite o... |
| gsumsgrpccat 18767 | Homomorphic property of no... |
| gsumccat 18768 | Homomorphic property of co... |
| gsumws2 18769 | Valuation of a pair in a m... |
| gsumccatsn 18770 | Homomorphic property of co... |
| gsumspl 18771 | The primary purpose of the... |
| gsumwmhm 18772 | Behavior of homomorphisms ... |
| gsumwspan 18773 | The submonoid generated by... |
| frmdval 18778 | Value of the free monoid c... |
| frmdbas 18779 | The base set of a free mon... |
| frmdelbas 18780 | An element of the base set... |
| frmdplusg 18781 | The monoid operation of a ... |
| frmdadd 18782 | Value of the monoid operat... |
| vrmdfval 18783 | The canonical injection fr... |
| vrmdval 18784 | The value of the generatin... |
| vrmdf 18785 | The mapping from the index... |
| frmdmnd 18786 | A free monoid is a monoid.... |
| frmd0 18787 | The identity of the free m... |
| frmdsssubm 18788 | The set of words taking va... |
| frmdgsum 18789 | Any word in a free monoid ... |
| frmdss2 18790 | A subset of generators is ... |
| frmdup1 18791 | Any assignment of the gene... |
| frmdup2 18792 | The evaluation map has the... |
| frmdup3lem 18793 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18794 | Universal property of the ... |
| efmnd 18797 | The monoid of endofunction... |
| efmndbas 18798 | The base set of the monoid... |
| efmndbasabf 18799 | The base set of the monoid... |
| elefmndbas 18800 | Two ways of saying a funct... |
| elefmndbas2 18801 | Two ways of saying a funct... |
| efmndbasf 18802 | Elements in the monoid of ... |
| efmndhash 18803 | The monoid of endofunction... |
| efmndbasfi 18804 | The monoid of endofunction... |
| efmndfv 18805 | The function value of an e... |
| efmndtset 18806 | The topology of the monoid... |
| efmndplusg 18807 | The group operation of a m... |
| efmndov 18808 | The value of the group ope... |
| efmndcl 18809 | The group operation of the... |
| efmndtopn 18810 | The topology of the monoid... |
| symggrplem 18811 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18812 | The monoid of endofunction... |
| efmndsgrp 18813 | The monoid of endofunction... |
| ielefmnd 18814 | The identity function rest... |
| efmndid 18815 | The identity function rest... |
| efmndmnd 18816 | The monoid of endofunction... |
| efmnd0nmnd 18817 | Even the monoid of endofun... |
| efmndbas0 18818 | The base set of the monoid... |
| efmnd1hash 18819 | The monoid of endofunction... |
| efmnd1bas 18820 | The monoid of endofunction... |
| efmnd2hash 18821 | The monoid of endofunction... |
| submefmnd 18822 | If the base set of a monoi... |
| sursubmefmnd 18823 | The set of surjective endo... |
| injsubmefmnd 18824 | The set of injective endof... |
| idressubmefmnd 18825 | The singleton containing o... |
| idresefmnd 18826 | The structure with the sin... |
| smndex1ibas 18827 | The modulo function ` I ` ... |
| smndex1iidm 18828 | The modulo function ` I ` ... |
| smndex1gbas 18829 | The constant functions ` (... |
| smndex1gid 18830 | The composition of a const... |
| smndex1igid 18831 | The composition of the mod... |
| smndex1basss 18832 | The modulo function ` I ` ... |
| smndex1bas 18833 | The base set of the monoid... |
| smndex1mgm 18834 | The monoid of endofunction... |
| smndex1sgrp 18835 | The monoid of endofunction... |
| smndex1mndlem 18836 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18837 | The monoid of endofunction... |
| smndex1id 18838 | The modulo function ` I ` ... |
| smndex1n0mnd 18839 | The identity of the monoid... |
| nsmndex1 18840 | The base set ` B ` of the ... |
| smndex2dbas 18841 | The doubling function ` D ... |
| smndex2dnrinv 18842 | The doubling function ` D ... |
| smndex2hbas 18843 | The halving functions ` H ... |
| smndex2dlinvh 18844 | The halving functions ` H ... |
| mgm2nsgrplem1 18845 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18846 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18847 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18848 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18849 | A small magma (with two el... |
| sgrp2nmndlem1 18850 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18851 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18852 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18853 | A small semigroup (with tw... |
| sgrp2rid2ex 18854 | A small semigroup (with tw... |
| sgrp2nmndlem4 18855 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18856 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18857 | A small semigroup (with tw... |
| mgmnsgrpex 18858 | There is a magma which is ... |
| sgrpnmndex 18859 | There is a semigroup which... |
| sgrpssmgm 18860 | The class of all semigroup... |
| mndsssgrp 18861 | The class of all monoids i... |
| pwmndgplus 18862 | The operation of the monoi... |
| pwmndid 18863 | The identity of the monoid... |
| pwmnd 18864 | The power set of a class `... |
| isgrp 18871 | The predicate "is a group"... |
| grpmnd 18872 | A group is a monoid. (Con... |
| grpcl 18873 | Closure of the operation o... |
| grpass 18874 | A group operation is assoc... |
| grpinvex 18875 | Every member of a group ha... |
| grpideu 18876 | The two-sided identity ele... |
| grpassd 18877 | A group operation is assoc... |
| grpmndd 18878 | A group is a monoid. (Con... |
| grpcld 18879 | Closure of the operation o... |
| grpplusf 18880 | The group addition operati... |
| grpplusfo 18881 | The group addition operati... |
| resgrpplusfrn 18882 | The underlying set of a gr... |
| grppropd 18883 | If two structures have the... |
| grpprop 18884 | If two structures have the... |
| grppropstr 18885 | Generalize a specific 2-el... |
| grpss 18886 | Show that a structure exte... |
| isgrpd2e 18887 | Deduce a group from its pr... |
| isgrpd2 18888 | Deduce a group from its pr... |
| isgrpde 18889 | Deduce a group from its pr... |
| isgrpd 18890 | Deduce a group from its pr... |
| isgrpi 18891 | Properties that determine ... |
| grpsgrp 18892 | A group is a semigroup. (... |
| grpmgmd 18893 | A group is a magma, deduct... |
| dfgrp2 18894 | Alternate definition of a ... |
| dfgrp2e 18895 | Alternate definition of a ... |
| isgrpix 18896 | Properties that determine ... |
| grpidcl 18897 | The identity element of a ... |
| grpbn0 18898 | The base set of a group is... |
| grplid 18899 | The identity element of a ... |
| grprid 18900 | The identity element of a ... |
| grplidd 18901 | The identity element of a ... |
| grpridd 18902 | The identity element of a ... |
| grpn0 18903 | A group is not empty. (Co... |
| hashfingrpnn 18904 | A finite group has positiv... |
| grprcan 18905 | Right cancellation law for... |
| grpinveu 18906 | The left inverse element o... |
| grpid 18907 | Two ways of saying that an... |
| isgrpid2 18908 | Properties showing that an... |
| grpidd2 18909 | Deduce the identity elemen... |
| grpinvfval 18910 | The inverse function of a ... |
| grpinvfvalALT 18911 | Shorter proof of ~ grpinvf... |
| grpinvval 18912 | The inverse of a group ele... |
| grpinvfn 18913 | Functionality of the group... |
| grpinvfvi 18914 | The group inverse function... |
| grpsubfval 18915 | Group subtraction (divisio... |
| grpsubfvalALT 18916 | Shorter proof of ~ grpsubf... |
| grpsubval 18917 | Group subtraction (divisio... |
| grpinvf 18918 | The group inversion operat... |
| grpinvcl 18919 | A group element's inverse ... |
| grpinvcld 18920 | A group element's inverse ... |
| grplinv 18921 | The left inverse of a grou... |
| grprinv 18922 | The right inverse of a gro... |
| grpinvid1 18923 | The inverse of a group ele... |
| grpinvid2 18924 | The inverse of a group ele... |
| isgrpinv 18925 | Properties showing that a ... |
| grplinvd 18926 | The left inverse of a grou... |
| grprinvd 18927 | The right inverse of a gro... |
| grplrinv 18928 | In a group, every member h... |
| grpidinv2 18929 | A group's properties using... |
| grpidinv 18930 | A group has a left and rig... |
| grpinvid 18931 | The inverse of the identit... |
| grplcan 18932 | Left cancellation law for ... |
| grpasscan1 18933 | An associative cancellatio... |
| grpasscan2 18934 | An associative cancellatio... |
| grpidrcan 18935 | If right adding an element... |
| grpidlcan 18936 | If left adding an element ... |
| grpinvinv 18937 | Double inverse law for gro... |
| grpinvcnv 18938 | The group inverse is its o... |
| grpinv11 18939 | The group inverse is one-t... |
| grpinv11OLD 18940 | Obsolete version of ~ grpi... |
| grpinvf1o 18941 | The group inverse is a one... |
| grpinvnz 18942 | The inverse of a nonzero g... |
| grpinvnzcl 18943 | The inverse of a nonzero g... |
| grpsubinv 18944 | Subtraction of an inverse.... |
| grplmulf1o 18945 | Left multiplication by a g... |
| grpraddf1o 18946 | Right addition by a group ... |
| grpinvpropd 18947 | If two structures have the... |
| grpidssd 18948 | If the base set of a group... |
| grpinvssd 18949 | If the base set of a group... |
| grpinvadd 18950 | The inverse of the group o... |
| grpsubf 18951 | Functionality of group sub... |
| grpsubcl 18952 | Closure of group subtracti... |
| grpsubrcan 18953 | Right cancellation law for... |
| grpinvsub 18954 | Inverse of a group subtrac... |
| grpinvval2 18955 | A ~ df-neg -like equation ... |
| grpsubid 18956 | Subtraction of a group ele... |
| grpsubid1 18957 | Subtraction of the identit... |
| grpsubeq0 18958 | If the difference between ... |
| grpsubadd0sub 18959 | Subtraction expressed as a... |
| grpsubadd 18960 | Relationship between group... |
| grpsubsub 18961 | Double group subtraction. ... |
| grpaddsubass 18962 | Associative-type law for g... |
| grppncan 18963 | Cancellation law for subtr... |
| grpnpcan 18964 | Cancellation law for subtr... |
| grpsubsub4 18965 | Double group subtraction (... |
| grppnpcan2 18966 | Cancellation law for mixed... |
| grpnpncan 18967 | Cancellation law for group... |
| grpnpncan0 18968 | Cancellation law for group... |
| grpnnncan2 18969 | Cancellation law for group... |
| dfgrp3lem 18970 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18971 | Alternate definition of a ... |
| dfgrp3e 18972 | Alternate definition of a ... |
| grplactfval 18973 | The left group action of e... |
| grplactval 18974 | The value of the left grou... |
| grplactcnv 18975 | The left group action of e... |
| grplactf1o 18976 | The left group action of e... |
| grpsubpropd 18977 | Weak property deduction fo... |
| grpsubpropd2 18978 | Strong property deduction ... |
| grp1 18979 | The (smallest) structure r... |
| grp1inv 18980 | The inverse function of th... |
| prdsinvlem 18981 | Characterization of invers... |
| prdsgrpd 18982 | The product of a family of... |
| prdsinvgd 18983 | Negation in a product of g... |
| pwsgrp 18984 | A structure power of a gro... |
| pwsinvg 18985 | Negation in a group power.... |
| pwssub 18986 | Subtraction in a group pow... |
| imasgrp2 18987 | The image structure of a g... |
| imasgrp 18988 | The image structure of a g... |
| imasgrpf1 18989 | The image of a group under... |
| qusgrp2 18990 | Prove that a quotient stru... |
| xpsgrp 18991 | The binary product of grou... |
| xpsinv 18992 | Value of the negation oper... |
| xpsgrpsub 18993 | Value of the subtraction o... |
| mhmlem 18994 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18995 | A surjective monoid morphi... |
| mhmmnd 18996 | The image of a monoid ` G ... |
| mhmfmhm 18997 | The function fulfilling th... |
| ghmgrp 18998 | The image of a group ` G `... |
| mulgfval 19001 | Group multiple (exponentia... |
| mulgfvalALT 19002 | Shorter proof of ~ mulgfva... |
| mulgval 19003 | Value of the group multipl... |
| mulgfn 19004 | Functionality of the group... |
| mulgfvi 19005 | The group multiple operati... |
| mulg0 19006 | Group multiple (exponentia... |
| mulgnn 19007 | Group multiple (exponentia... |
| ressmulgnn 19008 | Values for the group multi... |
| ressmulgnn0 19009 | Values for the group multi... |
| ressmulgnnd 19010 | Values for the group multi... |
| mulgnngsum 19011 | Group multiple (exponentia... |
| mulgnn0gsum 19012 | Group multiple (exponentia... |
| mulg1 19013 | Group multiple (exponentia... |
| mulgnnp1 19014 | Group multiple (exponentia... |
| mulg2 19015 | Group multiple (exponentia... |
| mulgnegnn 19016 | Group multiple (exponentia... |
| mulgnn0p1 19017 | Group multiple (exponentia... |
| mulgnnsubcl 19018 | Closure of the group multi... |
| mulgnn0subcl 19019 | Closure of the group multi... |
| mulgsubcl 19020 | Closure of the group multi... |
| mulgnncl 19021 | Closure of the group multi... |
| mulgnn0cl 19022 | Closure of the group multi... |
| mulgcl 19023 | Closure of the group multi... |
| mulgneg 19024 | Group multiple (exponentia... |
| mulgnegneg 19025 | The inverse of a negative ... |
| mulgm1 19026 | Group multiple (exponentia... |
| mulgnn0cld 19027 | Closure of the group multi... |
| mulgcld 19028 | Deduction associated with ... |
| mulgaddcomlem 19029 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19030 | The group multiple operato... |
| mulginvcom 19031 | The group multiple operato... |
| mulginvinv 19032 | The group multiple operato... |
| mulgnn0z 19033 | A group multiple of the id... |
| mulgz 19034 | A group multiple of the id... |
| mulgnndir 19035 | Sum of group multiples, fo... |
| mulgnn0dir 19036 | Sum of group multiples, ge... |
| mulgdirlem 19037 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19038 | Sum of group multiples, ge... |
| mulgp1 19039 | Group multiple (exponentia... |
| mulgneg2 19040 | Group multiple (exponentia... |
| mulgnnass 19041 | Product of group multiples... |
| mulgnn0ass 19042 | Product of group multiples... |
| mulgass 19043 | Product of group multiples... |
| mulgassr 19044 | Reversed product of group ... |
| mulgmodid 19045 | Casting out multiples of t... |
| mulgsubdir 19046 | Distribution of group mult... |
| mhmmulg 19047 | A homomorphism of monoids ... |
| mulgpropd 19048 | Two structures with the sa... |
| submmulgcl 19049 | Closure of the group multi... |
| submmulg 19050 | A group multiple is the sa... |
| pwsmulg 19051 | Value of a group multiple ... |
| issubg 19058 | The subgroup predicate. (... |
| subgss 19059 | A subgroup is a subset. (... |
| subgid 19060 | A group is a subgroup of i... |
| subggrp 19061 | A subgroup is a group. (C... |
| subgbas 19062 | The base of the restricted... |
| subgrcl 19063 | Reverse closure for the su... |
| subg0 19064 | A subgroup of a group must... |
| subginv 19065 | The inverse of an element ... |
| subg0cl 19066 | The group identity is an e... |
| subginvcl 19067 | The inverse of an element ... |
| subgcl 19068 | A subgroup is closed under... |
| subgsubcl 19069 | A subgroup is closed under... |
| subgsub 19070 | The subtraction of element... |
| subgmulgcl 19071 | Closure of the group multi... |
| subgmulg 19072 | A group multiple is the sa... |
| issubg2 19073 | Characterize the subgroups... |
| issubgrpd2 19074 | Prove a subgroup by closur... |
| issubgrpd 19075 | Prove a subgroup by closur... |
| issubg3 19076 | A subgroup is a symmetric ... |
| issubg4 19077 | A subgroup is a nonempty s... |
| grpissubg 19078 | If the base set of a group... |
| resgrpisgrp 19079 | If the base set of a group... |
| subgsubm 19080 | A subgroup is a submonoid.... |
| subsubg 19081 | A subgroup of a subgroup i... |
| subgint 19082 | The intersection of a none... |
| 0subg 19083 | The zero subgroup of an ar... |
| trivsubgd 19084 | The only subgroup of a tri... |
| trivsubgsnd 19085 | The only subgroup of a tri... |
| isnsg 19086 | Property of being a normal... |
| isnsg2 19087 | Weaken the condition of ~ ... |
| nsgbi 19088 | Defining property of a nor... |
| nsgsubg 19089 | A normal subgroup is a sub... |
| nsgconj 19090 | The conjugation of an elem... |
| isnsg3 19091 | A subgroup is normal iff t... |
| subgacs 19092 | Subgroups are an algebraic... |
| nsgacs 19093 | Normal subgroups form an a... |
| elnmz 19094 | Elementhood in the normali... |
| nmzbi 19095 | Defining property of the n... |
| nmzsubg 19096 | The normalizer N_G(S) of a... |
| ssnmz 19097 | A subgroup is a subset of ... |
| isnsg4 19098 | A subgroup is normal iff i... |
| nmznsg 19099 | Any subgroup is a normal s... |
| 0nsg 19100 | The zero subgroup is norma... |
| nsgid 19101 | The whole group is a norma... |
| 0idnsgd 19102 | The whole group and the ze... |
| trivnsgd 19103 | The only normal subgroup o... |
| triv1nsgd 19104 | A trivial group has exactl... |
| 1nsgtrivd 19105 | A group with exactly one n... |
| releqg 19106 | The left coset equivalence... |
| eqgfval 19107 | Value of the subgroup left... |
| eqgval 19108 | Value of the subgroup left... |
| eqger 19109 | The subgroup coset equival... |
| eqglact 19110 | A left coset can be expres... |
| eqgid 19111 | The left coset containing ... |
| eqgen 19112 | Each coset is equipotent t... |
| eqgcpbl 19113 | The subgroup coset equival... |
| eqg0el 19114 | Equivalence class of a quo... |
| quselbas 19115 | Membership in the base set... |
| quseccl0 19116 | Closure of the quotient ma... |
| qusgrp 19117 | If ` Y ` is a normal subgr... |
| quseccl 19118 | Closure of the quotient ma... |
| qusadd 19119 | Value of the group operati... |
| qus0 19120 | Value of the group identit... |
| qusinv 19121 | Value of the group inverse... |
| qussub 19122 | Value of the group subtrac... |
| ecqusaddd 19123 | Addition of equivalence cl... |
| ecqusaddcl 19124 | Closure of the addition in... |
| lagsubg2 19125 | Lagrange's theorem for fin... |
| lagsubg 19126 | Lagrange's theorem for Gro... |
| eqg0subg 19127 | The coset equivalence rela... |
| eqg0subgecsn 19128 | The equivalence classes mo... |
| qus0subgbas 19129 | The base set of a quotient... |
| qus0subgadd 19130 | The addition in a quotient... |
| cycsubmel 19131 | Characterization of an ele... |
| cycsubmcl 19132 | The set of nonnegative int... |
| cycsubm 19133 | The set of nonnegative int... |
| cyccom 19134 | Condition for an operation... |
| cycsubmcom 19135 | The operation of a monoid ... |
| cycsubggend 19136 | The cyclic subgroup genera... |
| cycsubgcl 19137 | The set of integer powers ... |
| cycsubgss 19138 | The cyclic subgroup genera... |
| cycsubg 19139 | The cyclic group generated... |
| cycsubgcld 19140 | The cyclic subgroup genera... |
| cycsubg2 19141 | The subgroup generated by ... |
| cycsubg2cl 19142 | Any multiple of an element... |
| reldmghm 19145 | Lemma for group homomorphi... |
| isghm 19146 | Property of being a homomo... |
| isghmOLD 19147 | Obsolete version of ~ isgh... |
| isghm3 19148 | Property of a group homomo... |
| ghmgrp1 19149 | A group homomorphism is on... |
| ghmgrp2 19150 | A group homomorphism is on... |
| ghmf 19151 | A group homomorphism is a ... |
| ghmlin 19152 | A homomorphism of groups i... |
| ghmid 19153 | A homomorphism of groups p... |
| ghminv 19154 | A homomorphism of groups p... |
| ghmsub 19155 | Linearity of subtraction t... |
| isghmd 19156 | Deduction for a group homo... |
| ghmmhm 19157 | A group homomorphism is a ... |
| ghmmhmb 19158 | Group homomorphisms and mo... |
| ghmmulg 19159 | A group homomorphism prese... |
| ghmrn 19160 | The range of a homomorphis... |
| 0ghm 19161 | The constant zero linear f... |
| idghm 19162 | The identity homomorphism ... |
| resghm 19163 | Restriction of a homomorph... |
| resghm2 19164 | One direction of ~ resghm2... |
| resghm2b 19165 | Restriction of the codomai... |
| ghmghmrn 19166 | A group homomorphism from ... |
| ghmco 19167 | The composition of group h... |
| ghmima 19168 | The image of a subgroup un... |
| ghmpreima 19169 | The inverse image of a sub... |
| ghmeql 19170 | The equalizer of two group... |
| ghmnsgima 19171 | The image of a normal subg... |
| ghmnsgpreima 19172 | The inverse image of a nor... |
| ghmker 19173 | The kernel of a homomorphi... |
| ghmeqker 19174 | Two source points map to t... |
| pwsdiagghm 19175 | Diagonal homomorphism into... |
| f1ghm0to0 19176 | If a group homomorphism ` ... |
| ghmf1 19177 | Two ways of saying a group... |
| kerf1ghm 19178 | A group homomorphism ` F `... |
| ghmf1o 19179 | A bijective group homomorp... |
| conjghm 19180 | Conjugation is an automorp... |
| conjsubg 19181 | A conjugated subgroup is a... |
| conjsubgen 19182 | A conjugated subgroup is e... |
| conjnmz 19183 | A subgroup is unchanged un... |
| conjnmzb 19184 | Alternative condition for ... |
| conjnsg 19185 | A normal subgroup is uncha... |
| qusghm 19186 | If ` Y ` is a normal subgr... |
| ghmpropd 19187 | Group homomorphism depends... |
| gimfn 19192 | The group isomorphism func... |
| isgim 19193 | An isomorphism of groups i... |
| gimf1o 19194 | An isomorphism of groups i... |
| gimghm 19195 | An isomorphism of groups i... |
| isgim2 19196 | A group isomorphism is a h... |
| subggim 19197 | Behavior of subgroups unde... |
| gimcnv 19198 | The converse of a group is... |
| gimco 19199 | The composition of group i... |
| gim0to0 19200 | A group isomorphism maps t... |
| brgic 19201 | The relation "is isomorphi... |
| brgici 19202 | Prove isomorphic by an exp... |
| gicref 19203 | Isomorphism is reflexive. ... |
| giclcl 19204 | Isomorphism implies the le... |
| gicrcl 19205 | Isomorphism implies the ri... |
| gicsym 19206 | Isomorphism is symmetric. ... |
| gictr 19207 | Isomorphism is transitive.... |
| gicer 19208 | Isomorphism is an equivale... |
| gicen 19209 | Isomorphic groups have equ... |
| gicsubgen 19210 | A less trivial example of ... |
| ghmqusnsglem1 19211 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19212 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19213 | The mapping ` H ` induced ... |
| ghmquskerlem1 19214 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19215 | In the case of theorem ~ g... |
| ghmquskerlem2 19216 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19217 | The mapping ` H ` induced ... |
| ghmqusker 19218 | A surjective group homomor... |
| gicqusker 19219 | The image ` H ` of a group... |
| isga 19222 | The predicate "is a (left)... |
| gagrp 19223 | The left argument of a gro... |
| gaset 19224 | The right argument of a gr... |
| gagrpid 19225 | The identity of the group ... |
| gaf 19226 | The mapping of the group a... |
| gafo 19227 | A group action is onto its... |
| gaass 19228 | An "associative" property ... |
| ga0 19229 | The action of a group on t... |
| gaid 19230 | The trivial action of a gr... |
| subgga 19231 | A subgroup acts on its par... |
| gass 19232 | A subset of a group action... |
| gasubg 19233 | The restriction of a group... |
| gaid2 19234 | A group operation is a lef... |
| galcan 19235 | The action of a particular... |
| gacan 19236 | Group inverses cancel in a... |
| gapm 19237 | The action of a particular... |
| gaorb 19238 | The orbit equivalence rela... |
| gaorber 19239 | The orbit equivalence rela... |
| gastacl 19240 | The stabilizer subgroup in... |
| gastacos 19241 | Write the coset relation f... |
| orbstafun 19242 | Existence and uniqueness f... |
| orbstaval 19243 | Value of the function at a... |
| orbsta 19244 | The Orbit-Stabilizer theor... |
| orbsta2 19245 | Relation between the size ... |
| cntrval 19250 | Substitute definition of t... |
| cntzfval 19251 | First level substitution f... |
| cntzval 19252 | Definition substitution fo... |
| elcntz 19253 | Elementhood in the central... |
| cntzel 19254 | Membership in a centralize... |
| cntzsnval 19255 | Special substitution for t... |
| elcntzsn 19256 | Value of the centralizer o... |
| sscntz 19257 | A centralizer expression f... |
| cntzrcl 19258 | Reverse closure for elemen... |
| cntzssv 19259 | The centralizer is uncondi... |
| cntzi 19260 | Membership in a centralize... |
| elcntr 19261 | Elementhood in the center ... |
| cntrss 19262 | The center is a subset of ... |
| cntri 19263 | Defining property of the c... |
| resscntz 19264 | Centralizer in a substruct... |
| cntzsgrpcl 19265 | Centralizers are closed un... |
| cntz2ss 19266 | Centralizers reverse the s... |
| cntzrec 19267 | Reciprocity relationship f... |
| cntziinsn 19268 | Express any centralizer as... |
| cntzsubm 19269 | Centralizers in a monoid a... |
| cntzsubg 19270 | Centralizers in a group ar... |
| cntzidss 19271 | If the elements of ` S ` c... |
| cntzmhm 19272 | Centralizers in a monoid a... |
| cntzmhm2 19273 | Centralizers in a monoid a... |
| cntrsubgnsg 19274 | A central subgroup is norm... |
| cntrnsg 19275 | The center of a group is a... |
| oppgval 19278 | Value of the opposite grou... |
| oppgplusfval 19279 | Value of the addition oper... |
| oppgplus 19280 | Value of the addition oper... |
| setsplusg 19281 | The other components of an... |
| oppgbas 19282 | Base set of an opposite gr... |
| oppgtset 19283 | Topology of an opposite gr... |
| oppgtopn 19284 | Topology of an opposite gr... |
| oppgmnd 19285 | The opposite of a monoid i... |
| oppgmndb 19286 | Bidirectional form of ~ op... |
| oppgid 19287 | Zero in a monoid is a symm... |
| oppggrp 19288 | The opposite of a group is... |
| oppggrpb 19289 | Bidirectional form of ~ op... |
| oppginv 19290 | Inverses in a group are a ... |
| invoppggim 19291 | The inverse is an antiauto... |
| oppggic 19292 | Every group is (naturally)... |
| oppgsubm 19293 | Being a submonoid is a sym... |
| oppgsubg 19294 | Being a subgroup is a symm... |
| oppgcntz 19295 | A centralizer in a group i... |
| oppgcntr 19296 | The center of a group is t... |
| gsumwrev 19297 | A sum in an opposite monoi... |
| oppgle 19298 | less-than relation of an o... |
| oppglt 19299 | less-than relation of an o... |
| symgval 19302 | The value of the symmetric... |
| symgbas 19303 | The base set of the symmet... |
| elsymgbas2 19304 | Two ways of saying a funct... |
| elsymgbas 19305 | Two ways of saying a funct... |
| symgbasf1o 19306 | Elements in the symmetric ... |
| symgbasf 19307 | A permutation (element of ... |
| symgbasmap 19308 | A permutation (element of ... |
| symghash 19309 | The symmetric group on ` n... |
| symgbasfi 19310 | The symmetric group on a f... |
| symgfv 19311 | The function value of a pe... |
| symgfvne 19312 | The function values of a p... |
| symgressbas 19313 | The symmetric group on ` A... |
| symgplusg 19314 | The group operation of a s... |
| symgov 19315 | The value of the group ope... |
| symgcl 19316 | The group operation of the... |
| idresperm 19317 | The identity function rest... |
| symgmov1 19318 | For a permutation of a set... |
| symgmov2 19319 | For a permutation of a set... |
| symgbas0 19320 | The base set of the symmet... |
| symg1hash 19321 | The symmetric group on a s... |
| symg1bas 19322 | The symmetric group on a s... |
| symg2hash 19323 | The symmetric group on a (... |
| symg2bas 19324 | The symmetric group on a p... |
| 0symgefmndeq 19325 | The symmetric group on the... |
| snsymgefmndeq 19326 | The symmetric group on a s... |
| symgpssefmnd 19327 | For a set ` A ` with more ... |
| symgvalstruct 19328 | The value of the symmetric... |
| symgsubmefmnd 19329 | The symmetric group on a s... |
| symgtset 19330 | The topology of the symmet... |
| symggrp 19331 | The symmetric group on a s... |
| symgid 19332 | The group identity element... |
| symginv 19333 | The group inverse in the s... |
| symgsubmefmndALT 19334 | The symmetric group on a s... |
| galactghm 19335 | The currying of a group ac... |
| lactghmga 19336 | The converse of ~ galactgh... |
| symgtopn 19337 | The topology of the symmet... |
| symgga 19338 | The symmetric group induce... |
| pgrpsubgsymgbi 19339 | Every permutation group is... |
| pgrpsubgsymg 19340 | Every permutation group is... |
| idressubgsymg 19341 | The singleton containing o... |
| idrespermg 19342 | The structure with the sin... |
| cayleylem1 19343 | Lemma for ~ cayley . (Con... |
| cayleylem2 19344 | Lemma for ~ cayley . (Con... |
| cayley 19345 | Cayley's Theorem (construc... |
| cayleyth 19346 | Cayley's Theorem (existenc... |
| symgfix2 19347 | If a permutation does not ... |
| symgextf 19348 | The extension of a permuta... |
| symgextfv 19349 | The function value of the ... |
| symgextfve 19350 | The function value of the ... |
| symgextf1lem 19351 | Lemma for ~ symgextf1 . (... |
| symgextf1 19352 | The extension of a permuta... |
| symgextfo 19353 | The extension of a permuta... |
| symgextf1o 19354 | The extension of a permuta... |
| symgextsymg 19355 | The extension of a permuta... |
| symgextres 19356 | The restriction of the ext... |
| gsumccatsymgsn 19357 | Homomorphic property of co... |
| gsmsymgrfixlem1 19358 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19359 | The composition of permuta... |
| fvcosymgeq 19360 | The values of two composit... |
| gsmsymgreqlem1 19361 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19362 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19363 | Two combination of permuta... |
| symgfixelq 19364 | A permutation of a set fix... |
| symgfixels 19365 | The restriction of a permu... |
| symgfixelsi 19366 | The restriction of a permu... |
| symgfixf 19367 | The mapping of a permutati... |
| symgfixf1 19368 | The mapping of a permutati... |
| symgfixfolem1 19369 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19370 | The mapping of a permutati... |
| symgfixf1o 19371 | The mapping of a permutati... |
| f1omvdmvd 19374 | A permutation of any class... |
| f1omvdcnv 19375 | A permutation and its inve... |
| mvdco 19376 | Composing two permutations... |
| f1omvdconj 19377 | Conjugation of a permutati... |
| f1otrspeq 19378 | A transposition is charact... |
| f1omvdco2 19379 | If exactly one of two perm... |
| f1omvdco3 19380 | If a point is moved by exa... |
| pmtrfval 19381 | The function generating tr... |
| pmtrval 19382 | A generated transposition,... |
| pmtrfv 19383 | General value of mapping a... |
| pmtrprfv 19384 | In a transposition of two ... |
| pmtrprfv3 19385 | In a transposition of two ... |
| pmtrf 19386 | Functionality of a transpo... |
| pmtrmvd 19387 | A transposition moves prec... |
| pmtrrn 19388 | Transposing two points giv... |
| pmtrfrn 19389 | A transposition (as a kind... |
| pmtrffv 19390 | Mapping of a point under a... |
| pmtrrn2 19391 | For any transposition ther... |
| pmtrfinv 19392 | A transposition function i... |
| pmtrfmvdn0 19393 | A transposition moves at l... |
| pmtrff1o 19394 | A transposition function i... |
| pmtrfcnv 19395 | A transposition function i... |
| pmtrfb 19396 | An intrinsic characterizat... |
| pmtrfconj 19397 | Any conjugate of a transpo... |
| symgsssg 19398 | The symmetric group has su... |
| symgfisg 19399 | The symmetric group has a ... |
| symgtrf 19400 | Transpositions are element... |
| symggen 19401 | The span of the transposit... |
| symggen2 19402 | A finite permutation group... |
| symgtrinv 19403 | To invert a permutation re... |
| pmtr3ncomlem1 19404 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19405 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19406 | Transpositions over sets w... |
| pmtrdifellem1 19407 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19408 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19409 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19410 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19411 | A transposition of element... |
| pmtrdifwrdellem1 19412 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19413 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19414 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19415 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19416 | A sequence of transpositio... |
| pmtrdifwrdel2 19417 | A sequence of transpositio... |
| pmtrprfval 19418 | The transpositions on a pa... |
| pmtrprfvalrn 19419 | The range of the transposi... |
| psgnunilem1 19424 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19425 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19426 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19427 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19428 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19429 | Addition and subtraction o... |
| psgnuni 19430 | If the same permutation ca... |
| psgnfval 19431 | Function definition of the... |
| psgnfn 19432 | Functionality and domain o... |
| psgndmsubg 19433 | The finitary permutations ... |
| psgneldm 19434 | Property of being a finita... |
| psgneldm2 19435 | The finitary permutations ... |
| psgneldm2i 19436 | A sequence of transpositio... |
| psgneu 19437 | A finitary permutation has... |
| psgnval 19438 | Value of the permutation s... |
| psgnvali 19439 | A finitary permutation has... |
| psgnvalii 19440 | Any representation of a pe... |
| psgnpmtr 19441 | All transpositions are odd... |
| psgn0fv0 19442 | The permutation sign funct... |
| sygbasnfpfi 19443 | The class of non-fixed poi... |
| psgnfvalfi 19444 | Function definition of the... |
| psgnvalfi 19445 | Value of the permutation s... |
| psgnran 19446 | The range of the permutati... |
| gsmtrcl 19447 | The group sum of transposi... |
| psgnfitr 19448 | A permutation of a finite ... |
| psgnfieu 19449 | A permutation of a finite ... |
| pmtrsn 19450 | The value of the transposi... |
| psgnsn 19451 | The permutation sign funct... |
| psgnprfval 19452 | The permutation sign funct... |
| psgnprfval1 19453 | The permutation sign of th... |
| psgnprfval2 19454 | The permutation sign of th... |
| odfval 19463 | Value of the order functio... |
| odfvalALT 19464 | Shorter proof of ~ odfval ... |
| odval 19465 | Second substitution for th... |
| odlem1 19466 | The group element order is... |
| odcl 19467 | The order of a group eleme... |
| odf 19468 | Functionality of the group... |
| odid 19469 | Any element to the power o... |
| odlem2 19470 | Any positive annihilator o... |
| odmodnn0 19471 | Reduce the argument of a g... |
| mndodconglem 19472 | Lemma for ~ mndodcong . (... |
| mndodcong 19473 | If two multipliers are con... |
| mndodcongi 19474 | If two multipliers are con... |
| oddvdsnn0 19475 | The only multiples of ` A ... |
| odnncl 19476 | If a nonzero multiple of a... |
| odmod 19477 | Reduce the argument of a g... |
| oddvds 19478 | The only multiples of ` A ... |
| oddvdsi 19479 | Any group element is annih... |
| odcong 19480 | If two multipliers are con... |
| odeq 19481 | The ~ oddvds property uniq... |
| odval2 19482 | A non-conditional definiti... |
| odcld 19483 | The order of a group eleme... |
| odm1inv 19484 | The (order-1)th multiple o... |
| odmulgid 19485 | A relationship between the... |
| odmulg2 19486 | The order of a multiple di... |
| odmulg 19487 | Relationship between the o... |
| odmulgeq 19488 | A multiple of a point of f... |
| odbezout 19489 | If ` N ` is coprime to the... |
| od1 19490 | The order of the group ide... |
| odeq1 19491 | The group identity is the ... |
| odinv 19492 | The order of the inverse o... |
| odf1 19493 | The multiples of an elemen... |
| odinf 19494 | The multiples of an elemen... |
| dfod2 19495 | An alternative definition ... |
| odcl2 19496 | The order of an element of... |
| oddvds2 19497 | The order of an element of... |
| finodsubmsubg 19498 | A submonoid whose elements... |
| 0subgALT 19499 | A shorter proof of ~ 0subg... |
| submod 19500 | The order of an element is... |
| subgod 19501 | The order of an element is... |
| odsubdvds 19502 | The order of an element of... |
| odf1o1 19503 | An element with zero order... |
| odf1o2 19504 | An element with nonzero or... |
| odhash 19505 | An element of zero order g... |
| odhash2 19506 | If an element has nonzero ... |
| odhash3 19507 | An element which generates... |
| odngen 19508 | A cyclic subgroup of size ... |
| gexval 19509 | Value of the exponent of a... |
| gexlem1 19510 | The group element order is... |
| gexcl 19511 | The exponent of a group is... |
| gexid 19512 | Any element to the power o... |
| gexlem2 19513 | Any positive annihilator o... |
| gexdvdsi 19514 | Any group element is annih... |
| gexdvds 19515 | The only ` N ` that annihi... |
| gexdvds2 19516 | An integer divides the gro... |
| gexod 19517 | Any group element is annih... |
| gexcl3 19518 | If the order of every grou... |
| gexnnod 19519 | Every group element has fi... |
| gexcl2 19520 | The exponent of a finite g... |
| gexdvds3 19521 | The exponent of a finite g... |
| gex1 19522 | A group or monoid has expo... |
| ispgp 19523 | A group is a ` P ` -group ... |
| pgpprm 19524 | Reverse closure for the fi... |
| pgpgrp 19525 | Reverse closure for the se... |
| pgpfi1 19526 | A finite group with order ... |
| pgp0 19527 | The identity subgroup is a... |
| subgpgp 19528 | A subgroup of a p-group is... |
| sylow1lem1 19529 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19530 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19531 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19532 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19533 | Lemma for ~ sylow1 . Usin... |
| sylow1 19534 | Sylow's first theorem. If... |
| odcau 19535 | Cauchy's theorem for the o... |
| pgpfi 19536 | The converse to ~ pgpfi1 .... |
| pgpfi2 19537 | Alternate version of ~ pgp... |
| pgphash 19538 | The order of a p-group. (... |
| isslw 19539 | The property of being a Sy... |
| slwprm 19540 | Reverse closure for the fi... |
| slwsubg 19541 | A Sylow ` P ` -subgroup is... |
| slwispgp 19542 | Defining property of a Syl... |
| slwpss 19543 | A proper superset of a Syl... |
| slwpgp 19544 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19545 | Every ` P ` -subgroup is c... |
| slwn0 19546 | Every finite group contain... |
| subgslw 19547 | A Sylow subgroup that is c... |
| sylow2alem1 19548 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19549 | Lemma for ~ sylow2a . All... |
| sylow2a 19550 | A named lemma of Sylow's s... |
| sylow2blem1 19551 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19552 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19553 | Sylow's second theorem. P... |
| sylow2b 19554 | Sylow's second theorem. A... |
| slwhash 19555 | A sylow subgroup has cardi... |
| fislw 19556 | The sylow subgroups of a f... |
| sylow2 19557 | Sylow's second theorem. S... |
| sylow3lem1 19558 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19559 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19560 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19561 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19562 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19563 | Lemma for ~ sylow3 , secon... |
| sylow3 19564 | Sylow's third theorem. Th... |
| lsmfval 19569 | The subgroup sum function ... |
| lsmvalx 19570 | Subspace sum value (for a ... |
| lsmelvalx 19571 | Subspace sum membership (f... |
| lsmelvalix 19572 | Subspace sum membership (f... |
| oppglsm 19573 | The subspace sum operation... |
| lsmssv 19574 | Subgroup sum is a subset o... |
| lsmless1x 19575 | Subset implies subgroup su... |
| lsmless2x 19576 | Subset implies subgroup su... |
| lsmub1x 19577 | Subgroup sum is an upper b... |
| lsmub2x 19578 | Subgroup sum is an upper b... |
| lsmval 19579 | Subgroup sum value (for a ... |
| lsmelval 19580 | Subgroup sum membership (f... |
| lsmelvali 19581 | Subgroup sum membership (f... |
| lsmelvalm 19582 | Subgroup sum membership an... |
| lsmelvalmi 19583 | Membership of vector subtr... |
| lsmsubm 19584 | The sum of two commuting s... |
| lsmsubg 19585 | The sum of two commuting s... |
| lsmcom2 19586 | Subgroup sum commutes. (C... |
| smndlsmidm 19587 | The direct product is idem... |
| lsmub1 19588 | Subgroup sum is an upper b... |
| lsmub2 19589 | Subgroup sum is an upper b... |
| lsmunss 19590 | Union of subgroups is a su... |
| lsmless1 19591 | Subset implies subgroup su... |
| lsmless2 19592 | Subset implies subgroup su... |
| lsmless12 19593 | Subset implies subgroup su... |
| lsmidm 19594 | Subgroup sum is idempotent... |
| lsmlub 19595 | The least upper bound prop... |
| lsmss1 19596 | Subgroup sum with a subset... |
| lsmss1b 19597 | Subgroup sum with a subset... |
| lsmss2 19598 | Subgroup sum with a subset... |
| lsmss2b 19599 | Subgroup sum with a subset... |
| lsmass 19600 | Subgroup sum is associativ... |
| mndlsmidm 19601 | Subgroup sum is idempotent... |
| lsm01 19602 | Subgroup sum with the zero... |
| lsm02 19603 | Subgroup sum with the zero... |
| subglsm 19604 | The subgroup sum evaluated... |
| lssnle 19605 | Equivalent expressions for... |
| lsmmod 19606 | The modular law holds for ... |
| lsmmod2 19607 | Modular law dual for subgr... |
| lsmpropd 19608 | If two structures have the... |
| cntzrecd 19609 | Commute the "subgroups com... |
| lsmcntz 19610 | The "subgroups commute" pr... |
| lsmcntzr 19611 | The "subgroups commute" pr... |
| lsmdisj 19612 | Disjointness from a subgro... |
| lsmdisj2 19613 | Association of the disjoin... |
| lsmdisj3 19614 | Association of the disjoin... |
| lsmdisjr 19615 | Disjointness from a subgro... |
| lsmdisj2r 19616 | Association of the disjoin... |
| lsmdisj3r 19617 | Association of the disjoin... |
| lsmdisj2a 19618 | Association of the disjoin... |
| lsmdisj2b 19619 | Association of the disjoin... |
| lsmdisj3a 19620 | Association of the disjoin... |
| lsmdisj3b 19621 | Association of the disjoin... |
| subgdisj1 19622 | Vectors belonging to disjo... |
| subgdisj2 19623 | Vectors belonging to disjo... |
| subgdisjb 19624 | Vectors belonging to disjo... |
| pj1fval 19625 | The left projection functi... |
| pj1val 19626 | The left projection functi... |
| pj1eu 19627 | Uniqueness of a left proje... |
| pj1f 19628 | The left projection functi... |
| pj2f 19629 | The right projection funct... |
| pj1id 19630 | Any element of a direct su... |
| pj1eq 19631 | Any element of a direct su... |
| pj1lid 19632 | The left projection functi... |
| pj1rid 19633 | The left projection functi... |
| pj1ghm 19634 | The left projection functi... |
| pj1ghm2 19635 | The left projection functi... |
| lsmhash 19636 | The order of the direct pr... |
| efgmval 19643 | Value of the formal invers... |
| efgmf 19644 | The formal inverse operati... |
| efgmnvl 19645 | The inversion function on ... |
| efgrcl 19646 | Lemma for ~ efgval . (Con... |
| efglem 19647 | Lemma for ~ efgval . (Con... |
| efgval 19648 | Value of the free group co... |
| efger 19649 | Value of the free group co... |
| efgi 19650 | Value of the free group co... |
| efgi0 19651 | Value of the free group co... |
| efgi1 19652 | Value of the free group co... |
| efgtf 19653 | Value of the free group co... |
| efgtval 19654 | Value of the extension fun... |
| efgval2 19655 | Value of the free group co... |
| efgi2 19656 | Value of the free group co... |
| efgtlen 19657 | Value of the free group co... |
| efginvrel2 19658 | The inverse of the reverse... |
| efginvrel1 19659 | The inverse of the reverse... |
| efgsf 19660 | Value of the auxiliary fun... |
| efgsdm 19661 | Elementhood in the domain ... |
| efgsval 19662 | Value of the auxiliary fun... |
| efgsdmi 19663 | Property of the last link ... |
| efgsval2 19664 | Value of the auxiliary fun... |
| efgsrel 19665 | The start and end of any e... |
| efgs1 19666 | A singleton of an irreduci... |
| efgs1b 19667 | Every extension sequence e... |
| efgsp1 19668 | If ` F ` is an extension s... |
| efgsres 19669 | An initial segment of an e... |
| efgsfo 19670 | For any word, there is a s... |
| efgredlema 19671 | The reduced word that form... |
| efgredlemf 19672 | Lemma for ~ efgredleme . ... |
| efgredlemg 19673 | Lemma for ~ efgred . (Con... |
| efgredleme 19674 | Lemma for ~ efgred . (Con... |
| efgredlemd 19675 | The reduced word that form... |
| efgredlemc 19676 | The reduced word that form... |
| efgredlemb 19677 | The reduced word that form... |
| efgredlem 19678 | The reduced word that form... |
| efgred 19679 | The reduced word that form... |
| efgrelexlema 19680 | If two words ` A , B ` are... |
| efgrelexlemb 19681 | If two words ` A , B ` are... |
| efgrelex 19682 | If two words ` A , B ` are... |
| efgredeu 19683 | There is a unique reduced ... |
| efgred2 19684 | Two extension sequences ha... |
| efgcpbllema 19685 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19686 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19687 | Two extension sequences ha... |
| efgcpbl2 19688 | Two extension sequences ha... |
| frgpval 19689 | Value of the free group co... |
| frgpcpbl 19690 | Compatibility of the group... |
| frgp0 19691 | The free group is a group.... |
| frgpeccl 19692 | Closure of the quotient ma... |
| frgpgrp 19693 | The free group is a group.... |
| frgpadd 19694 | Addition in the free group... |
| frgpinv 19695 | The inverse of an element ... |
| frgpmhm 19696 | The "natural map" from wor... |
| vrgpfval 19697 | The canonical injection fr... |
| vrgpval 19698 | The value of the generatin... |
| vrgpf 19699 | The mapping from the index... |
| vrgpinv 19700 | The inverse of a generatin... |
| frgpuptf 19701 | Any assignment of the gene... |
| frgpuptinv 19702 | Any assignment of the gene... |
| frgpuplem 19703 | Any assignment of the gene... |
| frgpupf 19704 | Any assignment of the gene... |
| frgpupval 19705 | Any assignment of the gene... |
| frgpup1 19706 | Any assignment of the gene... |
| frgpup2 19707 | The evaluation map has the... |
| frgpup3lem 19708 | The evaluation map has the... |
| frgpup3 19709 | Universal property of the ... |
| 0frgp 19710 | The free group on zero gen... |
| isabl 19715 | The predicate "is an Abeli... |
| ablgrp 19716 | An Abelian group is a grou... |
| ablgrpd 19717 | An Abelian group is a grou... |
| ablcmn 19718 | An Abelian group is a comm... |
| ablcmnd 19719 | An Abelian group is a comm... |
| iscmn 19720 | The predicate "is a commut... |
| isabl2 19721 | The predicate "is an Abeli... |
| cmnpropd 19722 | If two structures have the... |
| ablpropd 19723 | If two structures have the... |
| ablprop 19724 | If two structures have the... |
| iscmnd 19725 | Properties that determine ... |
| isabld 19726 | Properties that determine ... |
| isabli 19727 | Properties that determine ... |
| cmnmnd 19728 | A commutative monoid is a ... |
| cmncom 19729 | A commutative monoid is co... |
| ablcom 19730 | An Abelian group operation... |
| cmn32 19731 | Commutative/associative la... |
| cmn4 19732 | Commutative/associative la... |
| cmn12 19733 | Commutative/associative la... |
| abl32 19734 | Commutative/associative la... |
| cmnmndd 19735 | A commutative monoid is a ... |
| cmnbascntr 19736 | The base set of a commutat... |
| rinvmod 19737 | Uniqueness of a right inve... |
| ablinvadd 19738 | The inverse of an Abelian ... |
| ablsub2inv 19739 | Abelian group subtraction ... |
| ablsubadd 19740 | Relationship between Abeli... |
| ablsub4 19741 | Commutative/associative su... |
| abladdsub4 19742 | Abelian group addition/sub... |
| abladdsub 19743 | Associative-type law for g... |
| ablsubadd23 19744 | Commutative/associative la... |
| ablsubaddsub 19745 | Double subtraction and add... |
| ablpncan2 19746 | Cancellation law for subtr... |
| ablpncan3 19747 | A cancellation law for Abe... |
| ablsubsub 19748 | Law for double subtraction... |
| ablsubsub4 19749 | Law for double subtraction... |
| ablpnpcan 19750 | Cancellation law for mixed... |
| ablnncan 19751 | Cancellation law for group... |
| ablsub32 19752 | Swap the second and third ... |
| ablnnncan 19753 | Cancellation law for group... |
| ablnnncan1 19754 | Cancellation law for group... |
| ablsubsub23 19755 | Swap subtrahend and result... |
| mulgnn0di 19756 | Group multiple of a sum, f... |
| mulgdi 19757 | Group multiple of a sum. ... |
| mulgmhm 19758 | The map from ` x ` to ` n ... |
| mulgghm 19759 | The map from ` x ` to ` n ... |
| mulgsubdi 19760 | Group multiple of a differ... |
| ghmfghm 19761 | The function fulfilling th... |
| ghmcmn 19762 | The image of a commutative... |
| ghmabl 19763 | The image of an abelian gr... |
| invghm 19764 | The inversion map is a gro... |
| eqgabl 19765 | Value of the subgroup cose... |
| qusecsub 19766 | Two subgroup cosets are eq... |
| subgabl 19767 | A subgroup of an abelian g... |
| subcmn 19768 | A submonoid of a commutati... |
| submcmn 19769 | A submonoid of a commutati... |
| submcmn2 19770 | A submonoid is commutative... |
| cntzcmn 19771 | The centralizer of any sub... |
| cntzcmnss 19772 | Any subset in a commutativ... |
| cntrcmnd 19773 | The center of a monoid is ... |
| cntrabl 19774 | The center of a group is a... |
| cntzspan 19775 | If the generators commute,... |
| cntzcmnf 19776 | Discharge the centralizer ... |
| ghmplusg 19777 | The pointwise sum of two l... |
| ablnsg 19778 | Every subgroup of an abeli... |
| odadd1 19779 | The order of a product in ... |
| odadd2 19780 | The order of a product in ... |
| odadd 19781 | The order of a product is ... |
| gex2abl 19782 | A group with exponent 2 (o... |
| gexexlem 19783 | Lemma for ~ gexex . (Cont... |
| gexex 19784 | In an abelian group with f... |
| torsubg 19785 | The set of all elements of... |
| oddvdssubg 19786 | The set of all elements wh... |
| lsmcomx 19787 | Subgroup sum commutes (ext... |
| ablcntzd 19788 | All subgroups in an abelia... |
| lsmcom 19789 | Subgroup sum commutes. (C... |
| lsmsubg2 19790 | The sum of two subgroups i... |
| lsm4 19791 | Commutative/associative la... |
| prdscmnd 19792 | The product of a family of... |
| prdsabld 19793 | The product of a family of... |
| pwscmn 19794 | The structure power on a c... |
| pwsabl 19795 | The structure power on an ... |
| qusabl 19796 | If ` Y ` is a subgroup of ... |
| abl1 19797 | The (smallest) structure r... |
| abln0 19798 | Abelian groups (and theref... |
| cnaddablx 19799 | The complex numbers are an... |
| cnaddabl 19800 | The complex numbers are an... |
| cnaddid 19801 | The group identity element... |
| cnaddinv 19802 | Value of the group inverse... |
| zaddablx 19803 | The integers are an Abelia... |
| frgpnabllem1 19804 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19805 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19806 | The free group on two or m... |
| imasabl 19807 | The image structure of an ... |
| iscyg 19810 | Definition of a cyclic gro... |
| iscyggen 19811 | The property of being a cy... |
| iscyggen2 19812 | The property of being a cy... |
| iscyg2 19813 | A cyclic group is a group ... |
| cyggeninv 19814 | The inverse of a cyclic ge... |
| cyggenod 19815 | An element is the generato... |
| cyggenod2 19816 | In an infinite cyclic grou... |
| iscyg3 19817 | Definition of a cyclic gro... |
| iscygd 19818 | Definition of a cyclic gro... |
| iscygodd 19819 | Show that a group with an ... |
| cycsubmcmn 19820 | The set of nonnegative int... |
| cyggrp 19821 | A cyclic group is a group.... |
| cygabl 19822 | A cyclic group is abelian.... |
| cygctb 19823 | A cyclic group is countabl... |
| 0cyg 19824 | The trivial group is cycli... |
| prmcyg 19825 | A group with prime order i... |
| lt6abl 19826 | A group with fewer than ` ... |
| ghmcyg 19827 | The image of a cyclic grou... |
| cyggex2 19828 | The exponent of a cyclic g... |
| cyggex 19829 | The exponent of a finite c... |
| cyggexb 19830 | A finite abelian group is ... |
| giccyg 19831 | Cyclicity is a group prope... |
| cycsubgcyg 19832 | The cyclic subgroup genera... |
| cycsubgcyg2 19833 | The cyclic subgroup genera... |
| gsumval3a 19834 | Value of the group sum ope... |
| gsumval3eu 19835 | The group sum as defined i... |
| gsumval3lem1 19836 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19837 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19838 | Value of the group sum ope... |
| gsumcllem 19839 | Lemma for ~ gsumcl and rel... |
| gsumzres 19840 | Extend a finite group sum ... |
| gsumzcl2 19841 | Closure of a finite group ... |
| gsumzcl 19842 | Closure of a finite group ... |
| gsumzf1o 19843 | Re-index a finite group su... |
| gsumres 19844 | Extend a finite group sum ... |
| gsumcl2 19845 | Closure of a finite group ... |
| gsumcl 19846 | Closure of a finite group ... |
| gsumf1o 19847 | Re-index a finite group su... |
| gsumreidx 19848 | Re-index a finite group su... |
| gsumzsubmcl 19849 | Closure of a group sum in ... |
| gsumsubmcl 19850 | Closure of a group sum in ... |
| gsumsubgcl 19851 | Closure of a group sum in ... |
| gsumzaddlem 19852 | The sum of two group sums.... |
| gsumzadd 19853 | The sum of two group sums.... |
| gsumadd 19854 | The sum of two group sums.... |
| gsummptfsadd 19855 | The sum of two group sums ... |
| gsummptfidmadd 19856 | The sum of two group sums ... |
| gsummptfidmadd2 19857 | The sum of two group sums ... |
| gsumzsplit 19858 | Split a group sum into two... |
| gsumsplit 19859 | Split a group sum into two... |
| gsumsplit2 19860 | Split a group sum into two... |
| gsummptfidmsplit 19861 | Split a group sum expresse... |
| gsummptfidmsplitres 19862 | Split a group sum expresse... |
| gsummptfzsplit 19863 | Split a group sum expresse... |
| gsummptfzsplitl 19864 | Split a group sum expresse... |
| gsumconst 19865 | Sum of a constant series. ... |
| gsumconstf 19866 | Sum of a constant series. ... |
| gsummptshft 19867 | Index shift of a finite gr... |
| gsumzmhm 19868 | Apply a group homomorphism... |
| gsummhm 19869 | Apply a group homomorphism... |
| gsummhm2 19870 | Apply a group homomorphism... |
| gsummptmhm 19871 | Apply a group homomorphism... |
| gsummulglem 19872 | Lemma for ~ gsummulg and ~... |
| gsummulg 19873 | Nonnegative multiple of a ... |
| gsummulgz 19874 | Integer multiple of a grou... |
| gsumzoppg 19875 | The opposite of a group su... |
| gsumzinv 19876 | Inverse of a group sum. (... |
| gsuminv 19877 | Inverse of a group sum. (... |
| gsummptfidminv 19878 | Inverse of a group sum exp... |
| gsumsub 19879 | The difference of two grou... |
| gsummptfssub 19880 | The difference of two grou... |
| gsummptfidmsub 19881 | The difference of two grou... |
| gsumsnfd 19882 | Group sum of a singleton, ... |
| gsumsnd 19883 | Group sum of a singleton, ... |
| gsumsnf 19884 | Group sum of a singleton, ... |
| gsumsn 19885 | Group sum of a singleton. ... |
| gsumpr 19886 | Group sum of a pair. (Con... |
| gsumzunsnd 19887 | Append an element to a fin... |
| gsumunsnfd 19888 | Append an element to a fin... |
| gsumunsnd 19889 | Append an element to a fin... |
| gsumunsnf 19890 | Append an element to a fin... |
| gsumunsn 19891 | Append an element to a fin... |
| gsumdifsnd 19892 | Extract a summand from a f... |
| gsumpt 19893 | Sum of a family that is no... |
| gsummptf1o 19894 | Re-index a finite group su... |
| gsummptun 19895 | Group sum of a disjoint un... |
| gsummpt1n0 19896 | If only one summand in a f... |
| gsummptif1n0 19897 | If only one summand in a f... |
| gsummptcl 19898 | Closure of a finite group ... |
| gsummptfif1o 19899 | Re-index a finite group su... |
| gsummptfzcl 19900 | Closure of a finite group ... |
| gsum2dlem1 19901 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19902 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19903 | Write a sum over a two-dim... |
| gsum2d2lem 19904 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19905 | Write a group sum over a t... |
| gsumcom2 19906 | Two-dimensional commutatio... |
| gsumxp 19907 | Write a group sum over a c... |
| gsumcom 19908 | Commute the arguments of a... |
| gsumcom3 19909 | A commutative law for fini... |
| gsumcom3fi 19910 | A commutative law for fini... |
| gsumxp2 19911 | Write a group sum over a c... |
| prdsgsum 19912 | Finite commutative sums in... |
| pwsgsum 19913 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19914 | Replacing a finitely suppo... |
| nn0gsumfz 19915 | Replacing a finitely suppo... |
| nn0gsumfz0 19916 | Replacing a finitely suppo... |
| gsummptnn0fz 19917 | A final group sum over a f... |
| gsummptnn0fzfv 19918 | A final group sum over a f... |
| telgsumfzslem 19919 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19920 | Telescoping group sum rang... |
| telgsumfz 19921 | Telescoping group sum rang... |
| telgsumfz0s 19922 | Telescoping finite group s... |
| telgsumfz0 19923 | Telescoping finite group s... |
| telgsums 19924 | Telescoping finitely suppo... |
| telgsum 19925 | Telescoping finitely suppo... |
| reldmdprd 19930 | The domain of the internal... |
| dmdprd 19931 | The domain of definition o... |
| dmdprdd 19932 | Show that a given family i... |
| dprddomprc 19933 | A family of subgroups inde... |
| dprddomcld 19934 | If a family of subgroups i... |
| dprdval0prc 19935 | The internal direct produc... |
| dprdval 19936 | The value of the internal ... |
| eldprd 19937 | A class ` A ` is an intern... |
| dprdgrp 19938 | Reverse closure for the in... |
| dprdf 19939 | The function ` S ` is a fa... |
| dprdf2 19940 | The function ` S ` is a fa... |
| dprdcntz 19941 | The function ` S ` is a fa... |
| dprddisj 19942 | The function ` S ` is a fa... |
| dprdw 19943 | The property of being a fi... |
| dprdwd 19944 | A mapping being a finitely... |
| dprdff 19945 | A finitely supported funct... |
| dprdfcl 19946 | A finitely supported funct... |
| dprdffsupp 19947 | A finitely supported funct... |
| dprdfcntz 19948 | A function on the elements... |
| dprdssv 19949 | The internal direct produc... |
| dprdfid 19950 | A function mapping all but... |
| eldprdi 19951 | The domain of definition o... |
| dprdfinv 19952 | Take the inverse of a grou... |
| dprdfadd 19953 | Take the sum of group sums... |
| dprdfsub 19954 | Take the difference of gro... |
| dprdfeq0 19955 | The zero function is the o... |
| dprdf11 19956 | Two group sums over a dire... |
| dprdsubg 19957 | The internal direct produc... |
| dprdub 19958 | Each factor is a subset of... |
| dprdlub 19959 | The direct product is smal... |
| dprdspan 19960 | The direct product is the ... |
| dprdres 19961 | Restriction of a direct pr... |
| dprdss 19962 | Create a direct product by... |
| dprdz 19963 | A family consisting entire... |
| dprd0 19964 | The empty family is an int... |
| dprdf1o 19965 | Rearrange the index set of... |
| dprdf1 19966 | Rearrange the index set of... |
| subgdmdprd 19967 | A direct product in a subg... |
| subgdprd 19968 | A direct product in a subg... |
| dprdsn 19969 | A singleton family is an i... |
| dmdprdsplitlem 19970 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19971 | The function ` S ` is a fa... |
| dprddisj2 19972 | The function ` S ` is a fa... |
| dprd2dlem2 19973 | The direct product of a co... |
| dprd2dlem1 19974 | The direct product of a co... |
| dprd2da 19975 | The direct product of a co... |
| dprd2db 19976 | The direct product of a co... |
| dprd2d2 19977 | The direct product of a co... |
| dmdprdsplit2lem 19978 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19979 | The direct product splits ... |
| dmdprdsplit 19980 | The direct product splits ... |
| dprdsplit 19981 | The direct product is the ... |
| dmdprdpr 19982 | A singleton family is an i... |
| dprdpr 19983 | A singleton family is an i... |
| dpjlem 19984 | Lemma for theorems about d... |
| dpjcntz 19985 | The two subgroups that app... |
| dpjdisj 19986 | The two subgroups that app... |
| dpjlsm 19987 | The two subgroups that app... |
| dpjfval 19988 | Value of the direct produc... |
| dpjval 19989 | Value of the direct produc... |
| dpjf 19990 | The ` X ` -th index projec... |
| dpjidcl 19991 | The key property of projec... |
| dpjeq 19992 | Decompose a group sum into... |
| dpjid 19993 | The key property of projec... |
| dpjlid 19994 | The ` X ` -th index projec... |
| dpjrid 19995 | The ` Y ` -th index projec... |
| dpjghm 19996 | The direct product is the ... |
| dpjghm2 19997 | The direct product is the ... |
| ablfacrplem 19998 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 19999 | A finite abelian group who... |
| ablfacrp2 20000 | The factors ` K , L ` of ~... |
| ablfac1lem 20001 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20002 | The factors of ~ ablfac1b ... |
| ablfac1b 20003 | Any abelian group is the d... |
| ablfac1c 20004 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20005 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20006 | The factorization of ~ abl... |
| pgpfac1lem1 20007 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20008 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20009 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20010 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20011 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20012 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20013 | Factorization of a finite ... |
| pgpfaclem1 20014 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20015 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20016 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20017 | Full factorization of a fi... |
| ablfaclem1 20018 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20019 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20020 | Lemma for ~ ablfac . (Con... |
| ablfac 20021 | The Fundamental Theorem of... |
| ablfac2 20022 | Choose generators for each... |
| issimpg 20025 | The predicate "is a simple... |
| issimpgd 20026 | Deduce a simple group from... |
| simpggrp 20027 | A simple group is a group.... |
| simpggrpd 20028 | A simple group is a group.... |
| simpg2nsg 20029 | A simple group has two nor... |
| trivnsimpgd 20030 | Trivial groups are not sim... |
| simpgntrivd 20031 | Simple groups are nontrivi... |
| simpgnideld 20032 | A simple group contains a ... |
| simpgnsgd 20033 | The only normal subgroups ... |
| simpgnsgeqd 20034 | A normal subgroup of a sim... |
| 2nsgsimpgd 20035 | If any normal subgroup of ... |
| simpgnsgbid 20036 | A nontrivial group is simp... |
| ablsimpnosubgd 20037 | A subgroup of an abelian s... |
| ablsimpg1gend 20038 | An abelian simple group is... |
| ablsimpgcygd 20039 | An abelian simple group is... |
| ablsimpgfindlem1 20040 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20041 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20042 | Relationship between the o... |
| ablsimpgfind 20043 | An abelian simple group is... |
| fincygsubgd 20044 | The subgroup referenced in... |
| fincygsubgodd 20045 | Calculate the order of a s... |
| fincygsubgodexd 20046 | A finite cyclic group has ... |
| prmgrpsimpgd 20047 | A group of prime order is ... |
| ablsimpgprmd 20048 | An abelian simple group ha... |
| ablsimpgd 20049 | An abelian group is simple... |
| isomnd 20054 | A (left) ordered monoid is... |
| isogrp 20055 | A (left-)ordered group is ... |
| ogrpgrp 20056 | A left-ordered group is a ... |
| omndmnd 20057 | A left-ordered monoid is a... |
| omndtos 20058 | A left-ordered monoid is a... |
| omndadd 20059 | In an ordered monoid, the ... |
| omndaddr 20060 | In a right ordered monoid,... |
| omndadd2d 20061 | In a commutative left orde... |
| omndadd2rd 20062 | In a left- and right- orde... |
| submomnd 20063 | A submonoid of an ordered ... |
| omndmul2 20064 | In an ordered monoid, the ... |
| omndmul3 20065 | In an ordered monoid, the ... |
| omndmul 20066 | In a commutative ordered m... |
| ogrpinv0le 20067 | In an ordered group, the o... |
| ogrpsub 20068 | In an ordered group, the o... |
| ogrpaddlt 20069 | In an ordered group, stric... |
| ogrpaddltbi 20070 | In a right ordered group, ... |
| ogrpaddltrd 20071 | In a right ordered group, ... |
| ogrpaddltrbid 20072 | In a right ordered group, ... |
| ogrpsublt 20073 | In an ordered group, stric... |
| ogrpinv0lt 20074 | In an ordered group, the o... |
| ogrpinvlt 20075 | In an ordered group, the o... |
| gsumle 20076 | A finite sum in an ordered... |
| fnmgp 20079 | The multiplicative group o... |
| mgpval 20080 | Value of the multiplicatio... |
| mgpplusg 20081 | Value of the group operati... |
| mgpbas 20082 | Base set of the multiplica... |
| mgpsca 20083 | The multiplication monoid ... |
| mgptset 20084 | Topology component of the ... |
| mgptopn 20085 | Topology of the multiplica... |
| mgpds 20086 | Distance function of the m... |
| mgpress 20087 | Subgroup commutes with the... |
| prdsmgp 20088 | The multiplicative monoid ... |
| isrng 20091 | The predicate "is a non-un... |
| rngabl 20092 | A non-unital ring is an (a... |
| rngmgp 20093 | A non-unital ring is a sem... |
| rngmgpf 20094 | Restricted functionality o... |
| rnggrp 20095 | A non-unital ring is a (ad... |
| rngass 20096 | Associative law for the mu... |
| rngdi 20097 | Distributive law for the m... |
| rngdir 20098 | Distributive law for the m... |
| rngacl 20099 | Closure of the addition op... |
| rng0cl 20100 | The zero element of a non-... |
| rngcl 20101 | Closure of the multiplicat... |
| rnglz 20102 | The zero of a non-unital r... |
| rngrz 20103 | The zero of a non-unital r... |
| rngmneg1 20104 | Negation of a product in a... |
| rngmneg2 20105 | Negation of a product in a... |
| rngm2neg 20106 | Double negation of a produ... |
| rngansg 20107 | Every additive subgroup of... |
| rngsubdi 20108 | Ring multiplication distri... |
| rngsubdir 20109 | Ring multiplication distri... |
| isrngd 20110 | Properties that determine ... |
| rngpropd 20111 | If two structures have the... |
| prdsmulrngcl 20112 | Closure of the multiplicat... |
| prdsrngd 20113 | A product of non-unital ri... |
| imasrng 20114 | The image structure of a n... |
| imasrngf1 20115 | The image of a non-unital ... |
| xpsrngd 20116 | A product of two non-unita... |
| qusrng 20117 | The quotient structure of ... |
| ringidval 20120 | The value of the unity ele... |
| dfur2 20121 | The multiplicative identit... |
| ringurd 20122 | Deduce the unity element o... |
| issrg 20125 | The predicate "is a semiri... |
| srgcmn 20126 | A semiring is a commutativ... |
| srgmnd 20127 | A semiring is a monoid. (... |
| srgmgp 20128 | A semiring is a monoid und... |
| srgdilem 20129 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20130 | Closure of the multiplicat... |
| srgass 20131 | Associative law for the mu... |
| srgideu 20132 | The unity element of a sem... |
| srgfcl 20133 | Functionality of the multi... |
| srgdi 20134 | Distributive law for the m... |
| srgdir 20135 | Distributive law for the m... |
| srgidcl 20136 | The unity element of a sem... |
| srg0cl 20137 | The zero element of a semi... |
| srgidmlem 20138 | Lemma for ~ srglidm and ~ ... |
| srglidm 20139 | The unity element of a sem... |
| srgridm 20140 | The unity element of a sem... |
| issrgid 20141 | Properties showing that an... |
| srgacl 20142 | Closure of the addition op... |
| srgcom 20143 | Commutativity of the addit... |
| srgrz 20144 | The zero of a semiring is ... |
| srglz 20145 | The zero of a semiring is ... |
| srgisid 20146 | In a semiring, the only le... |
| o2timesd 20147 | An element of a ring-like ... |
| rglcom4d 20148 | Restricted commutativity o... |
| srgo2times 20149 | A semiring element plus it... |
| srgcom4lem 20150 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20151 | Restricted commutativity o... |
| srg1zr 20152 | The only semiring with a b... |
| srgen1zr 20153 | The only semiring with one... |
| srgmulgass 20154 | An associative property be... |
| srgpcomp 20155 | If two elements of a semir... |
| srgpcompp 20156 | If two elements of a semir... |
| srgpcomppsc 20157 | If two elements of a semir... |
| srglmhm 20158 | Left-multiplication in a s... |
| srgrmhm 20159 | Right-multiplication in a ... |
| srgsummulcr 20160 | A finite semiring sum mult... |
| sgsummulcl 20161 | A finite semiring sum mult... |
| srg1expzeq1 20162 | The exponentiation (by a n... |
| srgbinomlem1 20163 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20164 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20165 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20166 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20167 | Lemma for ~ srgbinom . In... |
| srgbinom 20168 | The binomial theorem for c... |
| csrgbinom 20169 | The binomial theorem for c... |
| isring 20174 | The predicate "is a (unita... |
| ringgrp 20175 | A ring is a group. (Contr... |
| ringmgp 20176 | A ring is a monoid under m... |
| iscrng 20177 | A commutative ring is a ri... |
| crngmgp 20178 | A commutative ring's multi... |
| ringgrpd 20179 | A ring is a group. (Contr... |
| ringmnd 20180 | A ring is a monoid under a... |
| ringmgm 20181 | A ring is a magma. (Contr... |
| crngring 20182 | A commutative ring is a ri... |
| crngringd 20183 | A commutative ring is a ri... |
| crnggrpd 20184 | A commutative ring is a gr... |
| mgpf 20185 | Restricted functionality o... |
| ringdilem 20186 | Properties of a unital rin... |
| ringcl 20187 | Closure of the multiplicat... |
| crngcom 20188 | A commutative ring's multi... |
| iscrng2 20189 | A commutative ring is a ri... |
| ringass 20190 | Associative law for multip... |
| ringideu 20191 | The unity element of a rin... |
| crngcomd 20192 | Multiplication is commutat... |
| crngbascntr 20193 | The base set of a commutat... |
| ringassd 20194 | Associative law for multip... |
| crng12d 20195 | Commutative/associative la... |
| crng32d 20196 | Commutative/associative la... |
| ringcld 20197 | Closure of the multiplicat... |
| ringdi 20198 | Distributive law for the m... |
| ringdir 20199 | Distributive law for the m... |
| ringdid 20200 | Distributive law for the m... |
| ringdird 20201 | Distributive law for the m... |
| ringidcl 20202 | The unity element of a rin... |
| ringidcld 20203 | The unity element of a rin... |
| ring0cl 20204 | The zero element of a ring... |
| ringidmlem 20205 | Lemma for ~ ringlidm and ~... |
| ringlidm 20206 | The unity element of a rin... |
| ringridm 20207 | The unity element of a rin... |
| isringid 20208 | Properties showing that an... |
| ringlidmd 20209 | The unity element of a rin... |
| ringridmd 20210 | The unity element of a rin... |
| ringid 20211 | The multiplication operati... |
| ringo2times 20212 | A ring element plus itself... |
| ringadd2 20213 | A ring element plus itself... |
| ringidss 20214 | A subset of the multiplica... |
| ringacl 20215 | Closure of the addition op... |
| ringcomlem 20216 | Lemma for ~ ringcom . Thi... |
| ringcom 20217 | Commutativity of the addit... |
| ringabl 20218 | A ring is an Abelian group... |
| ringcmn 20219 | A ring is a commutative mo... |
| ringabld 20220 | A ring is an Abelian group... |
| ringcmnd 20221 | A ring is a commutative mo... |
| ringrng 20222 | A unital ring is a non-uni... |
| ringssrng 20223 | The unital rings are non-u... |
| isringrng 20224 | The predicate "is a unital... |
| ringpropd 20225 | If two structures have the... |
| crngpropd 20226 | If two structures have the... |
| ringprop 20227 | If two structures have the... |
| isringd 20228 | Properties that determine ... |
| iscrngd 20229 | Properties that determine ... |
| ringlz 20230 | The zero of a unital ring ... |
| ringrz 20231 | The zero of a unital ring ... |
| ringlzd 20232 | The zero of a unital ring ... |
| ringrzd 20233 | The zero of a unital ring ... |
| ringsrg 20234 | Any ring is also a semirin... |
| ring1eq0 20235 | If one and zero are equal,... |
| ring1ne0 20236 | If a ring has at least two... |
| ringinvnz1ne0 20237 | In a unital ring, a left i... |
| ringinvnzdiv 20238 | In a unital ring, a left i... |
| ringnegl 20239 | Negation in a ring is the ... |
| ringnegr 20240 | Negation in a ring is the ... |
| ringmneg1 20241 | Negation of a product in a... |
| ringmneg2 20242 | Negation of a product in a... |
| ringm2neg 20243 | Double negation of a produ... |
| ringsubdi 20244 | Ring multiplication distri... |
| ringsubdir 20245 | Ring multiplication distri... |
| mulgass2 20246 | An associative property be... |
| ring1 20247 | The (smallest) structure r... |
| ringn0 20248 | Rings exist. (Contributed... |
| ringlghm 20249 | Left-multiplication in a r... |
| ringrghm 20250 | Right-multiplication in a ... |
| gsummulc1OLD 20251 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20252 | Obsolete version of ~ gsum... |
| gsummulc1 20253 | A finite ring sum multipli... |
| gsummulc2 20254 | A finite ring sum multipli... |
| gsummgp0 20255 | If one factor in a finite ... |
| gsumdixp 20256 | Distribute a binary produc... |
| prdsmulrcl 20257 | A structure product of rin... |
| prdsringd 20258 | A product of rings is a ri... |
| prdscrngd 20259 | A product of commutative r... |
| prds1 20260 | Value of the ring unity in... |
| pwsring 20261 | A structure power of a rin... |
| pws1 20262 | Value of the ring unity in... |
| pwscrng 20263 | A structure power of a com... |
| pwsmgp 20264 | The multiplicative group o... |
| pwspjmhmmgpd 20265 | The projection given by ~ ... |
| pwsexpg 20266 | Value of a group exponenti... |
| pwsgprod 20267 | Finite products in a power... |
| imasring 20268 | The image structure of a r... |
| imasringf1 20269 | The image of a ring under ... |
| xpsringd 20270 | A product of two rings is ... |
| xpsring1d 20271 | The multiplicative identit... |
| qusring2 20272 | The quotient structure of ... |
| crngbinom 20273 | The binomial theorem for c... |
| opprval 20276 | Value of the opposite ring... |
| opprmulfval 20277 | Value of the multiplicatio... |
| opprmul 20278 | Value of the multiplicatio... |
| crngoppr 20279 | In a commutative ring, the... |
| opprlem 20280 | Lemma for ~ opprbas and ~ ... |
| opprbas 20281 | Base set of an opposite ri... |
| oppradd 20282 | Addition operation of an o... |
| opprrng 20283 | An opposite non-unital rin... |
| opprrngb 20284 | A class is a non-unital ri... |
| opprring 20285 | An opposite ring is a ring... |
| opprringb 20286 | Bidirectional form of ~ op... |
| oppr0 20287 | Additive identity of an op... |
| oppr1 20288 | Multiplicative identity of... |
| opprneg 20289 | The negative function in a... |
| opprsubg 20290 | Being a subgroup is a symm... |
| mulgass3 20291 | An associative property be... |
| reldvdsr 20298 | The divides relation is a ... |
| dvdsrval 20299 | Value of the divides relat... |
| dvdsr 20300 | Value of the divides relat... |
| dvdsr2 20301 | Value of the divides relat... |
| dvdsrmul 20302 | A left-multiple of ` X ` i... |
| dvdsrcl 20303 | Closure of a dividing elem... |
| dvdsrcl2 20304 | Closure of a dividing elem... |
| dvdsrid 20305 | An element in a (unital) r... |
| dvdsrtr 20306 | Divisibility is transitive... |
| dvdsrmul1 20307 | The divisibility relation ... |
| dvdsrneg 20308 | An element divides its neg... |
| dvdsr01 20309 | In a ring, zero is divisib... |
| dvdsr02 20310 | Only zero is divisible by ... |
| isunit 20311 | Property of being a unit o... |
| 1unit 20312 | The multiplicative identit... |
| unitcl 20313 | A unit is an element of th... |
| unitss 20314 | The set of units is contai... |
| opprunit 20315 | Being a unit is a symmetri... |
| crngunit 20316 | Property of being a unit i... |
| dvdsunit 20317 | A divisor of a unit is a u... |
| unitmulcl 20318 | The product of units is a ... |
| unitmulclb 20319 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20320 | The base set of the group ... |
| unitgrp 20321 | The group of units is a gr... |
| unitabl 20322 | The group of units of a co... |
| unitgrpid 20323 | The identity of the group ... |
| unitsubm 20324 | The group of units is a su... |
| invrfval 20327 | Multiplicative inverse fun... |
| unitinvcl 20328 | The inverse of a unit exis... |
| unitinvinv 20329 | The inverse of the inverse... |
| ringinvcl 20330 | The inverse of a unit is a... |
| unitlinv 20331 | A unit times its inverse i... |
| unitrinv 20332 | A unit times its inverse i... |
| 1rinv 20333 | The inverse of the ring un... |
| 0unit 20334 | The additive identity is a... |
| unitnegcl 20335 | The negative of a unit is ... |
| ringunitnzdiv 20336 | In a unitary ring, a unit ... |
| ring1nzdiv 20337 | In a unitary ring, the rin... |
| dvrfval 20340 | Division operation in a ri... |
| dvrval 20341 | Division operation in a ri... |
| dvrcl 20342 | Closure of division operat... |
| unitdvcl 20343 | The units are closed under... |
| dvrid 20344 | A ring element divided by ... |
| dvr1 20345 | A ring element divided by ... |
| dvrass 20346 | An associative law for div... |
| dvrcan1 20347 | A cancellation law for div... |
| dvrcan3 20348 | A cancellation law for div... |
| dvreq1 20349 | Equality in terms of ratio... |
| dvrdir 20350 | Distributive law for the d... |
| rdivmuldivd 20351 | Multiplication of two rati... |
| ringinvdv 20352 | Write the inverse function... |
| rngidpropd 20353 | The ring unity depends onl... |
| dvdsrpropd 20354 | The divisibility relation ... |
| unitpropd 20355 | The set of units depends o... |
| invrpropd 20356 | The ring inverse function ... |
| isirred 20357 | An irreducible element of ... |
| isnirred 20358 | The property of being a no... |
| isirred2 20359 | Expand out the class diffe... |
| opprirred 20360 | Irreducibility is symmetri... |
| irredn0 20361 | The additive identity is n... |
| irredcl 20362 | An irreducible element is ... |
| irrednu 20363 | An irreducible element is ... |
| irredn1 20364 | The multiplicative identit... |
| irredrmul 20365 | The product of an irreduci... |
| irredlmul 20366 | The product of a unit and ... |
| irredmul 20367 | If product of two elements... |
| irredneg 20368 | The negative of an irreduc... |
| irrednegb 20369 | An element is irreducible ... |
| rnghmrcl 20376 | Reverse closure of a non-u... |
| rnghmfn 20377 | The mapping of two non-uni... |
| rnghmval 20378 | The set of the non-unital ... |
| isrnghm 20379 | A function is a non-unital... |
| isrnghmmul 20380 | A function is a non-unital... |
| rnghmmgmhm 20381 | A non-unital ring homomorp... |
| rnghmval2 20382 | The non-unital ring homomo... |
| isrngim 20383 | An isomorphism of non-unit... |
| rngimrcl 20384 | Reverse closure for an iso... |
| rnghmghm 20385 | A non-unital ring homomorp... |
| rnghmf 20386 | A ring homomorphism is a f... |
| rnghmmul 20387 | A homomorphism of non-unit... |
| isrnghm2d 20388 | Demonstration of non-unita... |
| isrnghmd 20389 | Demonstration of non-unita... |
| rnghmf1o 20390 | A non-unital ring homomorp... |
| isrngim2 20391 | An isomorphism of non-unit... |
| rngimf1o 20392 | An isomorphism of non-unit... |
| rngimrnghm 20393 | An isomorphism of non-unit... |
| rngimcnv 20394 | The converse of an isomorp... |
| rnghmco 20395 | The composition of non-uni... |
| idrnghm 20396 | The identity homomorphism ... |
| c0mgm 20397 | The constant mapping to ze... |
| c0mhm 20398 | The constant mapping to ze... |
| c0ghm 20399 | The constant mapping to ze... |
| c0snmgmhm 20400 | The constant mapping to ze... |
| c0snmhm 20401 | The constant mapping to ze... |
| c0snghm 20402 | The constant mapping to ze... |
| rngisomfv1 20403 | If there is a non-unital r... |
| rngisom1 20404 | If there is a non-unital r... |
| rngisomring 20405 | If there is a non-unital r... |
| rngisomring1 20406 | If there is a non-unital r... |
| dfrhm2 20412 | The property of a ring hom... |
| rhmrcl1 20414 | Reverse closure of a ring ... |
| rhmrcl2 20415 | Reverse closure of a ring ... |
| isrhm 20416 | A function is a ring homom... |
| rhmmhm 20417 | A ring homomorphism is a h... |
| rhmisrnghm 20418 | Each unital ring homomorph... |
| rimrcl 20419 | Reverse closure for an iso... |
| isrim0 20420 | A ring isomorphism is a ho... |
| rhmghm 20421 | A ring homomorphism is an ... |
| rhmf 20422 | A ring homomorphism is a f... |
| rhmmul 20423 | A homomorphism of rings pr... |
| isrhm2d 20424 | Demonstration of ring homo... |
| isrhmd 20425 | Demonstration of ring homo... |
| rhm1 20426 | Ring homomorphisms are req... |
| idrhm 20427 | The identity homomorphism ... |
| rhmf1o 20428 | A ring homomorphism is bij... |
| isrim 20429 | An isomorphism of rings is... |
| rimf1o 20430 | An isomorphism of rings is... |
| rimrhm 20431 | A ring isomorphism is a ho... |
| rimgim 20432 | An isomorphism of rings is... |
| rimisrngim 20433 | Each unital ring isomorphi... |
| rhmfn 20434 | The mapping of two rings t... |
| rhmval 20435 | The ring homomorphisms bet... |
| rhmco 20436 | The composition of ring ho... |
| pwsco1rhm 20437 | Right composition with a f... |
| pwsco2rhm 20438 | Left composition with a ri... |
| brric 20439 | The relation "is isomorphi... |
| brrici 20440 | Prove isomorphic by an exp... |
| brric2 20441 | The relation "is isomorphi... |
| ricgic 20442 | If two rings are (ring) is... |
| rhmdvdsr 20443 | A ring homomorphism preser... |
| rhmopp 20444 | A ring homomorphism is als... |
| elrhmunit 20445 | Ring homomorphisms preserv... |
| rhmunitinv 20446 | Ring homomorphisms preserv... |
| isnzr 20449 | Property of a nonzero ring... |
| nzrnz 20450 | One and zero are different... |
| nzrring 20451 | A nonzero ring is a ring. ... |
| nzrringOLD 20452 | Obsolete version of ~ nzrr... |
| isnzr2 20453 | Equivalent characterizatio... |
| isnzr2hash 20454 | Equivalent characterizatio... |
| nzrpropd 20455 | If two structures have the... |
| opprnzrb 20456 | The opposite of a nonzero ... |
| opprnzr 20457 | The opposite of a nonzero ... |
| ringelnzr 20458 | A ring is nonzero if it ha... |
| nzrunit 20459 | A unit is nonzero in any n... |
| 0ringnnzr 20460 | A ring is a zero ring iff ... |
| 0ring 20461 | If a ring has only one ele... |
| 0ringdif 20462 | A zero ring is a ring whic... |
| 0ringbas 20463 | The base set of a zero rin... |
| 0ring01eq 20464 | In a ring with only one el... |
| 01eq0ring 20465 | If the zero and the identi... |
| 01eq0ringOLD 20466 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20467 | In a unital ring the zero ... |
| 0ring1eq0 20468 | In a zero ring, a ring whi... |
| c0rhm 20469 | The constant mapping to ze... |
| c0rnghm 20470 | The constant mapping to ze... |
| zrrnghm 20471 | The constant mapping to ze... |
| nrhmzr 20472 | There is no ring homomorph... |
| islring 20475 | The predicate "is a local ... |
| lringnzr 20476 | A local ring is a nonzero ... |
| lringring 20477 | A local ring is a ring. (... |
| lringnz 20478 | A local ring is a nonzero ... |
| lringuplu 20479 | If the sum of two elements... |
| issubrng 20482 | The subring of non-unital ... |
| subrngss 20483 | A subring is a subset. (C... |
| subrngid 20484 | Every non-unital ring is a... |
| subrngrng 20485 | A subring is a non-unital ... |
| subrngrcl 20486 | Reverse closure for a subr... |
| subrngsubg 20487 | A subring is a subgroup. ... |
| subrngringnsg 20488 | A subring is a normal subg... |
| subrngbas 20489 | Base set of a subring stru... |
| subrng0 20490 | A subring always has the s... |
| subrngacl 20491 | A subring is closed under ... |
| subrngmcl 20492 | A subring is closed under ... |
| issubrng2 20493 | Characterize the subrings ... |
| opprsubrng 20494 | Being a subring is a symme... |
| subrngint 20495 | The intersection of a none... |
| subrngin 20496 | The intersection of two su... |
| subrngmre 20497 | The subrings of a non-unit... |
| subsubrng 20498 | A subring of a subring is ... |
| subsubrng2 20499 | The set of subrings of a s... |
| rhmimasubrnglem 20500 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20501 | The homomorphic image of a... |
| cntzsubrng 20502 | Centralizers in a non-unit... |
| subrngpropd 20503 | If two structures have the... |
| issubrg 20506 | The subring predicate. (C... |
| subrgss 20507 | A subring is a subset. (C... |
| subrgid 20508 | Every ring is a subring of... |
| subrgring 20509 | A subring is a ring. (Con... |
| subrgcrng 20510 | A subring of a commutative... |
| subrgrcl 20511 | Reverse closure for a subr... |
| subrgsubg 20512 | A subring is a subgroup. ... |
| subrgsubrng 20513 | A subring of a unital ring... |
| subrg0 20514 | A subring always has the s... |
| subrg1cl 20515 | A subring contains the mul... |
| subrgbas 20516 | Base set of a subring stru... |
| subrg1 20517 | A subring always has the s... |
| subrgacl 20518 | A subring is closed under ... |
| subrgmcl 20519 | A subring is closed under ... |
| subrgsubm 20520 | A subring is a submonoid o... |
| subrgdvds 20521 | If an element divides anot... |
| subrguss 20522 | A unit of a subring is a u... |
| subrginv 20523 | A subring always has the s... |
| subrgdv 20524 | A subring always has the s... |
| subrgunit 20525 | An element of a ring is a ... |
| subrgugrp 20526 | The units of a subring for... |
| issubrg2 20527 | Characterize the subrings ... |
| opprsubrg 20528 | Being a subring is a symme... |
| subrgnzr 20529 | A subring of a nonzero rin... |
| subrgint 20530 | The intersection of a none... |
| subrgin 20531 | The intersection of two su... |
| subrgmre 20532 | The subrings of a ring are... |
| subsubrg 20533 | A subring of a subring is ... |
| subsubrg2 20534 | The set of subrings of a s... |
| issubrg3 20535 | A subring is an additive s... |
| resrhm 20536 | Restriction of a ring homo... |
| resrhm2b 20537 | Restriction of the codomai... |
| rhmeql 20538 | The equalizer of two ring ... |
| rhmima 20539 | The homomorphic image of a... |
| rnrhmsubrg 20540 | The range of a ring homomo... |
| cntzsubr 20541 | Centralizers in a ring are... |
| pwsdiagrhm 20542 | Diagonal homomorphism into... |
| subrgpropd 20543 | If two structures have the... |
| rhmpropd 20544 | Ring homomorphism depends ... |
| rgspnval 20547 | Value of the ring-span of ... |
| rgspncl 20548 | The ring-span of a set is ... |
| rgspnssid 20549 | The ring-span of a set con... |
| rgspnmin 20550 | The ring-span is contained... |
| rngcval 20553 | Value of the category of n... |
| rnghmresfn 20554 | The class of non-unital ri... |
| rnghmresel 20555 | An element of the non-unit... |
| rngcbas 20556 | Set of objects of the cate... |
| rngchomfval 20557 | Set of arrows of the categ... |
| rngchom 20558 | Set of arrows of the categ... |
| elrngchom 20559 | A morphism of non-unital r... |
| rngchomfeqhom 20560 | The functionalized Hom-set... |
| rngccofval 20561 | Composition in the categor... |
| rngcco 20562 | Composition in the categor... |
| dfrngc2 20563 | Alternate definition of th... |
| rnghmsscmap2 20564 | The non-unital ring homomo... |
| rnghmsscmap 20565 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20566 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20567 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20568 | The non-unital ring homomo... |
| rngccat 20569 | The category of non-unital... |
| rngcid 20570 | The identity arrow in the ... |
| rngcsect 20571 | A section in the category ... |
| rngcinv 20572 | An inverse in the category... |
| rngciso 20573 | An isomorphism in the cate... |
| rngcifuestrc 20574 | The "inclusion functor" fr... |
| funcrngcsetc 20575 | The "natural forgetful fun... |
| funcrngcsetcALT 20576 | Alternate proof of ~ funcr... |
| zrinitorngc 20577 | The zero ring is an initia... |
| zrtermorngc 20578 | The zero ring is a termina... |
| zrzeroorngc 20579 | The zero ring is a zero ob... |
| ringcval 20582 | Value of the category of u... |
| rhmresfn 20583 | The class of unital ring h... |
| rhmresel 20584 | An element of the unital r... |
| ringcbas 20585 | Set of objects of the cate... |
| ringchomfval 20586 | Set of arrows of the categ... |
| ringchom 20587 | Set of arrows of the categ... |
| elringchom 20588 | A morphism of unital rings... |
| ringchomfeqhom 20589 | The functionalized Hom-set... |
| ringccofval 20590 | Composition in the categor... |
| ringcco 20591 | Composition in the categor... |
| dfringc2 20592 | Alternate definition of th... |
| rhmsscmap2 20593 | The unital ring homomorphi... |
| rhmsscmap 20594 | The unital ring homomorphi... |
| rhmsubcsetclem1 20595 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20596 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20597 | The unital ring homomorphi... |
| ringccat 20598 | The category of unital rin... |
| ringcid 20599 | The identity arrow in the ... |
| rhmsscrnghm 20600 | The unital ring homomorphi... |
| rhmsubcrngclem1 20601 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20602 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20603 | The unital ring homomorphi... |
| rngcresringcat 20604 | The restriction of the cat... |
| ringcsect 20605 | A section in the category ... |
| ringcinv 20606 | An inverse in the category... |
| ringciso 20607 | An isomorphism in the cate... |
| ringcbasbas 20608 | An element of the base set... |
| funcringcsetc 20609 | The "natural forgetful fun... |
| zrtermoringc 20610 | The zero ring is a termina... |
| zrninitoringc 20611 | The zero ring is not an in... |
| srhmsubclem1 20612 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20613 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20614 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20615 | According to ~ df-subc , t... |
| sringcat 20616 | The restriction of the cat... |
| crhmsubc 20617 | According to ~ df-subc , t... |
| cringcat 20618 | The restriction of the cat... |
| rngcrescrhm 20619 | The category of non-unital... |
| rhmsubclem1 20620 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20621 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20622 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20623 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20624 | According to ~ df-subc , t... |
| rhmsubccat 20625 | The restriction of the cat... |
| rrgval 20632 | Value of the set or left-r... |
| isrrg 20633 | Membership in the set of l... |
| rrgeq0i 20634 | Property of a left-regular... |
| rrgeq0 20635 | Left-multiplication by a l... |
| rrgsupp 20636 | Left multiplication by a l... |
| rrgss 20637 | Left-regular elements are ... |
| unitrrg 20638 | Units are regular elements... |
| rrgnz 20639 | In a nonzero ring, the zer... |
| isdomn 20640 | Expand definition of a dom... |
| domnnzr 20641 | A domain is a nonzero ring... |
| domnring 20642 | A domain is a ring. (Cont... |
| domneq0 20643 | In a domain, a product is ... |
| domnmuln0 20644 | In a domain, a product of ... |
| isdomn5 20645 | The equivalence between th... |
| isdomn2 20646 | A ring is a domain iff all... |
| isdomn2OLD 20647 | Obsolete version of ~ isdo... |
| domnrrg 20648 | In a domain, a nonzero ele... |
| isdomn6 20649 | A ring is a domain iff the... |
| isdomn3 20650 | Nonzero elements form a mu... |
| isdomn4 20651 | A ring is a domain iff it ... |
| opprdomnb 20652 | A class is a domain if and... |
| opprdomn 20653 | The opposite of a domain i... |
| isdomn4r 20654 | A ring is a domain iff it ... |
| domnlcanb 20655 | Left-cancellation law for ... |
| domnlcan 20656 | Left-cancellation law for ... |
| domnrcanb 20657 | Right-cancellation law for... |
| domnrcan 20658 | Right-cancellation law for... |
| domneq0r 20659 | Right multiplication by a ... |
| isidom 20660 | An integral domain is a co... |
| idomdomd 20661 | An integral domain is a do... |
| idomcringd 20662 | An integral domain is a co... |
| idomringd 20663 | An integral domain is a ri... |
| isdrng 20668 | The predicate "is a divisi... |
| drngunit 20669 | Elementhood in the set of ... |
| drngui 20670 | The set of units of a divi... |
| drngring 20671 | A division ring is a ring.... |
| drngringd 20672 | A division ring is a ring.... |
| drnggrpd 20673 | A division ring is a group... |
| drnggrp 20674 | A division ring is a group... |
| isfld 20675 | A field is a commutative d... |
| flddrngd 20676 | A field is a division ring... |
| fldcrngd 20677 | A field is a commutative r... |
| isdrng2 20678 | A division ring can equiva... |
| drngprop 20679 | If two structures have the... |
| drngmgp 20680 | A division ring contains a... |
| drngid 20681 | A division ring's unity is... |
| drngunz 20682 | A division ring's unity is... |
| drngnzr 20683 | A division ring is a nonze... |
| drngdomn 20684 | A division ring is a domai... |
| drngmcl 20685 | The product of two nonzero... |
| drngmclOLD 20686 | Obsolete version of ~ drng... |
| drngid2 20687 | Properties showing that an... |
| drnginvrcl 20688 | Closure of the multiplicat... |
| drnginvrn0 20689 | The multiplicative inverse... |
| drnginvrcld 20690 | Closure of the multiplicat... |
| drnginvrl 20691 | Property of the multiplica... |
| drnginvrr 20692 | Property of the multiplica... |
| drnginvrld 20693 | Property of the multiplica... |
| drnginvrrd 20694 | Property of the multiplica... |
| drngmul0or 20695 | A product is zero iff one ... |
| drngmul0orOLD 20696 | Obsolete version of ~ drng... |
| drngmulne0 20697 | A product is nonzero iff b... |
| drngmuleq0 20698 | An element is zero iff its... |
| opprdrng 20699 | The opposite of a division... |
| isdrngd 20700 | Properties that characteri... |
| isdrngrd 20701 | Properties that characteri... |
| isdrngdOLD 20702 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20703 | Obsolete version of ~ isdr... |
| drngpropd 20704 | If two structures have the... |
| fldpropd 20705 | If two structures have the... |
| fldidom 20706 | A field is an integral dom... |
| fidomndrnglem 20707 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20708 | A finite domain is a divis... |
| fiidomfld 20709 | A finite integral domain i... |
| rng1nnzr 20710 | The (smallest) structure r... |
| ring1zr 20711 | The only (unital) ring wit... |
| rngen1zr 20712 | The only (unital) ring wit... |
| ringen1zr 20713 | The only unital ring with ... |
| rng1nfld 20714 | The zero ring is not a fie... |
| issubdrg 20715 | Characterize the subfields... |
| drhmsubc 20716 | According to ~ df-subc , t... |
| drngcat 20717 | The restriction of the cat... |
| fldcat 20718 | The restriction of the cat... |
| fldc 20719 | The restriction of the cat... |
| fldhmsubc 20720 | According to ~ df-subc , t... |
| issdrg 20723 | Property of a division sub... |
| sdrgrcl 20724 | Reverse closure for a sub-... |
| sdrgdrng 20725 | A sub-division-ring is a d... |
| sdrgsubrg 20726 | A sub-division-ring is a s... |
| sdrgid 20727 | Every division ring is a d... |
| sdrgss 20728 | A division subring is a su... |
| sdrgbas 20729 | Base set of a sub-division... |
| issdrg2 20730 | Property of a division sub... |
| sdrgunit 20731 | A unit of a sub-division-r... |
| imadrhmcl 20732 | The image of a (nontrivial... |
| fldsdrgfld 20733 | A sub-division-ring of a f... |
| acsfn1p 20734 | Construction of a closure ... |
| subrgacs 20735 | Closure property of subrin... |
| sdrgacs 20736 | Closure property of divisi... |
| cntzsdrg 20737 | Centralizers in division r... |
| subdrgint 20738 | The intersection of a none... |
| sdrgint 20739 | The intersection of a none... |
| primefld 20740 | The smallest sub division ... |
| primefld0cl 20741 | The prime field contains t... |
| primefld1cl 20742 | The prime field contains t... |
| abvfval 20745 | Value of the set of absolu... |
| isabv 20746 | Elementhood in the set of ... |
| isabvd 20747 | Properties that determine ... |
| abvrcl 20748 | Reverse closure for the ab... |
| abvfge0 20749 | An absolute value is a fun... |
| abvf 20750 | An absolute value is a fun... |
| abvcl 20751 | An absolute value is a fun... |
| abvge0 20752 | The absolute value of a nu... |
| abveq0 20753 | The value of an absolute v... |
| abvne0 20754 | The absolute value of a no... |
| abvgt0 20755 | The absolute value of a no... |
| abvmul 20756 | An absolute value distribu... |
| abvtri 20757 | An absolute value satisfie... |
| abv0 20758 | The absolute value of zero... |
| abv1z 20759 | The absolute value of one ... |
| abv1 20760 | The absolute value of one ... |
| abvneg 20761 | The absolute value of a ne... |
| abvsubtri 20762 | An absolute value satisfie... |
| abvrec 20763 | The absolute value distrib... |
| abvdiv 20764 | The absolute value distrib... |
| abvdom 20765 | Any ring with an absolute ... |
| abvres 20766 | The restriction of an abso... |
| abvtrivd 20767 | The trivial absolute value... |
| abvtrivg 20768 | The trivial absolute value... |
| abvtriv 20769 | The trivial absolute value... |
| abvpropd 20770 | If two structures have the... |
| abvn0b 20771 | Another characterization o... |
| staffval 20776 | The functionalization of t... |
| stafval 20777 | The functionalization of t... |
| staffn 20778 | The functionalization is e... |
| issrng 20779 | The predicate "is a star r... |
| srngrhm 20780 | The involution function in... |
| srngring 20781 | A star ring is a ring. (C... |
| srngcnv 20782 | The involution function in... |
| srngf1o 20783 | The involution function in... |
| srngcl 20784 | The involution function in... |
| srngnvl 20785 | The involution function in... |
| srngadd 20786 | The involution function in... |
| srngmul 20787 | The involution function in... |
| srng1 20788 | The conjugate of the ring ... |
| srng0 20789 | The conjugate of the ring ... |
| issrngd 20790 | Properties that determine ... |
| idsrngd 20791 | A commutative ring is a st... |
| isorng 20796 | An ordered ring is a ring ... |
| orngring 20797 | An ordered ring is a ring.... |
| orngogrp 20798 | An ordered ring is an orde... |
| isofld 20799 | An ordered field is a fiel... |
| orngmul 20800 | In an ordered ring, the or... |
| orngsqr 20801 | In an ordered ring, all sq... |
| ornglmulle 20802 | In an ordered ring, multip... |
| orngrmulle 20803 | In an ordered ring, multip... |
| ornglmullt 20804 | In an ordered ring, multip... |
| orngrmullt 20805 | In an ordered ring, multip... |
| orngmullt 20806 | In an ordered ring, the st... |
| ofldfld 20807 | An ordered field is a fiel... |
| ofldtos 20808 | An ordered field is a tota... |
| orng0le1 20809 | In an ordered ring, the ri... |
| ofldlt1 20810 | In an ordered field, the r... |
| suborng 20811 | Every subring of an ordere... |
| subofld 20812 | Every subfield of an order... |
| islmod 20817 | The predicate "is a left m... |
| lmodlema 20818 | Lemma for properties of a ... |
| islmodd 20819 | Properties that determine ... |
| lmodgrp 20820 | A left module is a group. ... |
| lmodring 20821 | The scalar component of a ... |
| lmodfgrp 20822 | The scalar component of a ... |
| lmodgrpd 20823 | A left module is a group. ... |
| lmodbn0 20824 | The base set of a left mod... |
| lmodacl 20825 | Closure of ring addition f... |
| lmodmcl 20826 | Closure of ring multiplica... |
| lmodsn0 20827 | The set of scalars in a le... |
| lmodvacl 20828 | Closure of vector addition... |
| lmodass 20829 | Left module vector sum is ... |
| lmodlcan 20830 | Left cancellation law for ... |
| lmodvscl 20831 | Closure of scalar product ... |
| lmodvscld 20832 | Closure of scalar product ... |
| scaffval 20833 | The scalar multiplication ... |
| scafval 20834 | The scalar multiplication ... |
| scafeq 20835 | If the scalar multiplicati... |
| scaffn 20836 | The scalar multiplication ... |
| lmodscaf 20837 | The scalar multiplication ... |
| lmodvsdi 20838 | Distributive law for scala... |
| lmodvsdir 20839 | Distributive law for scala... |
| lmodvsass 20840 | Associative law for scalar... |
| lmod0cl 20841 | The ring zero in a left mo... |
| lmod1cl 20842 | The ring unity in a left m... |
| lmodvs1 20843 | Scalar product with the ri... |
| lmod0vcl 20844 | The zero vector is a vecto... |
| lmod0vlid 20845 | Left identity law for the ... |
| lmod0vrid 20846 | Right identity law for the... |
| lmod0vid 20847 | Identity equivalent to the... |
| lmod0vs 20848 | Zero times a vector is the... |
| lmodvs0 20849 | Anything times the zero ve... |
| lmodvsmmulgdi 20850 | Distributive law for a gro... |
| lmodfopnelem1 20851 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20852 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20853 | The (functionalized) opera... |
| lcomf 20854 | A linear-combination sum i... |
| lcomfsupp 20855 | A linear-combination sum i... |
| lmodvnegcl 20856 | Closure of vector negative... |
| lmodvnegid 20857 | Addition of a vector with ... |
| lmodvneg1 20858 | Minus 1 times a vector is ... |
| lmodvsneg 20859 | Multiplication of a vector... |
| lmodvsubcl 20860 | Closure of vector subtract... |
| lmodcom 20861 | Left module vector sum is ... |
| lmodabl 20862 | A left module is an abelia... |
| lmodcmn 20863 | A left module is a commuta... |
| lmodnegadd 20864 | Distribute negation throug... |
| lmod4 20865 | Commutative/associative la... |
| lmodvsubadd 20866 | Relationship between vecto... |
| lmodvaddsub4 20867 | Vector addition/subtractio... |
| lmodvpncan 20868 | Addition/subtraction cance... |
| lmodvnpcan 20869 | Cancellation law for vecto... |
| lmodvsubval2 20870 | Value of vector subtractio... |
| lmodsubvs 20871 | Subtraction of a scalar pr... |
| lmodsubdi 20872 | Scalar multiplication dist... |
| lmodsubdir 20873 | Scalar multiplication dist... |
| lmodsubeq0 20874 | If the difference between ... |
| lmodsubid 20875 | Subtraction of a vector fr... |
| lmodvsghm 20876 | Scalar multiplication of t... |
| lmodprop2d 20877 | If two structures have the... |
| lmodpropd 20878 | If two structures have the... |
| gsumvsmul 20879 | Pull a scalar multiplicati... |
| mptscmfsupp0 20880 | A mapping to a scalar prod... |
| mptscmfsuppd 20881 | A function mapping to a sc... |
| rmodislmodlem 20882 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20883 | The right module ` R ` ind... |
| lssset 20886 | The set of all (not necess... |
| islss 20887 | The predicate "is a subspa... |
| islssd 20888 | Properties that determine ... |
| lssss 20889 | A subspace is a set of vec... |
| lssel 20890 | A subspace member is a vec... |
| lss1 20891 | The set of vectors in a le... |
| lssuni 20892 | The union of all subspaces... |
| lssn0 20893 | A subspace is not empty. ... |
| 00lss 20894 | The empty structure has no... |
| lsscl 20895 | Closure property of a subs... |
| lssvacl 20896 | Closure of vector addition... |
| lssvsubcl 20897 | Closure of vector subtract... |
| lssvancl1 20898 | Non-closure: if one vector... |
| lssvancl2 20899 | Non-closure: if one vector... |
| lss0cl 20900 | The zero vector belongs to... |
| lsssn0 20901 | The singleton of the zero ... |
| lss0ss 20902 | The zero subspace is inclu... |
| lssle0 20903 | No subspace is smaller tha... |
| lssne0 20904 | A nonzero subspace has a n... |
| lssvneln0 20905 | A vector ` X ` which doesn... |
| lssneln0 20906 | A vector ` X ` which doesn... |
| lssssr 20907 | Conclude subspace ordering... |
| lssvscl 20908 | Closure of scalar product ... |
| lssvnegcl 20909 | Closure of negative vector... |
| lsssubg 20910 | All subspaces are subgroup... |
| lsssssubg 20911 | All subspaces are subgroup... |
| islss3 20912 | A linear subspace of a mod... |
| lsslmod 20913 | A submodule is a module. ... |
| lsslss 20914 | The subspaces of a subspac... |
| islss4 20915 | A linear subspace is a sub... |
| lss1d 20916 | One-dimensional subspace (... |
| lssintcl 20917 | The intersection of a none... |
| lssincl 20918 | The intersection of two su... |
| lssmre 20919 | The subspaces of a module ... |
| lssacs 20920 | Submodules are an algebrai... |
| prdsvscacl 20921 | Pointwise scalar multiplic... |
| prdslmodd 20922 | The product of a family of... |
| pwslmod 20923 | A structure power of a lef... |
| lspfval 20926 | The span function for a le... |
| lspf 20927 | The span function on a lef... |
| lspval 20928 | The span of a set of vecto... |
| lspcl 20929 | The span of a set of vecto... |
| lspsncl 20930 | The span of a singleton is... |
| lspprcl 20931 | The span of a pair is a su... |
| lsptpcl 20932 | The span of an unordered t... |
| lspsnsubg 20933 | The span of a singleton is... |
| 00lsp 20934 | ~ fvco4i lemma for linear ... |
| lspid 20935 | The span of a subspace is ... |
| lspssv 20936 | A span is a set of vectors... |
| lspss 20937 | Span preserves subset orde... |
| lspssid 20938 | A set of vectors is a subs... |
| lspidm 20939 | The span of a set of vecto... |
| lspun 20940 | The span of union is the s... |
| lspssp 20941 | If a set of vectors is a s... |
| mrclsp 20942 | Moore closure generalizes ... |
| lspsnss 20943 | The span of the singleton ... |
| ellspsn3 20944 | A member of the span of th... |
| lspprss 20945 | The span of a pair of vect... |
| lspsnid 20946 | A vector belongs to the sp... |
| ellspsn6 20947 | Relationship between a vec... |
| ellspsn5b 20948 | Relationship between a vec... |
| ellspsn5 20949 | Relationship between a vec... |
| lspprid1 20950 | A member of a pair of vect... |
| lspprid2 20951 | A member of a pair of vect... |
| lspprvacl 20952 | The sum of two vectors bel... |
| lssats2 20953 | A way to express atomistic... |
| ellspsni 20954 | A scalar product with a ve... |
| lspsn 20955 | Span of the singleton of a... |
| ellspsn 20956 | Member of span of the sing... |
| lspsnvsi 20957 | Span of a scalar product o... |
| lspsnss2 20958 | Comparable spans of single... |
| lspsnneg 20959 | Negation does not change t... |
| lspsnsub 20960 | Swapping subtraction order... |
| lspsn0 20961 | Span of the singleton of t... |
| lsp0 20962 | Span of the empty set. (C... |
| lspuni0 20963 | Union of the span of the e... |
| lspun0 20964 | The span of a union with t... |
| lspsneq0 20965 | Span of the singleton is t... |
| lspsneq0b 20966 | Equal singleton spans impl... |
| lmodindp1 20967 | Two independent (non-colin... |
| lsslsp 20968 | Spans in submodules corres... |
| lsslspOLD 20969 | Obsolete version of ~ lssl... |
| lss0v 20970 | The zero vector in a submo... |
| lsspropd 20971 | If two structures have the... |
| lsppropd 20972 | If two structures have the... |
| reldmlmhm 20979 | Lemma for module homomorph... |
| lmimfn 20980 | Lemma for module isomorphi... |
| islmhm 20981 | Property of being a homomo... |
| islmhm3 20982 | Property of a module homom... |
| lmhmlem 20983 | Non-quantified consequence... |
| lmhmsca 20984 | A homomorphism of left mod... |
| lmghm 20985 | A homomorphism of left mod... |
| lmhmlmod2 20986 | A homomorphism of left mod... |
| lmhmlmod1 20987 | A homomorphism of left mod... |
| lmhmf 20988 | A homomorphism of left mod... |
| lmhmlin 20989 | A homomorphism of left mod... |
| lmodvsinv 20990 | Multiplication of a vector... |
| lmodvsinv2 20991 | Multiplying a negated vect... |
| islmhm2 20992 | A one-equation proof of li... |
| islmhmd 20993 | Deduction for a module hom... |
| 0lmhm 20994 | The constant zero linear f... |
| idlmhm 20995 | The identity function on a... |
| invlmhm 20996 | The negative function on a... |
| lmhmco 20997 | The composition of two mod... |
| lmhmplusg 20998 | The pointwise sum of two l... |
| lmhmvsca 20999 | The pointwise scalar produ... |
| lmhmf1o 21000 | A bijective module homomor... |
| lmhmima 21001 | The image of a subspace un... |
| lmhmpreima 21002 | The inverse image of a sub... |
| lmhmlsp 21003 | Homomorphisms preserve spa... |
| lmhmrnlss 21004 | The range of a homomorphis... |
| lmhmkerlss 21005 | The kernel of a homomorphi... |
| reslmhm 21006 | Restriction of a homomorph... |
| reslmhm2 21007 | Expansion of the codomain ... |
| reslmhm2b 21008 | Expansion of the codomain ... |
| lmhmeql 21009 | The equalizer of two modul... |
| lspextmo 21010 | A linear function is compl... |
| pwsdiaglmhm 21011 | Diagonal homomorphism into... |
| pwssplit0 21012 | Splitting for structure po... |
| pwssplit1 21013 | Splitting for structure po... |
| pwssplit2 21014 | Splitting for structure po... |
| pwssplit3 21015 | Splitting for structure po... |
| islmim 21016 | An isomorphism of left mod... |
| lmimf1o 21017 | An isomorphism of left mod... |
| lmimlmhm 21018 | An isomorphism of modules ... |
| lmimgim 21019 | An isomorphism of modules ... |
| islmim2 21020 | An isomorphism of left mod... |
| lmimcnv 21021 | The converse of a bijectiv... |
| brlmic 21022 | The relation "is isomorphi... |
| brlmici 21023 | Prove isomorphic by an exp... |
| lmiclcl 21024 | Isomorphism implies the le... |
| lmicrcl 21025 | Isomorphism implies the ri... |
| lmicsym 21026 | Module isomorphism is symm... |
| lmhmpropd 21027 | Module homomorphism depend... |
| islbs 21030 | The predicate " ` B ` is a... |
| lbsss 21031 | A basis is a set of vector... |
| lbsel 21032 | An element of a basis is a... |
| lbssp 21033 | The span of a basis is the... |
| lbsind 21034 | A basis is linearly indepe... |
| lbsind2 21035 | A basis is linearly indepe... |
| lbspss 21036 | No proper subset of a basi... |
| lsmcl 21037 | The sum of two subspaces i... |
| lsmspsn 21038 | Member of subspace sum of ... |
| lsmelval2 21039 | Subspace sum membership in... |
| lsmsp 21040 | Subspace sum in terms of s... |
| lsmsp2 21041 | Subspace sum of spans of s... |
| lsmssspx 21042 | Subspace sum (in its exten... |
| lsmpr 21043 | The span of a pair of vect... |
| lsppreli 21044 | A vector expressed as a su... |
| lsmelpr 21045 | Two ways to say that a vec... |
| lsppr0 21046 | The span of a vector paire... |
| lsppr 21047 | Span of a pair of vectors.... |
| lspprel 21048 | Member of the span of a pa... |
| lspprabs 21049 | Absorption of vector sum i... |
| lspvadd 21050 | The span of a vector sum i... |
| lspsntri 21051 | Triangle-type inequality f... |
| lspsntrim 21052 | Triangle-type inequality f... |
| lbspropd 21053 | If two structures have the... |
| pj1lmhm 21054 | The left projection functi... |
| pj1lmhm2 21055 | The left projection functi... |
| islvec 21058 | The predicate "is a left v... |
| lvecdrng 21059 | The set of scalars of a le... |
| lveclmod 21060 | A left vector space is a l... |
| lveclmodd 21061 | A vector space is a left m... |
| lvecgrpd 21062 | A vector space is a group.... |
| lsslvec 21063 | A vector subspace is a vec... |
| lmhmlvec 21064 | The property for modules t... |
| lvecvs0or 21065 | If a scalar product is zer... |
| lvecvsn0 21066 | A scalar product is nonzer... |
| lssvs0or 21067 | If a scalar product belong... |
| lvecvscan 21068 | Cancellation law for scala... |
| lvecvscan2 21069 | Cancellation law for scala... |
| lvecinv 21070 | Invert coefficient of scal... |
| lspsnvs 21071 | A nonzero scalar product d... |
| lspsneleq 21072 | Membership relation that i... |
| lspsncmp 21073 | Comparable spans of nonzer... |
| lspsnne1 21074 | Two ways to express that v... |
| lspsnne2 21075 | Two ways to express that v... |
| lspsnnecom 21076 | Swap two vectors with diff... |
| lspabs2 21077 | Absorption law for span of... |
| lspabs3 21078 | Absorption law for span of... |
| lspsneq 21079 | Equal spans of singletons ... |
| lspsneu 21080 | Nonzero vectors with equal... |
| ellspsn4 21081 | A member of the span of th... |
| lspdisj 21082 | The span of a vector not i... |
| lspdisjb 21083 | A nonzero vector is not in... |
| lspdisj2 21084 | Unequal spans are disjoint... |
| lspfixed 21085 | Show membership in the spa... |
| lspexch 21086 | Exchange property for span... |
| lspexchn1 21087 | Exchange property for span... |
| lspexchn2 21088 | Exchange property for span... |
| lspindpi 21089 | Partial independence prope... |
| lspindp1 21090 | Alternate way to say 3 vec... |
| lspindp2l 21091 | Alternate way to say 3 vec... |
| lspindp2 21092 | Alternate way to say 3 vec... |
| lspindp3 21093 | Independence of 2 vectors ... |
| lspindp4 21094 | (Partial) independence of ... |
| lvecindp 21095 | Compute the ` X ` coeffici... |
| lvecindp2 21096 | Sums of independent vector... |
| lspsnsubn0 21097 | Unequal singleton spans im... |
| lsmcv 21098 | Subspace sum has the cover... |
| lspsolvlem 21099 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21100 | If ` X ` is in the span of... |
| lssacsex 21101 | In a vector space, subspac... |
| lspsnat 21102 | There is no subspace stric... |
| lspsncv0 21103 | The span of a singleton co... |
| lsppratlem1 21104 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21105 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21106 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21107 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21108 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21109 | Lemma for ~ lspprat . Neg... |
| lspprat 21110 | A proper subspace of the s... |
| islbs2 21111 | An equivalent formulation ... |
| islbs3 21112 | An equivalent formulation ... |
| lbsacsbs 21113 | Being a basis in a vector ... |
| lvecdim 21114 | The dimension theorem for ... |
| lbsextlem1 21115 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21116 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21117 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21118 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21119 | For any linearly independe... |
| lbsext 21120 | For any linearly independe... |
| lbsexg 21121 | Every vector space has a b... |
| lbsex 21122 | Every vector space has a b... |
| lvecprop2d 21123 | If two structures have the... |
| lvecpropd 21124 | If two structures have the... |
| sraval 21129 | Lemma for ~ srabase throug... |
| sralem 21130 | Lemma for ~ srabase and si... |
| srabase 21131 | Base set of a subring alge... |
| sraaddg 21132 | Additive operation of a su... |
| sramulr 21133 | Multiplicative operation o... |
| srasca 21134 | The set of scalars of a su... |
| sravsca 21135 | The scalar product operati... |
| sraip 21136 | The inner product operatio... |
| sratset 21137 | Topology component of a su... |
| sratopn 21138 | Topology component of a su... |
| srads 21139 | Distance function of a sub... |
| sraring 21140 | Condition for a subring al... |
| sralmod 21141 | The subring algebra is a l... |
| sralmod0 21142 | The subring module inherit... |
| issubrgd 21143 | Prove a subring by closure... |
| rlmfn 21144 | ` ringLMod ` is a function... |
| rlmval 21145 | Value of the ring module. ... |
| rlmval2 21146 | Value of the ring module e... |
| rlmbas 21147 | Base set of the ring modul... |
| rlmplusg 21148 | Vector addition in the rin... |
| rlm0 21149 | Zero vector in the ring mo... |
| rlmsub 21150 | Subtraction in the ring mo... |
| rlmmulr 21151 | Ring multiplication in the... |
| rlmsca 21152 | Scalars in the ring module... |
| rlmsca2 21153 | Scalars in the ring module... |
| rlmvsca 21154 | Scalar multiplication in t... |
| rlmtopn 21155 | Topology component of the ... |
| rlmds 21156 | Metric component of the ri... |
| rlmlmod 21157 | The ring module is a modul... |
| rlmlvec 21158 | The ring module over a div... |
| rlmlsm 21159 | Subgroup sum of the ring m... |
| rlmvneg 21160 | Vector negation in the rin... |
| rlmscaf 21161 | Functionalized scalar mult... |
| ixpsnbasval 21162 | The value of an infinite C... |
| lidlval 21167 | Value of the set of ring i... |
| rspval 21168 | Value of the ring span fun... |
| lidlss 21169 | An ideal is a subset of th... |
| lidlssbas 21170 | The base set of the restri... |
| lidlbas 21171 | A (left) ideal of a ring i... |
| islidl 21172 | Predicate of being a (left... |
| rnglidlmcl 21173 | A (left) ideal containing ... |
| rngridlmcl 21174 | A right ideal (which is a ... |
| dflidl2rng 21175 | Alternate (the usual textb... |
| isridlrng 21176 | A right ideal is a left id... |
| lidl0cl 21177 | An ideal contains 0. (Con... |
| lidlacl 21178 | An ideal is closed under a... |
| lidlnegcl 21179 | An ideal contains negative... |
| lidlsubg 21180 | An ideal is a subgroup of ... |
| lidlsubcl 21181 | An ideal is closed under s... |
| lidlmcl 21182 | An ideal is closed under l... |
| lidl1el 21183 | An ideal contains 1 iff it... |
| dflidl2 21184 | Alternate (the usual textb... |
| lidl0ALT 21185 | Alternate proof for ~ lidl... |
| rnglidl0 21186 | Every non-unital ring cont... |
| lidl0 21187 | Every ring contains a zero... |
| lidl1ALT 21188 | Alternate proof for ~ lidl... |
| rnglidl1 21189 | The base set of every non-... |
| lidl1 21190 | Every ring contains a unit... |
| lidlacs 21191 | The ideal system is an alg... |
| rspcl 21192 | The span of a set of ring ... |
| rspssid 21193 | The span of a set of ring ... |
| rsp1 21194 | The span of the identity e... |
| rsp0 21195 | The span of the zero eleme... |
| rspssp 21196 | The ideal span of a set of... |
| elrspsn 21197 | Membership in a principal ... |
| mrcrsp 21198 | Moore closure generalizes ... |
| lidlnz 21199 | A nonzero ideal contains a... |
| drngnidl 21200 | A division ring has only t... |
| lidlrsppropd 21201 | The left ideals and ring s... |
| rnglidlmmgm 21202 | The multiplicative group o... |
| rnglidlmsgrp 21203 | The multiplicative group o... |
| rnglidlrng 21204 | A (left) ideal of a non-un... |
| lidlnsg 21205 | An ideal is a normal subgr... |
| 2idlval 21208 | Definition of a two-sided ... |
| isridl 21209 | A right ideal is a left id... |
| 2idlelb 21210 | Membership in a two-sided ... |
| 2idllidld 21211 | A two-sided ideal is a lef... |
| 2idlridld 21212 | A two-sided ideal is a rig... |
| df2idl2rng 21213 | Alternate (the usual textb... |
| df2idl2 21214 | Alternate (the usual textb... |
| ridl0 21215 | Every ring contains a zero... |
| ridl1 21216 | Every ring contains a unit... |
| 2idl0 21217 | Every ring contains a zero... |
| 2idl1 21218 | Every ring contains a unit... |
| 2idlss 21219 | A two-sided ideal is a sub... |
| 2idlbas 21220 | The base set of a two-side... |
| 2idlelbas 21221 | The base set of a two-side... |
| rng2idlsubrng 21222 | A two-sided ideal of a non... |
| rng2idlnsg 21223 | A two-sided ideal of a non... |
| rng2idl0 21224 | The zero (additive identit... |
| rng2idlsubgsubrng 21225 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21226 | A two-sided ideal of a non... |
| rng2idlsubg0 21227 | The zero (additive identit... |
| 2idlcpblrng 21228 | The coset equivalence rela... |
| 2idlcpbl 21229 | The coset equivalence rela... |
| qus2idrng 21230 | The quotient of a non-unit... |
| qus1 21231 | The multiplicative identit... |
| qusring 21232 | If ` S ` is a two-sided id... |
| qusrhm 21233 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21234 | The preimage of an ideal b... |
| kerlidl 21235 | The kernel of a ring homom... |
| qusmul2idl 21236 | Value of the ring operatio... |
| crngridl 21237 | In a commutative ring, the... |
| crng2idl 21238 | In a commutative ring, a t... |
| qusmulrng 21239 | Value of the multiplicatio... |
| quscrng 21240 | The quotient of a commutat... |
| qusmulcrng 21241 | Value of the ring operatio... |
| rhmqusnsg 21242 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21243 | The ring unity of a two-si... |
| rngqiprngghmlem1 21244 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21245 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21246 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21247 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21248 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21249 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21250 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21251 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21252 | The base set of the produc... |
| rngqiprng 21253 | The product of the quotien... |
| rngqiprngimf 21254 | ` F ` is a function from (... |
| rngqiprngimfv 21255 | The value of the function ... |
| rngqiprngghm 21256 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21257 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21258 | ` F ` is a function from (... |
| rngqiprnglin 21259 | ` F ` is linear with respe... |
| rngqiprngho 21260 | ` F ` is a homomorphism of... |
| rngqiprngim 21261 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21262 | The unity of a two-sided i... |
| rngringbdlem1 21263 | In a unital ring, the quot... |
| rngringbdlem2 21264 | A non-unital ring is unita... |
| rngringbd 21265 | A non-unital ring is unita... |
| ring2idlqus 21266 | For every unital ring ther... |
| ring2idlqusb 21267 | A non-unital ring is unita... |
| rngqiprngfulem1 21268 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21269 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21270 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21271 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21272 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21273 | The ring unity of the prod... |
| rngqiprngfu 21274 | The function value of ` F ... |
| rngqiprngu 21275 | If a non-unital ring has a... |
| ring2idlqus1 21276 | If a non-unital ring has a... |
| lpival 21281 | Value of the set of princi... |
| islpidl 21282 | Property of being a princi... |
| lpi0 21283 | The zero ideal is always p... |
| lpi1 21284 | The unit ideal is always p... |
| islpir 21285 | Principal ideal rings are ... |
| lpiss 21286 | Principal ideals are a sub... |
| islpir2 21287 | Principal ideal rings are ... |
| lpirring 21288 | Principal ideal rings are ... |
| drnglpir 21289 | Division rings are princip... |
| rspsn 21290 | Membership in principal id... |
| lidldvgen 21291 | An element generates an id... |
| lpigen 21292 | An ideal is principal iff ... |
| cnfldstr 21313 | The field of complex numbe... |
| cnfldex 21314 | The field of complex numbe... |
| cnfldbas 21315 | The base set of the field ... |
| mpocnfldadd 21316 | The addition operation of ... |
| cnfldadd 21317 | The addition operation of ... |
| mpocnfldmul 21318 | The multiplication operati... |
| cnfldmul 21319 | The multiplication operati... |
| cnfldcj 21320 | The conjugation operation ... |
| cnfldtset 21321 | The topology component of ... |
| cnfldle 21322 | The ordering of the field ... |
| cnfldds 21323 | The metric of the field of... |
| cnfldunif 21324 | The uniform structure comp... |
| cnfldfun 21325 | The field of complex numbe... |
| cnfldfunALT 21326 | The field of complex numbe... |
| dfcnfldOLD 21327 | Obsolete version of ~ df-c... |
| cnfldstrOLD 21328 | Obsolete version of ~ cnfl... |
| cnfldexOLD 21329 | Obsolete version of ~ cnfl... |
| cnfldbasOLD 21330 | Obsolete version of ~ cnfl... |
| cnfldaddOLD 21331 | Obsolete version of ~ cnfl... |
| cnfldmulOLD 21332 | Obsolete version of ~ cnfl... |
| cnfldcjOLD 21333 | Obsolete version of ~ cnfl... |
| cnfldtsetOLD 21334 | Obsolete version of ~ cnfl... |
| cnfldleOLD 21335 | Obsolete version of ~ cnfl... |
| cnflddsOLD 21336 | Obsolete version of ~ cnfl... |
| cnfldunifOLD 21337 | Obsolete version of ~ cnfl... |
| cnfldfunOLD 21338 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLD 21339 | Obsolete version of ~ cnfl... |
| xrsstr 21340 | The extended real structur... |
| xrsex 21341 | The extended real structur... |
| xrsadd 21342 | The addition operation of ... |
| xrsmul 21343 | The multiplication operati... |
| xrstset 21344 | The topology component of ... |
| cncrng 21345 | The complex numbers form a... |
| cncrngOLD 21346 | Obsolete version of ~ cncr... |
| cnring 21347 | The complex numbers form a... |
| xrsmcmn 21348 | The "multiplicative group"... |
| cnfld0 21349 | Zero is the zero element o... |
| cnfld1 21350 | One is the unity element o... |
| cnfld1OLD 21351 | Obsolete version of ~ cnfl... |
| cnfldneg 21352 | The additive inverse in th... |
| cnfldplusf 21353 | The functionalized additio... |
| cnfldsub 21354 | The subtraction operator i... |
| cndrng 21355 | The complex numbers form a... |
| cndrngOLD 21356 | Obsolete version of ~ cndr... |
| cnflddiv 21357 | The division operation in ... |
| cnflddivOLD 21358 | Obsolete version of ~ cnfl... |
| cnfldinv 21359 | The multiplicative inverse... |
| cnfldmulg 21360 | The group multiple functio... |
| cnfldexp 21361 | The exponentiation operato... |
| cnsrng 21362 | The complex numbers form a... |
| xrsmgm 21363 | The "additive group" of th... |
| xrsnsgrp 21364 | The "additive group" of th... |
| xrsmgmdifsgrp 21365 | The "additive group" of th... |
| xrsds 21366 | The metric of the extended... |
| xrsdsval 21367 | The metric of the extended... |
| xrsdsreval 21368 | The metric of the extended... |
| xrsdsreclblem 21369 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21370 | The metric of the extended... |
| cnsubmlem 21371 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21372 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21373 | Lemma for ~ resubdrg and f... |
| cnsubrglemOLD 21374 | Obsolete version of ~ cnsu... |
| cnsubdrglem 21375 | Lemma for ~ resubdrg and f... |
| qsubdrg 21376 | The rational numbers form ... |
| zsubrg 21377 | The integers form a subrin... |
| gzsubrg 21378 | The gaussian integers form... |
| nn0subm 21379 | The nonnegative integers f... |
| rege0subm 21380 | The nonnegative reals form... |
| absabv 21381 | The regular absolute value... |
| zsssubrg 21382 | The integers are a subset ... |
| qsssubdrg 21383 | The rational numbers are a... |
| cnsubrg 21384 | There are no subrings of t... |
| cnmgpabl 21385 | The unit group of the comp... |
| cnmgpid 21386 | The group identity element... |
| cnmsubglem 21387 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21388 | The positive reals form a ... |
| gzrngunitlem 21389 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21390 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21391 | Relate a group sum on ` CC... |
| regsumfsum 21392 | Relate a group sum on ` ( ... |
| expmhm 21393 | Exponentiation is a monoid... |
| nn0srg 21394 | The nonnegative integers f... |
| rge0srg 21395 | The nonnegative real numbe... |
| xrge0plusg 21396 | The additive law of the ex... |
| xrs1mnd 21397 | The extended real numbers,... |
| xrs10 21398 | The zero of the extended r... |
| xrs1cmn 21399 | The extended real numbers ... |
| xrge0subm 21400 | The nonnegative extended r... |
| xrge0cmn 21401 | The nonnegative extended r... |
| xrge0omnd 21402 | The nonnegative extended r... |
| zringcrng 21405 | The ring of integers is a ... |
| zringring 21406 | The ring of integers is a ... |
| zringrng 21407 | The ring of integers is a ... |
| zringabl 21408 | The ring of integers is an... |
| zringgrp 21409 | The ring of integers is an... |
| zringbas 21410 | The integers are the base ... |
| zringplusg 21411 | The addition operation of ... |
| zringsub 21412 | The subtraction of element... |
| zringmulg 21413 | The multiplication (group ... |
| zringmulr 21414 | The multiplication operati... |
| zring0 21415 | The zero element of the ri... |
| zring1 21416 | The unity element of the r... |
| zringnzr 21417 | The ring of integers is a ... |
| dvdsrzring 21418 | Ring divisibility in the r... |
| zringlpirlem1 21419 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21420 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21421 | Lemma for ~ zringlpir . A... |
| zringinvg 21422 | The additive inverse of an... |
| zringunit 21423 | The units of ` ZZ ` are th... |
| zringlpir 21424 | The integers are a princip... |
| zringndrg 21425 | The integers are not a div... |
| zringcyg 21426 | The integers are a cyclic ... |
| zringsubgval 21427 | Subtraction in the ring of... |
| zringmpg 21428 | The multiplicative group o... |
| prmirredlem 21429 | A positive integer is irre... |
| dfprm2 21430 | The positive irreducible e... |
| prmirred 21431 | The irreducible elements o... |
| expghm 21432 | Exponentiation is a group ... |
| mulgghm2 21433 | The powers of a group elem... |
| mulgrhm 21434 | The powers of the element ... |
| mulgrhm2 21435 | The powers of the element ... |
| irinitoringc 21436 | The ring of integers is an... |
| nzerooringczr 21437 | There is no zero object in... |
| pzriprnglem1 21438 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21439 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21440 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21441 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21442 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21443 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21444 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21445 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21446 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21447 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21448 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21449 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21450 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21451 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21452 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21453 | The ring unity of the ring... |
| pzriprng 21454 | The non-unital ring ` ( ZZ... |
| pzriprng1 21455 | The ring unity of the ring... |
| zrhval 21464 | Define the unique homomorp... |
| zrhval2 21465 | Alternate value of the ` Z... |
| zrhmulg 21466 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21467 | The ` ZRHom ` homomorphism... |
| zrhrhm 21468 | The ` ZRHom ` homomorphism... |
| zrh1 21469 | Interpretation of 1 in a r... |
| zrh0 21470 | Interpretation of 0 in a r... |
| zrhpropd 21471 | The ` ZZ ` ring homomorphi... |
| zlmval 21472 | Augment an abelian group w... |
| zlmlem 21473 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21474 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21475 | Group operation of a ` ZZ ... |
| zlmmulr 21476 | Ring operation of a ` ZZ `... |
| zlmsca 21477 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21478 | Scalar multiplication oper... |
| zlmlmod 21479 | The ` ZZ ` -module operati... |
| chrval 21480 | Definition substitution of... |
| chrcl 21481 | Closure of the characteris... |
| chrid 21482 | The canonical ` ZZ ` ring ... |
| chrdvds 21483 | The ` ZZ ` ring homomorphi... |
| chrcong 21484 | If two integers are congru... |
| dvdschrmulg 21485 | In a ring, any multiple of... |
| fermltlchr 21486 | A generalization of Fermat... |
| chrnzr 21487 | Nonzero rings are precisel... |
| chrrhm 21488 | The characteristic restric... |
| domnchr 21489 | The characteristic of a do... |
| znlidl 21490 | The set ` n ZZ ` is an ide... |
| zncrng2 21491 | Making a commutative ring ... |
| znval 21492 | The value of the ` Z/nZ ` ... |
| znle 21493 | The value of the ` Z/nZ ` ... |
| znval2 21494 | Self-referential expressio... |
| znbaslem 21495 | Lemma for ~ znbas . (Cont... |
| znbas2 21496 | The base set of ` Z/nZ ` i... |
| znadd 21497 | The additive structure of ... |
| znmul 21498 | The multiplicative structu... |
| znzrh 21499 | The ` ZZ ` ring homomorphi... |
| znbas 21500 | The base set of ` Z/nZ ` s... |
| zncrng 21501 | ` Z/nZ ` is a commutative ... |
| znzrh2 21502 | The ` ZZ ` ring homomorphi... |
| znzrhval 21503 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21504 | The ` ZZ ` ring homomorphi... |
| zncyg 21505 | The group ` ZZ / n ZZ ` is... |
| zndvds 21506 | Express equality of equiva... |
| zndvds0 21507 | Special case of ~ zndvds w... |
| znf1o 21508 | The function ` F ` enumera... |
| zzngim 21509 | The ` ZZ ` ring homomorphi... |
| znle2 21510 | The ordering of the ` Z/nZ... |
| znleval 21511 | The ordering of the ` Z/nZ... |
| znleval2 21512 | The ordering of the ` Z/nZ... |
| zntoslem 21513 | Lemma for ~ zntos . (Cont... |
| zntos 21514 | The ` Z/nZ ` structure is ... |
| znhash 21515 | The ` Z/nZ ` structure has... |
| znfi 21516 | The ` Z/nZ ` structure is ... |
| znfld 21517 | The ` Z/nZ ` structure is ... |
| znidomb 21518 | The ` Z/nZ ` structure is ... |
| znchr 21519 | Cyclic rings are defined b... |
| znunit 21520 | The units of ` Z/nZ ` are ... |
| znunithash 21521 | The size of the unit group... |
| znrrg 21522 | The regular elements of ` ... |
| cygznlem1 21523 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21524 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21525 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21526 | A cyclic group with ` n ` ... |
| cygzn 21527 | A cyclic group with ` n ` ... |
| cygth 21528 | The "fundamental theorem o... |
| cyggic 21529 | Cyclic groups are isomorph... |
| frgpcyg 21530 | A free group is cyclic iff... |
| freshmansdream 21531 | For a prime number ` P ` ,... |
| frobrhm 21532 | In a commutative ring with... |
| ofldchr 21533 | The characteristic of an o... |
| cnmsgnsubg 21534 | The signs form a multiplic... |
| cnmsgnbas 21535 | The base set of the sign s... |
| cnmsgngrp 21536 | The group of signs under m... |
| psgnghm 21537 | The sign is a homomorphism... |
| psgnghm2 21538 | The sign is a homomorphism... |
| psgninv 21539 | The sign of a permutation ... |
| psgnco 21540 | Multiplicativity of the pe... |
| zrhpsgnmhm 21541 | Embedding of permutation s... |
| zrhpsgninv 21542 | The embedded sign of a per... |
| evpmss 21543 | Even permutations are perm... |
| psgnevpmb 21544 | A class is an even permuta... |
| psgnodpm 21545 | A permutation which is odd... |
| psgnevpm 21546 | A permutation which is eve... |
| psgnodpmr 21547 | If a permutation has sign ... |
| zrhpsgnevpm 21548 | The sign of an even permut... |
| zrhpsgnodpm 21549 | The sign of an odd permuta... |
| cofipsgn 21550 | Composition of any class `... |
| zrhpsgnelbas 21551 | Embedding of permutation s... |
| zrhcopsgnelbas 21552 | Embedding of permutation s... |
| evpmodpmf1o 21553 | The function for performin... |
| pmtrodpm 21554 | A transposition is an odd ... |
| psgnfix1 21555 | A permutation of a finite ... |
| psgnfix2 21556 | A permutation of a finite ... |
| psgndiflemB 21557 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21558 | Lemma 2 for ~ psgndif . (... |
| psgndif 21559 | Embedding of permutation s... |
| copsgndif 21560 | Embedding of permutation s... |
| rebase 21563 | The base of the field of r... |
| remulg 21564 | The multiplication (group ... |
| resubdrg 21565 | The real numbers form a di... |
| resubgval 21566 | Subtraction in the field o... |
| replusg 21567 | The addition operation of ... |
| remulr 21568 | The multiplication operati... |
| re0g 21569 | The zero element of the fi... |
| re1r 21570 | The unity element of the f... |
| rele2 21571 | The ordering relation of t... |
| relt 21572 | The ordering relation of t... |
| reds 21573 | The distance of the field ... |
| redvr 21574 | The division operation of ... |
| retos 21575 | The real numbers are a tot... |
| refld 21576 | The real numbers form a fi... |
| refldcj 21577 | The conjugation operation ... |
| resrng 21578 | The real numbers form a st... |
| regsumsupp 21579 | The group sum over the rea... |
| rzgrp 21580 | The quotient group ` RR / ... |
| isphl 21585 | The predicate "is a genera... |
| phllvec 21586 | A pre-Hilbert space is a l... |
| phllmod 21587 | A pre-Hilbert space is a l... |
| phlsrng 21588 | The scalar ring of a pre-H... |
| phllmhm 21589 | The inner product of a pre... |
| ipcl 21590 | Closure of the inner produ... |
| ipcj 21591 | Conjugate of an inner prod... |
| iporthcom 21592 | Orthogonality (meaning inn... |
| ip0l 21593 | Inner product with a zero ... |
| ip0r 21594 | Inner product with a zero ... |
| ipeq0 21595 | The inner product of a vec... |
| ipdir 21596 | Distributive law for inner... |
| ipdi 21597 | Distributive law for inner... |
| ip2di 21598 | Distributive law for inner... |
| ipsubdir 21599 | Distributive law for inner... |
| ipsubdi 21600 | Distributive law for inner... |
| ip2subdi 21601 | Distributive law for inner... |
| ipass 21602 | Associative law for inner ... |
| ipassr 21603 | "Associative" law for seco... |
| ipassr2 21604 | "Associative" law for inne... |
| ipffval 21605 | The inner product operatio... |
| ipfval 21606 | The inner product operatio... |
| ipfeq 21607 | If the inner product opera... |
| ipffn 21608 | The inner product operatio... |
| phlipf 21609 | The inner product operatio... |
| ip2eq 21610 | Two vectors are equal iff ... |
| isphld 21611 | Properties that determine ... |
| phlpropd 21612 | If two structures have the... |
| ssipeq 21613 | The inner product on a sub... |
| phssipval 21614 | The inner product on a sub... |
| phssip 21615 | The inner product (as a fu... |
| phlssphl 21616 | A subspace of an inner pro... |
| ocvfval 21623 | The orthocomplement operat... |
| ocvval 21624 | Value of the orthocompleme... |
| elocv 21625 | Elementhood in the orthoco... |
| ocvi 21626 | Property of a member of th... |
| ocvss 21627 | The orthocomplement of a s... |
| ocvocv 21628 | A set is contained in its ... |
| ocvlss 21629 | The orthocomplement of a s... |
| ocv2ss 21630 | Orthocomplements reverse s... |
| ocvin 21631 | An orthocomplement has tri... |
| ocvsscon 21632 | Two ways to say that ` S `... |
| ocvlsp 21633 | The orthocomplement of a l... |
| ocv0 21634 | The orthocomplement of the... |
| ocvz 21635 | The orthocomplement of the... |
| ocv1 21636 | The orthocomplement of the... |
| unocv 21637 | The orthocomplement of a u... |
| iunocv 21638 | The orthocomplement of an ... |
| cssval 21639 | The set of closed subspace... |
| iscss 21640 | The predicate "is a closed... |
| cssi 21641 | Property of a closed subsp... |
| cssss 21642 | A closed subspace is a sub... |
| iscss2 21643 | It is sufficient to prove ... |
| ocvcss 21644 | The orthocomplement of any... |
| cssincl 21645 | The zero subspace is a clo... |
| css0 21646 | The zero subspace is a clo... |
| css1 21647 | The whole space is a close... |
| csslss 21648 | A closed subspace of a pre... |
| lsmcss 21649 | A subset of a pre-Hilbert ... |
| cssmre 21650 | The closed subspaces of a ... |
| mrccss 21651 | The Moore closure correspo... |
| thlval 21652 | Value of the Hilbert latti... |
| thlbas 21653 | Base set of the Hilbert la... |
| thlle 21654 | Ordering on the Hilbert la... |
| thlleval 21655 | Ordering on the Hilbert la... |
| thloc 21656 | Orthocomplement on the Hil... |
| pjfval 21663 | The value of the projectio... |
| pjdm 21664 | A subspace is in the domai... |
| pjpm 21665 | The projection map is a pa... |
| pjfval2 21666 | Value of the projection ma... |
| pjval 21667 | Value of the projection ma... |
| pjdm2 21668 | A subspace is in the domai... |
| pjff 21669 | A projection is a linear o... |
| pjf 21670 | A projection is a function... |
| pjf2 21671 | A projection is a function... |
| pjfo 21672 | A projection is a surjecti... |
| pjcss 21673 | A projection subspace is a... |
| ocvpj 21674 | The orthocomplement of a p... |
| ishil 21675 | The predicate "is a Hilber... |
| ishil2 21676 | The predicate "is a Hilber... |
| isobs 21677 | The predicate "is an ortho... |
| obsip 21678 | The inner product of two e... |
| obsipid 21679 | A basis element has length... |
| obsrcl 21680 | Reverse closure for an ort... |
| obsss 21681 | An orthonormal basis is a ... |
| obsne0 21682 | A basis element is nonzero... |
| obsocv 21683 | An orthonormal basis has t... |
| obs2ocv 21684 | The double orthocomplement... |
| obselocv 21685 | A basis element is in the ... |
| obs2ss 21686 | A basis has no proper subs... |
| obslbs 21687 | An orthogonal basis is a l... |
| reldmdsmm 21690 | The direct sum is a well-b... |
| dsmmval 21691 | Value of the module direct... |
| dsmmbase 21692 | Base set of the module dir... |
| dsmmval2 21693 | Self-referential definitio... |
| dsmmbas2 21694 | Base set of the direct sum... |
| dsmmfi 21695 | For finite products, the d... |
| dsmmelbas 21696 | Membership in the finitely... |
| dsmm0cl 21697 | The all-zero vector is con... |
| dsmmacl 21698 | The finite hull is closed ... |
| prdsinvgd2 21699 | Negation of a single coord... |
| dsmmsubg 21700 | The finite hull of a produ... |
| dsmmlss 21701 | The finite hull of a produ... |
| dsmmlmod 21702 | The direct sum of a family... |
| frlmval 21705 | Value of the "free module"... |
| frlmlmod 21706 | The free module is a modul... |
| frlmpws 21707 | The free module as a restr... |
| frlmlss 21708 | The base set of the free m... |
| frlmpwsfi 21709 | The finite free module is ... |
| frlmsca 21710 | The ring of scalars of a f... |
| frlm0 21711 | Zero in a free module (rin... |
| frlmbas 21712 | Base set of the free modul... |
| frlmelbas 21713 | Membership in the base set... |
| frlmrcl 21714 | If a free module is inhabi... |
| frlmbasfsupp 21715 | Elements of the free modul... |
| frlmbasmap 21716 | Elements of the free modul... |
| frlmbasf 21717 | Elements of the free modul... |
| frlmlvec 21718 | The free module over a div... |
| frlmfibas 21719 | The base set of the finite... |
| elfrlmbasn0 21720 | If the dimension of a free... |
| frlmplusgval 21721 | Addition in a free module.... |
| frlmsubgval 21722 | Subtraction in a free modu... |
| frlmvscafval 21723 | Scalar multiplication in a... |
| frlmvplusgvalc 21724 | Coordinates of a sum with ... |
| frlmvscaval 21725 | Coordinates of a scalar mu... |
| frlmplusgvalb 21726 | Addition in a free module ... |
| frlmvscavalb 21727 | Scalar multiplication in a... |
| frlmvplusgscavalb 21728 | Addition combined with sca... |
| frlmgsum 21729 | Finite commutative sums in... |
| frlmsplit2 21730 | Restriction is homomorphic... |
| frlmsslss 21731 | A subset of a free module ... |
| frlmsslss2 21732 | A subset of a free module ... |
| frlmbas3 21733 | An element of the base set... |
| mpofrlmd 21734 | Elements of the free modul... |
| frlmip 21735 | The inner product of a fre... |
| frlmipval 21736 | The inner product of a fre... |
| frlmphllem 21737 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21738 | Conditions for a free modu... |
| uvcfval 21741 | Value of the unit-vector g... |
| uvcval 21742 | Value of a single unit vec... |
| uvcvval 21743 | Value of a unit vector coo... |
| uvcvvcl 21744 | A coordinate of a unit vec... |
| uvcvvcl2 21745 | A unit vector coordinate i... |
| uvcvv1 21746 | The unit vector is one at ... |
| uvcvv0 21747 | The unit vector is zero at... |
| uvcff 21748 | Domain and codomain of the... |
| uvcf1 21749 | In a nonzero ring, each un... |
| uvcresum 21750 | Any element of a free modu... |
| frlmssuvc1 21751 | A scalar multiple of a uni... |
| frlmssuvc2 21752 | A nonzero scalar multiple ... |
| frlmsslsp 21753 | A subset of a free module ... |
| frlmlbs 21754 | The unit vectors comprise ... |
| frlmup1 21755 | Any assignment of unit vec... |
| frlmup2 21756 | The evaluation map has the... |
| frlmup3 21757 | The range of such an evalu... |
| frlmup4 21758 | Universal property of the ... |
| ellspd 21759 | The elements of the span o... |
| elfilspd 21760 | Simplified version of ~ el... |
| rellindf 21765 | The independent-family pre... |
| islinds 21766 | Property of an independent... |
| linds1 21767 | An independent set of vect... |
| linds2 21768 | An independent set of vect... |
| islindf 21769 | Property of an independent... |
| islinds2 21770 | Expanded property of an in... |
| islindf2 21771 | Property of an independent... |
| lindff 21772 | Functional property of a l... |
| lindfind 21773 | A linearly independent fam... |
| lindsind 21774 | A linearly independent set... |
| lindfind2 21775 | In a linearly independent ... |
| lindsind2 21776 | In a linearly independent ... |
| lindff1 21777 | A linearly independent fam... |
| lindfrn 21778 | The range of an independen... |
| f1lindf 21779 | Rearranging and deleting e... |
| lindfres 21780 | Any restriction of an inde... |
| lindsss 21781 | Any subset of an independe... |
| f1linds 21782 | A family constructed from ... |
| islindf3 21783 | In a nonzero ring, indepen... |
| lindfmm 21784 | Linear independence of a f... |
| lindsmm 21785 | Linear independence of a s... |
| lindsmm2 21786 | The monomorphic image of a... |
| lsslindf 21787 | Linear independence is unc... |
| lsslinds 21788 | Linear independence is unc... |
| islbs4 21789 | A basis is an independent ... |
| lbslinds 21790 | A basis is independent. (... |
| islinds3 21791 | A subset is linearly indep... |
| islinds4 21792 | A set is independent in a ... |
| lmimlbs 21793 | The isomorphic image of a ... |
| lmiclbs 21794 | Having a basis is an isomo... |
| islindf4 21795 | A family is independent if... |
| islindf5 21796 | A family is independent if... |
| indlcim 21797 | An independent, spanning f... |
| lbslcic 21798 | A module with a basis is i... |
| lmisfree 21799 | A module has a basis iff i... |
| lvecisfrlm 21800 | Every vector space is isom... |
| lmimco 21801 | The composition of two iso... |
| lmictra 21802 | Module isomorphism is tran... |
| uvcf1o 21803 | In a nonzero ring, the map... |
| uvcendim 21804 | In a nonzero ring, the num... |
| frlmisfrlm 21805 | A free module is isomorphi... |
| frlmiscvec 21806 | Every free module is isomo... |
| isassa 21813 | The properties of an assoc... |
| assalem 21814 | The properties of an assoc... |
| assaass 21815 | Left-associative property ... |
| assaassr 21816 | Right-associative property... |
| assalmod 21817 | An associative algebra is ... |
| assaring 21818 | An associative algebra is ... |
| assasca 21819 | The scalars of an associat... |
| assa2ass 21820 | Left- and right-associativ... |
| assa2ass2 21821 | Left- and right-associativ... |
| isassad 21822 | Sufficient condition for b... |
| issubassa3 21823 | A subring that is also a s... |
| issubassa 21824 | The subalgebras of an asso... |
| sraassab 21825 | A subring algebra is an as... |
| sraassa 21826 | The subring algebra over a... |
| sraassaOLD 21827 | Obsolete version of ~ sraa... |
| rlmassa 21828 | The ring module over a com... |
| assapropd 21829 | If two structures have the... |
| aspval 21830 | Value of the algebraic clo... |
| asplss 21831 | The algebraic span of a se... |
| aspid 21832 | The algebraic span of a su... |
| aspsubrg 21833 | The algebraic span of a se... |
| aspss 21834 | Span preserves subset orde... |
| aspssid 21835 | A set of vectors is a subs... |
| asclfval 21836 | Function value of the alge... |
| asclval 21837 | Value of a mapped algebra ... |
| asclfn 21838 | Unconditional functionalit... |
| asclf 21839 | The algebra scalar lifting... |
| asclghm 21840 | The algebra scalar lifting... |
| asclelbas 21841 | Lifted scalars are in the ... |
| ascl0 21842 | The scalar 0 embedded into... |
| ascl1 21843 | The scalar 1 embedded into... |
| asclmul1 21844 | Left multiplication by a l... |
| asclmul2 21845 | Right multiplication by a ... |
| ascldimul 21846 | The algebra scalar lifting... |
| asclinvg 21847 | The group inverse (negatio... |
| asclrhm 21848 | The algebra scalar lifting... |
| rnascl 21849 | The set of lifted scalars ... |
| issubassa2 21850 | A subring of a unital alge... |
| rnasclsubrg 21851 | The scalar multiples of th... |
| rnasclmulcl 21852 | (Vector) multiplication is... |
| rnasclassa 21853 | The scalar multiples of th... |
| ressascl 21854 | The lifting of scalars is ... |
| asclpropd 21855 | If two structures have the... |
| aspval2 21856 | The algebraic closure is t... |
| assamulgscmlem1 21857 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21858 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21859 | Exponentiation of a scalar... |
| asclmulg 21860 | Apply group multiplication... |
| zlmassa 21861 | The ` ZZ ` -module operati... |
| reldmpsr 21872 | The multivariate power ser... |
| psrval 21873 | Value of the multivariate ... |
| psrvalstr 21874 | The multivariate power ser... |
| psrbag 21875 | Elementhood in the set of ... |
| psrbagf 21876 | A finite bag is a function... |
| psrbagfsupp 21877 | Finite bags have finite su... |
| snifpsrbag 21878 | A bag containing one eleme... |
| fczpsrbag 21879 | The constant function equa... |
| psrbaglesupp 21880 | The support of a dominated... |
| psrbaglecl 21881 | The set of finite bags is ... |
| psrbagaddcl 21882 | The sum of two finite bags... |
| psrbagcon 21883 | The analogue of the statem... |
| psrbaglefi 21884 | There are finitely many ba... |
| psrbagconcl 21885 | The complement of a bag is... |
| psrbagleadd1 21886 | The analogue of " ` X <_ F... |
| psrbagconf1o 21887 | Bag complementation is a b... |
| gsumbagdiaglem 21888 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21889 | Two-dimensional commutatio... |
| psrass1lem 21890 | A group sum commutation us... |
| psrbas 21891 | The base set of the multiv... |
| psrelbas 21892 | An element of the set of p... |
| psrelbasfun 21893 | An element of the set of p... |
| psrplusg 21894 | The addition operation of ... |
| psradd 21895 | The addition operation of ... |
| psraddcl 21896 | Closure of the power serie... |
| psraddclOLD 21897 | Obsolete version of ~ psra... |
| rhmpsrlem1 21898 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21899 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21900 | The multiplication operati... |
| psrmulfval 21901 | The multiplication operati... |
| psrmulval 21902 | The multiplication operati... |
| psrmulcllem 21903 | Closure of the power serie... |
| psrmulcl 21904 | Closure of the power serie... |
| psrsca 21905 | The scalar field of the mu... |
| psrvscafval 21906 | The scalar multiplication ... |
| psrvsca 21907 | The scalar multiplication ... |
| psrvscaval 21908 | The scalar multiplication ... |
| psrvscacl 21909 | Closure of the power serie... |
| psr0cl 21910 | The zero element of the ri... |
| psr0lid 21911 | The zero element of the ri... |
| psrnegcl 21912 | The negative function in t... |
| psrlinv 21913 | The negative function in t... |
| psrgrp 21914 | The ring of power series i... |
| psr0 21915 | The zero element of the ri... |
| psrneg 21916 | The negative function of t... |
| psrlmod 21917 | The ring of power series i... |
| psr1cl 21918 | The identity element of th... |
| psrlidm 21919 | The identity element of th... |
| psrridm 21920 | The identity element of th... |
| psrass1 21921 | Associative identity for t... |
| psrdi 21922 | Distributive law for the r... |
| psrdir 21923 | Distributive law for the r... |
| psrass23l 21924 | Associative identity for t... |
| psrcom 21925 | Commutative law for the ri... |
| psrass23 21926 | Associative identities for... |
| psrring 21927 | The ring of power series i... |
| psr1 21928 | The identity element of th... |
| psrcrng 21929 | The ring of power series i... |
| psrassa 21930 | The ring of power series i... |
| resspsrbas 21931 | A restricted power series ... |
| resspsradd 21932 | A restricted power series ... |
| resspsrmul 21933 | A restricted power series ... |
| resspsrvsca 21934 | A restricted power series ... |
| subrgpsr 21935 | A subring of the base ring... |
| psrascl 21936 | Value of the scalar inject... |
| psrasclcl 21937 | A scalar is lifted into a ... |
| mvrfval 21938 | Value of the generating el... |
| mvrval 21939 | Value of the generating el... |
| mvrval2 21940 | Value of the generating el... |
| mvrid 21941 | The ` X i ` -th coefficien... |
| mvrf 21942 | The power series variable ... |
| mvrf1 21943 | The power series variable ... |
| mvrcl2 21944 | A power series variable is... |
| reldmmpl 21945 | The multivariate polynomia... |
| mplval 21946 | Value of the set of multiv... |
| mplbas 21947 | Base set of the set of mul... |
| mplelbas 21948 | Property of being a polyno... |
| mvrcl 21949 | A power series variable is... |
| mvrf2 21950 | The power series/polynomia... |
| mplrcl 21951 | Reverse closure for the po... |
| mplelsfi 21952 | A polynomial treated as a ... |
| mplval2 21953 | Self-referential expressio... |
| mplbasss 21954 | The set of polynomials is ... |
| mplelf 21955 | A polynomial is defined as... |
| mplsubglem 21956 | If ` A ` is an ideal of se... |
| mpllsslem 21957 | If ` A ` is an ideal of su... |
| mplsubglem2 21958 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21959 | The set of polynomials is ... |
| mpllss 21960 | The set of polynomials is ... |
| mplsubrglem 21961 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21962 | The set of polynomials is ... |
| mpl0 21963 | The zero polynomial. (Con... |
| mplplusg 21964 | Value of addition in a pol... |
| mplmulr 21965 | Value of multiplication in... |
| mpladd 21966 | The addition operation on ... |
| mplneg 21967 | The negative function on m... |
| mplmul 21968 | The multiplication operati... |
| mpl1 21969 | The identity element of th... |
| mplsca 21970 | The scalar field of a mult... |
| mplvsca2 21971 | The scalar multiplication ... |
| mplvsca 21972 | The scalar multiplication ... |
| mplvscaval 21973 | The scalar multiplication ... |
| mplgrp 21974 | The polynomial ring is a g... |
| mpllmod 21975 | The polynomial ring is a l... |
| mplring 21976 | The polynomial ring is a r... |
| mpllvec 21977 | The polynomial ring is a v... |
| mplcrng 21978 | The polynomial ring is a c... |
| mplassa 21979 | The polynomial ring is an ... |
| mplringd 21980 | The polynomial ring is a r... |
| mpllmodd 21981 | The polynomial ring is a l... |
| mplascl0 21982 | The zero scalar as a polyn... |
| mplascl1 21983 | The one scalar as a polyno... |
| ressmplbas2 21984 | The base set of a restrict... |
| ressmplbas 21985 | A restricted polynomial al... |
| ressmpladd 21986 | A restricted polynomial al... |
| ressmplmul 21987 | A restricted polynomial al... |
| ressmplvsca 21988 | A restricted power series ... |
| subrgmpl 21989 | A subring of the base ring... |
| subrgmvr 21990 | The variables in a subring... |
| subrgmvrf 21991 | The variables in a polynom... |
| mplmon 21992 | A monomial is a polynomial... |
| mplmonmul 21993 | The product of two monomia... |
| mplcoe1 21994 | Decompose a polynomial int... |
| mplcoe3 21995 | Decompose a monomial in on... |
| mplcoe5lem 21996 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21997 | Decompose a monomial into ... |
| mplcoe2 21998 | Decompose a monomial into ... |
| mplbas2 21999 | An alternative expression ... |
| ltbval 22000 | Value of the well-order on... |
| ltbwe 22001 | The finite bag order is a ... |
| reldmopsr 22002 | Lemma for ordered power se... |
| opsrval 22003 | The value of the "ordered ... |
| opsrle 22004 | An alternative expression ... |
| opsrval2 22005 | Self-referential expressio... |
| opsrbaslem 22006 | Get a component of the ord... |
| opsrbas 22007 | The base set of the ordere... |
| opsrplusg 22008 | The addition operation of ... |
| opsrmulr 22009 | The multiplication operati... |
| opsrvsca 22010 | The scalar product operati... |
| opsrsca 22011 | The scalar ring of the ord... |
| opsrtoslem1 22012 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22013 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22014 | The ordered power series s... |
| opsrso 22015 | The ordered power series s... |
| opsrcrng 22016 | The ring of ordered power ... |
| opsrassa 22017 | The ring of ordered power ... |
| mplmon2 22018 | Express a scaled monomial.... |
| psrbag0 22019 | The empty bag is a bag. (... |
| psrbagsn 22020 | A singleton bag is a bag. ... |
| mplascl 22021 | Value of the scalar inject... |
| mplasclf 22022 | The scalar injection is a ... |
| subrgascl 22023 | The scalar injection funct... |
| subrgasclcl 22024 | The scalars in a polynomia... |
| mplmon2cl 22025 | A scaled monomial is a pol... |
| mplmon2mul 22026 | Product of scaled monomial... |
| mplind 22027 | Prove a property of polyno... |
| mplcoe4 22028 | Decompose a polynomial int... |
| evlslem4 22033 | The support of a tensor pr... |
| psrbagev1 22034 | A bag of multipliers provi... |
| psrbagev2 22035 | Closure of a sum using a b... |
| evlslem2 22036 | A linear function on the p... |
| evlslem3 22037 | Lemma for ~ evlseu . Poly... |
| evlslem6 22038 | Lemma for ~ evlseu . Fini... |
| evlslem1 22039 | Lemma for ~ evlseu , give ... |
| evlseu 22040 | For a given interpretation... |
| reldmevls 22041 | Well-behaved binary operat... |
| mpfrcl 22042 | Reverse closure for the se... |
| evlsval 22043 | Value of the polynomial ev... |
| evlsval2 22044 | Characterizing properties ... |
| evlsrhm 22045 | Polynomial evaluation is a... |
| evlsval3 22046 | Give a formula for the pol... |
| evlsvval 22047 | Give a formula for the eva... |
| evlsvvvallem 22048 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 22049 | Lemma for theorems using ~... |
| evlsvvval 22050 | Give a formula for the eva... |
| evlssca 22051 | Polynomial evaluation maps... |
| evlsvar 22052 | Polynomial evaluation maps... |
| evlsgsumadd 22053 | Polynomial evaluation maps... |
| evlsgsummul 22054 | Polynomial evaluation maps... |
| evlspw 22055 | Polynomial evaluation for ... |
| evlsvarpw 22056 | Polynomial evaluation for ... |
| evlval 22057 | Value of the simple/same r... |
| evlrhm 22058 | The simple evaluation map ... |
| evlcl 22059 | A polynomial over the ring... |
| evladdval 22060 | Polynomial evaluation buil... |
| evlmulval 22061 | Polynomial evaluation buil... |
| evlsscasrng 22062 | The evaluation of a scalar... |
| evlsca 22063 | Simple polynomial evaluati... |
| evlsvarsrng 22064 | The evaluation of the vari... |
| evlvar 22065 | Simple polynomial evaluati... |
| mpfconst 22066 | Constants are multivariate... |
| mpfproj 22067 | Projections are multivaria... |
| mpfsubrg 22068 | Polynomial functions are a... |
| mpff 22069 | Polynomial functions are f... |
| mpfaddcl 22070 | The sum of multivariate po... |
| mpfmulcl 22071 | The product of multivariat... |
| mpfind 22072 | Prove a property of polyno... |
| selvffval 22078 | Value of the "variable sel... |
| selvfval 22079 | Value of the "variable sel... |
| selvval 22080 | Value of the "variable sel... |
| reldmmhp 22082 | The domain of the homogene... |
| mhpfval 22083 | Value of the "homogeneous ... |
| mhpval 22084 | Value of the "homogeneous ... |
| ismhp 22085 | Property of being a homoge... |
| ismhp2 22086 | Deduce a homogeneous polyn... |
| ismhp3 22087 | A polynomial is homogeneou... |
| mhprcl 22088 | Reverse closure for homoge... |
| mhpmpl 22089 | A homogeneous polynomial i... |
| mhpdeg 22090 | All nonzero terms of a hom... |
| mhp0cl 22091 | The zero polynomial is hom... |
| mhpsclcl 22092 | A scalar (or constant) pol... |
| mhpvarcl 22093 | A power series variable is... |
| mhpmulcl 22094 | A product of homogeneous p... |
| mhppwdeg 22095 | Degree of a homogeneous po... |
| mhpaddcl 22096 | Homogeneous polynomials ar... |
| mhpinvcl 22097 | Homogeneous polynomials ar... |
| mhpsubg 22098 | Homogeneous polynomials fo... |
| mhpvscacl 22099 | Homogeneous polynomials ar... |
| mhplss 22100 | Homogeneous polynomials fo... |
| psdffval 22102 | Value of the power series ... |
| psdfval 22103 | Give a map between power s... |
| psdval 22104 | Evaluate the partial deriv... |
| psdcoef 22105 | Coefficient of a term of t... |
| psdcl 22106 | The derivative of a power ... |
| psdmplcl 22107 | The derivative of a polyno... |
| psdadd 22108 | The derivative of a sum is... |
| psdvsca 22109 | The derivative of a scaled... |
| psdmullem 22110 | Lemma for ~ psdmul . Tran... |
| psdmul 22111 | Product rule for power ser... |
| psd1 22112 | The derivative of one is z... |
| psdascl 22113 | The derivative of a consta... |
| psdmvr 22114 | The partial derivative of ... |
| psdpw 22115 | Power rule for partial der... |
| psr1baslem 22127 | The set of finite bags on ... |
| psr1val 22128 | Value of the ring of univa... |
| psr1crng 22129 | The ring of univariate pow... |
| psr1assa 22130 | The ring of univariate pow... |
| psr1tos 22131 | The ordered power series s... |
| psr1bas2 22132 | The base set of the ring o... |
| psr1bas 22133 | The base set of the ring o... |
| vr1val 22134 | The value of the generator... |
| vr1cl2 22135 | The variable ` X ` is a me... |
| ply1val 22136 | The value of the set of un... |
| ply1bas 22137 | The value of the base set ... |
| ply1basOLD 22138 | Obsolete version of ~ ply1... |
| ply1lss 22139 | Univariate polynomials for... |
| ply1subrg 22140 | Univariate polynomials for... |
| ply1crng 22141 | The ring of univariate pol... |
| ply1assa 22142 | The ring of univariate pol... |
| psr1bascl 22143 | A univariate power series ... |
| psr1basf 22144 | Univariate power series ba... |
| ply1basf 22145 | Univariate polynomial base... |
| ply1bascl 22146 | A univariate polynomial is... |
| ply1bascl2 22147 | A univariate polynomial is... |
| coe1fval 22148 | Value of the univariate po... |
| coe1fv 22149 | Value of an evaluated coef... |
| fvcoe1 22150 | Value of a multivariate co... |
| coe1fval3 22151 | Univariate power series co... |
| coe1f2 22152 | Functionality of univariat... |
| coe1fval2 22153 | Univariate polynomial coef... |
| coe1f 22154 | Functionality of univariat... |
| coe1fvalcl 22155 | A coefficient of a univari... |
| coe1sfi 22156 | Finite support of univaria... |
| coe1fsupp 22157 | The coefficient vector of ... |
| mptcoe1fsupp 22158 | A mapping involving coeffi... |
| coe1ae0 22159 | The coefficient vector of ... |
| vr1cl 22160 | The generator of a univari... |
| opsr0 22161 | Zero in the ordered power ... |
| opsr1 22162 | One in the ordered power s... |
| psr1plusg 22163 | Value of addition in a uni... |
| psr1vsca 22164 | Value of scalar multiplica... |
| psr1mulr 22165 | Value of multiplication in... |
| ply1plusg 22166 | Value of addition in a uni... |
| ply1vsca 22167 | Value of scalar multiplica... |
| ply1mulr 22168 | Value of multiplication in... |
| ply1ass23l 22169 | Associative identity with ... |
| ressply1bas2 22170 | The base set of a restrict... |
| ressply1bas 22171 | A restricted polynomial al... |
| ressply1add 22172 | A restricted polynomial al... |
| ressply1mul 22173 | A restricted polynomial al... |
| ressply1vsca 22174 | A restricted power series ... |
| subrgply1 22175 | A subring of the base ring... |
| gsumply1subr 22176 | Evaluate a group sum in a ... |
| psrbaspropd 22177 | Property deduction for pow... |
| psrplusgpropd 22178 | Property deduction for pow... |
| mplbaspropd 22179 | Property deduction for pol... |
| psropprmul 22180 | Reversing multiplication i... |
| ply1opprmul 22181 | Reversing multiplication i... |
| 00ply1bas 22182 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22183 | Protection compatibility o... |
| ply1plusgfvi 22184 | Protection compatibility o... |
| ply1baspropd 22185 | Property deduction for uni... |
| ply1plusgpropd 22186 | Property deduction for uni... |
| opsrring 22187 | Ordered power series form ... |
| opsrlmod 22188 | Ordered power series form ... |
| psr1ring 22189 | Univariate power series fo... |
| ply1ring 22190 | Univariate polynomials for... |
| psr1lmod 22191 | Univariate power series fo... |
| psr1sca 22192 | Scalars of a univariate po... |
| psr1sca2 22193 | Scalars of a univariate po... |
| ply1lmod 22194 | Univariate polynomials for... |
| ply1sca 22195 | Scalars of a univariate po... |
| ply1sca2 22196 | Scalars of a univariate po... |
| ply1ascl0 22197 | The zero scalar as a polyn... |
| ply1ascl1 22198 | The multiplicative identit... |
| ply1mpl0 22199 | The univariate polynomial ... |
| ply10s0 22200 | Zero times a univariate po... |
| ply1mpl1 22201 | The univariate polynomial ... |
| ply1ascl 22202 | The univariate polynomial ... |
| subrg1ascl 22203 | The scalar injection funct... |
| subrg1asclcl 22204 | The scalars in a polynomia... |
| subrgvr1 22205 | The variables in a subring... |
| subrgvr1cl 22206 | The variables in a polynom... |
| coe1z 22207 | The coefficient vector of ... |
| coe1add 22208 | The coefficient vector of ... |
| coe1addfv 22209 | A particular coefficient o... |
| coe1subfv 22210 | A particular coefficient o... |
| coe1mul2lem1 22211 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22212 | An equivalence for ~ coe1m... |
| coe1mul2 22213 | The coefficient vector of ... |
| coe1mul 22214 | The coefficient vector of ... |
| ply1moncl 22215 | Closure of the expression ... |
| ply1tmcl 22216 | Closure of the expression ... |
| coe1tm 22217 | Coefficient vector of a po... |
| coe1tmfv1 22218 | Nonzero coefficient of a p... |
| coe1tmfv2 22219 | Zero coefficient of a poly... |
| coe1tmmul2 22220 | Coefficient vector of a po... |
| coe1tmmul 22221 | Coefficient vector of a po... |
| coe1tmmul2fv 22222 | Function value of a right-... |
| coe1pwmul 22223 | Coefficient vector of a po... |
| coe1pwmulfv 22224 | Function value of a right-... |
| ply1scltm 22225 | A scalar is a term with ze... |
| coe1sclmul 22226 | Coefficient vector of a po... |
| coe1sclmulfv 22227 | A single coefficient of a ... |
| coe1sclmul2 22228 | Coefficient vector of a po... |
| ply1sclf 22229 | A scalar polynomial is a p... |
| ply1sclcl 22230 | The value of the algebra s... |
| coe1scl 22231 | Coefficient vector of a sc... |
| ply1sclid 22232 | Recover the base scalar fr... |
| ply1sclf1 22233 | The polynomial scalar func... |
| ply1scl0 22234 | The zero scalar is zero. ... |
| ply1scl0OLD 22235 | Obsolete version of ~ ply1... |
| ply1scln0 22236 | Nonzero scalars create non... |
| ply1scl1 22237 | The one scalar is the unit... |
| ply1scl1OLD 22238 | Obsolete version of ~ ply1... |
| coe1id 22239 | Coefficient vector of the ... |
| ply1idvr1 22240 | The identity of a polynomi... |
| ply1idvr1OLD 22241 | Obsolete version of ~ ply1... |
| cply1mul 22242 | The product of two constan... |
| ply1coefsupp 22243 | The decomposition of a uni... |
| ply1coe 22244 | Decompose a univariate pol... |
| eqcoe1ply1eq 22245 | Two polynomials over the s... |
| ply1coe1eq 22246 | Two polynomials over the s... |
| cply1coe0 22247 | All but the first coeffici... |
| cply1coe0bi 22248 | A polynomial is constant (... |
| coe1fzgsumdlem 22249 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22250 | Value of an evaluated coef... |
| ply1scleq 22251 | Equality of a constant pol... |
| ply1chr 22252 | The characteristic of a po... |
| gsumsmonply1 22253 | A finite group sum of scal... |
| gsummoncoe1 22254 | A coefficient of the polyn... |
| gsumply1eq 22255 | Two univariate polynomials... |
| lply1binom 22256 | The binomial theorem for l... |
| lply1binomsc 22257 | The binomial theorem for l... |
| ply1fermltlchr 22258 | Fermat's little theorem fo... |
| reldmevls1 22263 | Well-behaved binary operat... |
| ply1frcl 22264 | Reverse closure for the se... |
| evls1fval 22265 | Value of the univariate po... |
| evls1val 22266 | Value of the univariate po... |
| evls1rhmlem 22267 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22268 | Polynomial evaluation is a... |
| evls1sca 22269 | Univariate polynomial eval... |
| evls1gsumadd 22270 | Univariate polynomial eval... |
| evls1gsummul 22271 | Univariate polynomial eval... |
| evls1pw 22272 | Univariate polynomial eval... |
| evls1varpw 22273 | Univariate polynomial eval... |
| evl1fval 22274 | Value of the simple/same r... |
| evl1val 22275 | Value of the simple/same r... |
| evl1fval1lem 22276 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22277 | Value of the simple/same r... |
| evl1rhm 22278 | Polynomial evaluation is a... |
| fveval1fvcl 22279 | The function value of the ... |
| evl1sca 22280 | Polynomial evaluation maps... |
| evl1scad 22281 | Polynomial evaluation buil... |
| evl1var 22282 | Polynomial evaluation maps... |
| evl1vard 22283 | Polynomial evaluation buil... |
| evls1var 22284 | Univariate polynomial eval... |
| evls1scasrng 22285 | The evaluation of a scalar... |
| evls1varsrng 22286 | The evaluation of the vari... |
| evl1addd 22287 | Polynomial evaluation buil... |
| evl1subd 22288 | Polynomial evaluation buil... |
| evl1muld 22289 | Polynomial evaluation buil... |
| evl1vsd 22290 | Polynomial evaluation buil... |
| evl1expd 22291 | Polynomial evaluation buil... |
| pf1const 22292 | Constants are polynomial f... |
| pf1id 22293 | The identity is a polynomi... |
| pf1subrg 22294 | Polynomial functions are a... |
| pf1rcl 22295 | Reverse closure for the se... |
| pf1f 22296 | Polynomial functions are f... |
| mpfpf1 22297 | Convert a multivariate pol... |
| pf1mpf 22298 | Convert a univariate polyn... |
| pf1addcl 22299 | The sum of multivariate po... |
| pf1mulcl 22300 | The product of multivariat... |
| pf1ind 22301 | Prove a property of polyno... |
| evl1gsumdlem 22302 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22303 | Polynomial evaluation buil... |
| evl1gsumadd 22304 | Univariate polynomial eval... |
| evl1gsumaddval 22305 | Value of a univariate poly... |
| evl1gsummul 22306 | Univariate polynomial eval... |
| evl1varpw 22307 | Univariate polynomial eval... |
| evl1varpwval 22308 | Value of a univariate poly... |
| evl1scvarpw 22309 | Univariate polynomial eval... |
| evl1scvarpwval 22310 | Value of a univariate poly... |
| evl1gsummon 22311 | Value of a univariate poly... |
| evls1scafv 22312 | Value of the univariate po... |
| evls1expd 22313 | Univariate polynomial eval... |
| evls1varpwval 22314 | Univariate polynomial eval... |
| evls1fpws 22315 | Evaluation of a univariate... |
| ressply1evl 22316 | Evaluation of a univariate... |
| evls1addd 22317 | Univariate polynomial eval... |
| evls1muld 22318 | Univariate polynomial eval... |
| evls1vsca 22319 | Univariate polynomial eval... |
| asclply1subcl 22320 | Closure of the algebra sca... |
| evls1fvcl 22321 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22322 | The function ` F ` mapping... |
| evls1maplmhm 22323 | The function ` F ` mapping... |
| evls1maprnss 22324 | The function ` F ` mapping... |
| evl1maprhm 22325 | The function ` F ` mapping... |
| mhmcompl 22326 | The composition of a monoi... |
| mhmcoaddmpl 22327 | Show that the ring homomor... |
| rhmcomulmpl 22328 | Show that the ring homomor... |
| rhmmpl 22329 | Provide a ring homomorphis... |
| ply1vscl 22330 | Closure of scalar multipli... |
| mhmcoply1 22331 | The composition of a monoi... |
| rhmply1 22332 | Provide a ring homomorphis... |
| rhmply1vr1 22333 | A ring homomorphism betwee... |
| rhmply1vsca 22334 | Apply a ring homomorphism ... |
| rhmply1mon 22335 | Apply a ring homomorphism ... |
| mamufval 22338 | Functional value of the ma... |
| mamuval 22339 | Multiplication of two matr... |
| mamufv 22340 | A cell in the multiplicati... |
| mamudm 22341 | The domain of the matrix m... |
| mamufacex 22342 | Every solution of the equa... |
| mamures 22343 | Rows in a matrix product a... |
| grpvlinv 22344 | Tuple-wise left inverse in... |
| grpvrinv 22345 | Tuple-wise right inverse i... |
| ringvcl 22346 | Tuple-wise multiplication ... |
| mamucl 22347 | Operation closure of matri... |
| mamuass 22348 | Matrix multiplication is a... |
| mamudi 22349 | Matrix multiplication dist... |
| mamudir 22350 | Matrix multiplication dist... |
| mamuvs1 22351 | Matrix multiplication dist... |
| mamuvs2 22352 | Matrix multiplication dist... |
| matbas0pc 22355 | There is no matrix with a ... |
| matbas0 22356 | There is no matrix for a n... |
| matval 22357 | Value of the matrix algebr... |
| matrcl 22358 | Reverse closure for the ma... |
| matbas 22359 | The matrix ring has the sa... |
| matplusg 22360 | The matrix ring has the sa... |
| matsca 22361 | The matrix ring has the sa... |
| matvsca 22362 | The matrix ring has the sa... |
| mat0 22363 | The matrix ring has the sa... |
| matinvg 22364 | The matrix ring has the sa... |
| mat0op 22365 | Value of a zero matrix as ... |
| matsca2 22366 | The scalars of the matrix ... |
| matbas2 22367 | The base set of the matrix... |
| matbas2i 22368 | A matrix is a function. (... |
| matbas2d 22369 | The base set of the matrix... |
| eqmat 22370 | Two square matrices of the... |
| matecl 22371 | Each entry (according to W... |
| matecld 22372 | Each entry (according to W... |
| matplusg2 22373 | Addition in the matrix rin... |
| matvsca2 22374 | Scalar multiplication in t... |
| matlmod 22375 | The matrix ring is a linea... |
| matgrp 22376 | The matrix ring is a group... |
| matvscl 22377 | Closure of the scalar mult... |
| matsubg 22378 | The matrix ring has the sa... |
| matplusgcell 22379 | Addition in the matrix rin... |
| matsubgcell 22380 | Subtraction in the matrix ... |
| matinvgcell 22381 | Additive inversion in the ... |
| matvscacell 22382 | Scalar multiplication in t... |
| matgsum 22383 | Finite commutative sums in... |
| matmulr 22384 | Multiplication in the matr... |
| mamumat1cl 22385 | The identity matrix (as op... |
| mat1comp 22386 | The components of the iden... |
| mamulid 22387 | The identity matrix (as op... |
| mamurid 22388 | The identity matrix (as op... |
| matring 22389 | Existence of the matrix ri... |
| matassa 22390 | Existence of the matrix al... |
| matmulcell 22391 | Multiplication in the matr... |
| mpomatmul 22392 | Multiplication of two N x ... |
| mat1 22393 | Value of an identity matri... |
| mat1ov 22394 | Entries of an identity mat... |
| mat1bas 22395 | The identity matrix is a m... |
| matsc 22396 | The identity matrix multip... |
| ofco2 22397 | Distribution law for the f... |
| oftpos 22398 | The transposition of the v... |
| mattposcl 22399 | The transpose of a square ... |
| mattpostpos 22400 | The transpose of the trans... |
| mattposvs 22401 | The transposition of a mat... |
| mattpos1 22402 | The transposition of the i... |
| tposmap 22403 | The transposition of an I ... |
| mamutpos 22404 | Behavior of transposes in ... |
| mattposm 22405 | Multiplying two transposed... |
| matgsumcl 22406 | Closure of a group sum ove... |
| madetsumid 22407 | The identity summand in th... |
| matepmcl 22408 | Each entry of a matrix wit... |
| matepm2cl 22409 | Each entry of a matrix wit... |
| madetsmelbas 22410 | A summand of the determina... |
| madetsmelbas2 22411 | A summand of the determina... |
| mat0dimbas0 22412 | The empty set is the one a... |
| mat0dim0 22413 | The zero of the algebra of... |
| mat0dimid 22414 | The identity of the algebr... |
| mat0dimscm 22415 | The scalar multiplication ... |
| mat0dimcrng 22416 | The algebra of matrices wi... |
| mat1dimelbas 22417 | A matrix with dimension 1 ... |
| mat1dimbas 22418 | A matrix with dimension 1 ... |
| mat1dim0 22419 | The zero of the algebra of... |
| mat1dimid 22420 | The identity of the algebr... |
| mat1dimscm 22421 | The scalar multiplication ... |
| mat1dimmul 22422 | The ring multiplication in... |
| mat1dimcrng 22423 | The algebra of matrices wi... |
| mat1f1o 22424 | There is a 1-1 function fr... |
| mat1rhmval 22425 | The value of the ring homo... |
| mat1rhmelval 22426 | The value of the ring homo... |
| mat1rhmcl 22427 | The value of the ring homo... |
| mat1f 22428 | There is a function from a... |
| mat1ghm 22429 | There is a group homomorph... |
| mat1mhm 22430 | There is a monoid homomorp... |
| mat1rhm 22431 | There is a ring homomorphi... |
| mat1rngiso 22432 | There is a ring isomorphis... |
| mat1ric 22433 | A ring is isomorphic to th... |
| dmatval 22438 | The set of ` N ` x ` N ` d... |
| dmatel 22439 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22440 | An ` N ` x ` N ` diagonal ... |
| dmatid 22441 | The identity matrix is a d... |
| dmatelnd 22442 | An extradiagonal entry of ... |
| dmatmul 22443 | The product of two diagona... |
| dmatsubcl 22444 | The difference of two diag... |
| dmatsgrp 22445 | The set of diagonal matric... |
| dmatmulcl 22446 | The product of two diagona... |
| dmatsrng 22447 | The set of diagonal matric... |
| dmatcrng 22448 | The subring of diagonal ma... |
| dmatscmcl 22449 | The multiplication of a di... |
| scmatval 22450 | The set of ` N ` x ` N ` s... |
| scmatel 22451 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22452 | A scalar matrix can be exp... |
| scmatscmide 22453 | An entry of a scalar matri... |
| scmatscmiddistr 22454 | Distributive law for scala... |
| scmatmat 22455 | An ` N ` x ` N ` scalar ma... |
| scmate 22456 | An entry of an ` N ` x ` N... |
| scmatmats 22457 | The set of an ` N ` x ` N ... |
| scmateALT 22458 | Alternate proof of ~ scmat... |
| scmatscm 22459 | The multiplication of a ma... |
| scmatid 22460 | The identity matrix is a s... |
| scmatdmat 22461 | A scalar matrix is a diago... |
| scmataddcl 22462 | The sum of two scalar matr... |
| scmatsubcl 22463 | The difference of two scal... |
| scmatmulcl 22464 | The product of two scalar ... |
| scmatsgrp 22465 | The set of scalar matrices... |
| scmatsrng 22466 | The set of scalar matrices... |
| scmatcrng 22467 | The subring of scalar matr... |
| scmatsgrp1 22468 | The set of scalar matrices... |
| scmatsrng1 22469 | The set of scalar matrices... |
| smatvscl 22470 | Closure of the scalar mult... |
| scmatlss 22471 | The set of scalar matrices... |
| scmatstrbas 22472 | The set of scalar matrices... |
| scmatrhmval 22473 | The value of the ring homo... |
| scmatrhmcl 22474 | The value of the ring homo... |
| scmatf 22475 | There is a function from a... |
| scmatfo 22476 | There is a function from a... |
| scmatf1 22477 | There is a 1-1 function fr... |
| scmatf1o 22478 | There is a bijection betwe... |
| scmatghm 22479 | There is a group homomorph... |
| scmatmhm 22480 | There is a monoid homomorp... |
| scmatrhm 22481 | There is a ring homomorphi... |
| scmatrngiso 22482 | There is a ring isomorphis... |
| scmatric 22483 | A ring is isomorphic to ev... |
| mat0scmat 22484 | The empty matrix over a ri... |
| mat1scmat 22485 | A 1-dimensional matrix ove... |
| mvmulfval 22488 | Functional value of the ma... |
| mvmulval 22489 | Multiplication of a vector... |
| mvmulfv 22490 | A cell/element in the vect... |
| mavmulval 22491 | Multiplication of a vector... |
| mavmulfv 22492 | A cell/element in the vect... |
| mavmulcl 22493 | Multiplication of an NxN m... |
| 1mavmul 22494 | Multiplication of the iden... |
| mavmulass 22495 | Associativity of the multi... |
| mavmuldm 22496 | The domain of the matrix v... |
| mavmulsolcl 22497 | Every solution of the equa... |
| mavmul0 22498 | Multiplication of a 0-dime... |
| mavmul0g 22499 | The result of the 0-dimens... |
| mvmumamul1 22500 | The multiplication of an M... |
| mavmumamul1 22501 | The multiplication of an N... |
| marrepfval 22506 | First substitution for the... |
| marrepval0 22507 | Second substitution for th... |
| marrepval 22508 | Third substitution for the... |
| marrepeval 22509 | An entry of a matrix with ... |
| marrepcl 22510 | Closure of the row replace... |
| marepvfval 22511 | First substitution for the... |
| marepvval0 22512 | Second substitution for th... |
| marepvval 22513 | Third substitution for the... |
| marepveval 22514 | An entry of a matrix with ... |
| marepvcl 22515 | Closure of the column repl... |
| ma1repvcl 22516 | Closure of the column repl... |
| ma1repveval 22517 | An entry of an identity ma... |
| mulmarep1el 22518 | Element by element multipl... |
| mulmarep1gsum1 22519 | The sum of element by elem... |
| mulmarep1gsum2 22520 | The sum of element by elem... |
| 1marepvmarrepid 22521 | Replacing the ith row by 0... |
| submabas 22524 | Any subset of the index se... |
| submafval 22525 | First substitution for a s... |
| submaval0 22526 | Second substitution for a ... |
| submaval 22527 | Third substitution for a s... |
| submaeval 22528 | An entry of a submatrix of... |
| 1marepvsma1 22529 | The submatrix of the ident... |
| mdetfval 22532 | First substitution for the... |
| mdetleib 22533 | Full substitution of our d... |
| mdetleib2 22534 | Leibniz' formula can also ... |
| nfimdetndef 22535 | The determinant is not def... |
| mdetfval1 22536 | First substitution of an a... |
| mdetleib1 22537 | Full substitution of an al... |
| mdet0pr 22538 | The determinant function f... |
| mdet0f1o 22539 | The determinant function f... |
| mdet0fv0 22540 | The determinant of the emp... |
| mdetf 22541 | Functionality of the deter... |
| mdetcl 22542 | The determinant evaluates ... |
| m1detdiag 22543 | The determinant of a 1-dim... |
| mdetdiaglem 22544 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22545 | The determinant of a diago... |
| mdetdiagid 22546 | The determinant of a diago... |
| mdet1 22547 | The determinant of the ide... |
| mdetrlin 22548 | The determinant function i... |
| mdetrsca 22549 | The determinant function i... |
| mdetrsca2 22550 | The determinant function i... |
| mdetr0 22551 | The determinant of a matri... |
| mdet0 22552 | The determinant of the zer... |
| mdetrlin2 22553 | The determinant function i... |
| mdetralt 22554 | The determinant function i... |
| mdetralt2 22555 | The determinant function i... |
| mdetero 22556 | The determinant function i... |
| mdettpos 22557 | Determinant is invariant u... |
| mdetunilem1 22558 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22559 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22560 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22561 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22562 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22563 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22564 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22565 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22566 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22567 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22568 | According to the definitio... |
| mdetmul 22569 | Multiplicativity of the de... |
| m2detleiblem1 22570 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22571 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22572 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22573 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22574 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22575 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22576 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22577 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22582 | Lemma for ~ maducoeval2 . ... |
| madufval 22583 | First substitution for the... |
| maduval 22584 | Second substitution for th... |
| maducoeval 22585 | An entry of the adjunct (c... |
| maducoeval2 22586 | An entry of the adjunct (c... |
| maduf 22587 | Creating the adjunct of ma... |
| madutpos 22588 | The adjuct of a transposed... |
| madugsum 22589 | The determinant of a matri... |
| madurid 22590 | Multiplying a matrix with ... |
| madulid 22591 | Multiplying the adjunct of... |
| minmar1fval 22592 | First substitution for the... |
| minmar1val0 22593 | Second substitution for th... |
| minmar1val 22594 | Third substitution for the... |
| minmar1eval 22595 | An entry of a matrix for a... |
| minmar1marrep 22596 | The minor matrix is a spec... |
| minmar1cl 22597 | Closure of the row replace... |
| maducoevalmin1 22598 | The coefficients of an adj... |
| symgmatr01lem 22599 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22600 | Applying a permutation tha... |
| gsummatr01lem1 22601 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22602 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22603 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22604 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22605 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22606 | Replacing a row of a squar... |
| smadiadetlem0 22607 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22608 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22609 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22610 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22611 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22612 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22613 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22614 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22615 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22616 | The determinant of a subma... |
| smadiadetglem1 22617 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22618 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22619 | The determinant of a squar... |
| smadiadetg0 22620 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22621 | The determinant of a squar... |
| invrvald 22622 | If a matrix multiplied wit... |
| matinv 22623 | The inverse of a matrix is... |
| matunit 22624 | A matrix is a unit in the ... |
| slesolvec 22625 | Every solution of a system... |
| slesolinv 22626 | The solution of a system o... |
| slesolinvbi 22627 | The solution of a system o... |
| slesolex 22628 | Every system of linear equ... |
| cramerimplem1 22629 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22630 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22631 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22632 | One direction of Cramer's ... |
| cramerlem1 22633 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22634 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22635 | Lemma 3 for ~ cramer . (C... |
| cramer0 22636 | Special case of Cramer's r... |
| cramer 22637 | Cramer's rule. According ... |
| pmatring 22638 | The set of polynomial matr... |
| pmatlmod 22639 | The set of polynomial matr... |
| pmatassa 22640 | The set of polynomial matr... |
| pmat0op 22641 | The zero polynomial matrix... |
| pmat1op 22642 | The identity polynomial ma... |
| pmat1ovd 22643 | Entries of the identity po... |
| pmat0opsc 22644 | The zero polynomial matrix... |
| pmat1opsc 22645 | The identity polynomial ma... |
| pmat1ovscd 22646 | Entries of the identity po... |
| pmatcoe1fsupp 22647 | For a polynomial matrix th... |
| 1pmatscmul 22648 | The scalar product of the ... |
| cpmat 22655 | Value of the constructor o... |
| cpmatpmat 22656 | A constant polynomial matr... |
| cpmatel 22657 | Property of a constant pol... |
| cpmatelimp 22658 | Implication of a set being... |
| cpmatel2 22659 | Another property of a cons... |
| cpmatelimp2 22660 | Another implication of a s... |
| 1elcpmat 22661 | The identity of the ring o... |
| cpmatacl 22662 | The set of all constant po... |
| cpmatinvcl 22663 | The set of all constant po... |
| cpmatmcllem 22664 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22665 | The set of all constant po... |
| cpmatsubgpmat 22666 | The set of all constant po... |
| cpmatsrgpmat 22667 | The set of all constant po... |
| 0elcpmat 22668 | The zero of the ring of al... |
| mat2pmatfval 22669 | Value of the matrix transf... |
| mat2pmatval 22670 | The result of a matrix tra... |
| mat2pmatvalel 22671 | A (matrix) element of the ... |
| mat2pmatbas 22672 | The result of a matrix tra... |
| mat2pmatbas0 22673 | The result of a matrix tra... |
| mat2pmatf 22674 | The matrix transformation ... |
| mat2pmatf1 22675 | The matrix transformation ... |
| mat2pmatghm 22676 | The transformation of matr... |
| mat2pmatmul 22677 | The transformation of matr... |
| mat2pmat1 22678 | The transformation of the ... |
| mat2pmatmhm 22679 | The transformation of matr... |
| mat2pmatrhm 22680 | The transformation of matr... |
| mat2pmatlin 22681 | The transformation of matr... |
| 0mat2pmat 22682 | The transformed zero matri... |
| idmatidpmat 22683 | The transformed identity m... |
| d0mat2pmat 22684 | The transformed empty set ... |
| d1mat2pmat 22685 | The transformation of a ma... |
| mat2pmatscmxcl 22686 | A transformed matrix multi... |
| m2cpm 22687 | The result of a matrix tra... |
| m2cpmf 22688 | The matrix transformation ... |
| m2cpmf1 22689 | The matrix transformation ... |
| m2cpmghm 22690 | The transformation of matr... |
| m2cpmmhm 22691 | The transformation of matr... |
| m2cpmrhm 22692 | The transformation of matr... |
| m2pmfzmap 22693 | The transformed values of ... |
| m2pmfzgsumcl 22694 | Closure of the sum of scal... |
| cpm2mfval 22695 | Value of the inverse matri... |
| cpm2mval 22696 | The result of an inverse m... |
| cpm2mvalel 22697 | A (matrix) element of the ... |
| cpm2mf 22698 | The inverse matrix transfo... |
| m2cpminvid 22699 | The inverse transformation... |
| m2cpminvid2lem 22700 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22701 | The transformation applied... |
| m2cpmfo 22702 | The matrix transformation ... |
| m2cpmf1o 22703 | The matrix transformation ... |
| m2cpmrngiso 22704 | The transformation of matr... |
| matcpmric 22705 | The ring of matrices over ... |
| m2cpminv 22706 | The inverse matrix transfo... |
| m2cpminv0 22707 | The inverse matrix transfo... |
| decpmatval0 22710 | The matrix consisting of t... |
| decpmatval 22711 | The matrix consisting of t... |
| decpmate 22712 | An entry of the matrix con... |
| decpmatcl 22713 | Closure of the decompositi... |
| decpmataa0 22714 | The matrix consisting of t... |
| decpmatfsupp 22715 | The mapping to the matrice... |
| decpmatid 22716 | The matrix consisting of t... |
| decpmatmullem 22717 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22718 | The matrix consisting of t... |
| decpmatmulsumfsupp 22719 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22720 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22721 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22722 | Write a polynomial matrix ... |
| pmatcollpw2lem 22723 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22724 | Write a polynomial matrix ... |
| monmatcollpw 22725 | The matrix consisting of t... |
| pmatcollpwlem 22726 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22727 | Write a polynomial matrix ... |
| pmatcollpwfi 22728 | Write a polynomial matrix ... |
| pmatcollpw3lem 22729 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22730 | Write a polynomial matrix ... |
| pmatcollpw3fi 22731 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22732 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22733 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22734 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22735 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22736 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22737 | Write a scalar matrix over... |
| pm2mpf1lem 22740 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22741 | Value of the transformatio... |
| pm2mpfval 22742 | A polynomial matrix transf... |
| pm2mpcl 22743 | The transformation of poly... |
| pm2mpf 22744 | The transformation of poly... |
| pm2mpf1 22745 | The transformation of poly... |
| pm2mpcoe1 22746 | A coefficient of the polyn... |
| idpm2idmp 22747 | The transformation of the ... |
| mptcoe1matfsupp 22748 | The mapping extracting the... |
| mply1topmatcllem 22749 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22750 | A polynomial over matrices... |
| mply1topmatcl 22751 | A polynomial over matrices... |
| mp2pm2mplem1 22752 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22753 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22754 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22755 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22756 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22757 | A polynomial over matrices... |
| pm2mpghmlem2 22758 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22759 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22760 | The transformation of poly... |
| pm2mpf1o 22761 | The transformation of poly... |
| pm2mpghm 22762 | The transformation of poly... |
| pm2mpgrpiso 22763 | The transformation of poly... |
| pm2mpmhmlem1 22764 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22765 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22766 | The transformation of poly... |
| pm2mprhm 22767 | The transformation of poly... |
| pm2mprngiso 22768 | The transformation of poly... |
| pmmpric 22769 | The ring of polynomial mat... |
| monmat2matmon 22770 | The transformation of a po... |
| pm2mp 22771 | The transformation of a su... |
| chmatcl 22774 | Closure of the characteris... |
| chmatval 22775 | The entries of the charact... |
| chpmatfval 22776 | Value of the characteristi... |
| chpmatval 22777 | The characteristic polynom... |
| chpmatply1 22778 | The characteristic polynom... |
| chpmatval2 22779 | The characteristic polynom... |
| chpmat0d 22780 | The characteristic polynom... |
| chpmat1dlem 22781 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22782 | The characteristic polynom... |
| chpdmatlem0 22783 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22784 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22785 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22786 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22787 | The characteristic polynom... |
| chpscmat 22788 | The characteristic polynom... |
| chpscmat0 22789 | The characteristic polynom... |
| chpscmatgsumbin 22790 | The characteristic polynom... |
| chpscmatgsummon 22791 | The characteristic polynom... |
| chp0mat 22792 | The characteristic polynom... |
| chpidmat 22793 | The characteristic polynom... |
| chmaidscmat 22794 | The characteristic polynom... |
| fvmptnn04if 22795 | The function values of a m... |
| fvmptnn04ifa 22796 | The function value of a ma... |
| fvmptnn04ifb 22797 | The function value of a ma... |
| fvmptnn04ifc 22798 | The function value of a ma... |
| fvmptnn04ifd 22799 | The function value of a ma... |
| chfacfisf 22800 | The "characteristic factor... |
| chfacfisfcpmat 22801 | The "characteristic factor... |
| chfacffsupp 22802 | The "characteristic factor... |
| chfacfscmulcl 22803 | Closure of a scaled value ... |
| chfacfscmul0 22804 | A scaled value of the "cha... |
| chfacfscmulfsupp 22805 | A mapping of scaled values... |
| chfacfscmulgsum 22806 | Breaking up a sum of value... |
| chfacfpmmulcl 22807 | Closure of the value of th... |
| chfacfpmmul0 22808 | The value of the "characte... |
| chfacfpmmulfsupp 22809 | A mapping of values of the... |
| chfacfpmmulgsum 22810 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22811 | Breaking up a sum of value... |
| cayhamlem1 22812 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22813 | The right-hand fundamental... |
| cpmidgsum 22814 | Representation of the iden... |
| cpmidgsumm2pm 22815 | Representation of the iden... |
| cpmidpmatlem1 22816 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22817 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22818 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22819 | Representation of the iden... |
| cpmadugsumlemB 22820 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22821 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22822 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22823 | The product of the charact... |
| cpmadugsum 22824 | The product of the charact... |
| cpmidgsum2 22825 | Representation of the iden... |
| cpmidg2sum 22826 | Equality of two sums repre... |
| cpmadumatpolylem1 22827 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22828 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22829 | The product of the charact... |
| cayhamlem2 22830 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22831 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22832 | The coefficients of the ch... |
| cayhamlem3 22833 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22834 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22835 | The Cayley-Hamilton theore... |
| cayleyhamilton 22836 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22837 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22838 | The Cayley-Hamilton theore... |
| istopg 22841 | Express the predicate " ` ... |
| istop2g 22842 | Express the predicate " ` ... |
| uniopn 22843 | The union of a subset of a... |
| iunopn 22844 | The indexed union of a sub... |
| inopn 22845 | The intersection of two op... |
| fitop 22846 | A topology is closed under... |
| fiinopn 22847 | The intersection of a none... |
| iinopn 22848 | The intersection of a none... |
| unopn 22849 | The union of two open sets... |
| 0opn 22850 | The empty set is an open s... |
| 0ntop 22851 | The empty set is not a top... |
| topopn 22852 | The underlying set of a to... |
| eltopss 22853 | A member of a topology is ... |
| riinopn 22854 | A finite indexed relative ... |
| rintopn 22855 | A finite relative intersec... |
| istopon 22858 | Property of being a topolo... |
| topontop 22859 | A topology on a given base... |
| toponuni 22860 | The base set of a topology... |
| topontopi 22861 | A topology on a given base... |
| toponunii 22862 | The base set of a topology... |
| toptopon 22863 | Alternative definition of ... |
| toptopon2 22864 | A topology is the same thi... |
| topontopon 22865 | A topology on a set is a t... |
| funtopon 22866 | The class ` TopOn ` is a f... |
| toponrestid 22867 | Given a topology on a set,... |
| toponsspwpw 22868 | The set of topologies on a... |
| dmtopon 22869 | The domain of ` TopOn ` is... |
| fntopon 22870 | The class ` TopOn ` is a f... |
| toprntopon 22871 | A topology is the same thi... |
| toponmax 22872 | The base set of a topology... |
| toponss 22873 | A member of a topology is ... |
| toponcom 22874 | If ` K ` is a topology on ... |
| toponcomb 22875 | Biconditional form of ~ to... |
| topgele 22876 | The topologies over the sa... |
| topsn 22877 | The only topology on a sin... |
| istps 22880 | Express the predicate "is ... |
| istps2 22881 | Express the predicate "is ... |
| tpsuni 22882 | The base set of a topologi... |
| tpstop 22883 | The topology extractor on ... |
| tpspropd 22884 | A topological space depend... |
| tpsprop2d 22885 | A topological space depend... |
| topontopn 22886 | Express the predicate "is ... |
| tsettps 22887 | If the topology component ... |
| istpsi 22888 | Properties that determine ... |
| eltpsg 22889 | Properties that determine ... |
| eltpsi 22890 | Properties that determine ... |
| isbasisg 22893 | Express the predicate "the... |
| isbasis2g 22894 | Express the predicate "the... |
| isbasis3g 22895 | Express the predicate "the... |
| basis1 22896 | Property of a basis. (Con... |
| basis2 22897 | Property of a basis. (Con... |
| fiinbas 22898 | If a set is closed under f... |
| basdif0 22899 | A basis is not affected by... |
| baspartn 22900 | A disjoint system of sets ... |
| tgval 22901 | The topology generated by ... |
| tgval2 22902 | Definition of a topology g... |
| eltg 22903 | Membership in a topology g... |
| eltg2 22904 | Membership in a topology g... |
| eltg2b 22905 | Membership in a topology g... |
| eltg4i 22906 | An open set in a topology ... |
| eltg3i 22907 | The union of a set of basi... |
| eltg3 22908 | Membership in a topology g... |
| tgval3 22909 | Alternate expression for t... |
| tg1 22910 | Property of a member of a ... |
| tg2 22911 | Property of a member of a ... |
| bastg 22912 | A member of a basis is a s... |
| unitg 22913 | The topology generated by ... |
| tgss 22914 | Subset relation for genera... |
| tgcl 22915 | Show that a basis generate... |
| tgclb 22916 | The property ~ tgcl can be... |
| tgtopon 22917 | A basis generates a topolo... |
| topbas 22918 | A topology is its own basi... |
| tgtop 22919 | A topology is its own basi... |
| eltop 22920 | Membership in a topology, ... |
| eltop2 22921 | Membership in a topology. ... |
| eltop3 22922 | Membership in a topology. ... |
| fibas 22923 | A collection of finite int... |
| tgdom 22924 | A space has no more open s... |
| tgiun 22925 | The indexed union of a set... |
| tgidm 22926 | The topology generator fun... |
| bastop 22927 | Two ways to express that a... |
| tgtop11 22928 | The topology generation fu... |
| 0top 22929 | The singleton of the empty... |
| en1top 22930 | ` { (/) } ` is the only to... |
| en2top 22931 | If a topology has two elem... |
| tgss3 22932 | A criterion for determinin... |
| tgss2 22933 | A criterion for determinin... |
| basgen 22934 | Given a topology ` J ` , s... |
| basgen2 22935 | Given a topology ` J ` , s... |
| 2basgen 22936 | Conditions that determine ... |
| tgfiss 22937 | If a subbase is included i... |
| tgdif0 22938 | A generated topology is no... |
| bastop1 22939 | A subset of a topology is ... |
| bastop2 22940 | A version of ~ bastop1 tha... |
| distop 22941 | The discrete topology on a... |
| topnex 22942 | The class of all topologie... |
| distopon 22943 | The discrete topology on a... |
| sn0topon 22944 | The singleton of the empty... |
| sn0top 22945 | The singleton of the empty... |
| indislem 22946 | A lemma to eliminate some ... |
| indistopon 22947 | The indiscrete topology on... |
| indistop 22948 | The indiscrete topology on... |
| indisuni 22949 | The base set of the indisc... |
| fctop 22950 | The finite complement topo... |
| fctop2 22951 | The finite complement topo... |
| cctop 22952 | The countable complement t... |
| ppttop 22953 | The particular point topol... |
| pptbas 22954 | The particular point topol... |
| epttop 22955 | The excluded point topolog... |
| indistpsx 22956 | The indiscrete topology on... |
| indistps 22957 | The indiscrete topology on... |
| indistps2 22958 | The indiscrete topology on... |
| indistpsALT 22959 | The indiscrete topology on... |
| indistps2ALT 22960 | The indiscrete topology on... |
| distps 22961 | The discrete topology on a... |
| fncld 22968 | The closed-set generator i... |
| cldval 22969 | The set of closed sets of ... |
| ntrfval 22970 | The interior function on t... |
| clsfval 22971 | The closure function on th... |
| cldrcl 22972 | Reverse closure of the clo... |
| iscld 22973 | The predicate "the class `... |
| iscld2 22974 | A subset of the underlying... |
| cldss 22975 | A closed set is a subset o... |
| cldss2 22976 | The set of closed sets is ... |
| cldopn 22977 | The complement of a closed... |
| isopn2 22978 | A subset of the underlying... |
| opncld 22979 | The complement of an open ... |
| difopn 22980 | The difference of a closed... |
| topcld 22981 | The underlying set of a to... |
| ntrval 22982 | The interior of a subset o... |
| clsval 22983 | The closure of a subset of... |
| 0cld 22984 | The empty set is closed. ... |
| iincld 22985 | The indexed intersection o... |
| intcld 22986 | The intersection of a set ... |
| uncld 22987 | The union of two closed se... |
| cldcls 22988 | A closed subset equals its... |
| incld 22989 | The intersection of two cl... |
| riincld 22990 | An indexed relative inters... |
| iuncld 22991 | A finite indexed union of ... |
| unicld 22992 | A finite union of closed s... |
| clscld 22993 | The closure of a subset of... |
| clsf 22994 | The closure function is a ... |
| ntropn 22995 | The interior of a subset o... |
| clsval2 22996 | Express closure in terms o... |
| ntrval2 22997 | Interior expressed in term... |
| ntrdif 22998 | An interior of a complemen... |
| clsdif 22999 | A closure of a complement ... |
| clsss 23000 | Subset relationship for cl... |
| ntrss 23001 | Subset relationship for in... |
| sscls 23002 | A subset of a topology's u... |
| ntrss2 23003 | A subset includes its inte... |
| ssntr 23004 | An open subset of a set is... |
| clsss3 23005 | The closure of a subset of... |
| ntrss3 23006 | The interior of a subset o... |
| ntrin 23007 | A pairwise intersection of... |
| cmclsopn 23008 | The complement of a closur... |
| cmntrcld 23009 | The complement of an inter... |
| iscld3 23010 | A subset is closed iff it ... |
| iscld4 23011 | A subset is closed iff it ... |
| isopn3 23012 | A subset is open iff it eq... |
| clsidm 23013 | The closure operation is i... |
| ntridm 23014 | The interior operation is ... |
| clstop 23015 | The closure of a topology'... |
| ntrtop 23016 | The interior of a topology... |
| 0ntr 23017 | A subset with an empty int... |
| clsss2 23018 | If a subset is included in... |
| elcls 23019 | Membership in a closure. ... |
| elcls2 23020 | Membership in a closure. ... |
| clsndisj 23021 | Any open set containing a ... |
| ntrcls0 23022 | A subset whose closure has... |
| ntreq0 23023 | Two ways to say that a sub... |
| cldmre 23024 | The closed sets of a topol... |
| mrccls 23025 | Moore closure generalizes ... |
| cls0 23026 | The closure of the empty s... |
| ntr0 23027 | The interior of the empty ... |
| isopn3i 23028 | An open subset equals its ... |
| elcls3 23029 | Membership in a closure in... |
| opncldf1 23030 | A bijection useful for con... |
| opncldf2 23031 | The values of the open-clo... |
| opncldf3 23032 | The values of the converse... |
| isclo 23033 | A set ` A ` is clopen iff ... |
| isclo2 23034 | A set ` A ` is clopen iff ... |
| discld 23035 | The open sets of a discret... |
| sn0cld 23036 | The closed sets of the top... |
| indiscld 23037 | The closed sets of an indi... |
| mretopd 23038 | A Moore collection which i... |
| toponmre 23039 | The topologies over a give... |
| cldmreon 23040 | The closed sets of a topol... |
| iscldtop 23041 | A family is the closed set... |
| mreclatdemoBAD 23042 | The closed subspaces of a ... |
| neifval 23045 | Value of the neighborhood ... |
| neif 23046 | The neighborhood function ... |
| neiss2 23047 | A set with a neighborhood ... |
| neival 23048 | Value of the set of neighb... |
| isnei 23049 | The predicate "the class `... |
| neiint 23050 | An intuitive definition of... |
| isneip 23051 | The predicate "the class `... |
| neii1 23052 | A neighborhood is included... |
| neisspw 23053 | The neighborhoods of any s... |
| neii2 23054 | Property of a neighborhood... |
| neiss 23055 | Any neighborhood of a set ... |
| ssnei 23056 | A set is included in any o... |
| elnei 23057 | A point belongs to any of ... |
| 0nnei 23058 | The empty set is not a nei... |
| neips 23059 | A neighborhood of a set is... |
| opnneissb 23060 | An open set is a neighborh... |
| opnssneib 23061 | Any superset of an open se... |
| ssnei2 23062 | Any subset ` M ` of ` X ` ... |
| neindisj 23063 | Any neighborhood of an ele... |
| opnneiss 23064 | An open set is a neighborh... |
| opnneip 23065 | An open set is a neighborh... |
| opnnei 23066 | A set is open iff it is a ... |
| tpnei 23067 | The underlying set of a to... |
| neiuni 23068 | The union of the neighborh... |
| neindisj2 23069 | A point ` P ` belongs to t... |
| topssnei 23070 | A finer topology has more ... |
| innei 23071 | The intersection of two ne... |
| opnneiid 23072 | Only an open set is a neig... |
| neissex 23073 | For any neighborhood ` N `... |
| 0nei 23074 | The empty set is a neighbo... |
| neipeltop 23075 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23076 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23077 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23078 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23079 | If, to each element ` P ` ... |
| lpfval 23084 | The limit point function o... |
| lpval 23085 | The set of limit points of... |
| islp 23086 | The predicate "the class `... |
| lpsscls 23087 | The limit points of a subs... |
| lpss 23088 | The limit points of a subs... |
| lpdifsn 23089 | ` P ` is a limit point of ... |
| lpss3 23090 | Subset relationship for li... |
| islp2 23091 | The predicate " ` P ` is a... |
| islp3 23092 | The predicate " ` P ` is a... |
| maxlp 23093 | A point is a limit point o... |
| clslp 23094 | The closure of a subset of... |
| islpi 23095 | A point belonging to a set... |
| cldlp 23096 | A subset of a topological ... |
| isperf 23097 | Definition of a perfect sp... |
| isperf2 23098 | Definition of a perfect sp... |
| isperf3 23099 | A perfect space is a topol... |
| perflp 23100 | The limit points of a perf... |
| perfi 23101 | Property of a perfect spac... |
| perftop 23102 | A perfect space is a topol... |
| restrcl 23103 | Reverse closure for the su... |
| restbas 23104 | A subspace topology basis ... |
| tgrest 23105 | A subspace can be generate... |
| resttop 23106 | A subspace topology is a t... |
| resttopon 23107 | A subspace topology is a t... |
| restuni 23108 | The underlying set of a su... |
| stoig 23109 | The topological space buil... |
| restco 23110 | Composition of subspaces. ... |
| restabs 23111 | Equivalence of being a sub... |
| restin 23112 | When the subspace region i... |
| restuni2 23113 | The underlying set of a su... |
| resttopon2 23114 | The underlying set of a su... |
| rest0 23115 | The subspace topology indu... |
| restsn 23116 | The only subspace topology... |
| restsn2 23117 | The subspace topology indu... |
| restcld 23118 | A closed set of a subspace... |
| restcldi 23119 | A closed set is closed in ... |
| restcldr 23120 | A set which is closed in t... |
| restopnb 23121 | If ` B ` is an open subset... |
| ssrest 23122 | If ` K ` is a finer topolo... |
| restopn2 23123 | If ` A ` is open, then ` B... |
| restdis 23124 | A subspace of a discrete t... |
| restfpw 23125 | The restriction of the set... |
| neitr 23126 | The neighborhood of a trac... |
| restcls 23127 | A closure in a subspace to... |
| restntr 23128 | An interior in a subspace ... |
| restlp 23129 | The limit points of a subs... |
| restperf 23130 | Perfection of a subspace. ... |
| perfopn 23131 | An open subset of a perfec... |
| resstopn 23132 | The topology of a restrict... |
| resstps 23133 | A restricted topological s... |
| ordtbaslem 23134 | Lemma for ~ ordtbas . In ... |
| ordtval 23135 | Value of the order topolog... |
| ordtuni 23136 | Value of the order topolog... |
| ordtbas2 23137 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23138 | In a total order, the fini... |
| ordttopon 23139 | Value of the order topolog... |
| ordtopn1 23140 | An upward ray ` ( P , +oo ... |
| ordtopn2 23141 | A downward ray ` ( -oo , P... |
| ordtopn3 23142 | An open interval ` ( A , B... |
| ordtcld1 23143 | A downward ray ` ( -oo , P... |
| ordtcld2 23144 | An upward ray ` [ P , +oo ... |
| ordtcld3 23145 | A closed interval ` [ A , ... |
| ordttop 23146 | The order topology is a to... |
| ordtcnv 23147 | The order dual generates t... |
| ordtrest 23148 | The subspace topology of a... |
| ordtrest2lem 23149 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23150 | An interval-closed set ` A... |
| letopon 23151 | The topology of the extend... |
| letop 23152 | The topology of the extend... |
| letopuni 23153 | The topology of the extend... |
| xrstopn 23154 | The topology component of ... |
| xrstps 23155 | The extended real number s... |
| leordtvallem1 23156 | Lemma for ~ leordtval . (... |
| leordtvallem2 23157 | Lemma for ~ leordtval . (... |
| leordtval2 23158 | The topology of the extend... |
| leordtval 23159 | The topology of the extend... |
| iccordt 23160 | A closed interval is close... |
| iocpnfordt 23161 | An unbounded above open in... |
| icomnfordt 23162 | An unbounded above open in... |
| iooordt 23163 | An open interval is open i... |
| reordt 23164 | The real numbers are an op... |
| lecldbas 23165 | The set of closed interval... |
| pnfnei 23166 | A neighborhood of ` +oo ` ... |
| mnfnei 23167 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23168 | The restriction of the les... |
| ordtresticc 23169 | The restriction of the les... |
| lmrel 23176 | The topological space conv... |
| lmrcl 23177 | Reverse closure for the co... |
| lmfval 23178 | The relation "sequence ` f... |
| cnfval 23179 | The set of all continuous ... |
| cnpfval 23180 | The function mapping the p... |
| iscn 23181 | The predicate "the class `... |
| cnpval 23182 | The set of all functions f... |
| iscnp 23183 | The predicate "the class `... |
| iscn2 23184 | The predicate "the class `... |
| iscnp2 23185 | The predicate "the class `... |
| cntop1 23186 | Reverse closure for a cont... |
| cntop2 23187 | Reverse closure for a cont... |
| cnptop1 23188 | Reverse closure for a func... |
| cnptop2 23189 | Reverse closure for a func... |
| iscnp3 23190 | The predicate "the class `... |
| cnprcl 23191 | Reverse closure for a func... |
| cnf 23192 | A continuous function is a... |
| cnpf 23193 | A continuous function at p... |
| cnpcl 23194 | The value of a continuous ... |
| cnf2 23195 | A continuous function is a... |
| cnpf2 23196 | A continuous function at p... |
| cnprcl2 23197 | Reverse closure for a func... |
| tgcn 23198 | The continuity predicate w... |
| tgcnp 23199 | The "continuous at a point... |
| subbascn 23200 | The continuity predicate w... |
| ssidcn 23201 | The identity function is a... |
| cnpimaex 23202 | Property of a function con... |
| idcn 23203 | A restricted identity func... |
| lmbr 23204 | Express the binary relatio... |
| lmbr2 23205 | Express the binary relatio... |
| lmbrf 23206 | Express the binary relatio... |
| lmconst 23207 | A constant sequence conver... |
| lmcvg 23208 | Convergence property of a ... |
| iscnp4 23209 | The predicate "the class `... |
| cnpnei 23210 | A condition for continuity... |
| cnima 23211 | An open subset of the codo... |
| cnco 23212 | The composition of two con... |
| cnpco 23213 | The composition of a funct... |
| cnclima 23214 | A closed subset of the cod... |
| iscncl 23215 | A characterization of a co... |
| cncls2i 23216 | Property of the preimage o... |
| cnntri 23217 | Property of the preimage o... |
| cnclsi 23218 | Property of the image of a... |
| cncls2 23219 | Continuity in terms of clo... |
| cncls 23220 | Continuity in terms of clo... |
| cnntr 23221 | Continuity in terms of int... |
| cnss1 23222 | If the topology ` K ` is f... |
| cnss2 23223 | If the topology ` K ` is f... |
| cncnpi 23224 | A continuous function is c... |
| cnsscnp 23225 | The set of continuous func... |
| cncnp 23226 | A continuous function is c... |
| cncnp2 23227 | A continuous function is c... |
| cnnei 23228 | Continuity in terms of nei... |
| cnconst2 23229 | A constant function is con... |
| cnconst 23230 | A constant function is con... |
| cnrest 23231 | Continuity of a restrictio... |
| cnrest2 23232 | Equivalence of continuity ... |
| cnrest2r 23233 | Equivalence of continuity ... |
| cnpresti 23234 | One direction of ~ cnprest... |
| cnprest 23235 | Equivalence of continuity ... |
| cnprest2 23236 | Equivalence of point-conti... |
| cndis 23237 | Every function is continuo... |
| cnindis 23238 | Every function is continuo... |
| cnpdis 23239 | If ` A ` is an isolated po... |
| paste 23240 | Pasting lemma. If ` A ` a... |
| lmfpm 23241 | If ` F ` converges, then `... |
| lmfss 23242 | Inclusion of a function ha... |
| lmcl 23243 | Closure of a limit. (Cont... |
| lmss 23244 | Limit on a subspace. (Con... |
| sslm 23245 | A finer topology has fewer... |
| lmres 23246 | A function converges iff i... |
| lmff 23247 | If ` F ` converges, there ... |
| lmcls 23248 | Any convergent sequence of... |
| lmcld 23249 | Any convergent sequence of... |
| lmcnp 23250 | The image of a convergent ... |
| lmcn 23251 | The image of a convergent ... |
| ist0 23266 | The predicate "is a T_0 sp... |
| ist1 23267 | The predicate "is a T_1 sp... |
| ishaus 23268 | The predicate "is a Hausdo... |
| iscnrm 23269 | The property of being comp... |
| t0sep 23270 | Any two topologically indi... |
| t0dist 23271 | Any two distinct points in... |
| t1sncld 23272 | In a T_1 space, singletons... |
| t1ficld 23273 | In a T_1 space, finite set... |
| hausnei 23274 | Neighborhood property of a... |
| t0top 23275 | A T_0 space is a topologic... |
| t1top 23276 | A T_1 space is a topologic... |
| haustop 23277 | A Hausdorff space is a top... |
| isreg 23278 | The predicate "is a regula... |
| regtop 23279 | A regular space is a topol... |
| regsep 23280 | In a regular space, every ... |
| isnrm 23281 | The predicate "is a normal... |
| nrmtop 23282 | A normal space is a topolo... |
| cnrmtop 23283 | A completely normal space ... |
| iscnrm2 23284 | The property of being comp... |
| ispnrm 23285 | The property of being perf... |
| pnrmnrm 23286 | A perfectly normal space i... |
| pnrmtop 23287 | A perfectly normal space i... |
| pnrmcld 23288 | A closed set in a perfectl... |
| pnrmopn 23289 | An open set in a perfectly... |
| ist0-2 23290 | The predicate "is a T_0 sp... |
| ist0-3 23291 | The predicate "is a T_0 sp... |
| cnt0 23292 | The preimage of a T_0 topo... |
| ist1-2 23293 | An alternate characterizat... |
| t1t0 23294 | A T_1 space is a T_0 space... |
| ist1-3 23295 | A space is T_1 iff every p... |
| cnt1 23296 | The preimage of a T_1 topo... |
| ishaus2 23297 | Express the predicate " ` ... |
| haust1 23298 | A Hausdorff space is a T_1... |
| hausnei2 23299 | The Hausdorff condition st... |
| cnhaus 23300 | The preimage of a Hausdorf... |
| nrmsep3 23301 | In a normal space, given a... |
| nrmsep2 23302 | In a normal space, any two... |
| nrmsep 23303 | In a normal space, disjoin... |
| isnrm2 23304 | An alternate characterizat... |
| isnrm3 23305 | A topological space is nor... |
| cnrmi 23306 | A subspace of a completely... |
| cnrmnrm 23307 | A completely normal space ... |
| restcnrm 23308 | A subspace of a completely... |
| resthauslem 23309 | Lemma for ~ resthaus and s... |
| lpcls 23310 | The limit points of the cl... |
| perfcls 23311 | A subset of a perfect spac... |
| restt0 23312 | A subspace of a T_0 topolo... |
| restt1 23313 | A subspace of a T_1 topolo... |
| resthaus 23314 | A subspace of a Hausdorff ... |
| t1sep2 23315 | Any two points in a T_1 sp... |
| t1sep 23316 | Any two distinct points in... |
| sncld 23317 | A singleton is closed in a... |
| sshauslem 23318 | Lemma for ~ sshaus and sim... |
| sst0 23319 | A topology finer than a T_... |
| sst1 23320 | A topology finer than a T_... |
| sshaus 23321 | A topology finer than a Ha... |
| regsep2 23322 | In a regular space, a clos... |
| isreg2 23323 | A topological space is reg... |
| dnsconst 23324 | If a continuous mapping to... |
| ordtt1 23325 | The order topology is T_1 ... |
| lmmo 23326 | A sequence in a Hausdorff ... |
| lmfun 23327 | The convergence relation i... |
| dishaus 23328 | A discrete topology is Hau... |
| ordthauslem 23329 | Lemma for ~ ordthaus . (C... |
| ordthaus 23330 | The order topology of a to... |
| xrhaus 23331 | The topology of the extend... |
| iscmp 23334 | The predicate "is a compac... |
| cmpcov 23335 | An open cover of a compact... |
| cmpcov2 23336 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23337 | Combine ~ cmpcov with ~ ac... |
| cncmp 23338 | Compactness is respected b... |
| fincmp 23339 | A finite topology is compa... |
| 0cmp 23340 | The singleton of the empty... |
| cmptop 23341 | A compact topology is a to... |
| rncmp 23342 | The image of a compact set... |
| imacmp 23343 | The image of a compact set... |
| discmp 23344 | A discrete topology is com... |
| cmpsublem 23345 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23346 | Two equivalent ways of des... |
| tgcmp 23347 | A topology generated by a ... |
| cmpcld 23348 | A closed subset of a compa... |
| uncmp 23349 | The union of two compact s... |
| fiuncmp 23350 | A finite union of compact ... |
| sscmp 23351 | A subset of a compact topo... |
| hauscmplem 23352 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23353 | A compact subspace of a T2... |
| cmpfi 23354 | If a topology is compact a... |
| cmpfii 23355 | In a compact topology, a s... |
| bwth 23356 | The glorious Bolzano-Weier... |
| isconn 23359 | The predicate ` J ` is a c... |
| isconn2 23360 | The predicate ` J ` is a c... |
| connclo 23361 | The only nonempty clopen s... |
| conndisj 23362 | If a topology is connected... |
| conntop 23363 | A connected topology is a ... |
| indisconn 23364 | The indiscrete topology (o... |
| dfconn2 23365 | An alternate definition of... |
| connsuba 23366 | Connectedness for a subspa... |
| connsub 23367 | Two equivalent ways of say... |
| cnconn 23368 | Connectedness is respected... |
| nconnsubb 23369 | Disconnectedness for a sub... |
| connsubclo 23370 | If a clopen set meets a co... |
| connima 23371 | The image of a connected s... |
| conncn 23372 | A continuous function from... |
| iunconnlem 23373 | Lemma for ~ iunconn . (Co... |
| iunconn 23374 | The indexed union of conne... |
| unconn 23375 | The union of two connected... |
| clsconn 23376 | The closure of a connected... |
| conncompid 23377 | The connected component co... |
| conncompconn 23378 | The connected component co... |
| conncompss 23379 | The connected component co... |
| conncompcld 23380 | The connected component co... |
| conncompclo 23381 | The connected component co... |
| t1connperf 23382 | A connected T_1 space is p... |
| is1stc 23387 | The predicate "is a first-... |
| is1stc2 23388 | An equivalent way of sayin... |
| 1stctop 23389 | A first-countable topology... |
| 1stcclb 23390 | A property of points in a ... |
| 1stcfb 23391 | For any point ` A ` in a f... |
| is2ndc 23392 | The property of being seco... |
| 2ndctop 23393 | A second-countable topolog... |
| 2ndci 23394 | A countable basis generate... |
| 2ndcsb 23395 | Having a countable subbase... |
| 2ndcredom 23396 | A second-countable space h... |
| 2ndc1stc 23397 | A second-countable space i... |
| 1stcrestlem 23398 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23399 | A subspace of a first-coun... |
| 2ndcrest 23400 | A subspace of a second-cou... |
| 2ndcctbss 23401 | If a topology is second-co... |
| 2ndcdisj 23402 | Any disjoint family of ope... |
| 2ndcdisj2 23403 | Any disjoint collection of... |
| 2ndcomap 23404 | A surjective continuous op... |
| 2ndcsep 23405 | A second-countable topolog... |
| dis2ndc 23406 | A discrete space is second... |
| 1stcelcls 23407 | A point belongs to the clo... |
| 1stccnp 23408 | A mapping is continuous at... |
| 1stccn 23409 | A mapping ` X --> Y ` , wh... |
| islly 23414 | The property of being a lo... |
| isnlly 23415 | The property of being an n... |
| llyeq 23416 | Equality theorem for the `... |
| nllyeq 23417 | Equality theorem for the `... |
| llytop 23418 | A locally ` A ` space is a... |
| nllytop 23419 | A locally ` A ` space is a... |
| llyi 23420 | The property of a locally ... |
| nllyi 23421 | The property of an n-local... |
| nlly2i 23422 | Eliminate the neighborhood... |
| llynlly 23423 | A locally ` A ` space is n... |
| llyssnlly 23424 | A locally ` A ` space is n... |
| llyss 23425 | The "locally" predicate re... |
| nllyss 23426 | The "n-locally" predicate ... |
| subislly 23427 | The property of a subspace... |
| restnlly 23428 | If the property ` A ` pass... |
| restlly 23429 | If the property ` A ` pass... |
| islly2 23430 | An alternative expression ... |
| llyrest 23431 | An open subspace of a loca... |
| nllyrest 23432 | An open subspace of an n-l... |
| loclly 23433 | If ` A ` is a local proper... |
| llyidm 23434 | Idempotence of the "locall... |
| nllyidm 23435 | Idempotence of the "n-loca... |
| toplly 23436 | A topology is locally a to... |
| topnlly 23437 | A topology is n-locally a ... |
| hauslly 23438 | A Hausdorff space is local... |
| hausnlly 23439 | A Hausdorff space is n-loc... |
| hausllycmp 23440 | A compact Hausdorff space ... |
| cldllycmp 23441 | A closed subspace of a loc... |
| lly1stc 23442 | First-countability is a lo... |
| dislly 23443 | The discrete space ` ~P X ... |
| disllycmp 23444 | A discrete space is locall... |
| dis1stc 23445 | A discrete space is first-... |
| hausmapdom 23446 | If ` X ` is a first-counta... |
| hauspwdom 23447 | Simplify the cardinal ` A ... |
| refrel 23454 | Refinement is a relation. ... |
| isref 23455 | The property of being a re... |
| refbas 23456 | A refinement covers the sa... |
| refssex 23457 | Every set in a refinement ... |
| ssref 23458 | A subcover is a refinement... |
| refref 23459 | Reflexivity of refinement.... |
| reftr 23460 | Refinement is transitive. ... |
| refun0 23461 | Adding the empty set prese... |
| isptfin 23462 | The statement "is a point-... |
| islocfin 23463 | The statement "is a locall... |
| finptfin 23464 | A finite cover is a point-... |
| ptfinfin 23465 | A point covered by a point... |
| finlocfin 23466 | A finite cover of a topolo... |
| locfintop 23467 | A locally finite cover cov... |
| locfinbas 23468 | A locally finite cover mus... |
| locfinnei 23469 | A point covered by a local... |
| lfinpfin 23470 | A locally finite cover is ... |
| lfinun 23471 | Adding a finite set preser... |
| locfincmp 23472 | For a compact space, the l... |
| unisngl 23473 | Taking the union of the se... |
| dissnref 23474 | The set of singletons is a... |
| dissnlocfin 23475 | The set of singletons is l... |
| locfindis 23476 | The locally finite covers ... |
| locfincf 23477 | A locally finite cover in ... |
| comppfsc 23478 | A space where every open c... |
| kgenval 23481 | Value of the compact gener... |
| elkgen 23482 | Value of the compact gener... |
| kgeni 23483 | Property of the open sets ... |
| kgentopon 23484 | The compact generator gene... |
| kgenuni 23485 | The base set of the compac... |
| kgenftop 23486 | The compact generator gene... |
| kgenf 23487 | The compact generator is a... |
| kgentop 23488 | A compactly generated spac... |
| kgenss 23489 | The compact generator gene... |
| kgenhaus 23490 | The compact generator gene... |
| kgencmp 23491 | The compact generator topo... |
| kgencmp2 23492 | The compact generator topo... |
| kgenidm 23493 | The compact generator is i... |
| iskgen2 23494 | A space is compactly gener... |
| iskgen3 23495 | Derive the usual definitio... |
| llycmpkgen2 23496 | A locally compact space is... |
| cmpkgen 23497 | A compact space is compact... |
| llycmpkgen 23498 | A locally compact space is... |
| 1stckgenlem 23499 | The one-point compactifica... |
| 1stckgen 23500 | A first-countable space is... |
| kgen2ss 23501 | The compact generator pres... |
| kgencn 23502 | A function from a compactl... |
| kgencn2 23503 | A function ` F : J --> K `... |
| kgencn3 23504 | The set of continuous func... |
| kgen2cn 23505 | A continuous function is a... |
| txval 23510 | Value of the binary topolo... |
| txuni2 23511 | The underlying set of the ... |
| txbasex 23512 | The basis for the product ... |
| txbas 23513 | The set of Cartesian produ... |
| eltx 23514 | A set in a product is open... |
| txtop 23515 | The product of two topolog... |
| ptval 23516 | The value of the product t... |
| ptpjpre1 23517 | The preimage of a projecti... |
| elpt 23518 | Elementhood in the bases o... |
| elptr 23519 | A basic open set in the pr... |
| elptr2 23520 | A basic open set in the pr... |
| ptbasid 23521 | The base set of the produc... |
| ptuni2 23522 | The base set for the produ... |
| ptbasin 23523 | The basis for a product to... |
| ptbasin2 23524 | The basis for a product to... |
| ptbas 23525 | The basis for a product to... |
| ptpjpre2 23526 | The basis for a product to... |
| ptbasfi 23527 | The basis for the product ... |
| pttop 23528 | The product topology is a ... |
| ptopn 23529 | A basic open set in the pr... |
| ptopn2 23530 | A sub-basic open set in th... |
| xkotf 23531 | Functionality of function ... |
| xkobval 23532 | Alternative expression for... |
| xkoval 23533 | Value of the compact-open ... |
| xkotop 23534 | The compact-open topology ... |
| xkoopn 23535 | A basic open set of the co... |
| txtopi 23536 | The product of two topolog... |
| txtopon 23537 | The underlying set of the ... |
| txuni 23538 | The underlying set of the ... |
| txunii 23539 | The underlying set of the ... |
| ptuni 23540 | The base set for the produ... |
| ptunimpt 23541 | Base set of a product topo... |
| pttopon 23542 | The base set for the produ... |
| pttoponconst 23543 | The base set for a product... |
| ptuniconst 23544 | The base set for a product... |
| xkouni 23545 | The base set of the compac... |
| xkotopon 23546 | The base set of the compac... |
| ptval2 23547 | The value of the product t... |
| txopn 23548 | The product of two open se... |
| txcld 23549 | The product of two closed ... |
| txcls 23550 | Closure of a rectangle in ... |
| txss12 23551 | Subset property of the top... |
| txbasval 23552 | It is sufficient to consid... |
| neitx 23553 | The Cartesian product of t... |
| txcnpi 23554 | Continuity of a two-argume... |
| tx1cn 23555 | Continuity of the first pr... |
| tx2cn 23556 | Continuity of the second p... |
| ptpjcn 23557 | Continuity of a projection... |
| ptpjopn 23558 | The projection map is an o... |
| ptcld 23559 | A closed box in the produc... |
| ptcldmpt 23560 | A closed box in the produc... |
| ptclsg 23561 | The closure of a box in th... |
| ptcls 23562 | The closure of a box in th... |
| dfac14lem 23563 | Lemma for ~ dfac14 . By e... |
| dfac14 23564 | Theorem ~ ptcls is an equi... |
| xkoccn 23565 | The "constant function" fu... |
| txcnp 23566 | If two functions are conti... |
| ptcnplem 23567 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23568 | If every projection of a f... |
| upxp 23569 | Universal property of the ... |
| txcnmpt 23570 | A map into the product of ... |
| uptx 23571 | Universal property of the ... |
| txcn 23572 | A map into the product of ... |
| ptcn 23573 | If every projection of a f... |
| prdstopn 23574 | Topology of a structure pr... |
| prdstps 23575 | A structure product of top... |
| pwstps 23576 | A structure power of a top... |
| txrest 23577 | The subspace of a topologi... |
| txdis 23578 | The topological product of... |
| txindislem 23579 | Lemma for ~ txindis . (Co... |
| txindis 23580 | The topological product of... |
| txdis1cn 23581 | A function is jointly cont... |
| txlly 23582 | If the property ` A ` is p... |
| txnlly 23583 | If the property ` A ` is p... |
| pthaus 23584 | The product of a collectio... |
| ptrescn 23585 | Restriction is a continuou... |
| txtube 23586 | The "tube lemma". If ` X ... |
| txcmplem1 23587 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23588 | Lemma for ~ txcmp . (Cont... |
| txcmp 23589 | The topological product of... |
| txcmpb 23590 | The topological product of... |
| hausdiag 23591 | A topology is Hausdorff if... |
| hauseqlcld 23592 | In a Hausdorff topology, t... |
| txhaus 23593 | The topological product of... |
| txlm 23594 | Two sequences converge iff... |
| lmcn2 23595 | The image of a convergent ... |
| tx1stc 23596 | The topological product of... |
| tx2ndc 23597 | The topological product of... |
| txkgen 23598 | The topological product of... |
| xkohaus 23599 | If the codomain space is H... |
| xkoptsub 23600 | The compact-open topology ... |
| xkopt 23601 | The compact-open topology ... |
| xkopjcn 23602 | Continuity of a projection... |
| xkoco1cn 23603 | If ` F ` is a continuous f... |
| xkoco2cn 23604 | If ` F ` is a continuous f... |
| xkococnlem 23605 | Continuity of the composit... |
| xkococn 23606 | Continuity of the composit... |
| cnmptid 23607 | The identity function is c... |
| cnmptc 23608 | A constant function is con... |
| cnmpt11 23609 | The composition of continu... |
| cnmpt11f 23610 | The composition of continu... |
| cnmpt1t 23611 | The composition of continu... |
| cnmpt12f 23612 | The composition of continu... |
| cnmpt12 23613 | The composition of continu... |
| cnmpt1st 23614 | The projection onto the fi... |
| cnmpt2nd 23615 | The projection onto the se... |
| cnmpt2c 23616 | A constant function is con... |
| cnmpt21 23617 | The composition of continu... |
| cnmpt21f 23618 | The composition of continu... |
| cnmpt2t 23619 | The composition of continu... |
| cnmpt22 23620 | The composition of continu... |
| cnmpt22f 23621 | The composition of continu... |
| cnmpt1res 23622 | The restriction of a conti... |
| cnmpt2res 23623 | The restriction of a conti... |
| cnmptcom 23624 | The argument converse of a... |
| cnmptkc 23625 | The curried first projecti... |
| cnmptkp 23626 | The evaluation of the inne... |
| cnmptk1 23627 | The composition of a curri... |
| cnmpt1k 23628 | The composition of a one-a... |
| cnmptkk 23629 | The composition of two cur... |
| xkofvcn 23630 | Joint continuity of the fu... |
| cnmptk1p 23631 | The evaluation of a currie... |
| cnmptk2 23632 | The uncurrying of a currie... |
| xkoinjcn 23633 | Continuity of "injection",... |
| cnmpt2k 23634 | The currying of a two-argu... |
| txconn 23635 | The topological product of... |
| imasnopn 23636 | If a relation graph is ope... |
| imasncld 23637 | If a relation graph is clo... |
| imasncls 23638 | If a relation graph is clo... |
| qtopval 23641 | Value of the quotient topo... |
| qtopval2 23642 | Value of the quotient topo... |
| elqtop 23643 | Value of the quotient topo... |
| qtopres 23644 | The quotient topology is u... |
| qtoptop2 23645 | The quotient topology is a... |
| qtoptop 23646 | The quotient topology is a... |
| elqtop2 23647 | Value of the quotient topo... |
| qtopuni 23648 | The base set of the quotie... |
| elqtop3 23649 | Value of the quotient topo... |
| qtoptopon 23650 | The base set of the quotie... |
| qtopid 23651 | A quotient map is a contin... |
| idqtop 23652 | The quotient topology indu... |
| qtopcmplem 23653 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23654 | A quotient of a compact sp... |
| qtopconn 23655 | A quotient of a connected ... |
| qtopkgen 23656 | A quotient of a compactly ... |
| basqtop 23657 | An injection maps bases to... |
| tgqtop 23658 | An injection maps generate... |
| qtopcld 23659 | The property of being a cl... |
| qtopcn 23660 | Universal property of a qu... |
| qtopss 23661 | A surjective continuous fu... |
| qtopeu 23662 | Universal property of the ... |
| qtoprest 23663 | If ` A ` is a saturated op... |
| qtopomap 23664 | If ` F ` is a surjective c... |
| qtopcmap 23665 | If ` F ` is a surjective c... |
| imastopn 23666 | The topology of an image s... |
| imastps 23667 | The image of a topological... |
| qustps 23668 | A quotient structure is a ... |
| kqfval 23669 | Value of the function appe... |
| kqfeq 23670 | Two points in the Kolmogor... |
| kqffn 23671 | The topological indistingu... |
| kqval 23672 | Value of the quotient topo... |
| kqtopon 23673 | The Kolmogorov quotient is... |
| kqid 23674 | The topological indistingu... |
| ist0-4 23675 | The topological indistingu... |
| kqfvima 23676 | When the image set is open... |
| kqsat 23677 | Any open set is saturated ... |
| kqdisj 23678 | A version of ~ imain for t... |
| kqcldsat 23679 | Any closed set is saturate... |
| kqopn 23680 | The topological indistingu... |
| kqcld 23681 | The topological indistingu... |
| kqt0lem 23682 | Lemma for ~ kqt0 . (Contr... |
| isr0 23683 | The property " ` J ` is an... |
| r0cld 23684 | The analogue of the T_1 ax... |
| regr1lem 23685 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23686 | A Kolmogorov quotient of a... |
| kqreglem1 23687 | A Kolmogorov quotient of a... |
| kqreglem2 23688 | If the Kolmogorov quotient... |
| kqnrmlem1 23689 | A Kolmogorov quotient of a... |
| kqnrmlem2 23690 | If the Kolmogorov quotient... |
| kqtop 23691 | The Kolmogorov quotient is... |
| kqt0 23692 | The Kolmogorov quotient is... |
| kqf 23693 | The Kolmogorov quotient is... |
| r0sep 23694 | The separation property of... |
| nrmr0reg 23695 | A normal R_0 space is also... |
| regr1 23696 | A regular space is R_1, wh... |
| kqreg 23697 | The Kolmogorov quotient of... |
| kqnrm 23698 | The Kolmogorov quotient of... |
| hmeofn 23703 | The set of homeomorphisms ... |
| hmeofval 23704 | The set of all the homeomo... |
| ishmeo 23705 | The predicate F is a homeo... |
| hmeocn 23706 | A homeomorphism is continu... |
| hmeocnvcn 23707 | The converse of a homeomor... |
| hmeocnv 23708 | The converse of a homeomor... |
| hmeof1o2 23709 | A homeomorphism is a 1-1-o... |
| hmeof1o 23710 | A homeomorphism is a 1-1-o... |
| hmeoima 23711 | The image of an open set b... |
| hmeoopn 23712 | Homeomorphisms preserve op... |
| hmeocld 23713 | Homeomorphisms preserve cl... |
| hmeocls 23714 | Homeomorphisms preserve cl... |
| hmeontr 23715 | Homeomorphisms preserve in... |
| hmeoimaf1o 23716 | The function mapping open ... |
| hmeores 23717 | The restriction of a homeo... |
| hmeoco 23718 | The composite of two homeo... |
| idhmeo 23719 | The identity function is a... |
| hmeocnvb 23720 | The converse of a homeomor... |
| hmeoqtop 23721 | A homeomorphism is a quoti... |
| hmph 23722 | Express the predicate ` J ... |
| hmphi 23723 | If there is a homeomorphis... |
| hmphtop 23724 | Reverse closure for the ho... |
| hmphtop1 23725 | The relation "being homeom... |
| hmphtop2 23726 | The relation "being homeom... |
| hmphref 23727 | "Is homeomorphic to" is re... |
| hmphsym 23728 | "Is homeomorphic to" is sy... |
| hmphtr 23729 | "Is homeomorphic to" is tr... |
| hmpher 23730 | "Is homeomorphic to" is an... |
| hmphen 23731 | Homeomorphisms preserve th... |
| hmphsymb 23732 | "Is homeomorphic to" is sy... |
| haushmphlem 23733 | Lemma for ~ haushmph and s... |
| cmphmph 23734 | Compactness is a topologic... |
| connhmph 23735 | Connectedness is a topolog... |
| t0hmph 23736 | T_0 is a topological prope... |
| t1hmph 23737 | T_1 is a topological prope... |
| haushmph 23738 | Hausdorff-ness is a topolo... |
| reghmph 23739 | Regularity is a topologica... |
| nrmhmph 23740 | Normality is a topological... |
| hmph0 23741 | A topology homeomorphic to... |
| hmphdis 23742 | Homeomorphisms preserve to... |
| hmphindis 23743 | Homeomorphisms preserve to... |
| indishmph 23744 | Equinumerous sets equipped... |
| hmphen2 23745 | Homeomorphisms preserve th... |
| cmphaushmeo 23746 | A continuous bijection fro... |
| ordthmeolem 23747 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23748 | An order isomorphism is a ... |
| txhmeo 23749 | Lift a pair of homeomorphi... |
| txswaphmeolem 23750 | Show inverse for the "swap... |
| txswaphmeo 23751 | There is a homeomorphism f... |
| pt1hmeo 23752 | The canonical homeomorphis... |
| ptuncnv 23753 | Exhibit the converse funct... |
| ptunhmeo 23754 | Define a homeomorphism fro... |
| xpstopnlem1 23755 | The function ` F ` used in... |
| xpstps 23756 | A binary product of topolo... |
| xpstopnlem2 23757 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23758 | The topology on a binary p... |
| ptcmpfi 23759 | A topological product of f... |
| xkocnv 23760 | The inverse of the "curryi... |
| xkohmeo 23761 | The Exponential Law for to... |
| qtopf1 23762 | If a quotient map is injec... |
| qtophmeo 23763 | If two functions on a base... |
| t0kq 23764 | A topological space is T_0... |
| kqhmph 23765 | A topological space is T_0... |
| ist1-5lem 23766 | Lemma for ~ ist1-5 and sim... |
| t1r0 23767 | A T_1 space is R_0. That ... |
| ist1-5 23768 | A topological space is T_1... |
| ishaus3 23769 | A topological space is Hau... |
| nrmreg 23770 | A normal T_1 space is regu... |
| reghaus 23771 | A regular T_0 space is Hau... |
| nrmhaus 23772 | A T_1 normal space is Haus... |
| elmptrab 23773 | Membership in a one-parame... |
| elmptrab2 23774 | Membership in a one-parame... |
| isfbas 23775 | The predicate " ` F ` is a... |
| fbasne0 23776 | There are no empty filter ... |
| 0nelfb 23777 | No filter base contains th... |
| fbsspw 23778 | A filter base on a set is ... |
| fbelss 23779 | An element of the filter b... |
| fbdmn0 23780 | The domain of a filter bas... |
| isfbas2 23781 | The predicate " ` F ` is a... |
| fbasssin 23782 | A filter base contains sub... |
| fbssfi 23783 | A filter base contains sub... |
| fbssint 23784 | A filter base contains sub... |
| fbncp 23785 | A filter base does not con... |
| fbun 23786 | A necessary and sufficient... |
| fbfinnfr 23787 | No filter base containing ... |
| opnfbas 23788 | The collection of open sup... |
| trfbas2 23789 | Conditions for the trace o... |
| trfbas 23790 | Conditions for the trace o... |
| isfil 23793 | The predicate "is a filter... |
| filfbas 23794 | A filter is a filter base.... |
| 0nelfil 23795 | The empty set doesn't belo... |
| fileln0 23796 | An element of a filter is ... |
| filsspw 23797 | A filter is a subset of th... |
| filelss 23798 | An element of a filter is ... |
| filss 23799 | A filter is closed under t... |
| filin 23800 | A filter is closed under t... |
| filtop 23801 | The underlying set belongs... |
| isfil2 23802 | Derive the standard axioms... |
| isfildlem 23803 | Lemma for ~ isfild . (Con... |
| isfild 23804 | Sufficient condition for a... |
| filfi 23805 | A filter is closed under t... |
| filinn0 23806 | The intersection of two el... |
| filintn0 23807 | A filter has the finite in... |
| filn0 23808 | The empty set is not a fil... |
| infil 23809 | The intersection of two fi... |
| snfil 23810 | A singleton is a filter. ... |
| fbasweak 23811 | A filter base on any set i... |
| snfbas 23812 | Condition for a singleton ... |
| fsubbas 23813 | A condition for a set to g... |
| fbasfip 23814 | A filter base has the fini... |
| fbunfip 23815 | A helpful lemma for showin... |
| fgval 23816 | The filter generating clas... |
| elfg 23817 | A condition for elements o... |
| ssfg 23818 | A filter base is a subset ... |
| fgss 23819 | A bigger base generates a ... |
| fgss2 23820 | A condition for a filter t... |
| fgfil 23821 | A filter generates itself.... |
| elfilss 23822 | An element belongs to a fi... |
| filfinnfr 23823 | No filter containing a fin... |
| fgcl 23824 | A generated filter is a fi... |
| fgabs 23825 | Absorption law for filter ... |
| neifil 23826 | The neighborhoods of a non... |
| filunibas 23827 | Recover the base set from ... |
| filunirn 23828 | Two ways to express a filt... |
| filconn 23829 | A filter gives rise to a c... |
| fbasrn 23830 | Given a filter on a domain... |
| filuni 23831 | The union of a nonempty se... |
| trfil1 23832 | Conditions for the trace o... |
| trfil2 23833 | Conditions for the trace o... |
| trfil3 23834 | Conditions for the trace o... |
| trfilss 23835 | If ` A ` is a member of th... |
| fgtr 23836 | If ` A ` is a member of th... |
| trfg 23837 | The trace operation and th... |
| trnei 23838 | The trace, over a set ` A ... |
| cfinfil 23839 | Relative complements of th... |
| csdfil 23840 | The set of all elements wh... |
| supfil 23841 | The supersets of a nonempt... |
| zfbas 23842 | The set of upper sets of i... |
| uzrest 23843 | The restriction of the set... |
| uzfbas 23844 | The set of upper sets of i... |
| isufil 23849 | The property of being an u... |
| ufilfil 23850 | An ultrafilter is a filter... |
| ufilss 23851 | For any subset of the base... |
| ufilb 23852 | The complement is in an ul... |
| ufilmax 23853 | Any filter finer than an u... |
| isufil2 23854 | The maximal property of an... |
| ufprim 23855 | An ultrafilter is a prime ... |
| trufil 23856 | Conditions for the trace o... |
| filssufilg 23857 | A filter is contained in s... |
| filssufil 23858 | A filter is contained in s... |
| isufl 23859 | Define the (strong) ultraf... |
| ufli 23860 | Property of a set that sat... |
| numufl 23861 | Consequence of ~ filssufil... |
| fiufl 23862 | A finite set satisfies the... |
| acufl 23863 | The axiom of choice implie... |
| ssufl 23864 | If ` Y ` is a subset of ` ... |
| ufileu 23865 | If the ultrafilter contain... |
| filufint 23866 | A filter is equal to the i... |
| uffix 23867 | Lemma for ~ fixufil and ~ ... |
| fixufil 23868 | The condition describing a... |
| uffixfr 23869 | An ultrafilter is either f... |
| uffix2 23870 | A classification of fixed ... |
| uffixsn 23871 | The singleton of the gener... |
| ufildom1 23872 | An ultrafilter is generate... |
| uffinfix 23873 | An ultrafilter containing ... |
| cfinufil 23874 | An ultrafilter is free iff... |
| ufinffr 23875 | An infinite subset is cont... |
| ufilen 23876 | Any infinite set has an ul... |
| ufildr 23877 | An ultrafilter gives rise ... |
| fin1aufil 23878 | There are no definable fre... |
| fmval 23889 | Introduce a function that ... |
| fmfil 23890 | A mapping filter is a filt... |
| fmf 23891 | Pushing-forward via a func... |
| fmss 23892 | A finer filter produces a ... |
| elfm 23893 | An element of a mapping fi... |
| elfm2 23894 | An element of a mapping fi... |
| fmfg 23895 | The image filter of a filt... |
| elfm3 23896 | An alternate formulation o... |
| imaelfm 23897 | An image of a filter eleme... |
| rnelfmlem 23898 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23899 | A condition for a filter t... |
| fmfnfmlem1 23900 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23901 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23902 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23903 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23904 | A filter finer than an ima... |
| fmufil 23905 | An image filter of an ultr... |
| fmid 23906 | The filter map applied to ... |
| fmco 23907 | Composition of image filte... |
| ufldom 23908 | The ultrafilter lemma prop... |
| flimval 23909 | The set of limit points of... |
| elflim2 23910 | The predicate "is a limit ... |
| flimtop 23911 | Reverse closure for the li... |
| flimneiss 23912 | A filter contains the neig... |
| flimnei 23913 | A filter contains all of t... |
| flimelbas 23914 | A limit point of a filter ... |
| flimfil 23915 | Reverse closure for the li... |
| flimtopon 23916 | Reverse closure for the li... |
| elflim 23917 | The predicate "is a limit ... |
| flimss2 23918 | A limit point of a filter ... |
| flimss1 23919 | A limit point of a filter ... |
| neiflim 23920 | A point is a limit point o... |
| flimopn 23921 | The condition for being a ... |
| fbflim 23922 | A condition for a filter t... |
| fbflim2 23923 | A condition for a filter b... |
| flimclsi 23924 | The convergent points of a... |
| hausflimlem 23925 | If ` A ` and ` B ` are bot... |
| hausflimi 23926 | One direction of ~ hausfli... |
| hausflim 23927 | A condition for a topology... |
| flimcf 23928 | Fineness is properly chara... |
| flimrest 23929 | The set of limit points in... |
| flimclslem 23930 | Lemma for ~ flimcls . (Co... |
| flimcls 23931 | Closure in terms of filter... |
| flimsncls 23932 | If ` A ` is a limit point ... |
| hauspwpwf1 23933 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23934 | If ` X ` is a Hausdorff sp... |
| flffval 23935 | Given a topology and a fil... |
| flfval 23936 | Given a function from a fi... |
| flfnei 23937 | The property of being a li... |
| flfneii 23938 | A neighborhood of a limit ... |
| isflf 23939 | The property of being a li... |
| flfelbas 23940 | A limit point of a functio... |
| flffbas 23941 | Limit points of a function... |
| flftg 23942 | Limit points of a function... |
| hausflf 23943 | If a function has its valu... |
| hausflf2 23944 | If a convergent function h... |
| cnpflfi 23945 | Forward direction of ~ cnp... |
| cnpflf2 23946 | ` F ` is continuous at poi... |
| cnpflf 23947 | Continuity of a function a... |
| cnflf 23948 | A function is continuous i... |
| cnflf2 23949 | A function is continuous i... |
| flfcnp 23950 | A continuous function pres... |
| lmflf 23951 | The topological limit rela... |
| txflf 23952 | Two sequences converge in ... |
| flfcnp2 23953 | The image of a convergent ... |
| fclsval 23954 | The set of all cluster poi... |
| isfcls 23955 | A cluster point of a filte... |
| fclsfil 23956 | Reverse closure for the cl... |
| fclstop 23957 | Reverse closure for the cl... |
| fclstopon 23958 | Reverse closure for the cl... |
| isfcls2 23959 | A cluster point of a filte... |
| fclsopn 23960 | Write the cluster point co... |
| fclsopni 23961 | An open neighborhood of a ... |
| fclselbas 23962 | A cluster point is in the ... |
| fclsneii 23963 | A neighborhood of a cluste... |
| fclssscls 23964 | The set of cluster points ... |
| fclsnei 23965 | Cluster points in terms of... |
| supnfcls 23966 | The filter of supersets of... |
| fclsbas 23967 | Cluster points in terms of... |
| fclsss1 23968 | A finer topology has fewer... |
| fclsss2 23969 | A finer filter has fewer c... |
| fclsrest 23970 | The set of cluster points ... |
| fclscf 23971 | Characterization of finene... |
| flimfcls 23972 | A limit point is a cluster... |
| fclsfnflim 23973 | A filter clusters at a poi... |
| flimfnfcls 23974 | A filter converges to a po... |
| fclscmpi 23975 | Forward direction of ~ fcl... |
| fclscmp 23976 | A space is compact iff eve... |
| uffclsflim 23977 | The cluster points of an u... |
| ufilcmp 23978 | A space is compact iff eve... |
| fcfval 23979 | The set of cluster points ... |
| isfcf 23980 | The property of being a cl... |
| fcfnei 23981 | The property of being a cl... |
| fcfelbas 23982 | A cluster point of a funct... |
| fcfneii 23983 | A neighborhood of a cluste... |
| flfssfcf 23984 | A limit point of a functio... |
| uffcfflf 23985 | If the domain filter is an... |
| cnpfcfi 23986 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23987 | A function ` F ` is contin... |
| cnfcf 23988 | Continuity of a function i... |
| flfcntr 23989 | A continuous function's va... |
| alexsublem 23990 | Lemma for ~ alexsub . (Co... |
| alexsub 23991 | The Alexander Subbase Theo... |
| alexsubb 23992 | Biconditional form of the ... |
| alexsubALTlem1 23993 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23994 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23995 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23996 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23997 | The Alexander Subbase Theo... |
| ptcmplem1 23998 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23999 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 24000 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 24001 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 24002 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24003 | Tychonoff's theorem: The ... |
| ptcmp 24004 | Tychonoff's theorem: The ... |
| cnextval 24007 | The function applying cont... |
| cnextfval 24008 | The continuous extension o... |
| cnextrel 24009 | In the general case, a con... |
| cnextfun 24010 | If the target space is Hau... |
| cnextfvval 24011 | The value of the continuou... |
| cnextf 24012 | Extension by continuity. ... |
| cnextcn 24013 | Extension by continuity. ... |
| cnextfres1 24014 | ` F ` and its extension by... |
| cnextfres 24015 | ` F ` and its extension by... |
| istmd 24020 | The predicate "is a topolo... |
| tmdmnd 24021 | A topological monoid is a ... |
| tmdtps 24022 | A topological monoid is a ... |
| istgp 24023 | The predicate "is a topolo... |
| tgpgrp 24024 | A topological group is a g... |
| tgptmd 24025 | A topological group is a t... |
| tgptps 24026 | A topological group is a t... |
| tmdtopon 24027 | The topology of a topologi... |
| tgptopon 24028 | The topology of a topologi... |
| tmdcn 24029 | In a topological monoid, t... |
| tgpcn 24030 | In a topological group, th... |
| tgpinv 24031 | In a topological group, th... |
| grpinvhmeo 24032 | The inverse function in a ... |
| cnmpt1plusg 24033 | Continuity of the group su... |
| cnmpt2plusg 24034 | Continuity of the group su... |
| tmdcn2 24035 | Write out the definition o... |
| tgpsubcn 24036 | In a topological group, th... |
| istgp2 24037 | A group with a topology is... |
| tmdmulg 24038 | In a topological monoid, t... |
| tgpmulg 24039 | In a topological group, th... |
| tgpmulg2 24040 | In a topological monoid, t... |
| tmdgsum 24041 | In a topological monoid, t... |
| tmdgsum2 24042 | For any neighborhood ` U `... |
| oppgtmd 24043 | The opposite of a topologi... |
| oppgtgp 24044 | The opposite of a topologi... |
| distgp 24045 | Any group equipped with th... |
| indistgp 24046 | Any group equipped with th... |
| efmndtmd 24047 | The monoid of endofunction... |
| tmdlactcn 24048 | The left group action of e... |
| tgplacthmeo 24049 | The left group action of e... |
| submtmd 24050 | A submonoid of a topologic... |
| subgtgp 24051 | A subgroup of a topologica... |
| symgtgp 24052 | The symmetric group is a t... |
| subgntr 24053 | A subgroup of a topologica... |
| opnsubg 24054 | An open subgroup of a topo... |
| clssubg 24055 | The closure of a subgroup ... |
| clsnsg 24056 | The closure of a normal su... |
| cldsubg 24057 | A subgroup of finite index... |
| tgpconncompeqg 24058 | The connected component co... |
| tgpconncomp 24059 | The identity component, th... |
| tgpconncompss 24060 | The identity component is ... |
| ghmcnp 24061 | A group homomorphism on to... |
| snclseqg 24062 | The coset of the closure o... |
| tgphaus 24063 | A topological group is Hau... |
| tgpt1 24064 | Hausdorff and T1 are equiv... |
| tgpt0 24065 | Hausdorff and T0 are equiv... |
| qustgpopn 24066 | A quotient map in a topolo... |
| qustgplem 24067 | Lemma for ~ qustgp . (Con... |
| qustgp 24068 | The quotient of a topologi... |
| qustgphaus 24069 | The quotient of a topologi... |
| prdstmdd 24070 | The product of a family of... |
| prdstgpd 24071 | The product of a family of... |
| tsmsfbas 24074 | The collection of all sets... |
| tsmslem1 24075 | The finite partial sums of... |
| tsmsval2 24076 | Definition of the topologi... |
| tsmsval 24077 | Definition of the topologi... |
| tsmspropd 24078 | The group sum depends only... |
| eltsms 24079 | The property of being a su... |
| tsmsi 24080 | The property of being a su... |
| tsmscl 24081 | A sum in a topological gro... |
| haustsms 24082 | In a Hausdorff topological... |
| haustsms2 24083 | In a Hausdorff topological... |
| tsmscls 24084 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24085 | The convergent points of a... |
| tsmsid 24086 | If a sum is finite, the us... |
| haustsmsid 24087 | In a Hausdorff topological... |
| tsms0 24088 | The sum of zero is zero. ... |
| tsmssubm 24089 | Evaluate an infinite group... |
| tsmsres 24090 | Extend an infinite group s... |
| tsmsf1o 24091 | Re-index an infinite group... |
| tsmsmhm 24092 | Apply a continuous group h... |
| tsmsadd 24093 | The sum of two infinite gr... |
| tsmsinv 24094 | Inverse of an infinite gro... |
| tsmssub 24095 | The difference of two infi... |
| tgptsmscls 24096 | A sum in a topological gro... |
| tgptsmscld 24097 | The set of limit points to... |
| tsmssplit 24098 | Split a topological group ... |
| tsmsxplem1 24099 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24100 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24101 | Write a sum over a two-dim... |
| istrg 24110 | Express the predicate " ` ... |
| trgtmd 24111 | The multiplicative monoid ... |
| istdrg 24112 | Express the predicate " ` ... |
| tdrgunit 24113 | The unit group of a topolo... |
| trgtgp 24114 | A topological ring is a to... |
| trgtmd2 24115 | A topological ring is a to... |
| trgtps 24116 | A topological ring is a to... |
| trgring 24117 | A topological ring is a ri... |
| trggrp 24118 | A topological ring is a gr... |
| tdrgtrg 24119 | A topological division rin... |
| tdrgdrng 24120 | A topological division rin... |
| tdrgring 24121 | A topological division rin... |
| tdrgtmd 24122 | A topological division rin... |
| tdrgtps 24123 | A topological division rin... |
| istdrg2 24124 | A topological-ring divisio... |
| mulrcn 24125 | The functionalization of t... |
| invrcn2 24126 | The multiplicative inverse... |
| invrcn 24127 | The multiplicative inverse... |
| cnmpt1mulr 24128 | Continuity of ring multipl... |
| cnmpt2mulr 24129 | Continuity of ring multipl... |
| dvrcn 24130 | The division function is c... |
| istlm 24131 | The predicate " ` W ` is a... |
| vscacn 24132 | The scalar multiplication ... |
| tlmtmd 24133 | A topological module is a ... |
| tlmtps 24134 | A topological module is a ... |
| tlmlmod 24135 | A topological module is a ... |
| tlmtrg 24136 | The scalar ring of a topol... |
| tlmscatps 24137 | The scalar ring of a topol... |
| istvc 24138 | A topological vector space... |
| tvctdrg 24139 | The scalar field of a topo... |
| cnmpt1vsca 24140 | Continuity of scalar multi... |
| cnmpt2vsca 24141 | Continuity of scalar multi... |
| tlmtgp 24142 | A topological vector space... |
| tvctlm 24143 | A topological vector space... |
| tvclmod 24144 | A topological vector space... |
| tvclvec 24145 | A topological vector space... |
| ustfn 24148 | The defined uniform struct... |
| ustval 24149 | The class of all uniform s... |
| isust 24150 | The predicate " ` U ` is a... |
| ustssxp 24151 | Entourages are subsets of ... |
| ustssel 24152 | A uniform structure is upw... |
| ustbasel 24153 | The full set is always an ... |
| ustincl 24154 | A uniform structure is clo... |
| ustdiag 24155 | The diagonal set is includ... |
| ustinvel 24156 | If ` V ` is an entourage, ... |
| ustexhalf 24157 | For each entourage ` V ` t... |
| ustrel 24158 | The elements of uniform st... |
| ustfilxp 24159 | A uniform structure on a n... |
| ustne0 24160 | A uniform structure cannot... |
| ustssco 24161 | In an uniform structure, a... |
| ustexsym 24162 | In an uniform structure, f... |
| ustex2sym 24163 | In an uniform structure, f... |
| ustex3sym 24164 | In an uniform structure, f... |
| ustref 24165 | Any element of the base se... |
| ust0 24166 | The unique uniform structu... |
| ustn0 24167 | The empty set is not an un... |
| ustund 24168 | If two intersecting sets `... |
| ustelimasn 24169 | Any point ` A ` is near en... |
| ustneism 24170 | For a point ` A ` in ` X `... |
| ustbas2 24171 | Second direction for ~ ust... |
| ustuni 24172 | The set union of a uniform... |
| ustbas 24173 | Recover the base of an uni... |
| ustimasn 24174 | Lemma for ~ ustuqtop . (C... |
| trust 24175 | The trace of a uniform str... |
| utopval 24178 | The topology induced by a ... |
| elutop 24179 | Open sets in the topology ... |
| utoptop 24180 | The topology induced by a ... |
| utopbas 24181 | The base of the topology i... |
| utoptopon 24182 | Topology induced by a unif... |
| restutop 24183 | Restriction of a topology ... |
| restutopopn 24184 | The restriction of the top... |
| ustuqtoplem 24185 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24186 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24187 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24188 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24189 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24190 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24191 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24192 | For a given uniform struct... |
| utopsnneiplem 24193 | The neighborhoods of a poi... |
| utopsnneip 24194 | The neighborhoods of a poi... |
| utopsnnei 24195 | Images of singletons by en... |
| utop2nei 24196 | For any symmetrical entour... |
| utop3cls 24197 | Relation between a topolog... |
| utopreg 24198 | All Hausdorff uniform spac... |
| ussval 24205 | The uniform structure on u... |
| ussid 24206 | In case the base of the ` ... |
| isusp 24207 | The predicate ` W ` is a u... |
| ressuss 24208 | Value of the uniform struc... |
| ressust 24209 | The uniform structure of a... |
| ressusp 24210 | The restriction of a unifo... |
| tusval 24211 | The value of the uniform s... |
| tuslem 24212 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24213 | The base set of a construc... |
| tusunif 24214 | The uniform structure of a... |
| tususs 24215 | The uniform structure of a... |
| tustopn 24216 | The topology induced by a ... |
| tususp 24217 | A constructed uniform spac... |
| tustps 24218 | A constructed uniform spac... |
| uspreg 24219 | If a uniform space is Haus... |
| ucnval 24222 | The set of all uniformly c... |
| isucn 24223 | The predicate " ` F ` is a... |
| isucn2 24224 | The predicate " ` F ` is a... |
| ucnimalem 24225 | Reformulate the ` G ` func... |
| ucnima 24226 | An equivalent statement of... |
| ucnprima 24227 | The preimage by a uniforml... |
| iducn 24228 | The identity is uniformly ... |
| cstucnd 24229 | A constant function is uni... |
| ucncn 24230 | Uniform continuity implies... |
| iscfilu 24233 | The predicate " ` F ` is a... |
| cfilufbas 24234 | A Cauchy filter base is a ... |
| cfiluexsm 24235 | For a Cauchy filter base a... |
| fmucndlem 24236 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24237 | The image of a Cauchy filt... |
| cfilufg 24238 | The filter generated by a ... |
| trcfilu 24239 | Condition for the trace of... |
| cfiluweak 24240 | A Cauchy filter base is al... |
| neipcfilu 24241 | In an uniform space, a nei... |
| iscusp 24244 | The predicate " ` W ` is a... |
| cuspusp 24245 | A complete uniform space i... |
| cuspcvg 24246 | In a complete uniform spac... |
| iscusp2 24247 | The predicate " ` W ` is a... |
| cnextucn 24248 | Extension by continuity. ... |
| ucnextcn 24249 | Extension by continuity. ... |
| ispsmet 24250 | Express the predicate " ` ... |
| psmetdmdm 24251 | Recover the base set from ... |
| psmetf 24252 | The distance function of a... |
| psmetcl 24253 | Closure of the distance fu... |
| psmet0 24254 | The distance function of a... |
| psmettri2 24255 | Triangle inequality for th... |
| psmetsym 24256 | The distance function of a... |
| psmettri 24257 | Triangle inequality for th... |
| psmetge0 24258 | The distance function of a... |
| psmetxrge0 24259 | The distance function of a... |
| psmetres2 24260 | Restriction of a pseudomet... |
| psmetlecl 24261 | Real closure of an extende... |
| distspace 24262 | A set ` X ` together with ... |
| ismet 24269 | Express the predicate " ` ... |
| isxmet 24270 | Express the predicate " ` ... |
| ismeti 24271 | Properties that determine ... |
| isxmetd 24272 | Properties that determine ... |
| isxmet2d 24273 | It is safe to only require... |
| metflem 24274 | Lemma for ~ metf and other... |
| xmetf 24275 | Mapping of the distance fu... |
| metf 24276 | Mapping of the distance fu... |
| xmetcl 24277 | Closure of the distance fu... |
| metcl 24278 | Closure of the distance fu... |
| ismet2 24279 | An extended metric is a me... |
| metxmet 24280 | A metric is an extended me... |
| xmetdmdm 24281 | Recover the base set from ... |
| metdmdm 24282 | Recover the base set from ... |
| xmetunirn 24283 | Two ways to express an ext... |
| xmeteq0 24284 | The value of an extended m... |
| meteq0 24285 | The value of a metric is z... |
| xmettri2 24286 | Triangle inequality for th... |
| mettri2 24287 | Triangle inequality for th... |
| xmet0 24288 | The distance function of a... |
| met0 24289 | The distance function of a... |
| xmetge0 24290 | The distance function of a... |
| metge0 24291 | The distance function of a... |
| xmetlecl 24292 | Real closure of an extende... |
| xmetsym 24293 | The distance function of a... |
| xmetpsmet 24294 | An extended metric is a ps... |
| xmettpos 24295 | The distance function of a... |
| metsym 24296 | The distance function of a... |
| xmettri 24297 | Triangle inequality for th... |
| mettri 24298 | Triangle inequality for th... |
| xmettri3 24299 | Triangle inequality for th... |
| mettri3 24300 | Triangle inequality for th... |
| xmetrtri 24301 | One half of the reverse tr... |
| xmetrtri2 24302 | The reverse triangle inequ... |
| metrtri 24303 | Reverse triangle inequalit... |
| xmetgt0 24304 | The distance function of a... |
| metgt0 24305 | The distance function of a... |
| metn0 24306 | A metric space is nonempty... |
| xmetres2 24307 | Restriction of an extended... |
| metreslem 24308 | Lemma for ~ metres . (Con... |
| metres2 24309 | Lemma for ~ metres . (Con... |
| xmetres 24310 | A restriction of an extend... |
| metres 24311 | A restriction of a metric ... |
| 0met 24312 | The empty metric. (Contri... |
| prdsdsf 24313 | The product metric is a fu... |
| prdsxmetlem 24314 | The product metric is an e... |
| prdsxmet 24315 | The product metric is an e... |
| prdsmet 24316 | The product metric is a me... |
| ressprdsds 24317 | Restriction of a product m... |
| resspwsds 24318 | Restriction of a power met... |
| imasdsf1olem 24319 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24320 | The distance function is t... |
| imasf1oxmet 24321 | The image of an extended m... |
| imasf1omet 24322 | The image of a metric is a... |
| xpsdsfn 24323 | Closure of the metric in a... |
| xpsdsfn2 24324 | Closure of the metric in a... |
| xpsxmetlem 24325 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24326 | A product metric of extend... |
| xpsdsval 24327 | Value of the metric in a b... |
| xpsmet 24328 | The direct product of two ... |
| blfvalps 24329 | The value of the ball func... |
| blfval 24330 | The value of the ball func... |
| blvalps 24331 | The ball around a point ` ... |
| blval 24332 | The ball around a point ` ... |
| elblps 24333 | Membership in a ball. (Co... |
| elbl 24334 | Membership in a ball. (Co... |
| elbl2ps 24335 | Membership in a ball. (Co... |
| elbl2 24336 | Membership in a ball. (Co... |
| elbl3ps 24337 | Membership in a ball, with... |
| elbl3 24338 | Membership in a ball, with... |
| blcomps 24339 | Commute the arguments to t... |
| blcom 24340 | Commute the arguments to t... |
| xblpnfps 24341 | The infinity ball in an ex... |
| xblpnf 24342 | The infinity ball in an ex... |
| blpnf 24343 | The infinity ball in a sta... |
| bldisj 24344 | Two balls are disjoint if ... |
| blgt0 24345 | A nonempty ball implies th... |
| bl2in 24346 | Two balls are disjoint if ... |
| xblss2ps 24347 | One ball is contained in a... |
| xblss2 24348 | One ball is contained in a... |
| blss2ps 24349 | One ball is contained in a... |
| blss2 24350 | One ball is contained in a... |
| blhalf 24351 | A ball of radius ` R / 2 `... |
| blfps 24352 | Mapping of a ball. (Contr... |
| blf 24353 | Mapping of a ball. (Contr... |
| blrnps 24354 | Membership in the range of... |
| blrn 24355 | Membership in the range of... |
| xblcntrps 24356 | A ball contains its center... |
| xblcntr 24357 | A ball contains its center... |
| blcntrps 24358 | A ball contains its center... |
| blcntr 24359 | A ball contains its center... |
| xbln0 24360 | A ball is nonempty iff the... |
| bln0 24361 | A ball is not empty. (Con... |
| blelrnps 24362 | A ball belongs to the set ... |
| blelrn 24363 | A ball belongs to the set ... |
| blssm 24364 | A ball is a subset of the ... |
| unirnblps 24365 | The union of the set of ba... |
| unirnbl 24366 | The union of the set of ba... |
| blin 24367 | The intersection of two ba... |
| ssblps 24368 | The size of a ball increas... |
| ssbl 24369 | The size of a ball increas... |
| blssps 24370 | Any point ` P ` in a ball ... |
| blss 24371 | Any point ` P ` in a ball ... |
| blssexps 24372 | Two ways to express the ex... |
| blssex 24373 | Two ways to express the ex... |
| ssblex 24374 | A nested ball exists whose... |
| blin2 24375 | Given any two balls and a ... |
| blbas 24376 | The balls of a metric spac... |
| blres 24377 | A ball in a restricted met... |
| xmeterval 24378 | Value of the "finitely sep... |
| xmeter 24379 | The "finitely separated" r... |
| xmetec 24380 | The equivalence classes un... |
| blssec 24381 | A ball centered at ` P ` i... |
| blpnfctr 24382 | The infinity ball in an ex... |
| xmetresbl 24383 | An extended metric restric... |
| mopnval 24384 | An open set is a subset of... |
| mopntopon 24385 | The set of open sets of a ... |
| mopntop 24386 | The set of open sets of a ... |
| mopnuni 24387 | The union of all open sets... |
| elmopn 24388 | The defining property of a... |
| mopnfss 24389 | The family of open sets of... |
| mopnm 24390 | The base set of a metric s... |
| elmopn2 24391 | A defining property of an ... |
| mopnss 24392 | An open set of a metric sp... |
| isxms 24393 | Express the predicate " ` ... |
| isxms2 24394 | Express the predicate " ` ... |
| isms 24395 | Express the predicate " ` ... |
| isms2 24396 | Express the predicate " ` ... |
| xmstopn 24397 | The topology component of ... |
| mstopn 24398 | The topology component of ... |
| xmstps 24399 | An extended metric space i... |
| msxms 24400 | A metric space is an exten... |
| mstps 24401 | A metric space is a topolo... |
| xmsxmet 24402 | The distance function, sui... |
| msmet 24403 | The distance function, sui... |
| msf 24404 | The distance function of a... |
| xmsxmet2 24405 | The distance function, sui... |
| msmet2 24406 | The distance function, sui... |
| mscl 24407 | Closure of the distance fu... |
| xmscl 24408 | Closure of the distance fu... |
| xmsge0 24409 | The distance function in a... |
| xmseq0 24410 | The distance between two p... |
| xmssym 24411 | The distance function in a... |
| xmstri2 24412 | Triangle inequality for th... |
| mstri2 24413 | Triangle inequality for th... |
| xmstri 24414 | Triangle inequality for th... |
| mstri 24415 | Triangle inequality for th... |
| xmstri3 24416 | Triangle inequality for th... |
| mstri3 24417 | Triangle inequality for th... |
| msrtri 24418 | Reverse triangle inequalit... |
| xmspropd 24419 | Property deduction for an ... |
| mspropd 24420 | Property deduction for a m... |
| setsmsbas 24421 | The base set of a construc... |
| setsmsds 24422 | The distance function of a... |
| setsmstset 24423 | The topology of a construc... |
| setsmstopn 24424 | The topology of a construc... |
| setsxms 24425 | The constructed metric spa... |
| setsms 24426 | The constructed metric spa... |
| tmsval 24427 | For any metric there is an... |
| tmslem 24428 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24429 | The base set of a construc... |
| tmsds 24430 | The metric of a constructe... |
| tmstopn 24431 | The topology of a construc... |
| tmsxms 24432 | The constructed metric spa... |
| tmsms 24433 | The constructed metric spa... |
| imasf1obl 24434 | The image of a metric spac... |
| imasf1oxms 24435 | The image of a metric spac... |
| imasf1oms 24436 | The image of a metric spac... |
| prdsbl 24437 | A ball in the product metr... |
| mopni 24438 | An open set of a metric sp... |
| mopni2 24439 | An open set of a metric sp... |
| mopni3 24440 | An open set of a metric sp... |
| blssopn 24441 | The balls of a metric spac... |
| unimopn 24442 | The union of a collection ... |
| mopnin 24443 | The intersection of two op... |
| mopn0 24444 | The empty set is an open s... |
| rnblopn 24445 | A ball of a metric space i... |
| blopn 24446 | A ball of a metric space i... |
| neibl 24447 | The neighborhoods around a... |
| blnei 24448 | A ball around a point is a... |
| lpbl 24449 | Every ball around a limit ... |
| blsscls2 24450 | A smaller closed ball is c... |
| blcld 24451 | A "closed ball" in a metri... |
| blcls 24452 | The closure of an open bal... |
| blsscls 24453 | If two concentric balls ha... |
| metss 24454 | Two ways of saying that me... |
| metequiv 24455 | Two ways of saying that tw... |
| metequiv2 24456 | If there is a sequence of ... |
| metss2lem 24457 | Lemma for ~ metss2 . (Con... |
| metss2 24458 | If the metric ` D ` is "st... |
| comet 24459 | The composition of an exte... |
| stdbdmetval 24460 | Value of the standard boun... |
| stdbdxmet 24461 | The standard bounded metri... |
| stdbdmet 24462 | The standard bounded metri... |
| stdbdbl 24463 | The standard bounded metri... |
| stdbdmopn 24464 | The standard bounded metri... |
| mopnex 24465 | The topology generated by ... |
| methaus 24466 | The topology generated by ... |
| met1stc 24467 | The topology generated by ... |
| met2ndci 24468 | A separable metric space (... |
| met2ndc 24469 | A metric space is second-c... |
| metrest 24470 | Two alternate formulations... |
| ressxms 24471 | The restriction of a metri... |
| ressms 24472 | The restriction of a metri... |
| prdsmslem1 24473 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24474 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24475 | Lemma for ~ prdsxms . The... |
| prdsxms 24476 | The indexed product struct... |
| prdsms 24477 | The indexed product struct... |
| pwsxms 24478 | A power of an extended met... |
| pwsms 24479 | A power of a metric space ... |
| xpsxms 24480 | A binary product of metric... |
| xpsms 24481 | A binary product of metric... |
| tmsxps 24482 | Express the product of two... |
| tmsxpsmopn 24483 | Express the product of two... |
| tmsxpsval 24484 | Value of the product of tw... |
| tmsxpsval2 24485 | Value of the product of tw... |
| metcnp3 24486 | Two ways to express that `... |
| metcnp 24487 | Two ways to say a mapping ... |
| metcnp2 24488 | Two ways to say a mapping ... |
| metcn 24489 | Two ways to say a mapping ... |
| metcnpi 24490 | Epsilon-delta property of ... |
| metcnpi2 24491 | Epsilon-delta property of ... |
| metcnpi3 24492 | Epsilon-delta property of ... |
| txmetcnp 24493 | Continuity of a binary ope... |
| txmetcn 24494 | Continuity of a binary ope... |
| metuval 24495 | Value of the uniform struc... |
| metustel 24496 | Define a filter base ` F `... |
| metustss 24497 | Range of the elements of t... |
| metustrel 24498 | Elements of the filter bas... |
| metustto 24499 | Any two elements of the fi... |
| metustid 24500 | The identity diagonal is i... |
| metustsym 24501 | Elements of the filter bas... |
| metustexhalf 24502 | For any element ` A ` of t... |
| metustfbas 24503 | The filter base generated ... |
| metust 24504 | The uniform structure gene... |
| cfilucfil 24505 | Given a metric ` D ` and a... |
| metuust 24506 | The uniform structure gene... |
| cfilucfil2 24507 | Given a metric ` D ` and a... |
| blval2 24508 | The ball around a point ` ... |
| elbl4 24509 | Membership in a ball, alte... |
| metuel 24510 | Elementhood in the uniform... |
| metuel2 24511 | Elementhood in the uniform... |
| metustbl 24512 | The "section" image of an ... |
| psmetutop 24513 | The topology induced by a ... |
| xmetutop 24514 | The topology induced by a ... |
| xmsusp 24515 | If the uniform set of a me... |
| restmetu 24516 | The uniform structure gene... |
| metucn 24517 | Uniform continuity in metr... |
| dscmet 24518 | The discrete metric on any... |
| dscopn 24519 | The discrete metric genera... |
| nrmmetd 24520 | Show that a group norm gen... |
| abvmet 24521 | An absolute value ` F ` ge... |
| nmfval 24534 | The value of the norm func... |
| nmval 24535 | The value of the norm as t... |
| nmfval0 24536 | The value of the norm func... |
| nmfval2 24537 | The value of the norm func... |
| nmval2 24538 | The value of the norm on a... |
| nmf2 24539 | The norm on a metric group... |
| nmpropd 24540 | Weak property deduction fo... |
| nmpropd2 24541 | Strong property deduction ... |
| isngp 24542 | The property of being a no... |
| isngp2 24543 | The property of being a no... |
| isngp3 24544 | The property of being a no... |
| ngpgrp 24545 | A normed group is a group.... |
| ngpms 24546 | A normed group is a metric... |
| ngpxms 24547 | A normed group is an exten... |
| ngptps 24548 | A normed group is a topolo... |
| ngpmet 24549 | The (induced) metric of a ... |
| ngpds 24550 | Value of the distance func... |
| ngpdsr 24551 | Value of the distance func... |
| ngpds2 24552 | Write the distance between... |
| ngpds2r 24553 | Write the distance between... |
| ngpds3 24554 | Write the distance between... |
| ngpds3r 24555 | Write the distance between... |
| ngprcan 24556 | Cancel right addition insi... |
| ngplcan 24557 | Cancel left addition insid... |
| isngp4 24558 | Express the property of be... |
| ngpinvds 24559 | Two elements are the same ... |
| ngpsubcan 24560 | Cancel right subtraction i... |
| nmf 24561 | The norm on a normed group... |
| nmcl 24562 | The norm of a normed group... |
| nmge0 24563 | The norm of a normed group... |
| nmeq0 24564 | The identity is the only e... |
| nmne0 24565 | The norm of a nonzero elem... |
| nmrpcl 24566 | The norm of a nonzero elem... |
| nminv 24567 | The norm of a negated elem... |
| nmmtri 24568 | The triangle inequality fo... |
| nmsub 24569 | The norm of the difference... |
| nmrtri 24570 | Reverse triangle inequalit... |
| nm2dif 24571 | Inequality for the differe... |
| nmtri 24572 | The triangle inequality fo... |
| nmtri2 24573 | Triangle inequality for th... |
| ngpi 24574 | The properties of a normed... |
| nm0 24575 | Norm of the identity eleme... |
| nmgt0 24576 | The norm of a nonzero elem... |
| sgrim 24577 | The induced metric on a su... |
| sgrimval 24578 | The induced metric on a su... |
| subgnm 24579 | The norm in a subgroup. (... |
| subgnm2 24580 | A substructure assigns the... |
| subgngp 24581 | A normed group restricted ... |
| ngptgp 24582 | A normed abelian group is ... |
| ngppropd 24583 | Property deduction for a n... |
| reldmtng 24584 | The function ` toNrmGrp ` ... |
| tngval 24585 | Value of the function whic... |
| tnglem 24586 | Lemma for ~ tngbas and sim... |
| tngbas 24587 | The base set of a structur... |
| tngplusg 24588 | The group addition of a st... |
| tng0 24589 | The group identity of a st... |
| tngmulr 24590 | The ring multiplication of... |
| tngsca 24591 | The scalar ring of a struc... |
| tngvsca 24592 | The scalar multiplication ... |
| tngip 24593 | The inner product operatio... |
| tngds 24594 | The metric function of a s... |
| tngtset 24595 | The topology generated by ... |
| tngtopn 24596 | The topology generated by ... |
| tngnm 24597 | The topology generated by ... |
| tngngp2 24598 | A norm turns a group into ... |
| tngngpd 24599 | Derive the axioms for a no... |
| tngngp 24600 | Derive the axioms for a no... |
| tnggrpr 24601 | If a structure equipped wi... |
| tngngp3 24602 | Alternate definition of a ... |
| nrmtngdist 24603 | The augmentation of a norm... |
| nrmtngnrm 24604 | The augmentation of a norm... |
| tngngpim 24605 | The induced metric of a no... |
| isnrg 24606 | A normed ring is a ring wi... |
| nrgabv 24607 | The norm of a normed ring ... |
| nrgngp 24608 | A normed ring is a normed ... |
| nrgring 24609 | A normed ring is a ring. ... |
| nmmul 24610 | The norm of a product in a... |
| nrgdsdi 24611 | Distribute a distance calc... |
| nrgdsdir 24612 | Distribute a distance calc... |
| nm1 24613 | The norm of one in a nonze... |
| unitnmn0 24614 | The norm of a unit is nonz... |
| nminvr 24615 | The norm of an inverse in ... |
| nmdvr 24616 | The norm of a division in ... |
| nrgdomn 24617 | A nonzero normed ring is a... |
| nrgtgp 24618 | A normed ring is a topolog... |
| subrgnrg 24619 | A normed ring restricted t... |
| tngnrg 24620 | Given any absolute value o... |
| isnlm 24621 | A normed (left) module is ... |
| nmvs 24622 | Defining property of a nor... |
| nlmngp 24623 | A normed module is a norme... |
| nlmlmod 24624 | A normed module is a left ... |
| nlmnrg 24625 | The scalar component of a ... |
| nlmngp2 24626 | The scalar component of a ... |
| nlmdsdi 24627 | Distribute a distance calc... |
| nlmdsdir 24628 | Distribute a distance calc... |
| nlmmul0or 24629 | If a scalar product is zer... |
| sranlm 24630 | The subring algebra over a... |
| nlmvscnlem2 24631 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24632 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24633 | The scalar multiplication ... |
| rlmnlm 24634 | The ring module over a nor... |
| rlmnm 24635 | The norm function in the r... |
| nrgtrg 24636 | A normed ring is a topolog... |
| nrginvrcnlem 24637 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24638 | The ring inverse function ... |
| nrgtdrg 24639 | A normed division ring is ... |
| nlmtlm 24640 | A normed module is a topol... |
| isnvc 24641 | A normed vector space is j... |
| nvcnlm 24642 | A normed vector space is a... |
| nvclvec 24643 | A normed vector space is a... |
| nvclmod 24644 | A normed vector space is a... |
| isnvc2 24645 | A normed vector space is j... |
| nvctvc 24646 | A normed vector space is a... |
| lssnlm 24647 | A subspace of a normed mod... |
| lssnvc 24648 | A subspace of a normed vec... |
| rlmnvc 24649 | The ring module over a nor... |
| ngpocelbl 24650 | Membership of an off-cente... |
| nmoffn 24657 | The function producing ope... |
| reldmnghm 24658 | Lemma for normed group hom... |
| reldmnmhm 24659 | Lemma for module homomorph... |
| nmofval 24660 | Value of the operator norm... |
| nmoval 24661 | Value of the operator norm... |
| nmogelb 24662 | Property of the operator n... |
| nmolb 24663 | Any upper bound on the val... |
| nmolb2d 24664 | Any upper bound on the val... |
| nmof 24665 | The operator norm is a fun... |
| nmocl 24666 | The operator norm of an op... |
| nmoge0 24667 | The operator norm of an op... |
| nghmfval 24668 | A normed group homomorphis... |
| isnghm 24669 | A normed group homomorphis... |
| isnghm2 24670 | A normed group homomorphis... |
| isnghm3 24671 | A normed group homomorphis... |
| bddnghm 24672 | A bounded group homomorphi... |
| nghmcl 24673 | A normed group homomorphis... |
| nmoi 24674 | The operator norm achieves... |
| nmoix 24675 | The operator norm is a bou... |
| nmoi2 24676 | The operator norm is a bou... |
| nmoleub 24677 | The operator norm, defined... |
| nghmrcl1 24678 | Reverse closure for a norm... |
| nghmrcl2 24679 | Reverse closure for a norm... |
| nghmghm 24680 | A normed group homomorphis... |
| nmo0 24681 | The operator norm of the z... |
| nmoeq0 24682 | The operator norm is zero ... |
| nmoco 24683 | An upper bound on the oper... |
| nghmco 24684 | The composition of normed ... |
| nmotri 24685 | Triangle inequality for th... |
| nghmplusg 24686 | The sum of two bounded lin... |
| 0nghm 24687 | The zero operator is a nor... |
| nmoid 24688 | The operator norm of the i... |
| idnghm 24689 | The identity operator is a... |
| nmods 24690 | Upper bound for the distan... |
| nghmcn 24691 | A normed group homomorphis... |
| isnmhm 24692 | A normed module homomorphi... |
| nmhmrcl1 24693 | Reverse closure for a norm... |
| nmhmrcl2 24694 | Reverse closure for a norm... |
| nmhmlmhm 24695 | A normed module homomorphi... |
| nmhmnghm 24696 | A normed module homomorphi... |
| nmhmghm 24697 | A normed module homomorphi... |
| isnmhm2 24698 | A normed module homomorphi... |
| nmhmcl 24699 | A normed module homomorphi... |
| idnmhm 24700 | The identity operator is a... |
| 0nmhm 24701 | The zero operator is a bou... |
| nmhmco 24702 | The composition of bounded... |
| nmhmplusg 24703 | The sum of two bounded lin... |
| qtopbaslem 24704 | The set of open intervals ... |
| qtopbas 24705 | The set of open intervals ... |
| retopbas 24706 | A basis for the standard t... |
| retop 24707 | The standard topology on t... |
| uniretop 24708 | The underlying set of the ... |
| retopon 24709 | The standard topology on t... |
| retps 24710 | The standard topological s... |
| iooretop 24711 | Open intervals are open se... |
| icccld 24712 | Closed intervals are close... |
| icopnfcld 24713 | Right-unbounded closed int... |
| iocmnfcld 24714 | Left-unbounded closed inte... |
| qdensere 24715 | ` QQ ` is dense in the sta... |
| cnmetdval 24716 | Value of the distance func... |
| cnmet 24717 | The absolute value metric ... |
| cnxmet 24718 | The absolute value metric ... |
| cnbl0 24719 | Two ways to write the open... |
| cnblcld 24720 | Two ways to write the clos... |
| cnfldms 24721 | The complex number field i... |
| cnfldxms 24722 | The complex number field i... |
| cnfldtps 24723 | The complex number field i... |
| cnfldnm 24724 | The norm of the field of c... |
| cnngp 24725 | The complex numbers form a... |
| cnnrg 24726 | The complex numbers form a... |
| cnfldtopn 24727 | The topology of the comple... |
| cnfldtopon 24728 | The topology of the comple... |
| cnfldtop 24729 | The topology of the comple... |
| cnfldhaus 24730 | The topology of the comple... |
| unicntop 24731 | The underlying set of the ... |
| cnopn 24732 | The set of complex numbers... |
| cnn0opn 24733 | The set of nonzero complex... |
| zringnrg 24734 | The ring of integers is a ... |
| remetdval 24735 | Value of the distance func... |
| remet 24736 | The absolute value metric ... |
| rexmet 24737 | The absolute value metric ... |
| bl2ioo 24738 | A ball in terms of an open... |
| ioo2bl 24739 | An open interval of reals ... |
| ioo2blex 24740 | An open interval of reals ... |
| blssioo 24741 | The balls of the standard ... |
| tgioo 24742 | The topology generated by ... |
| qdensere2 24743 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24744 | An open ball in the comple... |
| rehaus 24745 | The standard topology on t... |
| tgqioo 24746 | The topology generated by ... |
| re2ndc 24747 | The standard topology on t... |
| resubmet 24748 | The subspace topology indu... |
| tgioo2 24749 | The standard topology on t... |
| rerest 24750 | The subspace topology indu... |
| tgioo4 24751 | The standard topology on t... |
| tgioo3 24752 | The standard topology on t... |
| xrtgioo 24753 | The topology on the extend... |
| xrrest 24754 | The subspace topology indu... |
| xrrest2 24755 | The subspace topology indu... |
| xrsxmet 24756 | The metric on the extended... |
| xrsdsre 24757 | The metric on the extended... |
| xrsblre 24758 | Any ball of the metric of ... |
| xrsmopn 24759 | The metric on the extended... |
| zcld 24760 | The integers are a closed ... |
| recld2 24761 | The real numbers are a clo... |
| zcld2 24762 | The integers are a closed ... |
| zdis 24763 | The integers are a discret... |
| sszcld 24764 | Every subset of the intege... |
| reperflem 24765 | A subset of the real numbe... |
| reperf 24766 | The real numbers are a per... |
| cnperf 24767 | The complex numbers are a ... |
| iccntr 24768 | The interior of a closed i... |
| icccmplem1 24769 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24770 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24771 | Lemma for ~ icccmp . (Con... |
| icccmp 24772 | A closed interval in ` RR ... |
| reconnlem1 24773 | Lemma for ~ reconn . Conn... |
| reconnlem2 24774 | Lemma for ~ reconn . (Con... |
| reconn 24775 | A subset of the reals is c... |
| retopconn 24776 | Corollary of ~ reconn . T... |
| iccconn 24777 | A closed interval is conne... |
| opnreen 24778 | Every nonempty open set is... |
| rectbntr0 24779 | A countable subset of the ... |
| xrge0gsumle 24780 | A finite sum in the nonneg... |
| xrge0tsms 24781 | Any finite or infinite sum... |
| xrge0tsms2 24782 | Any finite or infinite sum... |
| metdcnlem 24783 | The metric function of a m... |
| xmetdcn2 24784 | The metric function of an ... |
| xmetdcn 24785 | The metric function of an ... |
| metdcn2 24786 | The metric function of a m... |
| metdcn 24787 | The metric function of a m... |
| msdcn 24788 | The metric function of a m... |
| cnmpt1ds 24789 | Continuity of the metric f... |
| cnmpt2ds 24790 | Continuity of the metric f... |
| nmcn 24791 | The norm of a normed group... |
| ngnmcncn 24792 | The norm of a normed group... |
| abscn 24793 | The absolute value functio... |
| metdsval 24794 | Value of the "distance to ... |
| metdsf 24795 | The distance from a point ... |
| metdsge 24796 | The distance from the poin... |
| metds0 24797 | If a point is in a set, it... |
| metdstri 24798 | A generalization of the tr... |
| metdsle 24799 | The distance from a point ... |
| metdsre 24800 | The distance from a point ... |
| metdseq0 24801 | The distance from a point ... |
| metdscnlem 24802 | Lemma for ~ metdscn . (Co... |
| metdscn 24803 | The function ` F ` which g... |
| metdscn2 24804 | The function ` F ` which g... |
| metnrmlem1a 24805 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24806 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24807 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24808 | Lemma for ~ metnrm . (Con... |
| metnrm 24809 | A metric space is normal. ... |
| metreg 24810 | A metric space is regular.... |
| addcnlem 24811 | Lemma for ~ addcn , ~ subc... |
| addcn 24812 | Complex number addition is... |
| subcn 24813 | Complex number subtraction... |
| mulcn 24814 | Complex number multiplicat... |
| divcnOLD 24815 | Obsolete version of ~ divc... |
| mpomulcn 24816 | Complex number multiplicat... |
| divcn 24817 | Complex number division is... |
| cnfldtgp 24818 | The complex numbers form a... |
| fsumcn 24819 | A finite sum of functions ... |
| fsum2cn 24820 | Version of ~ fsumcn for tw... |
| expcn 24821 | The power function on comp... |
| divccn 24822 | Division by a nonzero cons... |
| expcnOLD 24823 | Obsolete version of ~ expc... |
| divccnOLD 24824 | Obsolete version of ~ divc... |
| sqcn 24825 | The square function on com... |
| iitopon 24830 | The unit interval is a top... |
| iitop 24831 | The unit interval is a top... |
| iiuni 24832 | The base set of the unit i... |
| dfii2 24833 | Alternate definition of th... |
| dfii3 24834 | Alternate definition of th... |
| dfii4 24835 | Alternate definition of th... |
| dfii5 24836 | The unit interval expresse... |
| iicmp 24837 | The unit interval is compa... |
| iiconn 24838 | The unit interval is conne... |
| cncfval 24839 | The value of the continuou... |
| elcncf 24840 | Membership in the set of c... |
| elcncf2 24841 | Version of ~ elcncf with a... |
| cncfrss 24842 | Reverse closure of the con... |
| cncfrss2 24843 | Reverse closure of the con... |
| cncff 24844 | A continuous complex funct... |
| cncfi 24845 | Defining property of a con... |
| elcncf1di 24846 | Membership in the set of c... |
| elcncf1ii 24847 | Membership in the set of c... |
| rescncf 24848 | A continuous complex funct... |
| cncfcdm 24849 | Change the codomain of a c... |
| cncfss 24850 | The set of continuous func... |
| climcncf 24851 | Image of a limit under a c... |
| abscncf 24852 | Absolute value is continuo... |
| recncf 24853 | Real part is continuous. ... |
| imcncf 24854 | Imaginary part is continuo... |
| cjcncf 24855 | Complex conjugate is conti... |
| mulc1cncf 24856 | Multiplication by a consta... |
| divccncf 24857 | Division by a constant is ... |
| cncfco 24858 | The composition of two con... |
| cncfcompt2 24859 | Composition of continuous ... |
| cncfmet 24860 | Relate complex function co... |
| cncfcn 24861 | Relate complex function co... |
| cncfcn1 24862 | Relate complex function co... |
| cncfmptc 24863 | A constant function is a c... |
| cncfmptid 24864 | The identity function is a... |
| cncfmpt1f 24865 | Composition of continuous ... |
| cncfmpt2f 24866 | Composition of continuous ... |
| cncfmpt2ss 24867 | Composition of continuous ... |
| addccncf 24868 | Adding a constant is a con... |
| idcncf 24869 | The identity function is a... |
| sub1cncf 24870 | Subtracting a constant is ... |
| sub2cncf 24871 | Subtraction from a constan... |
| cdivcncf 24872 | Division with a constant n... |
| negcncf 24873 | The negative function is c... |
| negcncfOLD 24874 | Obsolete version of ~ negc... |
| negfcncf 24875 | The negative of a continuo... |
| abscncfALT 24876 | Absolute value is continuo... |
| cncfcnvcn 24877 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24878 | The power function on comp... |
| cnmptre 24879 | Lemma for ~ iirevcn and re... |
| cnmpopc 24880 | Piecewise definition of a ... |
| iirev 24881 | Reverse the unit interval.... |
| iirevcn 24882 | The reversion function is ... |
| iihalf1 24883 | Map the first half of ` II... |
| iihalf1cn 24884 | The first half function is... |
| iihalf1cnOLD 24885 | Obsolete version of ~ iiha... |
| iihalf2 24886 | Map the second half of ` I... |
| iihalf2cn 24887 | The second half function i... |
| iihalf2cnOLD 24888 | Obsolete version of ~ iiha... |
| elii1 24889 | Divide the unit interval i... |
| elii2 24890 | Divide the unit interval i... |
| iimulcl 24891 | The unit interval is close... |
| iimulcn 24892 | Multiplication is a contin... |
| iimulcnOLD 24893 | Obsolete version of ~ iimu... |
| icoopnst 24894 | A half-open interval start... |
| iocopnst 24895 | A half-open interval endin... |
| icchmeo 24896 | The natural bijection from... |
| icchmeoOLD 24897 | Obsolete version of ~ icch... |
| icopnfcnv 24898 | Define a bijection from ` ... |
| icopnfhmeo 24899 | The defined bijection from... |
| iccpnfcnv 24900 | Define a bijection from ` ... |
| iccpnfhmeo 24901 | The defined bijection from... |
| xrhmeo 24902 | The bijection from ` [ -u ... |
| xrhmph 24903 | The extended reals are hom... |
| xrcmp 24904 | The topology of the extend... |
| xrconn 24905 | The topology of the extend... |
| icccvx 24906 | A linear combination of tw... |
| oprpiece1res1 24907 | Restriction to the first p... |
| oprpiece1res2 24908 | Restriction to the second ... |
| cnrehmeo 24909 | The canonical bijection fr... |
| cnrehmeoOLD 24910 | Obsolete version of ~ cnre... |
| cnheiborlem 24911 | Lemma for ~ cnheibor . (C... |
| cnheibor 24912 | Heine-Borel theorem for co... |
| cnllycmp 24913 | The topology on the comple... |
| rellycmp 24914 | The topology on the reals ... |
| bndth 24915 | The Boundedness Theorem. ... |
| evth 24916 | The Extreme Value Theorem.... |
| evth2 24917 | The Extreme Value Theorem,... |
| lebnumlem1 24918 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24919 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24920 | Lemma for ~ lebnum . By t... |
| lebnum 24921 | The Lebesgue number lemma,... |
| xlebnum 24922 | Generalize ~ lebnum to ext... |
| lebnumii 24923 | Specialize the Lebesgue nu... |
| ishtpy 24929 | Membership in the class of... |
| htpycn 24930 | A homotopy is a continuous... |
| htpyi 24931 | A homotopy evaluated at it... |
| ishtpyd 24932 | Deduction for membership i... |
| htpycom 24933 | Given a homotopy from ` F ... |
| htpyid 24934 | A homotopy from a function... |
| htpyco1 24935 | Compose a homotopy with a ... |
| htpyco2 24936 | Compose a homotopy with a ... |
| htpycc 24937 | Concatenate two homotopies... |
| isphtpy 24938 | Membership in the class of... |
| phtpyhtpy 24939 | A path homotopy is a homot... |
| phtpycn 24940 | A path homotopy is a conti... |
| phtpyi 24941 | Membership in the class of... |
| phtpy01 24942 | Two path-homotopic paths h... |
| isphtpyd 24943 | Deduction for membership i... |
| isphtpy2d 24944 | Deduction for membership i... |
| phtpycom 24945 | Given a homotopy from ` F ... |
| phtpyid 24946 | A homotopy from a path to ... |
| phtpyco2 24947 | Compose a path homotopy wi... |
| phtpycc 24948 | Concatenate two path homot... |
| phtpcrel 24950 | The path homotopy relation... |
| isphtpc 24951 | The relation "is path homo... |
| phtpcer 24952 | Path homotopy is an equiva... |
| phtpc01 24953 | Path homotopic paths have ... |
| reparphti 24954 | Lemma for ~ reparpht . (C... |
| reparphtiOLD 24955 | Obsolete version of ~ repa... |
| reparpht 24956 | Reparametrization lemma. ... |
| phtpcco2 24957 | Compose a path homotopy wi... |
| pcofval 24968 | The value of the path conc... |
| pcoval 24969 | The concatenation of two p... |
| pcovalg 24970 | Evaluate the concatenation... |
| pcoval1 24971 | Evaluate the concatenation... |
| pco0 24972 | The starting point of a pa... |
| pco1 24973 | The ending point of a path... |
| pcoval2 24974 | Evaluate the concatenation... |
| pcocn 24975 | The concatenation of two p... |
| copco 24976 | The composition of a conca... |
| pcohtpylem 24977 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24978 | Homotopy invariance of pat... |
| pcoptcl 24979 | A constant function is a p... |
| pcopt 24980 | Concatenation with a point... |
| pcopt2 24981 | Concatenation with a point... |
| pcoass 24982 | Order of concatenation doe... |
| pcorevcl 24983 | Closure for a reversed pat... |
| pcorevlem 24984 | Lemma for ~ pcorev . Prov... |
| pcorev 24985 | Concatenation with the rev... |
| pcorev2 24986 | Concatenation with the rev... |
| pcophtb 24987 | The path homotopy equivale... |
| om1val 24988 | The definition of the loop... |
| om1bas 24989 | The base set of the loop s... |
| om1elbas 24990 | Elementhood in the base se... |
| om1addcl 24991 | Closure of the group opera... |
| om1plusg 24992 | The group operation (which... |
| om1tset 24993 | The topology of the loop s... |
| om1opn 24994 | The topology of the loop s... |
| pi1val 24995 | The definition of the fund... |
| pi1bas 24996 | The base set of the fundam... |
| pi1blem 24997 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24998 | Another way to write the l... |
| pi1bas2 24999 | The base set of the fundam... |
| pi1eluni 25000 | Elementhood in the base se... |
| pi1bas3 25001 | The base set of the fundam... |
| pi1cpbl 25002 | The group operation, loop ... |
| elpi1 25003 | The elements of the fundam... |
| elpi1i 25004 | The elements of the fundam... |
| pi1addf 25005 | The group operation of ` p... |
| pi1addval 25006 | The concatenation of two p... |
| pi1grplem 25007 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25008 | The fundamental group is a... |
| pi1id 25009 | The identity element of th... |
| pi1inv 25010 | An inverse in the fundamen... |
| pi1xfrf 25011 | Functionality of the loop ... |
| pi1xfrval 25012 | The value of the loop tran... |
| pi1xfr 25013 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25014 | Given a path ` F ` between... |
| pi1xfrcnv 25015 | Given a path ` F ` between... |
| pi1xfrgim 25016 | The mapping ` G ` between ... |
| pi1cof 25017 | Functionality of the loop ... |
| pi1coval 25018 | The value of the loop tran... |
| pi1coghm 25019 | The mapping ` G ` between ... |
| isclm 25022 | A subcomplex module is a l... |
| clmsca 25023 | The ring of scalars ` F ` ... |
| clmsubrg 25024 | The base set of the ring o... |
| clmlmod 25025 | A subcomplex module is a l... |
| clmgrp 25026 | A subcomplex module is an ... |
| clmabl 25027 | A subcomplex module is an ... |
| clmring 25028 | The scalar ring of a subco... |
| clmfgrp 25029 | The scalar ring of a subco... |
| clm0 25030 | The zero of the scalar rin... |
| clm1 25031 | The identity of the scalar... |
| clmadd 25032 | The addition of the scalar... |
| clmmul 25033 | The multiplication of the ... |
| clmcj 25034 | The conjugation of the sca... |
| isclmi 25035 | Reverse direction of ~ isc... |
| clmzss 25036 | The scalar ring of a subco... |
| clmsscn 25037 | The scalar ring of a subco... |
| clmsub 25038 | Subtraction in the scalar ... |
| clmneg 25039 | Negation in the scalar rin... |
| clmneg1 25040 | Minus one is in the scalar... |
| clmabs 25041 | Norm in the scalar ring of... |
| clmacl 25042 | Closure of ring addition f... |
| clmmcl 25043 | Closure of ring multiplica... |
| clmsubcl 25044 | Closure of ring subtractio... |
| lmhmclm 25045 | The domain of a linear ope... |
| clmvscl 25046 | Closure of scalar product ... |
| clmvsass 25047 | Associative law for scalar... |
| clmvscom 25048 | Commutative law for the sc... |
| clmvsdir 25049 | Distributive law for scala... |
| clmvsdi 25050 | Distributive law for scala... |
| clmvs1 25051 | Scalar product with ring u... |
| clmvs2 25052 | A vector plus itself is tw... |
| clm0vs 25053 | Zero times a vector is the... |
| clmopfne 25054 | The (functionalized) opera... |
| isclmp 25055 | The predicate "is a subcom... |
| isclmi0 25056 | Properties that determine ... |
| clmvneg1 25057 | Minus 1 times a vector is ... |
| clmvsneg 25058 | Multiplication of a vector... |
| clmmulg 25059 | The group multiple functio... |
| clmsubdir 25060 | Scalar multiplication dist... |
| clmpm1dir 25061 | Subtractive distributive l... |
| clmnegneg 25062 | Double negative of a vecto... |
| clmnegsubdi2 25063 | Distribution of negative o... |
| clmsub4 25064 | Rearrangement of 4 terms i... |
| clmvsrinv 25065 | A vector minus itself. (C... |
| clmvslinv 25066 | Minus a vector plus itself... |
| clmvsubval 25067 | Value of vector subtractio... |
| clmvsubval2 25068 | Value of vector subtractio... |
| clmvz 25069 | Two ways to express the ne... |
| zlmclm 25070 | The ` ZZ ` -module operati... |
| clmzlmvsca 25071 | The scalar product of a su... |
| nmoleub2lem 25072 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25073 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25074 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25075 | The operator norm is the s... |
| nmoleub2b 25076 | The operator norm is the s... |
| nmoleub3 25077 | The operator norm is the s... |
| nmhmcn 25078 | A linear operator over a n... |
| cmodscexp 25079 | The powers of ` _i ` belon... |
| cmodscmulexp 25080 | The scalar product of a ve... |
| cvslvec 25083 | A subcomplex vector space ... |
| cvsclm 25084 | A subcomplex vector space ... |
| iscvs 25085 | A subcomplex vector space ... |
| iscvsp 25086 | The predicate "is a subcom... |
| iscvsi 25087 | Properties that determine ... |
| cvsi 25088 | The properties of a subcom... |
| cvsunit 25089 | Unit group of the scalar r... |
| cvsdiv 25090 | Division of the scalar rin... |
| cvsdivcl 25091 | The scalar field of a subc... |
| cvsmuleqdivd 25092 | An equality involving rati... |
| cvsdiveqd 25093 | An equality involving rati... |
| cnlmodlem1 25094 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25095 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25096 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25097 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25098 | The set of complex numbers... |
| cnstrcvs 25099 | The set of complex numbers... |
| cnrbas 25100 | The set of complex numbers... |
| cnrlmod 25101 | The complex left module of... |
| cnrlvec 25102 | The complex left module of... |
| cncvs 25103 | The complex left module of... |
| recvs 25104 | The field of the real numb... |
| qcvs 25105 | The field of rational numb... |
| zclmncvs 25106 | The ring of integers as le... |
| isncvsngp 25107 | A normed subcomplex vector... |
| isncvsngpd 25108 | Properties that determine ... |
| ncvsi 25109 | The properties of a normed... |
| ncvsprp 25110 | Proportionality property o... |
| ncvsge0 25111 | The norm of a scalar produ... |
| ncvsm1 25112 | The norm of the opposite o... |
| ncvsdif 25113 | The norm of the difference... |
| ncvspi 25114 | The norm of a vector plus ... |
| ncvs1 25115 | From any nonzero vector of... |
| cnrnvc 25116 | The module of complex numb... |
| cnncvs 25117 | The module of complex numb... |
| cnnm 25118 | The norm of the normed sub... |
| ncvspds 25119 | Value of the distance func... |
| cnindmet 25120 | The metric induced on the ... |
| cnncvsaddassdemo 25121 | Derive the associative law... |
| cnncvsmulassdemo 25122 | Derive the associative law... |
| cnncvsabsnegdemo 25123 | Derive the absolute value ... |
| iscph 25128 | A subcomplex pre-Hilbert s... |
| cphphl 25129 | A subcomplex pre-Hilbert s... |
| cphnlm 25130 | A subcomplex pre-Hilbert s... |
| cphngp 25131 | A subcomplex pre-Hilbert s... |
| cphlmod 25132 | A subcomplex pre-Hilbert s... |
| cphlvec 25133 | A subcomplex pre-Hilbert s... |
| cphnvc 25134 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25135 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25136 | Lemma for ~ cphreccl . (C... |
| cphsca 25137 | A subcomplex pre-Hilbert s... |
| cphsubrg 25138 | The scalar field of a subc... |
| cphreccl 25139 | The scalar field of a subc... |
| cphdivcl 25140 | The scalar field of a subc... |
| cphcjcl 25141 | The scalar field of a subc... |
| cphsqrtcl 25142 | The scalar field of a subc... |
| cphabscl 25143 | The scalar field of a subc... |
| cphsqrtcl2 25144 | The scalar field of a subc... |
| cphsqrtcl3 25145 | If the scalar field of a s... |
| cphqss 25146 | The scalar field of a subc... |
| cphclm 25147 | A subcomplex pre-Hilbert s... |
| cphnmvs 25148 | Norm of a scalar product. ... |
| cphipcl 25149 | An inner product is a memb... |
| cphnmfval 25150 | The value of the norm in a... |
| cphnm 25151 | The square of the norm is ... |
| nmsq 25152 | The square of the norm is ... |
| cphnmf 25153 | The norm of a vector is a ... |
| cphnmcl 25154 | The norm of a vector is a ... |
| reipcl 25155 | An inner product of an ele... |
| ipge0 25156 | The inner product in a sub... |
| cphipcj 25157 | Conjugate of an inner prod... |
| cphipipcj 25158 | An inner product times its... |
| cphorthcom 25159 | Orthogonality (meaning inn... |
| cphip0l 25160 | Inner product with a zero ... |
| cphip0r 25161 | Inner product with a zero ... |
| cphipeq0 25162 | The inner product of a vec... |
| cphdir 25163 | Distributive law for inner... |
| cphdi 25164 | Distributive law for inner... |
| cph2di 25165 | Distributive law for inner... |
| cphsubdir 25166 | Distributive law for inner... |
| cphsubdi 25167 | Distributive law for inner... |
| cph2subdi 25168 | Distributive law for inner... |
| cphass 25169 | Associative law for inner ... |
| cphassr 25170 | "Associative" law for seco... |
| cph2ass 25171 | Move scalar multiplication... |
| cphassi 25172 | Associative law for the fi... |
| cphassir 25173 | "Associative" law for the ... |
| cphpyth 25174 | The pythagorean theorem fo... |
| tcphex 25175 | Lemma for ~ tcphbas and si... |
| tcphval 25176 | Define a function to augme... |
| tcphbas 25177 | The base set of a subcompl... |
| tchplusg 25178 | The addition operation of ... |
| tcphsub 25179 | The subtraction operation ... |
| tcphmulr 25180 | The ring operation of a su... |
| tcphsca 25181 | The scalar field of a subc... |
| tcphvsca 25182 | The scalar multiplication ... |
| tcphip 25183 | The inner product of a sub... |
| tcphtopn 25184 | The topology of a subcompl... |
| tcphphl 25185 | Augmentation of a subcompl... |
| tchnmfval 25186 | The norm of a subcomplex p... |
| tcphnmval 25187 | The norm of a subcomplex p... |
| cphtcphnm 25188 | The norm of a norm-augment... |
| tcphds 25189 | The distance of a pre-Hilb... |
| phclm 25190 | A pre-Hilbert space whose ... |
| tcphcphlem3 25191 | Lemma for ~ tcphcph : real... |
| ipcau2 25192 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25193 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25194 | Lemma for ~ tcphcph : homo... |
| tcphcph 25195 | The standard definition of... |
| ipcau 25196 | The Cauchy-Schwarz inequal... |
| nmparlem 25197 | Lemma for ~ nmpar . (Cont... |
| nmpar 25198 | A subcomplex pre-Hilbert s... |
| cphipval2 25199 | Value of the inner product... |
| 4cphipval2 25200 | Four times the inner produ... |
| cphipval 25201 | Value of the inner product... |
| ipcnlem2 25202 | The inner product operatio... |
| ipcnlem1 25203 | The inner product operatio... |
| ipcn 25204 | The inner product operatio... |
| cnmpt1ip 25205 | Continuity of inner produc... |
| cnmpt2ip 25206 | Continuity of inner produc... |
| csscld 25207 | A "closed subspace" in a s... |
| clsocv 25208 | The orthogonal complement ... |
| cphsscph 25209 | A subspace of a subcomplex... |
| lmmbr 25216 | Express the binary relatio... |
| lmmbr2 25217 | Express the binary relatio... |
| lmmbr3 25218 | Express the binary relatio... |
| lmmcvg 25219 | Convergence property of a ... |
| lmmbrf 25220 | Express the binary relatio... |
| lmnn 25221 | A condition that implies c... |
| cfilfval 25222 | The set of Cauchy filters ... |
| iscfil 25223 | The property of being a Ca... |
| iscfil2 25224 | The property of being a Ca... |
| cfilfil 25225 | A Cauchy filter is a filte... |
| cfili 25226 | Property of a Cauchy filte... |
| cfil3i 25227 | A Cauchy filter contains b... |
| cfilss 25228 | A filter finer than a Cauc... |
| fgcfil 25229 | The Cauchy filter conditio... |
| fmcfil 25230 | The Cauchy filter conditio... |
| iscfil3 25231 | A filter is Cauchy iff it ... |
| cfilfcls 25232 | Similar to ultrafilters ( ... |
| caufval 25233 | The set of Cauchy sequence... |
| iscau 25234 | Express the property " ` F... |
| iscau2 25235 | Express the property " ` F... |
| iscau3 25236 | Express the Cauchy sequenc... |
| iscau4 25237 | Express the property " ` F... |
| iscauf 25238 | Express the property " ` F... |
| caun0 25239 | A metric with a Cauchy seq... |
| caufpm 25240 | Inclusion of a Cauchy sequ... |
| caucfil 25241 | A Cauchy sequence predicat... |
| iscmet 25242 | The property " ` D ` is a ... |
| cmetcvg 25243 | The convergence of a Cauch... |
| cmetmet 25244 | A complete metric space is... |
| cmetmeti 25245 | A complete metric space is... |
| cmetcaulem 25246 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25247 | The convergence of a Cauch... |
| iscmet3lem3 25248 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25249 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25250 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25251 | The property " ` D ` is a ... |
| iscmet2 25252 | A metric ` D ` is complete... |
| cfilresi 25253 | A Cauchy filter on a metri... |
| cfilres 25254 | Cauchy filter on a metric ... |
| caussi 25255 | Cauchy sequence on a metri... |
| causs 25256 | Cauchy sequence on a metri... |
| equivcfil 25257 | If the metric ` D ` is "st... |
| equivcau 25258 | If the metric ` D ` is "st... |
| lmle 25259 | If the distance from each ... |
| nglmle 25260 | If the norm of each member... |
| lmclim 25261 | Relate a limit on the metr... |
| lmclimf 25262 | Relate a limit on the metr... |
| metelcls 25263 | A point belongs to the clo... |
| metcld 25264 | A subset of a metric space... |
| metcld2 25265 | A subset of a metric space... |
| caubl 25266 | Sufficient condition to en... |
| caublcls 25267 | The convergent point of a ... |
| metcnp4 25268 | Two ways to say a mapping ... |
| metcn4 25269 | Two ways to say a mapping ... |
| iscmet3i 25270 | Properties that determine ... |
| lmcau 25271 | Every convergent sequence ... |
| flimcfil 25272 | Every convergent filter in... |
| metsscmetcld 25273 | A complete subspace of a m... |
| cmetss 25274 | A subspace of a complete m... |
| equivcmet 25275 | If two metrics are strongl... |
| relcmpcmet 25276 | If ` D ` is a metric space... |
| cmpcmet 25277 | A compact metric space is ... |
| cfilucfil3 25278 | Given a metric ` D ` and a... |
| cfilucfil4 25279 | Given a metric ` D ` and a... |
| cncmet 25280 | The set of complex numbers... |
| recmet 25281 | The real numbers are a com... |
| bcthlem1 25282 | Lemma for ~ bcth . Substi... |
| bcthlem2 25283 | Lemma for ~ bcth . The ba... |
| bcthlem3 25284 | Lemma for ~ bcth . The li... |
| bcthlem4 25285 | Lemma for ~ bcth . Given ... |
| bcthlem5 25286 | Lemma for ~ bcth . The pr... |
| bcth 25287 | Baire's Category Theorem. ... |
| bcth2 25288 | Baire's Category Theorem, ... |
| bcth3 25289 | Baire's Category Theorem, ... |
| isbn 25296 | A Banach space is a normed... |
| bnsca 25297 | The scalar field of a Bana... |
| bnnvc 25298 | A Banach space is a normed... |
| bnnlm 25299 | A Banach space is a normed... |
| bnngp 25300 | A Banach space is a normed... |
| bnlmod 25301 | A Banach space is a left m... |
| bncms 25302 | A Banach space is a comple... |
| iscms 25303 | A complete metric space is... |
| cmscmet 25304 | The induced metric on a co... |
| bncmet 25305 | The induced metric on Bana... |
| cmsms 25306 | A complete metric space is... |
| cmspropd 25307 | Property deduction for a c... |
| cmssmscld 25308 | The restriction of a metri... |
| cmsss 25309 | The restriction of a compl... |
| lssbn 25310 | A subspace of a Banach spa... |
| cmetcusp1 25311 | If the uniform set of a co... |
| cmetcusp 25312 | The uniform space generate... |
| cncms 25313 | The field of complex numbe... |
| cnflduss 25314 | The uniform structure of t... |
| cnfldcusp 25315 | The field of complex numbe... |
| resscdrg 25316 | The real numbers are a sub... |
| cncdrg 25317 | The only complete subfield... |
| srabn 25318 | The subring algebra over a... |
| rlmbn 25319 | The ring module over a com... |
| ishl 25320 | The predicate "is a subcom... |
| hlbn 25321 | Every subcomplex Hilbert s... |
| hlcph 25322 | Every subcomplex Hilbert s... |
| hlphl 25323 | Every subcomplex Hilbert s... |
| hlcms 25324 | Every subcomplex Hilbert s... |
| hlprlem 25325 | Lemma for ~ hlpr . (Contr... |
| hlress 25326 | The scalar field of a subc... |
| hlpr 25327 | The scalar field of a subc... |
| ishl2 25328 | A Hilbert space is a compl... |
| cphssphl 25329 | A Banach subspace of a sub... |
| cmslssbn 25330 | A complete linear subspace... |
| cmscsscms 25331 | A closed subspace of a com... |
| bncssbn 25332 | A closed subspace of a Ban... |
| cssbn 25333 | A complete subspace of a n... |
| csschl 25334 | A complete subspace of a c... |
| cmslsschl 25335 | A complete linear subspace... |
| chlcsschl 25336 | A closed subspace of a sub... |
| retopn 25337 | The topology of the real n... |
| recms 25338 | The real numbers form a co... |
| reust 25339 | The Uniform structure of t... |
| recusp 25340 | The real numbers form a co... |
| rrxval 25345 | Value of the generalized E... |
| rrxbase 25346 | The base of the generalize... |
| rrxprds 25347 | Expand the definition of t... |
| rrxip 25348 | The inner product of the g... |
| rrxnm 25349 | The norm of the generalize... |
| rrxcph 25350 | Generalized Euclidean real... |
| rrxds 25351 | The distance over generali... |
| rrxvsca 25352 | The scalar product over ge... |
| rrxplusgvscavalb 25353 | The result of the addition... |
| rrxsca 25354 | The field of real numbers ... |
| rrx0 25355 | The zero ("origin") in a g... |
| rrx0el 25356 | The zero ("origin") in a g... |
| csbren 25357 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25358 | Triangle inequality in R^n... |
| rrxf 25359 | Euclidean vectors as funct... |
| rrxfsupp 25360 | Euclidean vectors are of f... |
| rrxsuppss 25361 | Support of Euclidean vecto... |
| rrxmvallem 25362 | Support of the function us... |
| rrxmval 25363 | The value of the Euclidean... |
| rrxmfval 25364 | The value of the Euclidean... |
| rrxmetlem 25365 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25366 | Euclidean space is a metri... |
| rrxdstprj1 25367 | The distance between two p... |
| rrxbasefi 25368 | The base of the generalize... |
| rrxdsfi 25369 | The distance over generali... |
| rrxmetfi 25370 | Euclidean space is a metri... |
| rrxdsfival 25371 | The value of the Euclidean... |
| ehlval 25372 | Value of the Euclidean spa... |
| ehlbase 25373 | The base of the Euclidean ... |
| ehl0base 25374 | The base of the Euclidean ... |
| ehl0 25375 | The Euclidean space of dim... |
| ehleudis 25376 | The Euclidean distance fun... |
| ehleudisval 25377 | The value of the Euclidean... |
| ehl1eudis 25378 | The Euclidean distance fun... |
| ehl1eudisval 25379 | The value of the Euclidean... |
| ehl2eudis 25380 | The Euclidean distance fun... |
| ehl2eudisval 25381 | The value of the Euclidean... |
| minveclem1 25382 | Lemma for ~ minvec . The ... |
| minveclem4c 25383 | Lemma for ~ minvec . The ... |
| minveclem2 25384 | Lemma for ~ minvec . Any ... |
| minveclem3a 25385 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25386 | Lemma for ~ minvec . The ... |
| minveclem3 25387 | Lemma for ~ minvec . The ... |
| minveclem4a 25388 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25389 | Lemma for ~ minvec . The ... |
| minveclem4 25390 | Lemma for ~ minvec . The ... |
| minveclem5 25391 | Lemma for ~ minvec . Disc... |
| minveclem6 25392 | Lemma for ~ minvec . Any ... |
| minveclem7 25393 | Lemma for ~ minvec . Sinc... |
| minvec 25394 | Minimizing vector theorem,... |
| pjthlem1 25395 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25396 | Lemma for ~ pjth . (Contr... |
| pjth 25397 | Projection Theorem: Any H... |
| pjth2 25398 | Projection Theorem with ab... |
| cldcss 25399 | Corollary of the Projectio... |
| cldcss2 25400 | Corollary of the Projectio... |
| hlhil 25401 | Corollary of the Projectio... |
| addcncf 25402 | The addition of two contin... |
| subcncf 25403 | The subtraction of two con... |
| mulcncf 25404 | The multiplication of two ... |
| mulcncfOLD 25405 | Obsolete version of ~ mulc... |
| divcncf 25406 | The quotient of two contin... |
| pmltpclem1 25407 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25408 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25409 | Any function on the reals ... |
| ivthlem1 25410 | Lemma for ~ ivth . The se... |
| ivthlem2 25411 | Lemma for ~ ivth . Show t... |
| ivthlem3 25412 | Lemma for ~ ivth , the int... |
| ivth 25413 | The intermediate value the... |
| ivth2 25414 | The intermediate value the... |
| ivthle 25415 | The intermediate value the... |
| ivthle2 25416 | The intermediate value the... |
| ivthicc 25417 | The interval between any t... |
| evthicc 25418 | Specialization of the Extr... |
| evthicc2 25419 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25420 | A continuous function on a... |
| ovolfcl 25425 | Closure for the interval e... |
| ovolfioo 25426 | Unpack the interval coveri... |
| ovolficc 25427 | Unpack the interval coveri... |
| ovolficcss 25428 | Any (closed) interval cove... |
| ovolfsval 25429 | The value of the interval ... |
| ovolfsf 25430 | Closure for the interval l... |
| ovolsf 25431 | Closure for the partial su... |
| ovolval 25432 | The value of the outer mea... |
| elovolmlem 25433 | Lemma for ~ elovolm and re... |
| elovolm 25434 | Elementhood in the set ` M... |
| elovolmr 25435 | Sufficient condition for e... |
| ovolmge0 25436 | The set ` M ` is composed ... |
| ovolcl 25437 | The volume of a set is an ... |
| ovollb 25438 | The outer volume is a lowe... |
| ovolgelb 25439 | The outer volume is the gr... |
| ovolge0 25440 | The volume of a set is alw... |
| ovolf 25441 | The domain and codomain of... |
| ovollecl 25442 | If an outer volume is boun... |
| ovolsslem 25443 | Lemma for ~ ovolss . (Con... |
| ovolss 25444 | The volume of a set is mon... |
| ovolsscl 25445 | If a set is contained in a... |
| ovolssnul 25446 | A subset of a nullset is n... |
| ovollb2lem 25447 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25448 | It is often more convenien... |
| ovolctb 25449 | The volume of a denumerabl... |
| ovolq 25450 | The rational numbers have ... |
| ovolctb2 25451 | The volume of a countable ... |
| ovol0 25452 | The empty set has 0 outer ... |
| ovolfi 25453 | A finite set has 0 outer L... |
| ovolsn 25454 | A singleton has 0 outer Le... |
| ovolunlem1a 25455 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25456 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25457 | Lemma for ~ ovolun . (Con... |
| ovolun 25458 | The Lebesgue outer measure... |
| ovolunnul 25459 | Adding a nullset does not ... |
| ovolfiniun 25460 | The Lebesgue outer measure... |
| ovoliunlem1 25461 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25462 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25463 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25464 | The Lebesgue outer measure... |
| ovoliun2 25465 | The Lebesgue outer measure... |
| ovoliunnul 25466 | A countable union of nulls... |
| shft2rab 25467 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25468 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25469 | Lemma for ~ ovolshft . (C... |
| ovolshft 25470 | The Lebesgue outer measure... |
| sca2rab 25471 | If ` B ` is a scale of ` A... |
| ovolscalem1 25472 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25473 | Lemma for ~ ovolshft . (C... |
| ovolsca 25474 | The Lebesgue outer measure... |
| ovolicc1 25475 | The measure of a closed in... |
| ovolicc2lem1 25476 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25477 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25478 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25479 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25480 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25481 | The measure of a closed in... |
| ovolicc 25482 | The measure of a closed in... |
| ovolicopnf 25483 | The measure of a right-unb... |
| ovolre 25484 | The measure of the real nu... |
| ismbl 25485 | The predicate " ` A ` is L... |
| ismbl2 25486 | From ~ ovolun , it suffice... |
| volres 25487 | A self-referencing abbrevi... |
| volf 25488 | The domain and codomain of... |
| mblvol 25489 | The volume of a measurable... |
| mblss 25490 | A measurable set is a subs... |
| mblsplit 25491 | The defining property of m... |
| volss 25492 | The Lebesgue measure is mo... |
| cmmbl 25493 | The complement of a measur... |
| nulmbl 25494 | A nullset is measurable. ... |
| nulmbl2 25495 | A set of outer measure zer... |
| unmbl 25496 | A union of measurable sets... |
| shftmbl 25497 | A shift of a measurable se... |
| 0mbl 25498 | The empty set is measurabl... |
| rembl 25499 | The set of all real number... |
| unidmvol 25500 | The union of the Lebesgue ... |
| inmbl 25501 | An intersection of measura... |
| difmbl 25502 | A difference of measurable... |
| finiunmbl 25503 | A finite union of measurab... |
| volun 25504 | The Lebesgue measure funct... |
| volinun 25505 | Addition of non-disjoint s... |
| volfiniun 25506 | The volume of a disjoint f... |
| iundisj 25507 | Rewrite a countable union ... |
| iundisj2 25508 | A disjoint union is disjoi... |
| voliunlem1 25509 | Lemma for ~ voliun . (Con... |
| voliunlem2 25510 | Lemma for ~ voliun . (Con... |
| voliunlem3 25511 | Lemma for ~ voliun . (Con... |
| iunmbl 25512 | The measurable sets are cl... |
| voliun 25513 | The Lebesgue measure funct... |
| volsuplem 25514 | Lemma for ~ volsup . (Con... |
| volsup 25515 | The volume of the limit of... |
| iunmbl2 25516 | The measurable sets are cl... |
| ioombl1lem1 25517 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25518 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25519 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25520 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25521 | An open right-unbounded in... |
| icombl1 25522 | A closed unbounded-above i... |
| icombl 25523 | A closed-below, open-above... |
| ioombl 25524 | An open real interval is m... |
| iccmbl 25525 | A closed real interval is ... |
| iccvolcl 25526 | A closed real interval has... |
| ovolioo 25527 | The measure of an open int... |
| volioo 25528 | The measure of an open int... |
| ioovolcl 25529 | An open real interval has ... |
| ovolfs2 25530 | Alternative expression for... |
| ioorcl2 25531 | An open interval with fini... |
| ioorf 25532 | Define a function from ope... |
| ioorval 25533 | Define a function from ope... |
| ioorinv2 25534 | The function ` F ` is an "... |
| ioorinv 25535 | The function ` F ` is an "... |
| ioorcl 25536 | The function ` F ` does no... |
| uniiccdif 25537 | A union of closed interval... |
| uniioovol 25538 | A disjoint union of open i... |
| uniiccvol 25539 | An almost-disjoint union o... |
| uniioombllem1 25540 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25541 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25542 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25543 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25544 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25545 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25546 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25547 | Lemma for ~ uniioombl . (... |
| uniioombl 25548 | A disjoint union of open i... |
| uniiccmbl 25549 | An almost-disjoint union o... |
| dyadf 25550 | The function ` F ` returns... |
| dyadval 25551 | Value of the dyadic ration... |
| dyadovol 25552 | Volume of a dyadic rationa... |
| dyadss 25553 | Two closed dyadic rational... |
| dyaddisjlem 25554 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25555 | Two closed dyadic rational... |
| dyadmaxlem 25556 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25557 | Any nonempty set of dyadic... |
| dyadmbllem 25558 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25559 | Any union of dyadic ration... |
| opnmbllem 25560 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25561 | All open sets are measurab... |
| opnmblALT 25562 | All open sets are measurab... |
| subopnmbl 25563 | Sets which are open in a m... |
| volsup2 25564 | The volume of ` A ` is the... |
| volcn 25565 | The function formed by res... |
| volivth 25566 | The Intermediate Value The... |
| vitalilem1 25567 | Lemma for ~ vitali . (Con... |
| vitalilem2 25568 | Lemma for ~ vitali . (Con... |
| vitalilem3 25569 | Lemma for ~ vitali . (Con... |
| vitalilem4 25570 | Lemma for ~ vitali . (Con... |
| vitalilem5 25571 | Lemma for ~ vitali . (Con... |
| vitali 25572 | If the reals can be well-o... |
| ismbf1 25583 | The predicate " ` F ` is a... |
| mbff 25584 | A measurable function is a... |
| mbfdm 25585 | The domain of a measurable... |
| mbfconstlem 25586 | Lemma for ~ mbfconst and r... |
| ismbf 25587 | The predicate " ` F ` is a... |
| ismbfcn 25588 | A complex function is meas... |
| mbfima 25589 | Definitional property of a... |
| mbfimaicc 25590 | The preimage of any closed... |
| mbfimasn 25591 | The preimage of a point un... |
| mbfconst 25592 | A constant function is mea... |
| mbf0 25593 | The empty function is meas... |
| mbfid 25594 | The identity function is m... |
| mbfmptcl 25595 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25596 | The domain of a measurable... |
| ismbfcn2 25597 | A complex function is meas... |
| ismbfd 25598 | Deduction to prove measura... |
| ismbf2d 25599 | Deduction to prove measura... |
| mbfeqalem1 25600 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25601 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25602 | If two functions are equal... |
| mbfres 25603 | The restriction of a measu... |
| mbfres2 25604 | Measurability of a piecewi... |
| mbfss 25605 | Change the domain of a mea... |
| mbfmulc2lem 25606 | Multiplication by a consta... |
| mbfmulc2re 25607 | Multiplication by a consta... |
| mbfmax 25608 | The maximum of two functio... |
| mbfneg 25609 | The negative of a measurab... |
| mbfpos 25610 | The positive part of a mea... |
| mbfposr 25611 | Converse to ~ mbfpos . (C... |
| mbfposb 25612 | A function is measurable i... |
| ismbf3d 25613 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25614 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25615 | The preimage of any open s... |
| mbfimaopn2 25616 | The preimage of any set op... |
| cncombf 25617 | The composition of a conti... |
| cnmbf 25618 | A continuous function is m... |
| mbfaddlem 25619 | The sum of two measurable ... |
| mbfadd 25620 | The sum of two measurable ... |
| mbfsub 25621 | The difference of two meas... |
| mbfmulc2 25622 | A complex constant times a... |
| mbfsup 25623 | The supremum of a sequence... |
| mbfinf 25624 | The infimum of a sequence ... |
| mbflimsup 25625 | The limit supremum of a se... |
| mbflimlem 25626 | The pointwise limit of a s... |
| mbflim 25627 | The pointwise limit of a s... |
| 0pval 25630 | The zero function evaluate... |
| 0plef 25631 | Two ways to say that the f... |
| 0pledm 25632 | Adjust the domain of the l... |
| isi1f 25633 | The predicate " ` F ` is a... |
| i1fmbf 25634 | Simple functions are measu... |
| i1ff 25635 | A simple function is a fun... |
| i1frn 25636 | A simple function has fini... |
| i1fima 25637 | Any preimage of a simple f... |
| i1fima2 25638 | Any preimage of a simple f... |
| i1fima2sn 25639 | Preimage of a singleton. ... |
| i1fd 25640 | A simplified set of assump... |
| i1f0rn 25641 | Any simple function takes ... |
| itg1val 25642 | The value of the integral ... |
| itg1val2 25643 | The value of the integral ... |
| itg1cl 25644 | Closure of the integral on... |
| itg1ge0 25645 | Closure of the integral on... |
| i1f0 25646 | The zero function is simpl... |
| itg10 25647 | The zero function has zero... |
| i1f1lem 25648 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25649 | Base case simple functions... |
| itg11 25650 | The integral of an indicat... |
| itg1addlem1 25651 | Decompose a preimage, whic... |
| i1faddlem 25652 | Decompose the preimage of ... |
| i1fmullem 25653 | Decompose the preimage of ... |
| i1fadd 25654 | The sum of two simple func... |
| i1fmul 25655 | The pointwise product of t... |
| itg1addlem2 25656 | Lemma for ~ itg1add . The... |
| itg1addlem3 25657 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25658 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 25659 | Lemma for ~ itg1add . (Co... |
| itg1add 25660 | The integral of a sum of s... |
| i1fmulclem 25661 | Decompose the preimage of ... |
| i1fmulc 25662 | A nonnegative constant tim... |
| itg1mulc 25663 | The integral of a constant... |
| i1fres 25664 | The "restriction" of a sim... |
| i1fpos 25665 | The positive part of a sim... |
| i1fposd 25666 | Deduction form of ~ i1fpos... |
| i1fsub 25667 | The difference of two simp... |
| itg1sub 25668 | The integral of a differen... |
| itg10a 25669 | The integral of a simple f... |
| itg1ge0a 25670 | The integral of an almost ... |
| itg1lea 25671 | Approximate version of ~ i... |
| itg1le 25672 | If one simple function dom... |
| itg1climres 25673 | Restricting the simple fun... |
| mbfi1fseqlem1 25674 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25675 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25676 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25677 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25678 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25679 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25680 | A characterization of meas... |
| mbfi1flimlem 25681 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25682 | Any real measurable functi... |
| mbfmullem2 25683 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25684 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25685 | The product of two measura... |
| itg2lcl 25686 | The set of lower sums is a... |
| itg2val 25687 | Value of the integral on n... |
| itg2l 25688 | Elementhood in the set ` L... |
| itg2lr 25689 | Sufficient condition for e... |
| xrge0f 25690 | A real function is a nonne... |
| itg2cl 25691 | The integral of a nonnegat... |
| itg2ub 25692 | The integral of a nonnegat... |
| itg2leub 25693 | Any upper bound on the int... |
| itg2ge0 25694 | The integral of a nonnegat... |
| itg2itg1 25695 | The integral of a nonnegat... |
| itg20 25696 | The integral of the zero f... |
| itg2lecl 25697 | If an ` S.2 ` integral is ... |
| itg2le 25698 | If one function dominates ... |
| itg2const 25699 | Integral of a constant fun... |
| itg2const2 25700 | When the base set of a con... |
| itg2seq 25701 | Definitional property of t... |
| itg2uba 25702 | Approximate version of ~ i... |
| itg2lea 25703 | Approximate version of ~ i... |
| itg2eqa 25704 | Approximate equality of in... |
| itg2mulclem 25705 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25706 | The integral of a nonnegat... |
| itg2splitlem 25707 | Lemma for ~ itg2split . (... |
| itg2split 25708 | The ` S.2 ` integral split... |
| itg2monolem1 25709 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25710 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25711 | Lemma for ~ itg2mono . (C... |
| itg2mono 25712 | The Monotone Convergence T... |
| itg2i1fseqle 25713 | Subject to the conditions ... |
| itg2i1fseq 25714 | Subject to the conditions ... |
| itg2i1fseq2 25715 | In an extension to the res... |
| itg2i1fseq3 25716 | Special case of ~ itg2i1fs... |
| itg2addlem 25717 | Lemma for ~ itg2add . (Co... |
| itg2add 25718 | The ` S.2 ` integral is li... |
| itg2gt0 25719 | If the function ` F ` is s... |
| itg2cnlem1 25720 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25721 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25722 | A sort of absolute continu... |
| ibllem 25723 | Conditioned equality theor... |
| isibl 25724 | The predicate " ` F ` is i... |
| isibl2 25725 | The predicate " ` F ` is i... |
| iblmbf 25726 | An integrable function is ... |
| iblitg 25727 | If a function is integrabl... |
| dfitg 25728 | Evaluate the class substit... |
| itgex 25729 | An integral is a set. (Co... |
| itgeq1f 25730 | Equality theorem for an in... |
| itgeq1fOLD 25731 | Obsolete version of ~ itge... |
| itgeq1 25732 | Equality theorem for an in... |
| nfitg1 25733 | Bound-variable hypothesis ... |
| nfitg 25734 | Bound-variable hypothesis ... |
| cbvitg 25735 | Change bound variable in a... |
| cbvitgv 25736 | Change bound variable in a... |
| itgeq2 25737 | Equality theorem for an in... |
| itgresr 25738 | The domain of an integral ... |
| itg0 25739 | The integral of anything o... |
| itgz 25740 | The integral of zero on an... |
| itgeq2dv 25741 | Equality theorem for an in... |
| itgmpt 25742 | Change bound variable in a... |
| itgcl 25743 | The integral of an integra... |
| itgvallem 25744 | Substitution lemma. (Cont... |
| itgvallem3 25745 | Lemma for ~ itgposval and ... |
| ibl0 25746 | The zero function is integ... |
| iblcnlem1 25747 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25748 | Expand out the universal q... |
| itgcnlem 25749 | Expand out the sum in ~ df... |
| iblrelem 25750 | Integrability of a real fu... |
| iblposlem 25751 | Lemma for ~ iblpos . (Con... |
| iblpos 25752 | Integrability of a nonnega... |
| iblre 25753 | Integrability of a real fu... |
| itgrevallem1 25754 | Lemma for ~ itgposval and ... |
| itgposval 25755 | The integral of a nonnegat... |
| itgreval 25756 | Decompose the integral of ... |
| itgrecl 25757 | Real closure of an integra... |
| iblcn 25758 | Integrability of a complex... |
| itgcnval 25759 | Decompose the integral of ... |
| itgre 25760 | Real part of an integral. ... |
| itgim 25761 | Imaginary part of an integ... |
| iblneg 25762 | The negative of an integra... |
| itgneg 25763 | Negation of an integral. ... |
| iblss 25764 | A subset of an integrable ... |
| iblss2 25765 | Change the domain of an in... |
| itgitg2 25766 | Transfer an integral using... |
| i1fibl 25767 | A simple function is integ... |
| itgitg1 25768 | Transfer an integral using... |
| itgle 25769 | Monotonicity of an integra... |
| itgge0 25770 | The integral of a positive... |
| itgss 25771 | Expand the set of an integ... |
| itgss2 25772 | Expand the set of an integ... |
| itgeqa 25773 | Approximate equality of in... |
| itgss3 25774 | Expand the set of an integ... |
| itgioo 25775 | Equality of integrals on o... |
| itgless 25776 | Expand the integral of a n... |
| iblconst 25777 | A constant function is int... |
| itgconst 25778 | Integral of a constant fun... |
| ibladdlem 25779 | Lemma for ~ ibladd . (Con... |
| ibladd 25780 | Add two integrals over the... |
| iblsub 25781 | Subtract two integrals ove... |
| itgaddlem1 25782 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25783 | Lemma for ~ itgadd . (Con... |
| itgadd 25784 | Add two integrals over the... |
| itgsub 25785 | Subtract two integrals ove... |
| itgfsum 25786 | Take a finite sum of integ... |
| iblabslem 25787 | Lemma for ~ iblabs . (Con... |
| iblabs 25788 | The absolute value of an i... |
| iblabsr 25789 | A measurable function is i... |
| iblmulc2 25790 | Multiply an integral by a ... |
| itgmulc2lem1 25791 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25792 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25793 | Multiply an integral by a ... |
| itgabs 25794 | The triangle inequality fo... |
| itgsplit 25795 | The ` S. ` integral splits... |
| itgspliticc 25796 | The ` S. ` integral splits... |
| itgsplitioo 25797 | The ` S. ` integral splits... |
| bddmulibl 25798 | A bounded function times a... |
| bddibl 25799 | A bounded function is inte... |
| cniccibl 25800 | A continuous function on a... |
| bddiblnc 25801 | Choice-free proof of ~ bdd... |
| cnicciblnc 25802 | Choice-free proof of ~ cni... |
| itggt0 25803 | The integral of a strictly... |
| itgcn 25804 | Transfer ~ itg2cn to the f... |
| ditgeq1 25807 | Equality theorem for the d... |
| ditgeq2 25808 | Equality theorem for the d... |
| ditgeq3 25809 | Equality theorem for the d... |
| ditgeq3dv 25810 | Equality theorem for the d... |
| ditgex 25811 | A directed integral is a s... |
| ditg0 25812 | Value of the directed inte... |
| cbvditg 25813 | Change bound variable in a... |
| cbvditgv 25814 | Change bound variable in a... |
| ditgpos 25815 | Value of the directed inte... |
| ditgneg 25816 | Value of the directed inte... |
| ditgcl 25817 | Closure of a directed inte... |
| ditgswap 25818 | Reverse a directed integra... |
| ditgsplitlem 25819 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25820 | This theorem is the raison... |
| reldv 25829 | The derivative function is... |
| limcvallem 25830 | Lemma for ~ ellimc . (Con... |
| limcfval 25831 | Value and set bounds on th... |
| ellimc 25832 | Value of the limit predica... |
| limcrcl 25833 | Reverse closure for the li... |
| limccl 25834 | Closure of the limit opera... |
| limcdif 25835 | It suffices to consider fu... |
| ellimc2 25836 | Write the definition of a ... |
| limcnlp 25837 | If ` B ` is not a limit po... |
| ellimc3 25838 | Write the epsilon-delta de... |
| limcflflem 25839 | Lemma for ~ limcflf . (Co... |
| limcflf 25840 | The limit operator can be ... |
| limcmo 25841 | If ` B ` is a limit point ... |
| limcmpt 25842 | Express the limit operator... |
| limcmpt2 25843 | Express the limit operator... |
| limcresi 25844 | Any limit of ` F ` is also... |
| limcres 25845 | If ` B ` is an interior po... |
| cnplimc 25846 | A function is continuous a... |
| cnlimc 25847 | ` F ` is a continuous func... |
| cnlimci 25848 | If ` F ` is a continuous f... |
| cnmptlimc 25849 | If ` F ` is a continuous f... |
| limccnp 25850 | If the limit of ` F ` at `... |
| limccnp2 25851 | The image of a convergent ... |
| limcco 25852 | Composition of two limits.... |
| limciun 25853 | A point is a limit of ` F ... |
| limcun 25854 | A point is a limit of ` F ... |
| dvlem 25855 | Closure for a difference q... |
| dvfval 25856 | Value and set bounds on th... |
| eldv 25857 | The differentiable predica... |
| dvcl 25858 | The derivative function ta... |
| dvbssntr 25859 | The set of differentiable ... |
| dvbss 25860 | The set of differentiable ... |
| dvbsss 25861 | The set of differentiable ... |
| perfdvf 25862 | The derivative is a functi... |
| recnprss 25863 | Both ` RR ` and ` CC ` are... |
| recnperf 25864 | Both ` RR ` and ` CC ` are... |
| dvfg 25865 | Explicitly write out the f... |
| dvf 25866 | The derivative is a functi... |
| dvfcn 25867 | The derivative is a functi... |
| dvreslem 25868 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25869 | Lemma for ~ dvres2 . (Con... |
| dvres 25870 | Restriction of a derivativ... |
| dvres2 25871 | Restriction of the base se... |
| dvres3 25872 | Restriction of a complex d... |
| dvres3a 25873 | Restriction of a complex d... |
| dvidlem 25874 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25875 | Derivative of a function r... |
| dvconst 25876 | Derivative of a constant f... |
| dvid 25877 | Derivative of the identity... |
| dvcnp 25878 | The difference quotient is... |
| dvcnp2 25879 | A function is continuous a... |
| dvcnp2OLD 25880 | Obsolete version of ~ dvcn... |
| dvcn 25881 | A differentiable function ... |
| dvnfval 25882 | Value of the iterated deri... |
| dvnff 25883 | The iterated derivative is... |
| dvn0 25884 | Zero times iterated deriva... |
| dvnp1 25885 | Successor iterated derivat... |
| dvn1 25886 | One times iterated derivat... |
| dvnf 25887 | The N-times derivative is ... |
| dvnbss 25888 | The set of N-times differe... |
| dvnadd 25889 | The ` N ` -th derivative o... |
| dvn2bss 25890 | An N-times differentiable ... |
| dvnres 25891 | Multiple derivative versio... |
| cpnfval 25892 | Condition for n-times cont... |
| fncpn 25893 | The ` C^n ` object is a fu... |
| elcpn 25894 | Condition for n-times cont... |
| cpnord 25895 | ` C^n ` conditions are ord... |
| cpncn 25896 | A ` C^n ` function is cont... |
| cpnres 25897 | The restriction of a ` C^n... |
| dvaddbr 25898 | The sum rule for derivativ... |
| dvmulbr 25899 | The product rule for deriv... |
| dvmulbrOLD 25900 | Obsolete version of ~ dvmu... |
| dvadd 25901 | The sum rule for derivativ... |
| dvmul 25902 | The product rule for deriv... |
| dvaddf 25903 | The sum rule for everywher... |
| dvmulf 25904 | The product rule for every... |
| dvcmul 25905 | The product rule when one ... |
| dvcmulf 25906 | The product rule when one ... |
| dvcobr 25907 | The chain rule for derivat... |
| dvcobrOLD 25908 | Obsolete version of ~ dvco... |
| dvco 25909 | The chain rule for derivat... |
| dvcof 25910 | The chain rule for everywh... |
| dvcjbr 25911 | The derivative of the conj... |
| dvcj 25912 | The derivative of the conj... |
| dvfre 25913 | The derivative of a real f... |
| dvnfre 25914 | The ` N ` -th derivative o... |
| dvexp 25915 | Derivative of a power func... |
| dvexp2 25916 | Derivative of an exponenti... |
| dvrec 25917 | Derivative of the reciproc... |
| dvmptres3 25918 | Function-builder for deriv... |
| dvmptid 25919 | Function-builder for deriv... |
| dvmptc 25920 | Function-builder for deriv... |
| dvmptcl 25921 | Closure lemma for ~ dvmptc... |
| dvmptadd 25922 | Function-builder for deriv... |
| dvmptmul 25923 | Function-builder for deriv... |
| dvmptres2 25924 | Function-builder for deriv... |
| dvmptres 25925 | Function-builder for deriv... |
| dvmptcmul 25926 | Function-builder for deriv... |
| dvmptdivc 25927 | Function-builder for deriv... |
| dvmptneg 25928 | Function-builder for deriv... |
| dvmptsub 25929 | Function-builder for deriv... |
| dvmptcj 25930 | Function-builder for deriv... |
| dvmptre 25931 | Function-builder for deriv... |
| dvmptim 25932 | Function-builder for deriv... |
| dvmptntr 25933 | Function-builder for deriv... |
| dvmptco 25934 | Function-builder for deriv... |
| dvrecg 25935 | Derivative of the reciproc... |
| dvmptdiv 25936 | Function-builder for deriv... |
| dvmptfsum 25937 | Function-builder for deriv... |
| dvcnvlem 25938 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25939 | A weak version of ~ dvcnvr... |
| dvexp3 25940 | Derivative of an exponenti... |
| dveflem 25941 | Derivative of the exponent... |
| dvef 25942 | Derivative of the exponent... |
| dvsincos 25943 | Derivative of the sine and... |
| dvsin 25944 | Derivative of the sine fun... |
| dvcos 25945 | Derivative of the cosine f... |
| dvferm1lem 25946 | Lemma for ~ dvferm . (Con... |
| dvferm1 25947 | One-sided version of ~ dvf... |
| dvferm2lem 25948 | Lemma for ~ dvferm . (Con... |
| dvferm2 25949 | One-sided version of ~ dvf... |
| dvferm 25950 | Fermat's theorem on statio... |
| rollelem 25951 | Lemma for ~ rolle . (Cont... |
| rolle 25952 | Rolle's theorem. If ` F `... |
| cmvth 25953 | Cauchy's Mean Value Theore... |
| cmvthOLD 25954 | Obsolete version of ~ cmvt... |
| mvth 25955 | The Mean Value Theorem. I... |
| dvlip 25956 | A function with derivative... |
| dvlipcn 25957 | A complex function with de... |
| dvlip2 25958 | Combine the results of ~ d... |
| c1liplem1 25959 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25960 | C^1 functions are Lipschit... |
| c1lip2 25961 | C^1 functions are Lipschit... |
| c1lip3 25962 | C^1 functions are Lipschit... |
| dveq0 25963 | If a continuous function h... |
| dv11cn 25964 | Two functions defined on a... |
| dvgt0lem1 25965 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25966 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25967 | A function on a closed int... |
| dvlt0 25968 | A function on a closed int... |
| dvge0 25969 | A function on a closed int... |
| dvle 25970 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25971 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25972 | Lemma for ~ dvivth . (Con... |
| dvivth 25973 | Darboux' theorem, or the i... |
| dvne0 25974 | A function on a closed int... |
| dvne0f1 25975 | A function on a closed int... |
| lhop1lem 25976 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25977 | L'Hôpital's Rule for... |
| lhop2 25978 | L'Hôpital's Rule for... |
| lhop 25979 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25980 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25981 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25982 | The derivative rule for in... |
| dvcvx 25983 | A real function with stric... |
| dvfsumle 25984 | Compare a finite sum to an... |
| dvfsumleOLD 25985 | Obsolete version of ~ dvfs... |
| dvfsumge 25986 | Compare a finite sum to an... |
| dvfsumabs 25987 | Compare a finite sum to an... |
| dvmptrecl 25988 | Real closure of a derivati... |
| dvfsumrlimf 25989 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25990 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25991 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2OLD 25992 | Obsolete version of ~ dvfs... |
| dvfsumlem3 25993 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25994 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25995 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25996 | Compare a finite sum to an... |
| dvfsumrlim2 25997 | Compare a finite sum to an... |
| dvfsumrlim3 25998 | Conjoin the statements of ... |
| dvfsum2 25999 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26000 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26001 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26002 | The Fundamental Theorem of... |
| ftc1lem3 26003 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26004 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26005 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26006 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26007 | The Fundamental Theorem of... |
| ftc1cn 26008 | Strengthen the assumptions... |
| ftc2 26009 | The Fundamental Theorem of... |
| ftc2ditglem 26010 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26011 | Directed integral analogue... |
| itgparts 26012 | Integration by parts. If ... |
| itgsubstlem 26013 | Lemma for ~ itgsubst . (C... |
| itgsubst 26014 | Integration by ` u ` -subs... |
| itgpowd 26015 | The integral of a monomial... |
| reldmmdeg 26020 | Multivariate degree is a b... |
| tdeglem1 26021 | Functionality of the total... |
| tdeglem3 26022 | Additivity of the total de... |
| tdeglem4 26023 | There is only one multi-in... |
| tdeglem2 26024 | Simplification of total de... |
| mdegfval 26025 | Value of the multivariate ... |
| mdegval 26026 | Value of the multivariate ... |
| mdegleb 26027 | Property of being of limit... |
| mdeglt 26028 | If there is an upper limit... |
| mdegldg 26029 | A nonzero polynomial has s... |
| mdegxrcl 26030 | Closure of polynomial degr... |
| mdegxrf 26031 | Functionality of polynomia... |
| mdegcl 26032 | Sharp closure for multivar... |
| mdeg0 26033 | Degree of the zero polynom... |
| mdegnn0cl 26034 | Degree of a nonzero polyno... |
| degltlem1 26035 | Theorem on arithmetic of e... |
| degltp1le 26036 | Theorem on arithmetic of e... |
| mdegaddle 26037 | The degree of a sum is at ... |
| mdegvscale 26038 | The degree of a scalar mul... |
| mdegvsca 26039 | The degree of a scalar mul... |
| mdegle0 26040 | A polynomial has nonpositi... |
| mdegmullem 26041 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26042 | The multivariate degree of... |
| deg1fval 26043 | Relate univariate polynomi... |
| deg1xrf 26044 | Functionality of univariat... |
| deg1xrcl 26045 | Closure of univariate poly... |
| deg1cl 26046 | Sharp closure of univariat... |
| mdegpropd 26047 | Property deduction for pol... |
| deg1fvi 26048 | Univariate polynomial degr... |
| deg1propd 26049 | Property deduction for pol... |
| deg1z 26050 | Degree of the zero univari... |
| deg1nn0cl 26051 | Degree of a nonzero univar... |
| deg1n0ima 26052 | Degree image of a set of p... |
| deg1nn0clb 26053 | A polynomial is nonzero if... |
| deg1lt0 26054 | A polynomial is zero iff i... |
| deg1ldg 26055 | A nonzero univariate polyn... |
| deg1ldgn 26056 | An index at which a polyno... |
| deg1ldgdomn 26057 | A nonzero univariate polyn... |
| deg1leb 26058 | Property of being of limit... |
| deg1val 26059 | Value of the univariate de... |
| deg1lt 26060 | If the degree of a univari... |
| deg1ge 26061 | Conversely, a nonzero coef... |
| coe1mul3 26062 | The coefficient vector of ... |
| coe1mul4 26063 | Value of the "leading" coe... |
| deg1addle 26064 | The degree of a sum is at ... |
| deg1addle2 26065 | If both factors have degre... |
| deg1add 26066 | Exact degree of a sum of t... |
| deg1vscale 26067 | The degree of a scalar tim... |
| deg1vsca 26068 | The degree of a scalar tim... |
| deg1invg 26069 | The degree of the negated ... |
| deg1suble 26070 | The degree of a difference... |
| deg1sub 26071 | Exact degree of a differen... |
| deg1mulle2 26072 | Produce a bound on the pro... |
| deg1sublt 26073 | Subtraction of two polynom... |
| deg1le0 26074 | A polynomial has nonpositi... |
| deg1sclle 26075 | A scalar polynomial has no... |
| deg1scl 26076 | A nonzero scalar polynomia... |
| deg1mul2 26077 | Degree of multiplication o... |
| deg1mul 26078 | Degree of multiplication o... |
| deg1mul3 26079 | Degree of multiplication o... |
| deg1mul3le 26080 | Degree of multiplication o... |
| deg1tmle 26081 | Limiting degree of a polyn... |
| deg1tm 26082 | Exact degree of a polynomi... |
| deg1pwle 26083 | Limiting degree of a varia... |
| deg1pw 26084 | Exact degree of a variable... |
| ply1nz 26085 | Univariate polynomials ove... |
| ply1nzb 26086 | Univariate polynomials are... |
| ply1domn 26087 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26088 | The ring of univariate pol... |
| ply1divmo 26099 | Uniqueness of a quotient i... |
| ply1divex 26100 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26101 | The division algorithm for... |
| ply1divalg2 26102 | Reverse the order of multi... |
| uc1pval 26103 | Value of the set of unitic... |
| isuc1p 26104 | Being a unitic polynomial.... |
| mon1pval 26105 | Value of the set of monic ... |
| ismon1p 26106 | Being a monic polynomial. ... |
| uc1pcl 26107 | Unitic polynomials are pol... |
| mon1pcl 26108 | Monic polynomials are poly... |
| uc1pn0 26109 | Unitic polynomials are not... |
| mon1pn0 26110 | Monic polynomials are not ... |
| uc1pdeg 26111 | Unitic polynomials have no... |
| uc1pldg 26112 | Unitic polynomials have un... |
| mon1pldg 26113 | Unitic polynomials have on... |
| mon1puc1p 26114 | Monic polynomials are unit... |
| uc1pmon1p 26115 | Make a unitic polynomial m... |
| deg1submon1p 26116 | The difference of two moni... |
| mon1pid 26117 | Monicity and degree of the... |
| q1pval 26118 | Value of the univariate po... |
| q1peqb 26119 | Characterizing property of... |
| q1pcl 26120 | Closure of the quotient by... |
| r1pval 26121 | Value of the polynomial re... |
| r1pcl 26122 | Closure of remainder follo... |
| r1pdeglt 26123 | The remainder has a degree... |
| r1pid 26124 | Express the original polyn... |
| r1pid2 26125 | Identity law for polynomia... |
| dvdsq1p 26126 | Divisibility in a polynomi... |
| dvdsr1p 26127 | Divisibility in a polynomi... |
| ply1remlem 26128 | A term of the form ` x - N... |
| ply1rem 26129 | The polynomial remainder t... |
| facth1 26130 | The factor theorem and its... |
| fta1glem1 26131 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26132 | Lemma for ~ fta1g . (Cont... |
| fta1g 26133 | The one-sided fundamental ... |
| fta1blem 26134 | Lemma for ~ fta1b . (Cont... |
| fta1b 26135 | The assumption that ` R ` ... |
| idomrootle 26136 | No element of an integral ... |
| drnguc1p 26137 | Over a division ring, all ... |
| ig1peu 26138 | There is a unique monic po... |
| ig1pval 26139 | Substitutions for the poly... |
| ig1pval2 26140 | Generator of the zero idea... |
| ig1pval3 26141 | Characterizing properties ... |
| ig1pcl 26142 | The monic generator of an ... |
| ig1pdvds 26143 | The monic generator of an ... |
| ig1prsp 26144 | Any ideal of polynomials o... |
| ply1lpir 26145 | The ring of polynomials ov... |
| ply1pid 26146 | The polynomials over a fie... |
| plyco0 26155 | Two ways to say that a fun... |
| plyval 26156 | Value of the polynomial se... |
| plybss 26157 | Reverse closure of the par... |
| elply 26158 | Definition of a polynomial... |
| elply2 26159 | The coefficient function c... |
| plyun0 26160 | The set of polynomials is ... |
| plyf 26161 | A polynomial is a function... |
| plyss 26162 | The polynomial set functio... |
| plyssc 26163 | Every polynomial ring is c... |
| elplyr 26164 | Sufficient condition for e... |
| elplyd 26165 | Sufficient condition for e... |
| ply1termlem 26166 | Lemma for ~ ply1term . (C... |
| ply1term 26167 | A one-term polynomial. (C... |
| plypow 26168 | A power is a polynomial. ... |
| plyconst 26169 | A constant function is a p... |
| ne0p 26170 | A test to show that a poly... |
| ply0 26171 | The zero function is a pol... |
| plyid 26172 | The identity function is a... |
| plyeq0lem 26173 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26174 | If a polynomial is zero at... |
| plypf1 26175 | Write the set of complex p... |
| plyaddlem1 26176 | Derive the coefficient fun... |
| plymullem1 26177 | Derive the coefficient fun... |
| plyaddlem 26178 | Lemma for ~ plyadd . (Con... |
| plymullem 26179 | Lemma for ~ plymul . (Con... |
| plyadd 26180 | The sum of two polynomials... |
| plymul 26181 | The product of two polynom... |
| plysub 26182 | The difference of two poly... |
| plyaddcl 26183 | The sum of two polynomials... |
| plymulcl 26184 | The product of two polynom... |
| plysubcl 26185 | The difference of two poly... |
| coeval 26186 | Value of the coefficient f... |
| coeeulem 26187 | Lemma for ~ coeeu . (Cont... |
| coeeu 26188 | Uniqueness of the coeffici... |
| coelem 26189 | Lemma for properties of th... |
| coeeq 26190 | If ` A ` satisfies the pro... |
| dgrval 26191 | Value of the degree functi... |
| dgrlem 26192 | Lemma for ~ dgrcl and simi... |
| coef 26193 | The domain and codomain of... |
| coef2 26194 | The domain and codomain of... |
| coef3 26195 | The domain and codomain of... |
| dgrcl 26196 | The degree of any polynomi... |
| dgrub 26197 | If the ` M ` -th coefficie... |
| dgrub2 26198 | All the coefficients above... |
| dgrlb 26199 | If all the coefficients ab... |
| coeidlem 26200 | Lemma for ~ coeid . (Cont... |
| coeid 26201 | Reconstruct a polynomial a... |
| coeid2 26202 | Reconstruct a polynomial a... |
| coeid3 26203 | Reconstruct a polynomial a... |
| plyco 26204 | The composition of two pol... |
| coeeq2 26205 | Compute the coefficient fu... |
| dgrle 26206 | Given an explicit expressi... |
| dgreq 26207 | If the highest term in a p... |
| 0dgr 26208 | A constant function has de... |
| 0dgrb 26209 | A function has degree zero... |
| dgrnznn 26210 | A nonzero polynomial with ... |
| coefv0 26211 | The result of evaluating a... |
| coeaddlem 26212 | Lemma for ~ coeadd and ~ d... |
| coemullem 26213 | Lemma for ~ coemul and ~ d... |
| coeadd 26214 | The coefficient function o... |
| coemul 26215 | A coefficient of a product... |
| coe11 26216 | The coefficient function i... |
| coemulhi 26217 | The leading coefficient of... |
| coemulc 26218 | The coefficient function i... |
| coe0 26219 | The coefficients of the ze... |
| coesub 26220 | The coefficient function o... |
| coe1termlem 26221 | The coefficient function o... |
| coe1term 26222 | The coefficient function o... |
| dgr1term 26223 | The degree of a monomial. ... |
| plycn 26224 | A polynomial is a continuo... |
| plycnOLD 26225 | Obsolete version of ~ plyc... |
| dgr0 26226 | The degree of the zero pol... |
| coeidp 26227 | The coefficients of the id... |
| dgrid 26228 | The degree of the identity... |
| dgreq0 26229 | The leading coefficient of... |
| dgrlt 26230 | Two ways to say that the d... |
| dgradd 26231 | The degree of a sum of pol... |
| dgradd2 26232 | The degree of a sum of pol... |
| dgrmul2 26233 | The degree of a product of... |
| dgrmul 26234 | The degree of a product of... |
| dgrmulc 26235 | Scalar multiplication by a... |
| dgrsub 26236 | The degree of a difference... |
| dgrcolem1 26237 | The degree of a compositio... |
| dgrcolem2 26238 | Lemma for ~ dgrco . (Cont... |
| dgrco 26239 | The degree of a compositio... |
| plycjlem 26240 | Lemma for ~ plycj and ~ co... |
| plycj 26241 | The double conjugation of ... |
| coecj 26242 | Double conjugation of a po... |
| plycjOLD 26243 | Obsolete version of ~ plyc... |
| coecjOLD 26244 | Obsolete version of ~ coec... |
| plyrecj 26245 | A polynomial with real coe... |
| plymul0or 26246 | Polynomial multiplication ... |
| ofmulrt 26247 | The set of roots of a prod... |
| plyreres 26248 | Real-coefficient polynomia... |
| dvply1 26249 | Derivative of a polynomial... |
| dvply2g 26250 | The derivative of a polyno... |
| dvply2gOLD 26251 | Obsolete version of ~ dvpl... |
| dvply2 26252 | The derivative of a polyno... |
| dvnply2 26253 | Polynomials have polynomia... |
| dvnply 26254 | Polynomials have polynomia... |
| plycpn 26255 | Polynomials are smooth. (... |
| quotval 26258 | Value of the quotient func... |
| plydivlem1 26259 | Lemma for ~ plydivalg . (... |
| plydivlem2 26260 | Lemma for ~ plydivalg . (... |
| plydivlem3 26261 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26262 | Lemma for ~ plydivex . In... |
| plydivex 26263 | Lemma for ~ plydivalg . (... |
| plydiveu 26264 | Lemma for ~ plydivalg . (... |
| plydivalg 26265 | The division algorithm on ... |
| quotlem 26266 | Lemma for properties of th... |
| quotcl 26267 | The quotient of two polyno... |
| quotcl2 26268 | Closure of the quotient fu... |
| quotdgr 26269 | Remainder property of the ... |
| plyremlem 26270 | Closure of a linear factor... |
| plyrem 26271 | The polynomial remainder t... |
| facth 26272 | The factor theorem. If a ... |
| fta1lem 26273 | Lemma for ~ fta1 . (Contr... |
| fta1 26274 | The easy direction of the ... |
| quotcan 26275 | Exact division with a mult... |
| vieta1lem1 26276 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26277 | Lemma for ~ vieta1 : induc... |
| vieta1 26278 | The first-order Vieta's fo... |
| plyexmo 26279 | An infinite set of values ... |
| elaa 26282 | Elementhood in the set of ... |
| aacn 26283 | An algebraic number is a c... |
| aasscn 26284 | The algebraic numbers are ... |
| elqaalem1 26285 | Lemma for ~ elqaa . The f... |
| elqaalem2 26286 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26287 | Lemma for ~ elqaa . (Cont... |
| elqaa 26288 | The set of numbers generat... |
| qaa 26289 | Every rational number is a... |
| qssaa 26290 | The rational numbers are c... |
| iaa 26291 | The imaginary unit is alge... |
| aareccl 26292 | The reciprocal of an algeb... |
| aacjcl 26293 | The conjugate of an algebr... |
| aannenlem1 26294 | Lemma for ~ aannen . (Con... |
| aannenlem2 26295 | Lemma for ~ aannen . (Con... |
| aannenlem3 26296 | The algebraic numbers are ... |
| aannen 26297 | The algebraic numbers are ... |
| aalioulem1 26298 | Lemma for ~ aaliou . An i... |
| aalioulem2 26299 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26300 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26301 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26302 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26303 | Lemma for ~ aaliou . (Con... |
| aaliou 26304 | Liouville's theorem on dio... |
| geolim3 26305 | Geometric series convergen... |
| aaliou2 26306 | Liouville's approximation ... |
| aaliou2b 26307 | Liouville's approximation ... |
| aaliou3lem1 26308 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26309 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26310 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26311 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26312 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26313 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26314 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26315 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26316 | Example of a "Liouville nu... |
| aaliou3 26317 | Example of a "Liouville nu... |
| taylfvallem1 26322 | Lemma for ~ taylfval . (C... |
| taylfvallem 26323 | Lemma for ~ taylfval . (C... |
| taylfval 26324 | Define the Taylor polynomi... |
| eltayl 26325 | Value of the Taylor series... |
| taylf 26326 | The Taylor series defines ... |
| tayl0 26327 | The Taylor series is alway... |
| taylplem1 26328 | Lemma for ~ taylpfval and ... |
| taylplem2 26329 | Lemma for ~ taylpfval and ... |
| taylpfval 26330 | Define the Taylor polynomi... |
| taylpf 26331 | The Taylor polynomial is a... |
| taylpval 26332 | Value of the Taylor polyno... |
| taylply2 26333 | The Taylor polynomial is a... |
| taylply2OLD 26334 | Obsolete version of ~ tayl... |
| taylply 26335 | The Taylor polynomial is a... |
| dvtaylp 26336 | The derivative of the Tayl... |
| dvntaylp 26337 | The ` M ` -th derivative o... |
| dvntaylp0 26338 | The first ` N ` derivative... |
| taylthlem1 26339 | Lemma for ~ taylth . This... |
| taylthlem2 26340 | Lemma for ~ taylth . (Con... |
| taylthlem2OLD 26341 | Obsolete version of ~ tayl... |
| taylth 26342 | Taylor's theorem. The Tay... |
| ulmrel 26345 | The uniform limit relation... |
| ulmscl 26346 | Closure of the base set in... |
| ulmval 26347 | Express the predicate: Th... |
| ulmcl 26348 | Closure of a uniform limit... |
| ulmf 26349 | Closure of a uniform limit... |
| ulmpm 26350 | Closure of a uniform limit... |
| ulmf2 26351 | Closure of a uniform limit... |
| ulm2 26352 | Simplify ~ ulmval when ` F... |
| ulmi 26353 | The uniform limit property... |
| ulmclm 26354 | A uniform limit of functio... |
| ulmres 26355 | A sequence of functions co... |
| ulmshftlem 26356 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26357 | A sequence of functions co... |
| ulm0 26358 | Every function converges u... |
| ulmuni 26359 | A sequence of functions un... |
| ulmdm 26360 | Two ways to express that a... |
| ulmcaulem 26361 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26362 | A sequence of functions co... |
| ulmcau2 26363 | A sequence of functions co... |
| ulmss 26364 | A uniform limit of functio... |
| ulmbdd 26365 | A uniform limit of bounded... |
| ulmcn 26366 | A uniform limit of continu... |
| ulmdvlem1 26367 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26368 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26369 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26370 | If ` F ` is a sequence of ... |
| mtest 26371 | The Weierstrass M-test. I... |
| mtestbdd 26372 | Given the hypotheses of th... |
| mbfulm 26373 | A uniform limit of measura... |
| iblulm 26374 | A uniform limit of integra... |
| itgulm 26375 | A uniform limit of integra... |
| itgulm2 26376 | A uniform limit of integra... |
| pserval 26377 | Value of the function ` G ... |
| pserval2 26378 | Value of the function ` G ... |
| psergf 26379 | The sequence of terms in t... |
| radcnvlem1 26380 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26381 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26382 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26383 | Zero is always a convergen... |
| radcnvcl 26384 | The radius of convergence ... |
| radcnvlt1 26385 | If ` X ` is within the ope... |
| radcnvlt2 26386 | If ` X ` is within the ope... |
| radcnvle 26387 | If ` X ` is a convergent p... |
| dvradcnv 26388 | The radius of convergence ... |
| pserulm 26389 | If ` S ` is a region conta... |
| psercn2 26390 | Since by ~ pserulm the ser... |
| psercn2OLD 26391 | Obsolete version of ~ pser... |
| psercnlem2 26392 | Lemma for ~ psercn . (Con... |
| psercnlem1 26393 | Lemma for ~ psercn . (Con... |
| psercn 26394 | An infinite series converg... |
| pserdvlem1 26395 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26396 | Lemma for ~ pserdv . (Con... |
| pserdv 26397 | The derivative of a power ... |
| pserdv2 26398 | The derivative of a power ... |
| abelthlem1 26399 | Lemma for ~ abelth . (Con... |
| abelthlem2 26400 | Lemma for ~ abelth . The ... |
| abelthlem3 26401 | Lemma for ~ abelth . (Con... |
| abelthlem4 26402 | Lemma for ~ abelth . (Con... |
| abelthlem5 26403 | Lemma for ~ abelth . (Con... |
| abelthlem6 26404 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26405 | Lemma for ~ abelth . (Con... |
| abelthlem7 26406 | Lemma for ~ abelth . (Con... |
| abelthlem8 26407 | Lemma for ~ abelth . (Con... |
| abelthlem9 26408 | Lemma for ~ abelth . By a... |
| abelth 26409 | Abel's theorem. If the po... |
| abelth2 26410 | Abel's theorem, restricted... |
| efcn 26411 | The exponential function i... |
| sincn 26412 | Sine is continuous. (Cont... |
| coscn 26413 | Cosine is continuous. (Co... |
| reeff1olem 26414 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26415 | The real exponential funct... |
| reefiso 26416 | The exponential function o... |
| efcvx 26417 | The exponential function o... |
| reefgim 26418 | The exponential function i... |
| pilem1 26419 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26420 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26421 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26422 | ` _pi ` is between 2 and 4... |
| sinpi 26423 | The sine of ` _pi ` is 0. ... |
| pire 26424 | ` _pi ` is a real number. ... |
| picn 26425 | ` _pi ` is a complex numbe... |
| pipos 26426 | ` _pi ` is positive. (Con... |
| pine0 26427 | ` _pi ` is nonzero. (Cont... |
| pirp 26428 | ` _pi ` is a positive real... |
| negpicn 26429 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26430 | Lemma for ~ sinhalfpi and ... |
| halfpire 26431 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26432 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26433 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26434 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26435 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26436 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26437 | The cosine of ` -u _pi / 2... |
| efhalfpi 26438 | The exponential of ` _i _p... |
| cospi 26439 | The cosine of ` _pi ` is `... |
| efipi 26440 | The exponential of ` _i x.... |
| eulerid 26441 | Euler's identity. (Contri... |
| sin2pi 26442 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26443 | The cosine of ` 2 _pi ` is... |
| ef2pi 26444 | The exponential of ` 2 _pi... |
| ef2kpi 26445 | If ` K ` is an integer, th... |
| efper 26446 | The exponential function i... |
| sinperlem 26447 | Lemma for ~ sinper and ~ c... |
| sinper 26448 | The sine function is perio... |
| cosper 26449 | The cosine function is per... |
| sin2kpi 26450 | If ` K ` is an integer, th... |
| cos2kpi 26451 | If ` K ` is an integer, th... |
| sin2pim 26452 | Sine of a number subtracte... |
| cos2pim 26453 | Cosine of a number subtrac... |
| sinmpi 26454 | Sine of a number less ` _p... |
| cosmpi 26455 | Cosine of a number less ` ... |
| sinppi 26456 | Sine of a number plus ` _p... |
| cosppi 26457 | Cosine of a number plus ` ... |
| efimpi 26458 | The exponential function a... |
| sinhalfpip 26459 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26460 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26461 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26462 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26463 | Ptolemy's Theorem. This t... |
| sincosq1lem 26464 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26465 | The signs of the sine and ... |
| sincosq2sgn 26466 | The signs of the sine and ... |
| sincosq3sgn 26467 | The signs of the sine and ... |
| sincosq4sgn 26468 | The signs of the sine and ... |
| coseq00topi 26469 | Location of the zeroes of ... |
| coseq0negpitopi 26470 | Location of the zeroes of ... |
| tanrpcl 26471 | Positive real closure of t... |
| tangtx 26472 | The tangent function is gr... |
| tanabsge 26473 | The tangent function is gr... |
| sinq12gt0 26474 | The sine of a number stric... |
| sinq12ge0 26475 | The sine of a number betwe... |
| sinq34lt0t 26476 | The sine of a number stric... |
| cosq14gt0 26477 | The cosine of a number str... |
| cosq14ge0 26478 | The cosine of a number bet... |
| sincosq1eq 26479 | Complementarity of the sin... |
| sincos4thpi 26480 | The sine and cosine of ` _... |
| tan4thpi 26481 | The tangent of ` _pi / 4 `... |
| tan4thpiOLD 26482 | Obsolete version of ~ tan4... |
| sincos6thpi 26483 | The sine and cosine of ` _... |
| sincos3rdpi 26484 | The sine and cosine of ` _... |
| pigt3 26485 | ` _pi ` is greater than 3.... |
| pige3 26486 | ` _pi ` is greater than or... |
| pige3ALT 26487 | Alternate proof of ~ pige3... |
| abssinper 26488 | The absolute value of sine... |
| sinkpi 26489 | The sine of an integer mul... |
| coskpi 26490 | The absolute value of the ... |
| sineq0 26491 | A complex number whose sin... |
| coseq1 26492 | A complex number whose cos... |
| cos02pilt1 26493 | Cosine is less than one be... |
| cosq34lt1 26494 | Cosine is less than one in... |
| efeq1 26495 | A complex number whose exp... |
| cosne0 26496 | The cosine function has no... |
| cosordlem 26497 | Lemma for ~ cosord . (Con... |
| cosord 26498 | Cosine is decreasing over ... |
| cos0pilt1 26499 | Cosine is between minus on... |
| cos11 26500 | Cosine is one-to-one over ... |
| sinord 26501 | Sine is increasing over th... |
| recosf1o 26502 | The cosine function is a b... |
| resinf1o 26503 | The sine function is a bij... |
| tanord1 26504 | The tangent function is st... |
| tanord 26505 | The tangent function is st... |
| tanregt0 26506 | The real part of the tange... |
| negpitopissre 26507 | The interval ` ( -u _pi (,... |
| efgh 26508 | The exponential function o... |
| efif1olem1 26509 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26510 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26511 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26512 | The exponential function o... |
| efif1o 26513 | The exponential function o... |
| efifo 26514 | The exponential function o... |
| eff1olem 26515 | The exponential function m... |
| eff1o 26516 | The exponential function m... |
| efabl 26517 | The image of a subgroup of... |
| efsubm 26518 | The image of a subgroup of... |
| circgrp 26519 | The circle group ` T ` is ... |
| circsubm 26520 | The circle group ` T ` is ... |
| logrn 26525 | The range of the natural l... |
| ellogrn 26526 | Write out the property ` A... |
| dflog2 26527 | The natural logarithm func... |
| relogrn 26528 | The range of the natural l... |
| logrncn 26529 | The range of the natural l... |
| eff1o2 26530 | The exponential function r... |
| logf1o 26531 | The natural logarithm func... |
| dfrelog 26532 | The natural logarithm func... |
| relogf1o 26533 | The natural logarithm func... |
| logrncl 26534 | Closure of the natural log... |
| logcl 26535 | Closure of the natural log... |
| logimcl 26536 | Closure of the imaginary p... |
| logcld 26537 | The logarithm of a nonzero... |
| logimcld 26538 | The imaginary part of the ... |
| logimclad 26539 | The imaginary part of the ... |
| abslogimle 26540 | The imaginary part of the ... |
| logrnaddcl 26541 | The range of the natural l... |
| relogcl 26542 | Closure of the natural log... |
| eflog 26543 | Relationship between the n... |
| logeq0im1 26544 | If the logarithm of a numb... |
| logccne0 26545 | The logarithm isn't 0 if i... |
| logne0 26546 | Logarithm of a non-1 posit... |
| reeflog 26547 | Relationship between the n... |
| logef 26548 | Relationship between the n... |
| relogef 26549 | Relationship between the n... |
| logeftb 26550 | Relationship between the n... |
| relogeftb 26551 | Relationship between the n... |
| log1 26552 | The natural logarithm of `... |
| loge 26553 | The natural logarithm of `... |
| logi 26554 | The natural logarithm of `... |
| logneg 26555 | The natural logarithm of a... |
| logm1 26556 | The natural logarithm of n... |
| lognegb 26557 | If a number has imaginary ... |
| relogoprlem 26558 | Lemma for ~ relogmul and ~... |
| relogmul 26559 | The natural logarithm of t... |
| relogdiv 26560 | The natural logarithm of t... |
| explog 26561 | Exponentiation of a nonzer... |
| reexplog 26562 | Exponentiation of a positi... |
| relogexp 26563 | The natural logarithm of p... |
| relog 26564 | Real part of a logarithm. ... |
| relogiso 26565 | The natural logarithm func... |
| reloggim 26566 | The natural logarithm is a... |
| logltb 26567 | The natural logarithm func... |
| logfac 26568 | The logarithm of a factori... |
| eflogeq 26569 | Solve an equation involvin... |
| logleb 26570 | Natural logarithm preserve... |
| rplogcl 26571 | Closure of the logarithm f... |
| logge0 26572 | The logarithm of a number ... |
| logcj 26573 | The natural logarithm dist... |
| efiarg 26574 | The exponential of the "ar... |
| cosargd 26575 | The cosine of the argument... |
| cosarg0d 26576 | The cosine of the argument... |
| argregt0 26577 | Closure of the argument of... |
| argrege0 26578 | Closure of the argument of... |
| argimgt0 26579 | Closure of the argument of... |
| argimlt0 26580 | Closure of the argument of... |
| logimul 26581 | Multiplying a number by ` ... |
| logneg2 26582 | The logarithm of the negat... |
| logmul2 26583 | Generalization of ~ relogm... |
| logdiv2 26584 | Generalization of ~ relogd... |
| abslogle 26585 | Bound on the magnitude of ... |
| tanarg 26586 | The basic relation between... |
| logdivlti 26587 | The ` log x / x ` function... |
| logdivlt 26588 | The ` log x / x ` function... |
| logdivle 26589 | The ` log x / x ` function... |
| relogcld 26590 | Closure of the natural log... |
| reeflogd 26591 | Relationship between the n... |
| relogmuld 26592 | The natural logarithm of t... |
| relogdivd 26593 | The natural logarithm of t... |
| logled 26594 | Natural logarithm preserve... |
| relogefd 26595 | Relationship between the n... |
| rplogcld 26596 | Closure of the logarithm f... |
| logge0d 26597 | The logarithm of a number ... |
| logge0b 26598 | The logarithm of a number ... |
| loggt0b 26599 | The logarithm of a number ... |
| logle1b 26600 | The logarithm of a number ... |
| loglt1b 26601 | The logarithm of a number ... |
| divlogrlim 26602 | The inverse logarithm func... |
| logno1 26603 | The logarithm function is ... |
| dvrelog 26604 | The derivative of the real... |
| relogcn 26605 | The real logarithm functio... |
| ellogdm 26606 | Elementhood in the "contin... |
| logdmn0 26607 | A number in the continuous... |
| logdmnrp 26608 | A number in the continuous... |
| logdmss 26609 | The continuity domain of `... |
| logcnlem2 26610 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26611 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26612 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26613 | Lemma for ~ logcn . (Cont... |
| logcn 26614 | The logarithm function is ... |
| dvloglem 26615 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26616 | The "continuous domain" of... |
| logf1o2 26617 | The logarithm maps its con... |
| dvlog 26618 | The derivative of the comp... |
| dvlog2lem 26619 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26620 | The derivative of the comp... |
| advlog 26621 | The antiderivative of the ... |
| advlogexp 26622 | The antiderivative of a po... |
| efopnlem1 26623 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26624 | Lemma for ~ efopn . (Cont... |
| efopn 26625 | The exponential map is an ... |
| logtayllem 26626 | Lemma for ~ logtayl . (Co... |
| logtayl 26627 | The Taylor series for ` -u... |
| logtaylsum 26628 | The Taylor series for ` -u... |
| logtayl2 26629 | Power series expression fo... |
| logccv 26630 | The natural logarithm func... |
| cxpval 26631 | Value of the complex power... |
| cxpef 26632 | Value of the complex power... |
| 0cxp 26633 | Value of the complex power... |
| cxpexpz 26634 | Relate the complex power f... |
| cxpexp 26635 | Relate the complex power f... |
| logcxp 26636 | Logarithm of a complex pow... |
| cxp0 26637 | Value of the complex power... |
| cxp1 26638 | Value of the complex power... |
| 1cxp 26639 | Value of the complex power... |
| ecxp 26640 | Write the exponential func... |
| cxpcl 26641 | Closure of the complex pow... |
| recxpcl 26642 | Real closure of the comple... |
| rpcxpcl 26643 | Positive real closure of t... |
| cxpne0 26644 | Complex exponentiation is ... |
| cxpeq0 26645 | Complex exponentiation is ... |
| cxpadd 26646 | Sum of exponents law for c... |
| cxpp1 26647 | Value of a nonzero complex... |
| cxpneg 26648 | Value of a complex number ... |
| cxpsub 26649 | Exponent subtraction law f... |
| cxpge0 26650 | Nonnegative exponentiation... |
| mulcxplem 26651 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26652 | Complex exponentiation of ... |
| cxprec 26653 | Complex exponentiation of ... |
| divcxp 26654 | Complex exponentiation of ... |
| cxpmul 26655 | Product of exponents law f... |
| cxpmul2 26656 | Product of exponents law f... |
| cxproot 26657 | The complex power function... |
| cxpmul2z 26658 | Generalize ~ cxpmul2 to ne... |
| abscxp 26659 | Absolute value of a power,... |
| abscxp2 26660 | Absolute value of a power,... |
| cxplt 26661 | Ordering property for comp... |
| cxple 26662 | Ordering property for comp... |
| cxplea 26663 | Ordering property for comp... |
| cxple2 26664 | Ordering property for comp... |
| cxplt2 26665 | Ordering property for comp... |
| cxple2a 26666 | Ordering property for comp... |
| cxplt3 26667 | Ordering property for comp... |
| cxple3 26668 | Ordering property for comp... |
| cxpsqrtlem 26669 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26670 | The complex exponential fu... |
| logsqrt 26671 | Logarithm of a square root... |
| cxp0d 26672 | Value of the complex power... |
| cxp1d 26673 | Value of the complex power... |
| 1cxpd 26674 | Value of the complex power... |
| cxpcld 26675 | Closure of the complex pow... |
| cxpmul2d 26676 | Product of exponents law f... |
| 0cxpd 26677 | Value of the complex power... |
| cxpexpzd 26678 | Relate the complex power f... |
| cxpefd 26679 | Value of the complex power... |
| cxpne0d 26680 | Complex exponentiation is ... |
| cxpp1d 26681 | Value of a nonzero complex... |
| cxpnegd 26682 | Value of a complex number ... |
| cxpmul2zd 26683 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26684 | Sum of exponents law for c... |
| cxpsubd 26685 | Exponent subtraction law f... |
| cxpltd 26686 | Ordering property for comp... |
| cxpled 26687 | Ordering property for comp... |
| cxplead 26688 | Ordering property for comp... |
| divcxpd 26689 | Complex exponentiation of ... |
| recxpcld 26690 | Positive real closure of t... |
| cxpge0d 26691 | Nonnegative exponentiation... |
| cxple2ad 26692 | Ordering property for comp... |
| cxplt2d 26693 | Ordering property for comp... |
| cxple2d 26694 | Ordering property for comp... |
| mulcxpd 26695 | Complex exponentiation of ... |
| recxpf1lem 26696 | Complex exponentiation on ... |
| cxpsqrtth 26697 | Square root theorem over t... |
| 2irrexpq 26698 | There exist irrational num... |
| cxprecd 26699 | Complex exponentiation of ... |
| rpcxpcld 26700 | Positive real closure of t... |
| logcxpd 26701 | Logarithm of a complex pow... |
| cxplt3d 26702 | Ordering property for comp... |
| cxple3d 26703 | Ordering property for comp... |
| cxpmuld 26704 | Product of exponents law f... |
| cxpgt0d 26705 | A positive real raised to ... |
| cxpcom 26706 | Commutative law for real e... |
| dvcxp1 26707 | The derivative of a comple... |
| dvcxp2 26708 | The derivative of a comple... |
| dvsqrt 26709 | The derivative of the real... |
| dvcncxp1 26710 | Derivative of complex powe... |
| dvcnsqrt 26711 | Derivative of square root ... |
| cxpcn 26712 | Domain of continuity of th... |
| cxpcnOLD 26713 | Obsolete version of ~ cxpc... |
| cxpcn2 26714 | Continuity of the complex ... |
| cxpcn3lem 26715 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26716 | Extend continuity of the c... |
| resqrtcn 26717 | Continuity of the real squ... |
| sqrtcn 26718 | Continuity of the square r... |
| cxpaddlelem 26719 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26720 | Ordering property for comp... |
| abscxpbnd 26721 | Bound on the absolute valu... |
| root1id 26722 | Property of an ` N ` -th r... |
| root1eq1 26723 | The only powers of an ` N ... |
| root1cj 26724 | Within the ` N ` -th roots... |
| cxpeq 26725 | Solve an equation involvin... |
| zrtelqelz 26726 | If the ` N ` -th root of a... |
| zrtdvds 26727 | A positive integer root di... |
| rtprmirr 26728 | The root of a prime number... |
| loglesqrt 26729 | An upper bound on the loga... |
| logreclem 26730 | Symmetry of the natural lo... |
| logrec 26731 | Logarithm of a reciprocal ... |
| logbval 26734 | Define the value of the ` ... |
| logbcl 26735 | General logarithm closure.... |
| logbid1 26736 | General logarithm is 1 whe... |
| logb1 26737 | The logarithm of ` 1 ` to ... |
| elogb 26738 | The general logarithm of a... |
| logbchbase 26739 | Change of base for logarit... |
| relogbval 26740 | Value of the general logar... |
| relogbcl 26741 | Closure of the general log... |
| relogbzcl 26742 | Closure of the general log... |
| relogbreexp 26743 | Power law for the general ... |
| relogbzexp 26744 | Power law for the general ... |
| relogbmul 26745 | The logarithm of the produ... |
| relogbmulexp 26746 | The logarithm of the produ... |
| relogbdiv 26747 | The logarithm of the quoti... |
| relogbexp 26748 | Identity law for general l... |
| nnlogbexp 26749 | Identity law for general l... |
| logbrec 26750 | Logarithm of a reciprocal ... |
| logbleb 26751 | The general logarithm func... |
| logblt 26752 | The general logarithm func... |
| relogbcxp 26753 | Identity law for the gener... |
| cxplogb 26754 | Identity law for the gener... |
| relogbcxpb 26755 | The logarithm is the inver... |
| logbmpt 26756 | The general logarithm to a... |
| logbf 26757 | The general logarithm to a... |
| logbfval 26758 | The general logarithm of a... |
| relogbf 26759 | The general logarithm to a... |
| logblog 26760 | The general logarithm to t... |
| logbgt0b 26761 | The logarithm of a positiv... |
| logbgcd1irr 26762 | The logarithm of an intege... |
| 2logb9irr 26763 | Example for ~ logbgcd1irr ... |
| logbprmirr 26764 | The logarithm of a prime t... |
| 2logb3irr 26765 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26766 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26767 | The square root of two to ... |
| 2irrexpqALT 26768 | Alternate proof of ~ 2irre... |
| angval 26769 | Define the angle function,... |
| angcan 26770 | Cancel a constant multipli... |
| angneg 26771 | Cancel a negative sign in ... |
| angvald 26772 | The (signed) angle between... |
| angcld 26773 | The (signed) angle between... |
| angrteqvd 26774 | Two vectors are at a right... |
| cosangneg2d 26775 | The cosine of the angle be... |
| angrtmuld 26776 | Perpendicularity of two ve... |
| ang180lem1 26777 | Lemma for ~ ang180 . Show... |
| ang180lem2 26778 | Lemma for ~ ang180 . Show... |
| ang180lem3 26779 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26780 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26781 | Lemma for ~ ang180 : Redu... |
| ang180 26782 | The sum of angles ` m A B ... |
| lawcoslem1 26783 | Lemma for ~ lawcos . Here... |
| lawcos 26784 | Law of cosines (also known... |
| pythag 26785 | Pythagorean theorem. Give... |
| isosctrlem1 26786 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26787 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26788 | Lemma for ~ isosctr . Cor... |
| isosctr 26789 | Isosceles triangle theorem... |
| ssscongptld 26790 | If two triangles have equa... |
| affineequiv 26791 | Equivalence between two wa... |
| affineequiv2 26792 | Equivalence between two wa... |
| affineequiv3 26793 | Equivalence between two wa... |
| affineequiv4 26794 | Equivalence between two wa... |
| affineequivne 26795 | Equivalence between two wa... |
| angpieqvdlem 26796 | Equivalence used in the pr... |
| angpieqvdlem2 26797 | Equivalence used in ~ angp... |
| angpined 26798 | If the angle at ABC is ` _... |
| angpieqvd 26799 | The angle ABC is ` _pi ` i... |
| chordthmlem 26800 | If ` M ` is the midpoint o... |
| chordthmlem2 26801 | If M is the midpoint of AB... |
| chordthmlem3 26802 | If M is the midpoint of AB... |
| chordthmlem4 26803 | If P is on the segment AB ... |
| chordthmlem5 26804 | If P is on the segment AB ... |
| chordthm 26805 | The intersecting chords th... |
| heron 26806 | Heron's formula gives the ... |
| quad2 26807 | The quadratic equation, wi... |
| quad 26808 | The quadratic equation. (... |
| 1cubrlem 26809 | The cube roots of unity. ... |
| 1cubr 26810 | The cube roots of unity. ... |
| dcubic1lem 26811 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26812 | Reverse direction of ~ dcu... |
| dcubic1 26813 | Forward direction of ~ dcu... |
| dcubic 26814 | Solutions to the depressed... |
| mcubic 26815 | Solutions to a monic cubic... |
| cubic2 26816 | The solution to the genera... |
| cubic 26817 | The cubic equation, which ... |
| binom4 26818 | Work out a quartic binomia... |
| dquartlem1 26819 | Lemma for ~ dquart . (Con... |
| dquartlem2 26820 | Lemma for ~ dquart . (Con... |
| dquart 26821 | Solve a depressed quartic ... |
| quart1cl 26822 | Closure lemmas for ~ quart... |
| quart1lem 26823 | Lemma for ~ quart1 . (Con... |
| quart1 26824 | Depress a quartic equation... |
| quartlem1 26825 | Lemma for ~ quart . (Cont... |
| quartlem2 26826 | Closure lemmas for ~ quart... |
| quartlem3 26827 | Closure lemmas for ~ quart... |
| quartlem4 26828 | Closure lemmas for ~ quart... |
| quart 26829 | The quartic equation, writ... |
| asinlem 26836 | The argument to the logari... |
| asinlem2 26837 | The argument to the logari... |
| asinlem3a 26838 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26839 | The argument to the logari... |
| asinf 26840 | Domain and codomain of the... |
| asincl 26841 | Closure for the arcsin fun... |
| acosf 26842 | Domain and codoamin of the... |
| acoscl 26843 | Closure for the arccos fun... |
| atandm 26844 | Since the property is a li... |
| atandm2 26845 | This form of ~ atandm is a... |
| atandm3 26846 | A compact form of ~ atandm... |
| atandm4 26847 | A compact form of ~ atandm... |
| atanf 26848 | Domain and codoamin of the... |
| atancl 26849 | Closure for the arctan fun... |
| asinval 26850 | Value of the arcsin functi... |
| acosval 26851 | Value of the arccos functi... |
| atanval 26852 | Value of the arctan functi... |
| atanre 26853 | A real number is in the do... |
| asinneg 26854 | The arcsine function is od... |
| acosneg 26855 | The negative symmetry rela... |
| efiasin 26856 | The exponential of the arc... |
| sinasin 26857 | The arcsine function is an... |
| cosacos 26858 | The arccosine function is ... |
| asinsinlem 26859 | Lemma for ~ asinsin . (Co... |
| asinsin 26860 | The arcsine function compo... |
| acoscos 26861 | The arccosine function is ... |
| asin1 26862 | The arcsine of ` 1 ` is ` ... |
| acos1 26863 | The arccosine of ` 1 ` is ... |
| reasinsin 26864 | The arcsine function compo... |
| asinsinb 26865 | Relationship between sine ... |
| acoscosb 26866 | Relationship between cosin... |
| asinbnd 26867 | The arcsine function has r... |
| acosbnd 26868 | The arccosine function has... |
| asinrebnd 26869 | Bounds on the arcsine func... |
| asinrecl 26870 | The arcsine function is re... |
| acosrecl 26871 | The arccosine function is ... |
| cosasin 26872 | The cosine of the arcsine ... |
| sinacos 26873 | The sine of the arccosine ... |
| atandmneg 26874 | The domain of the arctange... |
| atanneg 26875 | The arctangent function is... |
| atan0 26876 | The arctangent of zero is ... |
| atandmcj 26877 | The arctangent function di... |
| atancj 26878 | The arctangent function di... |
| atanrecl 26879 | The arctangent function is... |
| efiatan 26880 | Value of the exponential o... |
| atanlogaddlem 26881 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26882 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26883 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26884 | A variation on ~ atanlogad... |
| efiatan2 26885 | Value of the exponential o... |
| 2efiatan 26886 | Value of the exponential o... |
| tanatan 26887 | The arctangent function is... |
| atandmtan 26888 | The tangent function has r... |
| cosatan 26889 | The cosine of an arctangen... |
| cosatanne0 26890 | The arctangent function ha... |
| atantan 26891 | The arctangent function is... |
| atantanb 26892 | Relationship between tange... |
| atanbndlem 26893 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26894 | The arctangent function is... |
| atanord 26895 | The arctangent function is... |
| atan1 26896 | The arctangent of ` 1 ` is... |
| bndatandm 26897 | A point in the open unit d... |
| atans 26898 | The "domain of continuity"... |
| atans2 26899 | It suffices to show that `... |
| atansopn 26900 | The domain of continuity o... |
| atansssdm 26901 | The domain of continuity o... |
| ressatans 26902 | The real number line is a ... |
| dvatan 26903 | The derivative of the arct... |
| atancn 26904 | The arctangent is a contin... |
| atantayl 26905 | The Taylor series for ` ar... |
| atantayl2 26906 | The Taylor series for ` ar... |
| atantayl3 26907 | The Taylor series for ` ar... |
| leibpilem1 26908 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26909 | The Leibniz formula for ` ... |
| leibpi 26910 | The Leibniz formula for ` ... |
| leibpisum 26911 | The Leibniz formula for ` ... |
| log2cnv 26912 | Using the Taylor series fo... |
| log2tlbnd 26913 | Bound the error term in th... |
| log2ublem1 26914 | Lemma for ~ log2ub . The ... |
| log2ublem2 26915 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26916 | Lemma for ~ log2ub . In d... |
| log2ub 26917 | ` log 2 ` is less than ` 2... |
| log2le1 26918 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26919 | Lemma for ~ birthday . (C... |
| birthdaylem2 26920 | For general ` N ` and ` K ... |
| birthdaylem3 26921 | For general ` N ` and ` K ... |
| birthday 26922 | The Birthday Problem. The... |
| dmarea 26925 | The domain of the area fun... |
| areambl 26926 | The fibers of a measurable... |
| areass 26927 | A measurable region is a s... |
| dfarea 26928 | Rewrite ~ df-area self-ref... |
| areaf 26929 | Area measurement is a func... |
| areacl 26930 | The area of a measurable r... |
| areage0 26931 | The area of a measurable r... |
| areaval 26932 | The area of a measurable r... |
| rlimcnp 26933 | Relate a limit of a real-v... |
| rlimcnp2 26934 | Relate a limit of a real-v... |
| rlimcnp3 26935 | Relate a limit of a real-v... |
| xrlimcnp 26936 | Relate a limit of a real-v... |
| efrlim 26937 | The limit of the sequence ... |
| efrlimOLD 26938 | Obsolete version of ~ efrl... |
| dfef2 26939 | The limit of the sequence ... |
| cxplim 26940 | A power to a negative expo... |
| sqrtlim 26941 | The inverse square root fu... |
| rlimcxp 26942 | Any power to a positive ex... |
| o1cxp 26943 | An eventually bounded func... |
| cxp2limlem 26944 | A linear factor grows slow... |
| cxp2lim 26945 | Any power grows slower tha... |
| cxploglim 26946 | The logarithm grows slower... |
| cxploglim2 26947 | Every power of the logarit... |
| divsqrtsumlem 26948 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26949 | The function ` F ` used in... |
| divsqrsum 26950 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26951 | A bound on the distance of... |
| divsqrtsumo1 26952 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26953 | Closure of a 0-1 linear co... |
| scvxcvx 26954 | A strictly convex function... |
| jensenlem1 26955 | Lemma for ~ jensen . (Con... |
| jensenlem2 26956 | Lemma for ~ jensen . (Con... |
| jensen 26957 | Jensen's inequality, a fin... |
| amgmlem 26958 | Lemma for ~ amgm . (Contr... |
| amgm 26959 | Inequality of arithmetic a... |
| logdifbnd 26962 | Bound on the difference of... |
| logdiflbnd 26963 | Lower bound on the differe... |
| emcllem1 26964 | Lemma for ~ emcl . The se... |
| emcllem2 26965 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26966 | Lemma for ~ emcl . The fu... |
| emcllem4 26967 | Lemma for ~ emcl . The di... |
| emcllem5 26968 | Lemma for ~ emcl . The pa... |
| emcllem6 26969 | Lemma for ~ emcl . By the... |
| emcllem7 26970 | Lemma for ~ emcl and ~ har... |
| emcl 26971 | Closure and bounds for the... |
| harmonicbnd 26972 | A bound on the harmonic se... |
| harmonicbnd2 26973 | A bound on the harmonic se... |
| emre 26974 | The Euler-Mascheroni const... |
| emgt0 26975 | The Euler-Mascheroni const... |
| harmonicbnd3 26976 | A bound on the harmonic se... |
| harmoniclbnd 26977 | A bound on the harmonic se... |
| harmonicubnd 26978 | A bound on the harmonic se... |
| harmonicbnd4 26979 | The asymptotic behavior of... |
| fsumharmonic 26980 | Bound a finite sum based o... |
| zetacvg 26983 | The zeta series is converg... |
| eldmgm 26990 | Elementhood in the set of ... |
| dmgmaddn0 26991 | If ` A ` is not a nonposit... |
| dmlogdmgm 26992 | If ` A ` is in the continu... |
| rpdmgm 26993 | A positive real number is ... |
| dmgmn0 26994 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26995 | If ` A ` is not a nonposit... |
| dmgmdivn0 26996 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26997 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26998 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26999 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 27000 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 27001 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 27002 | The series ` G ` is unifor... |
| lgamgulm 27003 | The series ` G ` is unifor... |
| lgamgulm2 27004 | Rewrite the limit of the s... |
| lgambdd 27005 | The log-Gamma function is ... |
| lgamucov 27006 | The ` U ` regions used in ... |
| lgamucov2 27007 | The ` U ` regions used in ... |
| lgamcvglem 27008 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27009 | The log-Gamma function is ... |
| lgamf 27010 | The log-Gamma function is ... |
| gamf 27011 | The Gamma function is a co... |
| gamcl 27012 | The exponential of the log... |
| eflgam 27013 | The exponential of the log... |
| gamne0 27014 | The Gamma function is neve... |
| igamval 27015 | Value of the inverse Gamma... |
| igamz 27016 | Value of the inverse Gamma... |
| igamgam 27017 | Value of the inverse Gamma... |
| igamlgam 27018 | Value of the inverse Gamma... |
| igamf 27019 | Closure of the inverse Gam... |
| igamcl 27020 | Closure of the inverse Gam... |
| gamigam 27021 | The Gamma function is the ... |
| lgamcvg 27022 | The series ` G ` converges... |
| lgamcvg2 27023 | The series ` G ` converges... |
| gamcvg 27024 | The pointwise exponential ... |
| lgamp1 27025 | The functional equation of... |
| gamp1 27026 | The functional equation of... |
| gamcvg2lem 27027 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27028 | An infinite product expres... |
| regamcl 27029 | The Gamma function is real... |
| relgamcl 27030 | The log-Gamma function is ... |
| rpgamcl 27031 | The log-Gamma function is ... |
| lgam1 27032 | The log-Gamma function at ... |
| gam1 27033 | The log-Gamma function at ... |
| facgam 27034 | The Gamma function general... |
| gamfac 27035 | The Gamma function general... |
| wilthlem1 27036 | The only elements that are... |
| wilthlem2 27037 | Lemma for ~ wilth : induct... |
| wilthlem3 27038 | Lemma for ~ wilth . Here ... |
| wilth 27039 | Wilson's theorem. A numbe... |
| wilthimp 27040 | The forward implication of... |
| ftalem1 27041 | Lemma for ~ fta : "growth... |
| ftalem2 27042 | Lemma for ~ fta . There e... |
| ftalem3 27043 | Lemma for ~ fta . There e... |
| ftalem4 27044 | Lemma for ~ fta : Closure... |
| ftalem5 27045 | Lemma for ~ fta : Main pr... |
| ftalem6 27046 | Lemma for ~ fta : Dischar... |
| ftalem7 27047 | Lemma for ~ fta . Shift t... |
| fta 27048 | The Fundamental Theorem of... |
| basellem1 27049 | Lemma for ~ basel . Closu... |
| basellem2 27050 | Lemma for ~ basel . Show ... |
| basellem3 27051 | Lemma for ~ basel . Using... |
| basellem4 27052 | Lemma for ~ basel . By ~ ... |
| basellem5 27053 | Lemma for ~ basel . Using... |
| basellem6 27054 | Lemma for ~ basel . The f... |
| basellem7 27055 | Lemma for ~ basel . The f... |
| basellem8 27056 | Lemma for ~ basel . The f... |
| basellem9 27057 | Lemma for ~ basel . Since... |
| basel 27058 | The sum of the inverse squ... |
| efnnfsumcl 27071 | Finite sum closure in the ... |
| ppisval 27072 | The set of primes less tha... |
| ppisval2 27073 | The set of primes less tha... |
| ppifi 27074 | The set of primes less tha... |
| prmdvdsfi 27075 | The set of prime divisors ... |
| chtf 27076 | Domain and codoamin of the... |
| chtcl 27077 | Real closure of the Chebys... |
| chtval 27078 | Value of the Chebyshev fun... |
| efchtcl 27079 | The Chebyshev function is ... |
| chtge0 27080 | The Chebyshev function is ... |
| vmaval 27081 | Value of the von Mangoldt ... |
| isppw 27082 | Two ways to say that ` A `... |
| isppw2 27083 | Two ways to say that ` A `... |
| vmappw 27084 | Value of the von Mangoldt ... |
| vmaprm 27085 | Value of the von Mangoldt ... |
| vmacl 27086 | Closure for the von Mangol... |
| vmaf 27087 | Functionality of the von M... |
| efvmacl 27088 | The von Mangoldt is closed... |
| vmage0 27089 | The von Mangoldt function ... |
| chpval 27090 | Value of the second Chebys... |
| chpf 27091 | Functionality of the secon... |
| chpcl 27092 | Closure for the second Che... |
| efchpcl 27093 | The second Chebyshev funct... |
| chpge0 27094 | The second Chebyshev funct... |
| ppival 27095 | Value of the prime-countin... |
| ppival2 27096 | Value of the prime-countin... |
| ppival2g 27097 | Value of the prime-countin... |
| ppif 27098 | Domain and codomain of the... |
| ppicl 27099 | Real closure of the prime-... |
| muval 27100 | The value of the Möbi... |
| muval1 27101 | The value of the Möbi... |
| muval2 27102 | The value of the Möbi... |
| isnsqf 27103 | Two ways to say that a num... |
| issqf 27104 | Two ways to say that a num... |
| sqfpc 27105 | The prime count of a squar... |
| dvdssqf 27106 | A divisor of a squarefree ... |
| sqf11 27107 | A squarefree number is com... |
| muf 27108 | The Möbius function i... |
| mucl 27109 | Closure of the Möbius... |
| sgmval 27110 | The value of the divisor f... |
| sgmval2 27111 | The value of the divisor f... |
| 0sgm 27112 | The value of the sum-of-di... |
| sgmf 27113 | The divisor function is a ... |
| sgmcl 27114 | Closure of the divisor fun... |
| sgmnncl 27115 | Closure of the divisor fun... |
| mule1 27116 | The Möbius function t... |
| chtfl 27117 | The Chebyshev function doe... |
| chpfl 27118 | The second Chebyshev funct... |
| ppiprm 27119 | The prime-counting functio... |
| ppinprm 27120 | The prime-counting functio... |
| chtprm 27121 | The Chebyshev function at ... |
| chtnprm 27122 | The Chebyshev function at ... |
| chpp1 27123 | The second Chebyshev funct... |
| chtwordi 27124 | The Chebyshev function is ... |
| chpwordi 27125 | The second Chebyshev funct... |
| chtdif 27126 | The difference of the Cheb... |
| efchtdvds 27127 | The exponentiated Chebyshe... |
| ppifl 27128 | The prime-counting functio... |
| ppip1le 27129 | The prime-counting functio... |
| ppiwordi 27130 | The prime-counting functio... |
| ppidif 27131 | The difference of the prim... |
| ppi1 27132 | The prime-counting functio... |
| cht1 27133 | The Chebyshev function at ... |
| vma1 27134 | The von Mangoldt function ... |
| chp1 27135 | The second Chebyshev funct... |
| ppi1i 27136 | Inference form of ~ ppiprm... |
| ppi2i 27137 | Inference form of ~ ppinpr... |
| ppi2 27138 | The prime-counting functio... |
| ppi3 27139 | The prime-counting functio... |
| cht2 27140 | The Chebyshev function at ... |
| cht3 27141 | The Chebyshev function at ... |
| ppinncl 27142 | Closure of the prime-count... |
| chtrpcl 27143 | Closure of the Chebyshev f... |
| ppieq0 27144 | The prime-counting functio... |
| ppiltx 27145 | The prime-counting functio... |
| prmorcht 27146 | Relate the primorial (prod... |
| mumullem1 27147 | Lemma for ~ mumul . A mul... |
| mumullem2 27148 | Lemma for ~ mumul . The p... |
| mumul 27149 | The Möbius function i... |
| sqff1o 27150 | There is a bijection from ... |
| fsumdvdsdiaglem 27151 | A "diagonal commutation" o... |
| fsumdvdsdiag 27152 | A "diagonal commutation" o... |
| fsumdvdscom 27153 | A double commutation of di... |
| dvdsppwf1o 27154 | A bijection between the di... |
| dvdsflf1o 27155 | A bijection from the numbe... |
| dvdsflsumcom 27156 | A sum commutation from ` s... |
| fsumfldivdiaglem 27157 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27158 | The right-hand side of ~ d... |
| musum 27159 | The sum of the Möbius... |
| musumsum 27160 | Evaluate a collapsing sum ... |
| muinv 27161 | The Möbius inversion ... |
| mpodvdsmulf1o 27162 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27163 | Product of two divisor sum... |
| dvdsmulf1o 27164 | If ` M ` and ` N ` are two... |
| fsumdvdsmulOLD 27165 | Obsolete version of ~ fsum... |
| sgmppw 27166 | The value of the divisor f... |
| 0sgmppw 27167 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27168 | The sum of divisors for a ... |
| 1sgm2ppw 27169 | The sum of the divisors of... |
| sgmmul 27170 | The divisor function for f... |
| ppiublem1 27171 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27172 | A prime greater than ` 3 `... |
| ppiub 27173 | An upper bound on the prim... |
| vmalelog 27174 | The von Mangoldt function ... |
| chtlepsi 27175 | The first Chebyshev functi... |
| chprpcl 27176 | Closure of the second Cheb... |
| chpeq0 27177 | The second Chebyshev funct... |
| chteq0 27178 | The first Chebyshev functi... |
| chtleppi 27179 | Upper bound on the ` theta... |
| chtublem 27180 | Lemma for ~ chtub . (Cont... |
| chtub 27181 | An upper bound on the Cheb... |
| fsumvma 27182 | Rewrite a sum over the von... |
| fsumvma2 27183 | Apply ~ fsumvma for the co... |
| pclogsum 27184 | The logarithmic analogue o... |
| vmasum 27185 | The sum of the von Mangold... |
| logfac2 27186 | Another expression for the... |
| chpval2 27187 | Express the second Chebysh... |
| chpchtsum 27188 | The second Chebyshev funct... |
| chpub 27189 | An upper bound on the seco... |
| logfacubnd 27190 | A simple upper bound on th... |
| logfaclbnd 27191 | A lower bound on the logar... |
| logfacbnd3 27192 | Show the stronger statemen... |
| logfacrlim 27193 | Combine the estimates ~ lo... |
| logexprlim 27194 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27195 | Write out ~ logfacrlim as ... |
| mersenne 27196 | A Mersenne prime is a prim... |
| perfect1 27197 | Euclid's contribution to t... |
| perfectlem1 27198 | Lemma for ~ perfect . (Co... |
| perfectlem2 27199 | Lemma for ~ perfect . (Co... |
| perfect 27200 | The Euclid-Euler theorem, ... |
| dchrval 27203 | Value of the group of Diri... |
| dchrbas 27204 | Base set of the group of D... |
| dchrelbas 27205 | A Dirichlet character is a... |
| dchrelbas2 27206 | A Dirichlet character is a... |
| dchrelbas3 27207 | A Dirichlet character is a... |
| dchrelbasd 27208 | A Dirichlet character is a... |
| dchrrcl 27209 | Reverse closure for a Diri... |
| dchrmhm 27210 | A Dirichlet character is a... |
| dchrf 27211 | A Dirichlet character is a... |
| dchrelbas4 27212 | A Dirichlet character is a... |
| dchrzrh1 27213 | Value of a Dirichlet chara... |
| dchrzrhcl 27214 | A Dirichlet character take... |
| dchrzrhmul 27215 | A Dirichlet character is c... |
| dchrplusg 27216 | Group operation on the gro... |
| dchrmul 27217 | Group operation on the gro... |
| dchrmulcl 27218 | Closure of the group opera... |
| dchrn0 27219 | A Dirichlet character is n... |
| dchr1cl 27220 | Closure of the principal D... |
| dchrmullid 27221 | Left identity for the prin... |
| dchrinvcl 27222 | Closure of the group inver... |
| dchrabl 27223 | The set of Dirichlet chara... |
| dchrfi 27224 | The group of Dirichlet cha... |
| dchrghm 27225 | A Dirichlet character rest... |
| dchr1 27226 | Value of the principal Dir... |
| dchreq 27227 | A Dirichlet character is d... |
| dchrresb 27228 | A Dirichlet character is d... |
| dchrabs 27229 | A Dirichlet character take... |
| dchrinv 27230 | The inverse of a Dirichlet... |
| dchrabs2 27231 | A Dirichlet character take... |
| dchr1re 27232 | The principal Dirichlet ch... |
| dchrptlem1 27233 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27234 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27235 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27236 | For any element other than... |
| dchrsum2 27237 | An orthogonality relation ... |
| dchrsum 27238 | An orthogonality relation ... |
| sumdchr2 27239 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27240 | There are exactly ` phi ( ... |
| sumdchr 27241 | An orthogonality relation ... |
| dchr2sum 27242 | An orthogonality relation ... |
| sum2dchr 27243 | An orthogonality relation ... |
| bcctr 27244 | Value of the central binom... |
| pcbcctr 27245 | Prime count of a central b... |
| bcmono 27246 | The binomial coefficient i... |
| bcmax 27247 | The binomial coefficient t... |
| bcp1ctr 27248 | Ratio of two central binom... |
| bclbnd 27249 | A bound on the binomial co... |
| efexple 27250 | Convert a bound on a power... |
| bpos1lem 27251 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27252 | Bertrand's postulate, chec... |
| bposlem1 27253 | An upper bound on the prim... |
| bposlem2 27254 | There are no odd primes in... |
| bposlem3 27255 | Lemma for ~ bpos . Since ... |
| bposlem4 27256 | Lemma for ~ bpos . (Contr... |
| bposlem5 27257 | Lemma for ~ bpos . Bound ... |
| bposlem6 27258 | Lemma for ~ bpos . By usi... |
| bposlem7 27259 | Lemma for ~ bpos . The fu... |
| bposlem8 27260 | Lemma for ~ bpos . Evalua... |
| bposlem9 27261 | Lemma for ~ bpos . Derive... |
| bpos 27262 | Bertrand's postulate: ther... |
| zabsle1 27265 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27266 | When ` a ` is coprime to t... |
| lgslem2 27267 | The set ` Z ` of all integ... |
| lgslem3 27268 | The set ` Z ` of all integ... |
| lgslem4 27269 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27270 | Value of the Legendre symb... |
| lgsfval 27271 | Value of the function ` F ... |
| lgsfcl2 27272 | The function ` F ` is clos... |
| lgscllem 27273 | The Legendre symbol is an ... |
| lgsfcl 27274 | Closure of the function ` ... |
| lgsfle1 27275 | The function ` F ` has mag... |
| lgsval2lem 27276 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27277 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27278 | The Legendre symbol is an ... |
| lgs0 27279 | The Legendre symbol when t... |
| lgscl 27280 | The Legendre symbol is an ... |
| lgsle1 27281 | The Legendre symbol has ab... |
| lgsval2 27282 | The Legendre symbol at a p... |
| lgs2 27283 | The Legendre symbol at ` 2... |
| lgsval3 27284 | The Legendre symbol at an ... |
| lgsvalmod 27285 | The Legendre symbol is equ... |
| lgsval4 27286 | Restate ~ lgsval for nonze... |
| lgsfcl3 27287 | Closure of the function ` ... |
| lgsval4a 27288 | Same as ~ lgsval4 for posi... |
| lgscl1 27289 | The value of the Legendre ... |
| lgsneg 27290 | The Legendre symbol is eit... |
| lgsneg1 27291 | The Legendre symbol for no... |
| lgsmod 27292 | The Legendre (Jacobi) symb... |
| lgsdilem 27293 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27294 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27295 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27296 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27297 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27298 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27299 | The Legendre symbol is com... |
| lgsdirprm 27300 | The Legendre symbol is com... |
| lgsdir 27301 | The Legendre symbol is com... |
| lgsdilem2 27302 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27303 | The Legendre symbol is com... |
| lgsne0 27304 | The Legendre symbol is non... |
| lgsabs1 27305 | The Legendre symbol is non... |
| lgssq 27306 | The Legendre symbol at a s... |
| lgssq2 27307 | The Legendre symbol at a s... |
| lgsprme0 27308 | The Legendre symbol at any... |
| 1lgs 27309 | The Legendre symbol at ` 1... |
| lgs1 27310 | The Legendre symbol at ` 1... |
| lgsmodeq 27311 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27312 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27313 | Variation on ~ lgsdir vali... |
| lgsdinn0 27314 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27315 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27316 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27317 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27318 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27319 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27320 | The Legendre symbol for od... |
| lgsqrmod 27321 | If the Legendre symbol of ... |
| lgsqrmodndvds 27322 | If the Legendre symbol of ... |
| lgsdchrval 27323 | The Legendre symbol functi... |
| lgsdchr 27324 | The Legendre symbol functi... |
| gausslemma2dlem0a 27325 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27326 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27327 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27328 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27329 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27330 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27331 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27332 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27333 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27334 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27335 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27336 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27337 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27338 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27339 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27340 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27341 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27342 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27343 | Gauss' Lemma (see also the... |
| lgseisenlem1 27344 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27345 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27346 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27347 | Lemma for ~ lgseisen . (C... |
| lgseisen 27348 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27349 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27350 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27351 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27352 | The Law of Quadratic Recip... |
| lgsquad2lem1 27353 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27354 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27355 | Extend ~ lgsquad to coprim... |
| lgsquad3 27356 | Extend ~ lgsquad2 to integ... |
| m1lgs 27357 | The first supplement to th... |
| 2lgslem1a1 27358 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27359 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27360 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27361 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27362 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27363 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27364 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27365 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27366 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27367 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27368 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27369 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27370 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27371 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27372 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27373 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27374 | The Legendre symbol for ` ... |
| 2lgslem4 27375 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27376 | The second supplement to t... |
| 2lgsoddprmlem1 27377 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27378 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27379 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27380 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27381 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27382 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27383 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27384 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27385 | The second supplement to t... |
| 2sqlem1 27386 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27387 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27388 | Fibonacci's identity (actu... |
| 2sqlem3 27389 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27390 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27391 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27392 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27393 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27394 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27395 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27396 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27397 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27398 | Lemma for ~ 2sq . (Contri... |
| 2sq 27399 | All primes of the form ` 4... |
| 2sqblem 27400 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27401 | The converse to ~ 2sq . (... |
| 2sq2 27402 | ` 2 ` is the sum of square... |
| 2sqn0 27403 | If the sum of two squares ... |
| 2sqcoprm 27404 | If the sum of two squares ... |
| 2sqmod 27405 | Given two decompositions o... |
| 2sqmo 27406 | There exists at most one d... |
| 2sqnn0 27407 | All primes of the form ` 4... |
| 2sqnn 27408 | All primes of the form ` 4... |
| addsq2reu 27409 | For each complex number ` ... |
| addsqn2reu 27410 | For each complex number ` ... |
| addsqrexnreu 27411 | For each complex number, t... |
| addsqnreup 27412 | There is no unique decompo... |
| addsq2nreurex 27413 | For each complex number ` ... |
| addsqn2reurex2 27414 | For each complex number ` ... |
| 2sqreulem1 27415 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27416 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27417 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27418 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27419 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27420 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27421 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27422 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27423 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27424 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27425 | There exists a unique deco... |
| 2sqreunn 27426 | There exists a unique deco... |
| 2sqreult 27427 | There exists a unique deco... |
| 2sqreultb 27428 | There exists a unique deco... |
| 2sqreunnlt 27429 | There exists a unique deco... |
| 2sqreunnltb 27430 | There exists a unique deco... |
| 2sqreuop 27431 | There exists a unique deco... |
| 2sqreuopnn 27432 | There exists a unique deco... |
| 2sqreuoplt 27433 | There exists a unique deco... |
| 2sqreuopltb 27434 | There exists a unique deco... |
| 2sqreuopnnlt 27435 | There exists a unique deco... |
| 2sqreuopnnltb 27436 | There exists a unique deco... |
| 2sqreuopb 27437 | There exists a unique deco... |
| chebbnd1lem1 27438 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27439 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27440 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27441 | The Chebyshev bound: The ... |
| chtppilimlem1 27442 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27443 | Lemma for ~ chtppilim . (... |
| chtppilim 27444 | The ` theta ` function is ... |
| chto1ub 27445 | The ` theta ` function is ... |
| chebbnd2 27446 | The Chebyshev bound, part ... |
| chto1lb 27447 | The ` theta ` function is ... |
| chpchtlim 27448 | The ` psi ` and ` theta ` ... |
| chpo1ub 27449 | The ` psi ` function is up... |
| chpo1ubb 27450 | The ` psi ` function is up... |
| vmadivsum 27451 | The sum of the von Mangold... |
| vmadivsumb 27452 | Give a total bound on the ... |
| rplogsumlem1 27453 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27454 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27455 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27456 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27457 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27458 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27459 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27460 | Lemma for ~ dchrisum . Le... |
| dchrisum 27461 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27462 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27463 | The sum of the Möbius... |
| dchrvmasumlem1 27464 | An alternative expression ... |
| dchrvmasum2lem 27465 | Give an expression for ` l... |
| dchrvmasum2if 27466 | Combine the results of ~ d... |
| dchrvmasumlem2 27467 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27468 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27469 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27470 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27471 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27472 | An asymptotic approximatio... |
| dchrvmaeq0 27473 | The set ` W ` is the colle... |
| dchrisum0fval 27474 | Value of the function ` F ... |
| dchrisum0fmul 27475 | The function ` F ` , the d... |
| dchrisum0ff 27476 | The function ` F ` is a re... |
| dchrisum0flblem1 27477 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27478 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27479 | The divisor sum of a real ... |
| dchrisum0fno1 27480 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27481 | A partial result along the... |
| dchrisum0re 27482 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27483 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27484 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27485 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27486 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27487 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27488 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27489 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27490 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27491 | The sum of the Möbius... |
| dchrvmasumlem 27492 | The sum of the Möbius... |
| dchrmusum 27493 | The sum of the Möbius... |
| dchrvmasum 27494 | The sum of the von Mangold... |
| rpvmasum 27495 | The sum of the von Mangold... |
| rplogsum 27496 | The sum of ` log p / p ` o... |
| dirith2 27497 | Dirichlet's theorem: there... |
| dirith 27498 | Dirichlet's theorem: there... |
| mudivsum 27499 | Asymptotic formula for ` s... |
| mulogsumlem 27500 | Lemma for ~ mulogsum . (C... |
| mulogsum 27501 | Asymptotic formula for ... |
| logdivsum 27502 | Asymptotic analysis of ... |
| mulog2sumlem1 27503 | Asymptotic formula for ... |
| mulog2sumlem2 27504 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27505 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27506 | Asymptotic formula for ... |
| vmalogdivsum2 27507 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27508 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27509 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27510 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27511 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27512 | The Möbius inverse of... |
| log2sumbnd 27513 | Bound on the difference be... |
| selberglem1 27514 | Lemma for ~ selberg . Est... |
| selberglem2 27515 | Lemma for ~ selberg . (Co... |
| selberglem3 27516 | Lemma for ~ selberg . Est... |
| selberg 27517 | Selberg's symmetry formula... |
| selbergb 27518 | Convert eventual boundedne... |
| selberg2lem 27519 | Lemma for ~ selberg2 . Eq... |
| selberg2 27520 | Selberg's symmetry formula... |
| selberg2b 27521 | Convert eventual boundedne... |
| chpdifbndlem1 27522 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27523 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27524 | A bound on the difference ... |
| logdivbnd 27525 | A bound on a sum of logs, ... |
| selberg3lem1 27526 | Introduce a log weighting ... |
| selberg3lem2 27527 | Lemma for ~ selberg3 . Eq... |
| selberg3 27528 | Introduce a log weighting ... |
| selberg4lem1 27529 | Lemma for ~ selberg4 . Eq... |
| selberg4 27530 | The Selberg symmetry formu... |
| pntrval 27531 | Define the residual of the... |
| pntrf 27532 | Functionality of the resid... |
| pntrmax 27533 | There is a bound on the re... |
| pntrsumo1 27534 | A bound on a sum over ` R ... |
| pntrsumbnd 27535 | A bound on a sum over ` R ... |
| pntrsumbnd2 27536 | A bound on a sum over ` R ... |
| selbergr 27537 | Selberg's symmetry formula... |
| selberg3r 27538 | Selberg's symmetry formula... |
| selberg4r 27539 | Selberg's symmetry formula... |
| selberg34r 27540 | The sum of ~ selberg3r and... |
| pntsval 27541 | Define the "Selberg functi... |
| pntsf 27542 | Functionality of the Selbe... |
| selbergs 27543 | Selberg's symmetry formula... |
| selbergsb 27544 | Selberg's symmetry formula... |
| pntsval2 27545 | The Selberg function can b... |
| pntrlog2bndlem1 27546 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27547 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27548 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27549 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27550 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27551 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27552 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27553 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27554 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27555 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27556 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27557 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27558 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27559 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27560 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27561 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27562 | Lemma for ~ pnt . Establi... |
| pntlemd 27563 | Lemma for ~ pnt . Closure... |
| pntlemc 27564 | Lemma for ~ pnt . Closure... |
| pntlema 27565 | Lemma for ~ pnt . Closure... |
| pntlemb 27566 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27567 | Lemma for ~ pnt . Closure... |
| pntlemh 27568 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27569 | Lemma for ~ pnt . The "na... |
| pntlemq 27570 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27571 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27572 | Lemma for ~ pnt . The ind... |
| pntlemi 27573 | Lemma for ~ pnt . Elimina... |
| pntlemf 27574 | Lemma for ~ pnt . Add up ... |
| pntlemk 27575 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27576 | Lemma for ~ pnt . Combine... |
| pntleme 27577 | Lemma for ~ pnt . Package... |
| pntlem3 27578 | Lemma for ~ pnt . Equatio... |
| pntlemp 27579 | Lemma for ~ pnt . Wrappin... |
| pntleml 27580 | Lemma for ~ pnt . Equatio... |
| pnt3 27581 | The Prime Number Theorem, ... |
| pnt2 27582 | The Prime Number Theorem, ... |
| pnt 27583 | The Prime Number Theorem: ... |
| abvcxp 27584 | Raising an absolute value ... |
| padicfval 27585 | Value of the p-adic absolu... |
| padicval 27586 | Value of the p-adic absolu... |
| ostth2lem1 27587 | Lemma for ~ ostth2 , altho... |
| qrngbas 27588 | The base set of the field ... |
| qdrng 27589 | The rationals form a divis... |
| qrng0 27590 | The zero element of the fi... |
| qrng1 27591 | The unity element of the f... |
| qrngneg 27592 | The additive inverse in th... |
| qrngdiv 27593 | The division operation in ... |
| qabvle 27594 | By using induction on ` N ... |
| qabvexp 27595 | Induct the product rule ~ ... |
| ostthlem1 27596 | Lemma for ~ ostth . If tw... |
| ostthlem2 27597 | Lemma for ~ ostth . Refin... |
| qabsabv 27598 | The regular absolute value... |
| padicabv 27599 | The p-adic absolute value ... |
| padicabvf 27600 | The p-adic absolute value ... |
| padicabvcxp 27601 | All positive powers of the... |
| ostth1 27602 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27603 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27604 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27605 | Lemma for ~ ostth2 . (Con... |
| ostth2 27606 | - Lemma for ~ ostth : regu... |
| ostth3 27607 | - Lemma for ~ ostth : p-ad... |
| ostth 27608 | Ostrowski's theorem, which... |
| elno 27615 | Membership in the surreals... |
| elnoOLD 27616 | Obsolete version of ~ elno... |
| ltsval 27617 | The value of the surreal l... |
| bdayval 27618 | The value of the birthday ... |
| nofun 27619 | A surreal is a function. ... |
| nodmon 27620 | The domain of a surreal is... |
| norn 27621 | The range of a surreal is ... |
| nofnbday 27622 | A surreal is a function ov... |
| nodmord 27623 | The domain of a surreal ha... |
| elno2 27624 | An alternative condition f... |
| elno3 27625 | Another condition for memb... |
| ltsval2 27626 | Alternate expression for s... |
| nofv 27627 | The function value of a su... |
| nosgnn0 27628 | ` (/) ` is not a surreal s... |
| nosgnn0i 27629 | If ` X ` is a surreal sign... |
| noreson 27630 | The restriction of a surre... |
| ltsintdifex 27631 |
If ` A |
| ltsres 27632 | If the restrictions of two... |
| noxp1o 27633 | The Cartesian product of a... |
| noseponlem 27634 | Lemma for ~ nosepon . Con... |
| nosepon 27635 | Given two unequal surreals... |
| noextend 27636 | Extending a surreal by one... |
| noextendseq 27637 | Extend a surreal by a sequ... |
| noextenddif 27638 | Calculate the place where ... |
| noextendlt 27639 | Extending a surreal with a... |
| noextendgt 27640 | Extending a surreal with a... |
| nolesgn2o 27641 | Given ` A ` less-than or e... |
| nolesgn2ores 27642 | Given ` A ` less-than or e... |
| nogesgn1o 27643 | Given ` A ` greater than o... |
| nogesgn1ores 27644 | Given ` A ` greater than o... |
| ltssolem1 27645 | Lemma for ~ ltsso . The "... |
| ltsso 27646 | Less-than totally orders t... |
| bdayfo 27647 | The birthday function maps... |
| fvnobday 27648 | The value of a surreal at ... |
| nosepnelem 27649 | Lemma for ~ nosepne . (Co... |
| nosepne 27650 | The value of two non-equal... |
| nosep1o 27651 | If the value of a surreal ... |
| nosep2o 27652 | If the value of a surreal ... |
| nosepdmlem 27653 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27654 | The first place two surrea... |
| nosepeq 27655 | The values of two surreals... |
| nosepssdm 27656 | Given two non-equal surrea... |
| nodenselem4 27657 | Lemma for ~ nodense . Sho... |
| nodenselem5 27658 | Lemma for ~ nodense . If ... |
| nodenselem6 27659 | The restriction of a surre... |
| nodenselem7 27660 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27661 | Lemma for ~ nodense . Giv... |
| nodense 27662 | Given two distinct surreal... |
| bdayimaon 27663 | Lemma for full-eta propert... |
| nolt02olem 27664 | Lemma for ~ nolt02o . If ... |
| nolt02o 27665 | Given ` A ` less-than ` B ... |
| nogt01o 27666 | Given ` A ` greater than `... |
| noresle 27667 | Restriction law for surrea... |
| nomaxmo 27668 | A class of surreals has at... |
| nominmo 27669 | A class of surreals has at... |
| nosupprefixmo 27670 | In any class of surreals, ... |
| noinfprefixmo 27671 | In any class of surreals, ... |
| nosupcbv 27672 | Lemma to change bound vari... |
| nosupno 27673 | The next several theorems ... |
| nosupdm 27674 | The domain of the surreal ... |
| nosupbday 27675 | Birthday bounding law for ... |
| nosupfv 27676 | The value of surreal supre... |
| nosupres 27677 | A restriction law for surr... |
| nosupbnd1lem1 27678 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27679 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27680 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27681 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27682 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27683 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27684 | Bounding law from below fo... |
| nosupbnd2lem1 27685 | Bounding law from above wh... |
| nosupbnd2 27686 | Bounding law from above fo... |
| noinfcbv 27687 | Change bound variables for... |
| noinfno 27688 | The next several theorems ... |
| noinfdm 27689 | Next, we calculate the dom... |
| noinfbday 27690 | Birthday bounding law for ... |
| noinffv 27691 | The value of surreal infim... |
| noinfres 27692 | The restriction of surreal... |
| noinfbnd1lem1 27693 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27694 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27695 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27696 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27697 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27698 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27699 | Bounding law from above fo... |
| noinfbnd2lem1 27700 | Bounding law from below wh... |
| noinfbnd2 27701 | Bounding law from below fo... |
| nosupinfsep 27702 | Given two sets of surreals... |
| noetasuplem1 27703 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27704 | Lemma for ~ noeta . The r... |
| noetasuplem3 27705 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27706 | Lemma for ~ noeta . When ... |
| noetainflem1 27707 | Lemma for ~ noeta . Estab... |
| noetainflem2 27708 | Lemma for ~ noeta . The r... |
| noetainflem3 27709 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27710 | Lemma for ~ noeta . If ` ... |
| noetalem1 27711 | Lemma for ~ noeta . Eithe... |
| noetalem2 27712 | Lemma for ~ noeta . The f... |
| noeta 27713 | The full-eta axiom for the... |
| ltsirr 27716 | Surreal less-than is irref... |
| ltstr 27717 | Surreal less-than is trans... |
| ltsasym 27718 | Surreal less-than is asymm... |
| ltslin 27719 | Surreal less-than obeys tr... |
| ltstrieq2 27720 | Trichotomy law for surreal... |
| ltstrine 27721 | Trichotomy law for surreal... |
| lenlts 27722 | Surreal less-than or equal... |
| ltnles 27723 | Surreal less-than in terms... |
| lesloe 27724 | Surreal less-than or equal... |
| lestri3 27725 | Trichotomy law for surreal... |
| lesnltd 27726 | Surreal less-than or equal... |
| ltsnled 27727 | Surreal less-than in terms... |
| lesloed 27728 | Surreal less-than or equal... |
| lestri3d 27729 | Trichotomy law for surreal... |
| ltlestr 27730 | Surreal transitive law. (... |
| leltstr 27731 | Surreal transitive law. (... |
| lestr 27732 | Surreal transitive law. (... |
| ltstrd 27733 | Surreal less-than is trans... |
| ltlestrd 27734 | Surreal less-than is trans... |
| leltstrd 27735 | Surreal less-than is trans... |
| lestrd 27736 | Surreal less-than or equal... |
| lesid 27737 | Surreal less-than or equal... |
| lestric 27738 | Surreal trichotomy law. (... |
| maxs1 27739 | A surreal is less than or ... |
| maxs2 27740 | A surreal is less than or ... |
| mins1 27741 | The minimum of two surreal... |
| mins2 27742 | The minimum of two surreal... |
| ltlesd 27743 | Surreal less-than implies ... |
| ltsne 27744 | Surreal less-than implies ... |
| ltlesnd 27745 | Surreal less-than in terms... |
| bdayfun 27746 | The birthday function is a... |
| bdayfn 27747 | The birthday function is a... |
| bdaydm 27748 | The birthday function's do... |
| bdayrn 27749 | The birthday function's ra... |
| bdayon 27750 | The value of the birthday ... |
| nobdaymin 27751 | Any non-empty class of sur... |
| nocvxminlem 27752 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27753 | Given a nonempty convex cl... |
| noprc 27754 | The surreal numbers are a ... |
| noeta2 27759 | A version of ~ noeta with ... |
| brslts 27760 | Binary relation form of th... |
| sltsex1 27761 | The first argument of surr... |
| sltsex2 27762 | The second argument of sur... |
| sltsss1 27763 | The first argument of surr... |
| sltsss2 27764 | The second argument of sur... |
| sltssep 27765 | The separation property of... |
| sltsd 27766 | Deduce surreal set less-th... |
| sltssnb 27767 | Surreal set less-than of t... |
| sltssn 27768 | Surreal set less-than of t... |
| sltssepc 27769 | Two elements of separated ... |
| sltssepcd 27770 | Two elements of separated ... |
| ssslts1 27771 | Relation between surreal s... |
| ssslts2 27772 | Relation between surreal s... |
| nulslts 27773 | The empty set is less-than... |
| nulsgts 27774 | The empty set is greater t... |
| nulsltsd 27775 | The empty set is less-than... |
| nulsgtsd 27776 | The empty set is greater t... |
| conway 27777 | Conway's Simplicity Theore... |
| cutsval 27778 | The value of the surreal c... |
| cutcuts 27779 | Cut properties of the surr... |
| cutscl 27780 | Closure law for surreal cu... |
| cutscld 27781 | Closure law for surreal cu... |
| cutbday 27782 | The birthday of the surrea... |
| eqcuts 27783 | Condition for equality to ... |
| eqcuts2 27784 | Condition for equality to ... |
| sltstr 27785 | Transitive law for surreal... |
| sltsun1 27786 | Union law for surreal set ... |
| sltsun2 27787 | Union law for surreal set ... |
| cutsun12 27788 | Union law for surreal cuts... |
| dmcuts 27789 | The domain of the surreal ... |
| cutsf 27790 | Functionality statement fo... |
| etaslts 27791 | A restatement of ~ noeta u... |
| etaslts2 27792 | A version of ~ etaslts wit... |
| cutbdaybnd 27793 | An upper bound on the birt... |
| cutbdaybnd2 27794 | An upper bound on the birt... |
| cutbdaybnd2lim 27795 | An upper bound on the birt... |
| cutbdaylt 27796 | If a surreal lies in a gap... |
| lesrec 27797 | A comparison law for surre... |
| lesrecd 27798 | A comparison law for surre... |
| ltsrec 27799 | A comparison law for surre... |
| ltsrecd 27800 | A comparison law for surre... |
| sltsdisj 27801 | If ` A ` preceeds ` B ` , ... |
| eqcuts3 27802 | A variant of the simplicit... |
| 0no 27807 | Surreal zero is a surreal.... |
| 1no 27808 | Surreal one is a surreal. ... |
| bday0 27809 | Calculate the birthday of ... |
| 0lt1s 27810 | Surreal zero is less than ... |
| bday0b 27811 | The only surreal with birt... |
| bday1 27812 | The birthday of surreal on... |
| cuteq0 27813 | Condition for a surreal cu... |
| cutneg 27814 | The simplest number greate... |
| cuteq1 27815 | Condition for a surreal cu... |
| gt0ne0s 27816 | A positive surreal is not ... |
| gt0ne0sd 27817 | A positive surreal is not ... |
| 1ne0s 27818 | Surreal zero does not equa... |
| rightge0 27819 | A surreal is non-negative ... |
| madeval 27830 | The value of the made by f... |
| madeval2 27831 | Alternative characterizati... |
| oldval 27832 | The value of the old optio... |
| newval 27833 | The value of the new optio... |
| madef 27834 | The made function is a fun... |
| oldf 27835 | The older function is a fu... |
| newf 27836 | The new function is a func... |
| old0 27837 | No surreal is older than `... |
| madessno 27838 | Made sets are surreals. (... |
| oldssno 27839 | Old sets are surreals. (C... |
| newssno 27840 | New sets are surreals. (C... |
| madeno 27841 | An element of a made set i... |
| oldno 27842 | An element of an old set i... |
| newno 27843 | An element of a new set is... |
| madenod 27844 | An element of a made set i... |
| oldnod 27845 | An element of an old set i... |
| newnod 27846 | An element of a new set is... |
| leftval 27847 | The value of the left opti... |
| rightval 27848 | The value of the right opt... |
| elleft 27849 | Membership in the left set... |
| elright 27850 | Membership in the right se... |
| leftlt 27851 | A member of a surreal's le... |
| rightgt 27852 | A member of a surreal's ri... |
| leftf 27853 | The functionality of the l... |
| rightf 27854 | The functionality of the r... |
| elmade 27855 | Membership in the made fun... |
| elmade2 27856 | Membership in the made fun... |
| elold 27857 | Membership in an old set. ... |
| sltsleft 27858 | A surreal is greater than ... |
| sltsright 27859 | A surreal is less than its... |
| lltr 27860 | The left options of a surr... |
| made0 27861 | The only surreal made on d... |
| new0 27862 | The only surreal new on da... |
| old1 27863 | The only surreal older tha... |
| madess 27864 | If ` A ` is less than or e... |
| oldssmade 27865 | The older-than set is a su... |
| oldmade 27866 | An element of an old set i... |
| oldmaded 27867 | An element of an old set i... |
| oldss 27868 | If ` A ` is less than or e... |
| leftssold 27869 | The left options are a sub... |
| rightssold 27870 | The right options are a su... |
| leftssno 27871 | The left set of a surreal ... |
| rightssno 27872 | The right set of a surreal... |
| leftold 27873 | An element of a left set i... |
| rightold 27874 | An element of a right set ... |
| leftno 27875 | An element of a left set i... |
| rightno 27876 | An element of a right set ... |
| leftoldd 27877 | An element of a left set i... |
| leftnod 27878 | An element of a left set i... |
| rightoldd 27879 | An element of a right set ... |
| rightnod 27880 | An element of a right set ... |
| madecut 27881 | Given a section that is a ... |
| madeun 27882 | The made set is the union ... |
| madeoldsuc 27883 | The made set is the old se... |
| oldsuc 27884 | The value of the old set a... |
| oldlim 27885 | The value of the old set a... |
| madebdayim 27886 | If a surreal is a member o... |
| oldbdayim 27887 | If ` X ` is in the old set... |
| oldirr 27888 | No surreal is a member of ... |
| leftirr 27889 | No surreal is a member of ... |
| rightirr 27890 | No surreal is a member of ... |
| left0s 27891 | The left set of ` 0s ` is ... |
| right0s 27892 | The right set of ` 0s ` is... |
| left1s 27893 | The left set of ` 1s ` is ... |
| right1s 27894 | The right set of ` 1s ` is... |
| lrold 27895 | The union of the left and ... |
| madebdaylemold 27896 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27897 | Lemma for ~ madebday . If... |
| madebday 27898 | A surreal is part of the s... |
| oldbday 27899 | A surreal is part of the s... |
| newbday 27900 | A surreal is an element of... |
| newbdayim 27901 | One direction of the bicon... |
| lrcut 27902 | A surreal is equal to the ... |
| cutsfo 27903 | The surreal cut function i... |
| ltsn0 27904 | If ` X ` is less than ` Y ... |
| lruneq 27905 | If two surreals share a bi... |
| ltslpss 27906 | If two surreals share a bi... |
| leslss 27907 | If two surreals ` A ` and ... |
| 0elold 27908 | Zero is in the old set of ... |
| 0elleft 27909 | Zero is in the left set of... |
| 0elright 27910 | Zero is in the right set o... |
| madefi 27911 | The made set of an ordinal... |
| oldfi 27912 | The old set of an ordinal ... |
| bdayiun 27913 | The birthday of a surreal ... |
| bdayle 27914 | A condition for bounding a... |
| sltsbday 27915 | Birthday comparison rule f... |
| cofslts 27916 | If every element of ` A ` ... |
| coinitslts 27917 | If ` B ` is coinitial with... |
| cofcut1 27918 | If ` C ` is cofinal with `... |
| cofcut1d 27919 | If ` C ` is cofinal with `... |
| cofcut2 27920 | If ` A ` and ` C ` are mut... |
| cofcut2d 27921 | If ` A ` and ` C ` are mut... |
| cofcutr 27922 | If ` X ` is the cut of ` A... |
| cofcutr1d 27923 | If ` X ` is the cut of ` A... |
| cofcutr2d 27924 | If ` X ` is the cut of ` A... |
| cofcutrtime 27925 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27926 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27927 | If ` X ` is a timely cut o... |
| cofss 27928 | Cofinality for a subset. ... |
| coiniss 27929 | Coinitiality for a subset.... |
| cutlt 27930 | Eliminating all elements b... |
| cutpos 27931 | Reduce the elements of a c... |
| cutmax 27932 | If ` A ` has a maximum, th... |
| cutmin 27933 | If ` B ` has a minimum, th... |
| cutminmax 27934 | If the left set of ` X ` h... |
| lrrecval 27937 | The next step in the devel... |
| lrrecval2 27938 | Next, we establish an alte... |
| lrrecpo 27939 | Now, we establish that ` R... |
| lrrecse 27940 | Next, we show that ` R ` i... |
| lrrecfr 27941 | Now we show that ` R ` is ... |
| lrrecpred 27942 | Finally, we calculate the ... |
| noinds 27943 | Induction principle for a ... |
| norecfn 27944 | Surreal recursion over one... |
| norecov 27945 | Calculate the value of the... |
| noxpordpo 27948 | To get through most of the... |
| noxpordfr 27949 | Next we establish the foun... |
| noxpordse 27950 | Next we establish the set-... |
| noxpordpred 27951 | Next we calculate the pred... |
| no2indlesm 27952 | Double induction on surrea... |
| no2inds 27953 | Double induction on surrea... |
| norec2fn 27954 | The double-recursion opera... |
| norec2ov 27955 | The value of the double-re... |
| no3inds 27956 | Triple induction over surr... |
| addsfn 27959 | Surreal addition is a func... |
| addsval 27960 | The value of surreal addit... |
| addsval2 27961 | The value of surreal addit... |
| addsrid 27962 | Surreal addition to zero i... |
| addsridd 27963 | Surreal addition to zero i... |
| addscom 27964 | Surreal addition commutes.... |
| addscomd 27965 | Surreal addition commutes.... |
| addslid 27966 | Surreal addition to zero i... |
| addsproplem1 27967 | Lemma for surreal addition... |
| addsproplem2 27968 | Lemma for surreal addition... |
| addsproplem3 27969 | Lemma for surreal addition... |
| addsproplem4 27970 | Lemma for surreal addition... |
| addsproplem5 27971 | Lemma for surreal addition... |
| addsproplem6 27972 | Lemma for surreal addition... |
| addsproplem7 27973 | Lemma for surreal addition... |
| addsprop 27974 | Inductively show that surr... |
| addcutslem 27975 | Lemma for ~ addcuts . Sho... |
| addcuts 27976 | Demonstrate the cut proper... |
| addcuts2 27977 | Show that the cut involved... |
| addscld 27978 | Surreal numbers are closed... |
| addscl 27979 | Surreal numbers are closed... |
| addsf 27980 | Function statement for sur... |
| addsfo 27981 | Surreal addition is onto. ... |
| peano2no 27982 | A theorem for surreals tha... |
| ltadds1im 27983 | Surreal less-than is prese... |
| ltadds2im 27984 | Surreal less-than is prese... |
| leadds1im 27985 | Surreal less-than or equal... |
| leadds2im 27986 | Surreal less-than or equal... |
| leadds1 27987 | Addition to both sides of ... |
| leadds2 27988 | Addition to both sides of ... |
| ltadds2 27989 | Addition to both sides of ... |
| ltadds1 27990 | Addition to both sides of ... |
| addscan2 27991 | Cancellation law for surre... |
| addscan1 27992 | Cancellation law for surre... |
| leadds1d 27993 | Addition to both sides of ... |
| leadds2d 27994 | Addition to both sides of ... |
| ltadds2d 27995 | Addition to both sides of ... |
| ltadds1d 27996 | Addition to both sides of ... |
| addscan2d 27997 | Cancellation law for surre... |
| addscan1d 27998 | Cancellation law for surre... |
| addsuniflem 27999 | Lemma for ~ addsunif . St... |
| addsunif 28000 | Uniformity theorem for sur... |
| addsasslem1 28001 | Lemma for addition associa... |
| addsasslem2 28002 | Lemma for addition associa... |
| addsass 28003 | Surreal addition is associ... |
| addsassd 28004 | Surreal addition is associ... |
| adds32d 28005 | Commutative/associative la... |
| adds12d 28006 | Commutative/associative la... |
| adds4d 28007 | Rearrangement of four term... |
| adds42d 28008 | Rearrangement of four term... |
| ltaddspos1d 28009 | Addition of a positive num... |
| ltaddspos2d 28010 | Addition of a positive num... |
| lt2addsd 28011 | Adding both sides of two s... |
| addsgt0d 28012 | The sum of two positive su... |
| ltsp1d 28013 | A surreal is less than its... |
| addsge01d 28014 | A surreal is less-than or ... |
| addbdaylem 28015 | Lemma for ~ addbday . (Co... |
| addbday 28016 | The birthday of the sum of... |
| negsfn 28021 | Surreal negation is a func... |
| subsfn 28022 | Surreal subtraction is a f... |
| negsval 28023 | The value of the surreal n... |
| neg0s 28024 | Negative surreal zero is s... |
| neg1s 28025 | An expression for negative... |
| negsproplem1 28026 | Lemma for surreal negation... |
| negsproplem2 28027 | Lemma for surreal negation... |
| negsproplem3 28028 | Lemma for surreal negation... |
| negsproplem4 28029 | Lemma for surreal negation... |
| negsproplem5 28030 | Lemma for surreal negation... |
| negsproplem6 28031 | Lemma for surreal negation... |
| negsproplem7 28032 | Lemma for surreal negation... |
| negsprop 28033 | Show closure and ordering ... |
| negscl 28034 | The surreals are closed un... |
| negscld 28035 | The surreals are closed un... |
| ltnegsim 28036 | The forward direction of t... |
| negcut 28037 | The cut properties of surr... |
| negcut2 28038 | The cut that defines surre... |
| negsid 28039 | Surreal addition of a numb... |
| negsidd 28040 | Surreal addition of a numb... |
| negsex 28041 | Every surreal has a negati... |
| negnegs 28042 | A surreal is equal to the ... |
| ltnegs 28043 | Negative of both sides of ... |
| lenegs 28044 | Negative of both sides of ... |
| ltnegsd 28045 | Negative of both sides of ... |
| lenegsd 28046 | Negative of both sides of ... |
| negs11 28047 | Surreal negation is one-to... |
| negsdi 28048 | Distribution of surreal ne... |
| lt0negs2d 28049 | Comparison of a surreal an... |
| negsf 28050 | Function statement for sur... |
| negsfo 28051 | Function statement for sur... |
| negsf1o 28052 | Surreal negation is a bije... |
| negsunif 28053 | Uniformity property for su... |
| negbdaylem 28054 | Lemma for ~ negbday . Bou... |
| negbday 28055 | Negation of a surreal numb... |
| negleft 28056 | The left set of the negati... |
| negright 28057 | The right set of the negat... |
| subsval 28058 | The value of surreal subtr... |
| subsvald 28059 | The value of surreal subtr... |
| subscl 28060 | Closure law for surreal su... |
| subscld 28061 | Closure law for surreal su... |
| subsf 28062 | Function statement for sur... |
| subsfo 28063 | Surreal subtraction is an ... |
| negsval2 28064 | Surreal negation in terms ... |
| negsval2d 28065 | Surreal negation in terms ... |
| subsid1 28066 | Identity law for subtracti... |
| subsid 28067 | Subtraction of a surreal f... |
| subadds 28068 | Relationship between addit... |
| subaddsd 28069 | Relationship between addit... |
| pncans 28070 | Cancellation law for surre... |
| pncan3s 28071 | Subtraction and addition o... |
| pncan2s 28072 | Cancellation law for surre... |
| npcans 28073 | Cancellation law for surre... |
| ltsubs1 28074 | Subtraction from both side... |
| ltsubs2 28075 | Subtraction from both side... |
| ltsubs1d 28076 | Subtraction from both side... |
| ltsubs2d 28077 | Subtraction from both side... |
| negsubsdi2d 28078 | Distribution of negative o... |
| addsubsassd 28079 | Associative-type law for s... |
| addsubsd 28080 | Law for surreal addition a... |
| ltsubsubsbd 28081 | Equivalence for the surrea... |
| ltsubsubs2bd 28082 | Equivalence for the surrea... |
| ltsubsubs3bd 28083 | Equivalence for the surrea... |
| lesubsubsbd 28084 | Equivalence for the surrea... |
| lesubsubs2bd 28085 | Equivalence for the surrea... |
| lesubsubs3bd 28086 | Equivalence for the surrea... |
| ltsubaddsd 28087 | Surreal less-than relation... |
| ltsubadds2d 28088 | Surreal less-than relation... |
| ltaddsubsd 28089 | Surreal less-than relation... |
| ltaddsubs2d 28090 | Surreal less-than relation... |
| lesubaddsd 28091 | Surreal less-than or equal... |
| subsubs4d 28092 | Law for double surreal sub... |
| subsubs2d 28093 | Law for double surreal sub... |
| lesubsd 28094 | Swap subtrahends in a surr... |
| nncansd 28095 | Cancellation law for surre... |
| posdifsd 28096 | Comparison of two surreals... |
| ltsubsposd 28097 | Subtraction of a positive ... |
| subsge0d 28098 | Non-negative subtraction. ... |
| addsubs4d 28099 | Rearrangement of four term... |
| ltsm1d 28100 | A surreal is greater than ... |
| subscan1d 28101 | Cancellation law for surre... |
| subscan2d 28102 | Cancellation law for surre... |
| subseq0d 28103 | The difference between two... |
| mulsfn 28106 | Surreal multiplication is ... |
| mulsval 28107 | The value of surreal multi... |
| mulsval2lem 28108 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28109 | The value of surreal multi... |
| muls01 28110 | Surreal multiplication by ... |
| mulsrid 28111 | Surreal one is a right ide... |
| mulsridd 28112 | Surreal one is a right ide... |
| mulsproplemcbv 28113 | Lemma for surreal multipli... |
| mulsproplem1 28114 | Lemma for surreal multipli... |
| mulsproplem2 28115 | Lemma for surreal multipli... |
| mulsproplem3 28116 | Lemma for surreal multipli... |
| mulsproplem4 28117 | Lemma for surreal multipli... |
| mulsproplem5 28118 | Lemma for surreal multipli... |
| mulsproplem6 28119 | Lemma for surreal multipli... |
| mulsproplem7 28120 | Lemma for surreal multipli... |
| mulsproplem8 28121 | Lemma for surreal multipli... |
| mulsproplem9 28122 | Lemma for surreal multipli... |
| mulsproplem10 28123 | Lemma for surreal multipli... |
| mulsproplem11 28124 | Lemma for surreal multipli... |
| mulsproplem12 28125 | Lemma for surreal multipli... |
| mulsproplem13 28126 | Lemma for surreal multipli... |
| mulsproplem14 28127 | Lemma for surreal multipli... |
| mulsprop 28128 | Surreals are closed under ... |
| mulcutlem 28129 | Lemma for ~ mulcut . Stat... |
| mulcut 28130 | Show the cut properties of... |
| mulcut2 28131 | Show that the cut involved... |
| mulscl 28132 | The surreals are closed un... |
| mulscld 28133 | The surreals are closed un... |
| ltmuls 28134 | An ordering relationship f... |
| ltmulsd 28135 | An ordering relationship f... |
| lemulsd 28136 | An ordering relationship f... |
| mulscom 28137 | Surreal multiplication com... |
| mulscomd 28138 | Surreal multiplication com... |
| muls02 28139 | Surreal multiplication by ... |
| mulslid 28140 | Surreal one is a left iden... |
| mulslidd 28141 | Surreal one is a left iden... |
| mulsgt0 28142 | The product of two positiv... |
| mulsgt0d 28143 | The product of two positiv... |
| mulsge0d 28144 | The product of two non-neg... |
| sltmuls1 28145 | One surreal set less-than ... |
| sltmuls2 28146 | One surreal set less-than ... |
| mulsuniflem 28147 | Lemma for ~ mulsunif . St... |
| mulsunif 28148 | Surreal multiplication has... |
| addsdilem1 28149 | Lemma for surreal distribu... |
| addsdilem2 28150 | Lemma for surreal distribu... |
| addsdilem3 28151 | Lemma for ~ addsdi . Show... |
| addsdilem4 28152 | Lemma for ~ addsdi . Show... |
| addsdi 28153 | Distributive law for surre... |
| addsdid 28154 | Distributive law for surre... |
| addsdird 28155 | Distributive law for surre... |
| subsdid 28156 | Distribution of surreal mu... |
| subsdird 28157 | Distribution of surreal mu... |
| mulnegs1d 28158 | Product with negative is n... |
| mulnegs2d 28159 | Product with negative is n... |
| mul2negsd 28160 | Surreal product of two neg... |
| mulsasslem1 28161 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28162 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28163 | Lemma for ~ mulsass . Dem... |
| mulsass 28164 | Associative law for surrea... |
| mulsassd 28165 | Associative law for surrea... |
| muls4d 28166 | Rearrangement of four surr... |
| mulsunif2lem 28167 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28168 | Alternate expression for s... |
| ltmuls2 28169 | Multiplication of both sid... |
| ltmuls2d 28170 | Multiplication of both sid... |
| ltmuls1d 28171 | Multiplication of both sid... |
| lemuls2d 28172 | Multiplication of both sid... |
| lemuls1d 28173 | Multiplication of both sid... |
| ltmulnegs1d 28174 | Multiplication of both sid... |
| ltmulnegs2d 28175 | Multiplication of both sid... |
| mulscan2dlem 28176 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28177 | Cancellation of surreal mu... |
| mulscan1d 28178 | Cancellation of surreal mu... |
| muls12d 28179 | Commutative/associative la... |
| lemuls1ad 28180 | Multiplication of both sid... |
| ltmuls12ad 28181 | Comparison of the product ... |
| divsmo 28182 | Uniqueness of surreal inve... |
| muls0ord 28183 | If a surreal product is ze... |
| mulsne0bd 28184 | The product of two non-zer... |
| divsval 28187 | The value of surreal divis... |
| norecdiv 28188 | If a surreal has a recipro... |
| noreceuw 28189 | If a surreal has a recipro... |
| recsne0 28190 | If a surreal has a recipro... |
| divmulsw 28191 | Relationship between surre... |
| divmulswd 28192 | Relationship between surre... |
| divsclw 28193 | Weak division closure law.... |
| divsclwd 28194 | Weak division closure law.... |
| divscan2wd 28195 | A weak cancellation law fo... |
| divscan1wd 28196 | A weak cancellation law fo... |
| ltdivmulswd 28197 | Surreal less-than relation... |
| ltdivmuls2wd 28198 | Surreal less-than relation... |
| ltmuldivswd 28199 | Surreal less-than relation... |
| ltmuldivs2wd 28200 | Surreal less-than relation... |
| divsasswd 28201 | An associative law for sur... |
| divs1 28202 | A surreal divided by one i... |
| divs1d 28203 | A surreal divided by one i... |
| precsexlemcbv 28204 | Lemma for surreal reciproc... |
| precsexlem1 28205 | Lemma for surreal reciproc... |
| precsexlem2 28206 | Lemma for surreal reciproc... |
| precsexlem3 28207 | Lemma for surreal reciproc... |
| precsexlem4 28208 | Lemma for surreal reciproc... |
| precsexlem5 28209 | Lemma for surreal reciproc... |
| precsexlem6 28210 | Lemma for surreal reciproc... |
| precsexlem7 28211 | Lemma for surreal reciproc... |
| precsexlem8 28212 | Lemma for surreal reciproc... |
| precsexlem9 28213 | Lemma for surreal reciproc... |
| precsexlem10 28214 | Lemma for surreal reciproc... |
| precsexlem11 28215 | Lemma for surreal reciproc... |
| precsex 28216 | Every positive surreal has... |
| recsex 28217 | A non-zero surreal has a r... |
| recsexd 28218 | A non-zero surreal has a r... |
| divmuls 28219 | Relationship between surre... |
| divmulsd 28220 | Relationship between surre... |
| divscl 28221 | Surreal division closure l... |
| divscld 28222 | Surreal division closure l... |
| divscan2d 28223 | A cancellation law for sur... |
| divscan1d 28224 | A cancellation law for sur... |
| ltdivmulsd 28225 | Surreal less-than relation... |
| ltdivmuls2d 28226 | Surreal less-than relation... |
| ltmuldivsd 28227 | Surreal less-than relation... |
| ltmuldivs2d 28228 | Surreal less-than relation... |
| divsassd 28229 | An associative law for sur... |
| divmuldivsd 28230 | Multiplication of two surr... |
| divdivs1d 28231 | Surreal division into a fr... |
| divsrecd 28232 | Relationship between surre... |
| divsdird 28233 | Distribution of surreal di... |
| divscan3d 28234 | A cancellation law for sur... |
| abssval 28237 | The value of surreal absol... |
| absscl 28238 | Closure law for surreal ab... |
| abssid 28239 | The absolute value of a no... |
| abs0s 28240 | The absolute value of surr... |
| abssnid 28241 | For a negative surreal, it... |
| absmuls 28242 | Surreal absolute value dis... |
| abssge0 28243 | The absolute value of a su... |
| abssor 28244 | The absolute value of a su... |
| absnegs 28245 | Surreal absolute value of ... |
| leabss 28246 | A surreal is less than or ... |
| abslts 28247 | Surreal absolute value and... |
| abssubs 28248 | Swapping order of surreal ... |
| elons 28251 | Membership in the class of... |
| onssno 28252 | The surreal ordinals are a... |
| onno 28253 | A surreal ordinal is a sur... |
| 0ons 28254 | Surreal zero is a surreal ... |
| 1ons 28255 | Surreal one is a surreal o... |
| elons2 28256 | A surreal is ordinal iff i... |
| elons2d 28257 | The cut of any set of surr... |
| onleft 28258 | The left set of a surreal ... |
| ltonold 28259 | The class of ordinals less... |
| ltonsex 28260 | The class of ordinals less... |
| oncutleft 28261 | A surreal ordinal is equal... |
| oncutlt 28262 | A surreal ordinal is the s... |
| bday11on 28263 | The birthday function is o... |
| onnolt 28264 | If a surreal ordinal is le... |
| onlts 28265 | Less-than is the same as b... |
| onles 28266 | Less-than or equal is the ... |
| onltsd 28267 | Less-than is the same as b... |
| onlesd 28268 | Less-than or equal is the ... |
| oniso 28269 | The birthday function rest... |
| onswe 28270 | Surreal less-than well-ord... |
| onsse 28271 | Surreal less-than is set-l... |
| onsis 28272 | Transfinite induction sche... |
| ons2ind 28273 | Double induction schema fo... |
| bdayons 28274 | The birthday of a surreal ... |
| onaddscl 28275 | The surreal ordinals are c... |
| onmulscl 28276 | The surreal ordinals are c... |
| addonbday 28277 | The birthday of the sum of... |
| peano2ons 28278 | The successor of a surreal... |
| onsbnd 28279 | The surreals of a given bi... |
| onsbnd2 28280 | The surreals of a given bi... |
| seqsex 28283 | Existence of the surreal s... |
| seqseq123d 28284 | Equality deduction for the... |
| nfseqs 28285 | Hypothesis builder for the... |
| seqsval 28286 | The value of the surreal s... |
| noseqex 28287 | The next several theorems ... |
| noseq0 28288 | The surreal ` A ` is a mem... |
| noseqp1 28289 | One plus an element of ` Z... |
| noseqind 28290 | Peano's inductive postulat... |
| noseqinds 28291 | Induction schema for surre... |
| noseqssno 28292 | A surreal sequence is a su... |
| noseqno 28293 | An element of a surreal se... |
| om2noseq0 28294 | The mapping ` G ` is a one... |
| om2noseqsuc 28295 | The value of ` G ` at a su... |
| om2noseqfo 28296 | Function statement for ` G... |
| om2noseqlt 28297 | Surreal less-than relation... |
| om2noseqlt2 28298 | The mapping ` G ` preserve... |
| om2noseqf1o 28299 | ` G ` is a bijection. (Co... |
| om2noseqiso 28300 | ` G ` is an isomorphism fr... |
| om2noseqoi 28301 | An alternative definition ... |
| om2noseqrdg 28302 | A helper lemma for the val... |
| noseqrdglem 28303 | A helper lemma for the val... |
| noseqrdgfn 28304 | The recursive definition g... |
| noseqrdg0 28305 | Initial value of a recursi... |
| noseqrdgsuc 28306 | Successor value of a recur... |
| seqsfn 28307 | The surreal sequence build... |
| seqs1 28308 | The value of the surreal s... |
| seqsp1 28309 | The value of the surreal s... |
| n0sexg 28314 | The set of all non-negativ... |
| n0sex 28315 | The set of all non-negativ... |
| nnsex 28316 | The set of all positive su... |
| peano5n0s 28317 | Peano's inductive postulat... |
| n0ssno 28318 | The non-negative surreal i... |
| nnssn0s 28319 | The positive surreal integ... |
| nnssno 28320 | The positive surreal integ... |
| n0no 28321 | A non-negative surreal int... |
| nnno 28322 | A positive surreal integer... |
| n0nod 28323 | A non-negative surreal int... |
| nnnod 28324 | A positive surreal integer... |
| nnn0s 28325 | A positive surreal integer... |
| nnn0sd 28326 | A positive surreal integer... |
| 0n0s 28327 | Peano postulate: ` 0s ` is... |
| peano2n0s 28328 | Peano postulate: the succe... |
| peano2n0sd 28329 | Peano postulate: the succe... |
| dfn0s2 28330 | Alternate definition of th... |
| n0sind 28331 | Principle of Mathematical ... |
| n0cut 28332 | A cut form for non-negativ... |
| n0cut2 28333 | A cut form for the success... |
| n0on 28334 | A surreal natural is a sur... |
| nnne0s 28335 | A surreal positive integer... |
| n0sge0 28336 | A non-negative integer is ... |
| nnsgt0 28337 | A positive integer is grea... |
| elnns 28338 | Membership in the positive... |
| elnns2 28339 | A positive surreal integer... |
| n0s0suc 28340 | A non-negative surreal int... |
| nnsge1 28341 | A positive surreal integer... |
| n0addscl 28342 | The non-negative surreal i... |
| n0mulscl 28343 | The non-negative surreal i... |
| nnaddscl 28344 | The positive surreal integ... |
| nnmulscl 28345 | The positive surreal integ... |
| 1n0s 28346 | Surreal one is a non-negat... |
| 1nns 28347 | Surreal one is a positive ... |
| peano2nns 28348 | Peano postulate for positi... |
| nnsrecgt0d 28349 | The reciprocal of a positi... |
| n0bday 28350 | A non-negative surreal int... |
| n0ssoldg 28351 | The non-negative surreal i... |
| n0ssold 28352 | The non-negative surreal i... |
| n0fincut 28353 | The simplest number greate... |
| onsfi 28354 | A surreal ordinal with a f... |
| eln0s2 28355 | A non-negative surreal int... |
| onltn0s 28356 | A surreal ordinal that is ... |
| n0cutlt 28357 | A non-negative surreal int... |
| seqn0sfn 28358 | The surreal sequence build... |
| eln0s 28359 | A non-negative surreal int... |
| n0s0m1 28360 | Every non-negative surreal... |
| n0subs 28361 | Subtraction of non-negativ... |
| n0subs2 28362 | Subtraction of non-negativ... |
| n0ltsp1le 28363 | Non-negative surreal order... |
| n0lesltp1 28364 | Non-negative surreal order... |
| n0lesm1lt 28365 | Non-negative surreal order... |
| n0lts1e0 28366 | A non-negative surreal int... |
| bdayn0p1 28367 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28368 | The birthday function rest... |
| n0p1nns 28369 | One plus a non-negative su... |
| dfnns2 28370 | Alternate definition of th... |
| nnsind 28371 | Principle of Mathematical ... |
| nn1m1nns 28372 | Every positive surreal int... |
| nnm1n0s 28373 | A positive surreal integer... |
| eucliddivs 28374 | Euclid's division lemma fo... |
| oldfib 28375 | The old set of an ordinal ... |
| zsex 28378 | The surreal integers form ... |
| zssno 28379 | The surreal integers are a... |
| zno 28380 | A surreal integer is a sur... |
| znod 28381 | A surreal integer is a sur... |
| elzs 28382 | Membership in the set of s... |
| nnzsubs 28383 | The difference of two surr... |
| nnzs 28384 | A positive surreal integer... |
| nnzsd 28385 | A positive surreal integer... |
| 0zs 28386 | Zero is a surreal integer.... |
| n0zs 28387 | A non-negative surreal int... |
| n0zsd 28388 | A non-negative surreal int... |
| 1zs 28389 | One is a surreal integer. ... |
| znegscl 28390 | The surreal integers are c... |
| znegscld 28391 | The surreal integers are c... |
| zaddscl 28392 | The surreal integers are c... |
| zaddscld 28393 | The surreal integers are c... |
| zsubscld 28394 | The surreal integers are c... |
| zmulscld 28395 | The surreal integers are c... |
| elzn0s 28396 | A surreal integer is a sur... |
| elzs2 28397 | A surreal integer is eithe... |
| eln0zs 28398 | Non-negative surreal integ... |
| elnnzs 28399 | Positive surreal integer p... |
| elznns 28400 | Surreal integer property e... |
| zn0subs 28401 | The non-negative differenc... |
| peano5uzs 28402 | Peano's inductive postulat... |
| uzsind 28403 | Induction on the upper sur... |
| zsbday 28404 | A surreal integer has a fi... |
| zcuts 28405 | A cut expression for surre... |
| zcuts0 28406 | Either the left or right s... |
| zsoring 28407 | The surreal integers form ... |
| 1p1e2s 28414 | One plus one is two. Surr... |
| no2times 28415 | Version of ~ 2times for su... |
| 2nns 28416 | Surreal two is a surreal n... |
| 2no 28417 | Surreal two is a surreal n... |
| 2ne0s 28418 | Surreal two is non-zero. ... |
| n0seo 28419 | A non-negative surreal int... |
| zseo 28420 | A surreal integer is eithe... |
| twocut 28421 | Two times the cut of zero ... |
| nohalf 28422 | An explicit expression for... |
| expsval 28423 | The value of surreal expon... |
| expnnsval 28424 | Value of surreal exponenti... |
| exps0 28425 | Surreal exponentiation to ... |
| exps1 28426 | Surreal exponentiation to ... |
| expsp1 28427 | Value of a surreal number ... |
| expscllem 28428 | Lemma for proving non-nega... |
| expscl 28429 | Closure law for surreal ex... |
| n0expscl 28430 | Closure law for non-negati... |
| nnexpscl 28431 | Closure law for positive s... |
| zexpscl 28432 | Closure law for surreal in... |
| expadds 28433 | Sum of exponents law for s... |
| expsne0 28434 | A non-negative surreal int... |
| expsgt0 28435 | A non-negative surreal int... |
| pw2recs 28436 | Any power of two has a mul... |
| pw2divscld 28437 | Division closure for power... |
| pw2divmulsd 28438 | Relationship between surre... |
| pw2divscan3d 28439 | Cancellation law for surre... |
| pw2divscan2d 28440 | A cancellation law for sur... |
| pw2divsassd 28441 | An associative law for div... |
| pw2divscan4d 28442 | Cancellation law for divis... |
| pw2gt0divsd 28443 | Division of a positive sur... |
| pw2ge0divsd 28444 | Divison of a non-negative ... |
| pw2divsrecd 28445 | Relationship between surre... |
| pw2divsdird 28446 | Distribution of surreal di... |
| pw2divsnegd 28447 | Move negative sign inside ... |
| pw2ltdivmulsd 28448 | Surreal less-than relation... |
| pw2ltmuldivs2d 28449 | Surreal less-than relation... |
| pw2ltsdiv1d 28450 | Surreal less-than relation... |
| avglts1d 28451 | Ordering property for aver... |
| avglts2d 28452 | Ordering property for aver... |
| pw2divs0d 28453 | Division into zero is zero... |
| pw2divsidd 28454 | Identity law for division ... |
| pw2ltdivmuls2d 28455 | Surreal less-than relation... |
| halfcut 28456 | Relate the cut of twice of... |
| addhalfcut 28457 | The cut of a surreal non-n... |
| pw2cut 28458 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28459 | Simplify ~ pw2cut in the c... |
| pw2cut2 28460 | Cut expression for powers ... |
| bdaypw2n0bndlem 28461 | Lemma for ~ bdaypw2n0bnd .... |
| bdaypw2n0bnd 28462 | Upper bound for the birthd... |
| bdaypw2bnd 28463 | Birthday bounding rule for... |
| bdayfinbndcbv 28464 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem1 28465 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem2 28466 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbnd 28467 | Given a non-negative integ... |
| z12bdaylem1 28468 | Lemma for ~ z12bday . Pro... |
| z12bdaylem2 28469 | Lemma for ~ z12bday . Sho... |
| elz12s 28470 | Membership in the dyadic f... |
| elz12si 28471 | Inference form of membersh... |
| z12sex 28472 | The class of dyadic fracti... |
| zz12s 28473 | A surreal integer is a dya... |
| z12no 28474 | A dyadic is a surreal. (C... |
| z12addscl 28475 | The dyadics are closed und... |
| z12negscl 28476 | The dyadics are closed und... |
| z12subscl 28477 | The dyadics are closed und... |
| z12shalf 28478 | Half of a dyadic is a dyad... |
| z12negsclb 28479 | A surreal is a dyadic frac... |
| z12zsodd 28480 | A dyadic fraction is eithe... |
| z12sge0 28481 | An expression for non-nega... |
| z12bdaylem 28482 | Lemma for ~ z12bday . Han... |
| z12bday 28483 | A dyadic fraction has a fi... |
| bdayfinlem 28484 | Lemma for ~ bdayfin . Han... |
| bdayfin 28485 | A surreal has a finite bir... |
| dfz12s2 28486 | The set of dyadic fraction... |
| elreno 28489 | Membership in the set of s... |
| reno 28490 | A surreal real is a surrea... |
| renod 28491 | A surreal real is a surrea... |
| recut 28492 | The cut involved in defini... |
| elreno2 28493 | Alternate characterization... |
| 0reno 28494 | Surreal zero is a surreal ... |
| 1reno 28495 | Surreal one is a surreal r... |
| renegscl 28496 | The surreal reals are clos... |
| readdscl 28497 | The surreal reals are clos... |
| remulscllem1 28498 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28499 | Lemma for ~ remulscl . Bo... |
| remulscl 28500 | The surreal reals are clos... |
| itvndx 28511 | Index value of the Interva... |
| lngndx 28512 | Index value of the "line" ... |
| itvid 28513 | Utility theorem: index-ind... |
| lngid 28514 | Utility theorem: index-ind... |
| slotsinbpsd 28515 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28516 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28517 | The slot for the line is n... |
| trkgstr 28518 | Functionality of a Tarski ... |
| trkgbas 28519 | The base set of a Tarski g... |
| trkgdist 28520 | The measure of a distance ... |
| trkgitv 28521 | The congruence relation in... |
| istrkgc 28528 | Property of being a Tarski... |
| istrkgb 28529 | Property of being a Tarski... |
| istrkgcb 28530 | Property of being a Tarski... |
| istrkge 28531 | Property of fulfilling Euc... |
| istrkgl 28532 | Building lines from the se... |
| istrkgld 28533 | Property of fulfilling the... |
| istrkg2ld 28534 | Property of fulfilling the... |
| istrkg3ld 28535 | Property of fulfilling the... |
| axtgcgrrflx 28536 | Axiom of reflexivity of co... |
| axtgcgrid 28537 | Axiom of identity of congr... |
| axtgsegcon 28538 | Axiom of segment construct... |
| axtg5seg 28539 | Five segments axiom, Axiom... |
| axtgbtwnid 28540 | Identity of Betweenness. ... |
| axtgpasch 28541 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28542 | Axiom of Continuity. Axio... |
| axtgcont 28543 | Axiom of Continuity. Axio... |
| axtglowdim2 28544 | Lower dimension axiom for ... |
| axtgupdim2 28545 | Upper dimension axiom for ... |
| axtgeucl 28546 | Euclid's Axiom. Axiom A10... |
| tgjustf 28547 | Given any function ` F ` ,... |
| tgjustr 28548 | Given any equivalence rela... |
| tgjustc1 28549 | A justification for using ... |
| tgjustc2 28550 | A justification for using ... |
| tgcgrcomimp 28551 | Congruence commutes on the... |
| tgcgrcomr 28552 | Congruence commutes on the... |
| tgcgrcoml 28553 | Congruence commutes on the... |
| tgcgrcomlr 28554 | Congruence commutes on bot... |
| tgcgreqb 28555 | Congruence and equality. ... |
| tgcgreq 28556 | Congruence and equality. ... |
| tgcgrneq 28557 | Congruence and equality. ... |
| tgcgrtriv 28558 | Degenerate segments are co... |
| tgcgrextend 28559 | Link congruence over a pai... |
| tgsegconeq 28560 | Two points that satisfy th... |
| tgbtwntriv2 28561 | Betweenness always holds f... |
| tgbtwncom 28562 | Betweenness commutes. The... |
| tgbtwncomb 28563 | Betweenness commutes, bico... |
| tgbtwnne 28564 | Betweenness and inequality... |
| tgbtwntriv1 28565 | Betweenness always holds f... |
| tgbtwnswapid 28566 | If you can swap the first ... |
| tgbtwnintr 28567 | Inner transitivity law for... |
| tgbtwnexch3 28568 | Exchange the first endpoin... |
| tgbtwnouttr2 28569 | Outer transitivity law for... |
| tgbtwnexch2 28570 | Exchange the outer point o... |
| tgbtwnouttr 28571 | Outer transitivity law for... |
| tgbtwnexch 28572 | Outer transitivity law for... |
| tgtrisegint 28573 | A line segment between two... |
| tglowdim1 28574 | Lower dimension axiom for ... |
| tglowdim1i 28575 | Lower dimension axiom for ... |
| tgldimor 28576 | Excluded-middle like state... |
| tgldim0eq 28577 | In dimension zero, any two... |
| tgldim0itv 28578 | In dimension zero, any two... |
| tgldim0cgr 28579 | In dimension zero, any two... |
| tgbtwndiff 28580 | There is always a ` c ` di... |
| tgdim01 28581 | In geometries of dimension... |
| tgifscgr 28582 | Inner five segment congrue... |
| tgcgrsub 28583 | Removing identical parts f... |
| iscgrg 28586 | The congruence property fo... |
| iscgrgd 28587 | The property for two seque... |
| iscgrglt 28588 | The property for two seque... |
| trgcgrg 28589 | The property for two trian... |
| trgcgr 28590 | Triangle congruence. (Con... |
| ercgrg 28591 | The shape congruence relat... |
| tgcgrxfr 28592 | A line segment can be divi... |
| cgr3id 28593 | Reflexivity law for three-... |
| cgr3simp1 28594 | Deduce segment congruence ... |
| cgr3simp2 28595 | Deduce segment congruence ... |
| cgr3simp3 28596 | Deduce segment congruence ... |
| cgr3swap12 28597 | Permutation law for three-... |
| cgr3swap23 28598 | Permutation law for three-... |
| cgr3swap13 28599 | Permutation law for three-... |
| cgr3rotr 28600 | Permutation law for three-... |
| cgr3rotl 28601 | Permutation law for three-... |
| trgcgrcom 28602 | Commutative law for three-... |
| cgr3tr 28603 | Transitivity law for three... |
| tgbtwnxfr 28604 | A condition for extending ... |
| tgcgr4 28605 | Two quadrilaterals to be c... |
| isismt 28608 | Property of being an isome... |
| ismot 28609 | Property of being an isome... |
| motcgr 28610 | Property of a motion: dist... |
| idmot 28611 | The identity is a motion. ... |
| motf1o 28612 | Motions are bijections. (... |
| motcl 28613 | Closure of motions. (Cont... |
| motco 28614 | The composition of two mot... |
| cnvmot 28615 | The converse of a motion i... |
| motplusg 28616 | The operation for motions ... |
| motgrp 28617 | The motions of a geometry ... |
| motcgrg 28618 | Property of a motion: dist... |
| motcgr3 28619 | Property of a motion: dist... |
| tglng 28620 | Lines of a Tarski Geometry... |
| tglnfn 28621 | Lines as functions. (Cont... |
| tglnunirn 28622 | Lines are sets of points. ... |
| tglnpt 28623 | Lines are sets of points. ... |
| tglngne 28624 | It takes two different poi... |
| tglngval 28625 | The line going through poi... |
| tglnssp 28626 | Lines are subset of the ge... |
| tgellng 28627 | Property of lying on the l... |
| tgcolg 28628 | We choose the notation ` (... |
| btwncolg1 28629 | Betweenness implies coline... |
| btwncolg2 28630 | Betweenness implies coline... |
| btwncolg3 28631 | Betweenness implies coline... |
| colcom 28632 | Swapping the points defini... |
| colrot1 28633 | Rotating the points defini... |
| colrot2 28634 | Rotating the points defini... |
| ncolcom 28635 | Swapping non-colinear poin... |
| ncolrot1 28636 | Rotating non-colinear poin... |
| ncolrot2 28637 | Rotating non-colinear poin... |
| tgdim01ln 28638 | In geometries of dimension... |
| ncoltgdim2 28639 | If there are three non-col... |
| lnxfr 28640 | Transfer law for colineari... |
| lnext 28641 | Extend a line with a missi... |
| tgfscgr 28642 | Congruence law for the gen... |
| lncgr 28643 | Congruence rule for lines.... |
| lnid 28644 | Identity law for points on... |
| tgidinside 28645 | Law for finding a point in... |
| tgbtwnconn1lem1 28646 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28647 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28648 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28649 | Connectivity law for betwe... |
| tgbtwnconn2 28650 | Another connectivity law f... |
| tgbtwnconn3 28651 | Inner connectivity law for... |
| tgbtwnconnln3 28652 | Derive colinearity from be... |
| tgbtwnconn22 28653 | Double connectivity law fo... |
| tgbtwnconnln1 28654 | Derive colinearity from be... |
| tgbtwnconnln2 28655 | Derive colinearity from be... |
| legval 28658 | Value of the less-than rel... |
| legov 28659 | Value of the less-than rel... |
| legov2 28660 | An equivalent definition o... |
| legid 28661 | Reflexivity of the less-th... |
| btwnleg 28662 | Betweenness implies less-t... |
| legtrd 28663 | Transitivity of the less-t... |
| legtri3 28664 | Equality from the less-tha... |
| legtrid 28665 | Trichotomy law for the les... |
| leg0 28666 | Degenerated (zero-length) ... |
| legeq 28667 | Deduce equality from "less... |
| legbtwn 28668 | Deduce betweenness from "l... |
| tgcgrsub2 28669 | Removing identical parts f... |
| ltgseg 28670 | The set ` E ` denotes the ... |
| ltgov 28671 | Strict "shorter than" geom... |
| legov3 28672 | An equivalent definition o... |
| legso 28673 | The "shorter than" relatio... |
| ishlg 28676 | Rays : Definition 6.1 of ... |
| hlcomb 28677 | The half-line relation com... |
| hlcomd 28678 | The half-line relation com... |
| hlne1 28679 | The half-line relation imp... |
| hlne2 28680 | The half-line relation imp... |
| hlln 28681 | The half-line relation imp... |
| hleqnid 28682 | The endpoint does not belo... |
| hlid 28683 | The half-line relation is ... |
| hltr 28684 | The half-line relation is ... |
| hlbtwn 28685 | Betweenness is a sufficien... |
| btwnhl1 28686 | Deduce half-line from betw... |
| btwnhl2 28687 | Deduce half-line from betw... |
| btwnhl 28688 | Swap betweenness for a hal... |
| lnhl 28689 | Either a point ` C ` on th... |
| hlcgrex 28690 | Construct a point on a hal... |
| hlcgreulem 28691 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28692 | The point constructed in ~... |
| btwnlng1 28693 | Betweenness implies coline... |
| btwnlng2 28694 | Betweenness implies coline... |
| btwnlng3 28695 | Betweenness implies coline... |
| lncom 28696 | Swapping the points defini... |
| lnrot1 28697 | Rotating the points defini... |
| lnrot2 28698 | Rotating the points defini... |
| ncolne1 28699 | Non-colinear points are di... |
| ncolne2 28700 | Non-colinear points are di... |
| tgisline 28701 | The property of being a pr... |
| tglnne 28702 | It takes two different poi... |
| tglndim0 28703 | There are no lines in dime... |
| tgelrnln 28704 | The property of being a pr... |
| tglineeltr 28705 | Transitivity law for lines... |
| tglineelsb2 28706 | If ` S ` lies on PQ , then... |
| tglinerflx1 28707 | Reflexivity law for line m... |
| tglinerflx2 28708 | Reflexivity law for line m... |
| tglinecom 28709 | Commutativity law for line... |
| tglinethru 28710 | If ` A ` is a line contain... |
| tghilberti1 28711 | There is a line through an... |
| tghilberti2 28712 | There is at most one line ... |
| tglinethrueu 28713 | There is a unique line goi... |
| tglnne0 28714 | A line ` A ` has at least ... |
| tglnpt2 28715 | Find a second point on a l... |
| tglineintmo 28716 | Two distinct lines interse... |
| tglineineq 28717 | Two distinct lines interse... |
| tglineneq 28718 | Given three non-colinear p... |
| tglineinteq 28719 | Two distinct lines interse... |
| ncolncol 28720 | Deduce non-colinearity fro... |
| coltr 28721 | A transitivity law for col... |
| coltr3 28722 | A transitivity law for col... |
| colline 28723 | Three points are colinear ... |
| tglowdim2l 28724 | Reformulation of the lower... |
| tglowdim2ln 28725 | There is always one point ... |
| mirreu3 28728 | Existential uniqueness of ... |
| mirval 28729 | Value of the point inversi... |
| mirfv 28730 | Value of the point inversi... |
| mircgr 28731 | Property of the image by t... |
| mirbtwn 28732 | Property of the image by t... |
| ismir 28733 | Property of the image by t... |
| mirf 28734 | Point inversion as functio... |
| mircl 28735 | Closure of the point inver... |
| mirmir 28736 | The point inversion functi... |
| mircom 28737 | Variation on ~ mirmir . (... |
| mirreu 28738 | Any point has a unique ant... |
| mireq 28739 | Equality deduction for poi... |
| mirinv 28740 | The only invariant point o... |
| mirne 28741 | Mirror of non-center point... |
| mircinv 28742 | The center point is invari... |
| mirf1o 28743 | The point inversion functi... |
| miriso 28744 | The point inversion functi... |
| mirbtwni 28745 | Point inversion preserves ... |
| mirbtwnb 28746 | Point inversion preserves ... |
| mircgrs 28747 | Point inversion preserves ... |
| mirmir2 28748 | Point inversion of a point... |
| mirmot 28749 | Point investion is a motio... |
| mirln 28750 | If two points are on the s... |
| mirln2 28751 | If a point and its mirror ... |
| mirconn 28752 | Point inversion of connect... |
| mirhl 28753 | If two points ` X ` and ` ... |
| mirbtwnhl 28754 | If the center of the point... |
| mirhl2 28755 | Deduce half-line relation ... |
| mircgrextend 28756 | Link congruence over a pai... |
| mirtrcgr 28757 | Point inversion of one poi... |
| mirauto 28758 | Point inversion preserves ... |
| miduniq 28759 | Uniqueness of the middle p... |
| miduniq1 28760 | Uniqueness of the middle p... |
| miduniq2 28761 | If two point inversions co... |
| colmid 28762 | Colinearity and equidistan... |
| symquadlem 28763 | Lemma of the symetrial qua... |
| krippenlem 28764 | Lemma for ~ krippen . We ... |
| krippen 28765 | Krippenlemma (German for c... |
| midexlem 28766 | Lemma for the existence of... |
| israg 28771 | Property for 3 points A, B... |
| ragcom 28772 | Commutative rule for right... |
| ragcol 28773 | The right angle property i... |
| ragmir 28774 | Right angle property is pr... |
| mirrag 28775 | Right angle is conserved b... |
| ragtrivb 28776 | Trivial right angle. Theo... |
| ragflat2 28777 | Deduce equality from two r... |
| ragflat 28778 | Deduce equality from two r... |
| ragtriva 28779 | Trivial right angle. Theo... |
| ragflat3 28780 | Right angle and colinearit... |
| ragcgr 28781 | Right angle and colinearit... |
| motrag 28782 | Right angles are preserved... |
| ragncol 28783 | Right angle implies non-co... |
| perpln1 28784 | Derive a line from perpend... |
| perpln2 28785 | Derive a line from perpend... |
| isperp 28786 | Property for 2 lines A, B ... |
| perpcom 28787 | The "perpendicular" relati... |
| perpneq 28788 | Two perpendicular lines ar... |
| isperp2 28789 | Property for 2 lines A, B,... |
| isperp2d 28790 | One direction of ~ isperp2... |
| ragperp 28791 | Deduce that two lines are ... |
| footexALT 28792 | Alternative version of ~ f... |
| footexlem1 28793 | Lemma for ~ footex . (Con... |
| footexlem2 28794 | Lemma for ~ footex . (Con... |
| footex 28795 | From a point ` C ` outside... |
| foot 28796 | From a point ` C ` outside... |
| footne 28797 | Uniqueness of the foot poi... |
| footeq 28798 | Uniqueness of the foot poi... |
| hlperpnel 28799 | A point on a half-line whi... |
| perprag 28800 | Deduce a right angle from ... |
| perpdragALT 28801 | Deduce a right angle from ... |
| perpdrag 28802 | Deduce a right angle from ... |
| colperp 28803 | Deduce a perpendicularity ... |
| colperpexlem1 28804 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28805 | Lemma for ~ colperpex . S... |
| colperpexlem3 28806 | Lemma for ~ colperpex . C... |
| colperpex 28807 | In dimension 2 and above, ... |
| mideulem2 28808 | Lemma for ~ opphllem , whi... |
| opphllem 28809 | Lemma 8.24 of [Schwabhause... |
| mideulem 28810 | Lemma for ~ mideu . We ca... |
| midex 28811 | Existence of the midpoint,... |
| mideu 28812 | Existence and uniqueness o... |
| islnopp 28813 | The property for two point... |
| islnoppd 28814 | Deduce that ` A ` and ` B ... |
| oppne1 28815 | Points lying on opposite s... |
| oppne2 28816 | Points lying on opposite s... |
| oppne3 28817 | Points lying on opposite s... |
| oppcom 28818 | Commutativity rule for "op... |
| opptgdim2 28819 | If two points opposite to ... |
| oppnid 28820 | The "opposite to a line" r... |
| opphllem1 28821 | Lemma for ~ opphl . (Cont... |
| opphllem2 28822 | Lemma for ~ opphl . Lemma... |
| opphllem3 28823 | Lemma for ~ opphl : We as... |
| opphllem4 28824 | Lemma for ~ opphl . (Cont... |
| opphllem5 28825 | Second part of Lemma 9.4 o... |
| opphllem6 28826 | First part of Lemma 9.4 of... |
| oppperpex 28827 | Restating ~ colperpex usin... |
| opphl 28828 | If two points ` A ` and ` ... |
| outpasch 28829 | Axiom of Pasch, outer form... |
| hlpasch 28830 | An application of the axio... |
| ishpg 28833 | Value of the half-plane re... |
| hpgbr 28834 | Half-planes : property for... |
| hpgne1 28835 | Points on the open half pl... |
| hpgne2 28836 | Points on the open half pl... |
| lnopp2hpgb 28837 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28838 | If two points lie on the o... |
| hpgerlem 28839 | Lemma for the proof that t... |
| hpgid 28840 | The half-plane relation is... |
| hpgcom 28841 | The half-plane relation co... |
| hpgtr 28842 | The half-plane relation is... |
| colopp 28843 | Opposite sides of a line f... |
| colhp 28844 | Half-plane relation for co... |
| hphl 28845 | If two points are on the s... |
| midf 28850 | Midpoint as a function. (... |
| midcl 28851 | Closure of the midpoint. ... |
| ismidb 28852 | Property of the midpoint. ... |
| midbtwn 28853 | Betweenness of midpoint. ... |
| midcgr 28854 | Congruence of midpoint. (... |
| midid 28855 | Midpoint of a null segment... |
| midcom 28856 | Commutativity rule for the... |
| mirmid 28857 | Point inversion preserves ... |
| lmieu 28858 | Uniqueness of the line mir... |
| lmif 28859 | Line mirror as a function.... |
| lmicl 28860 | Closure of the line mirror... |
| islmib 28861 | Property of the line mirro... |
| lmicom 28862 | The line mirroring functio... |
| lmilmi 28863 | Line mirroring is an invol... |
| lmireu 28864 | Any point has a unique ant... |
| lmieq 28865 | Equality deduction for lin... |
| lmiinv 28866 | The invariants of the line... |
| lmicinv 28867 | The mirroring line is an i... |
| lmimid 28868 | If we have a right angle, ... |
| lmif1o 28869 | The line mirroring functio... |
| lmiisolem 28870 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28871 | The line mirroring functio... |
| lmimot 28872 | Line mirroring is a motion... |
| hypcgrlem1 28873 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28874 | Lemma for ~ hypcgr , case ... |
| hypcgr 28875 | If the catheti of two righ... |
| lmiopp 28876 | Line mirroring produces po... |
| lnperpex 28877 | Existence of a perpendicul... |
| trgcopy 28878 | Triangle construction: a c... |
| trgcopyeulem 28879 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28880 | Triangle construction: a c... |
| iscgra 28883 | Property for two angles AB... |
| iscgra1 28884 | A special version of ~ isc... |
| iscgrad 28885 | Sufficient conditions for ... |
| cgrane1 28886 | Angles imply inequality. ... |
| cgrane2 28887 | Angles imply inequality. ... |
| cgrane3 28888 | Angles imply inequality. ... |
| cgrane4 28889 | Angles imply inequality. ... |
| cgrahl1 28890 | Angle congruence is indepe... |
| cgrahl2 28891 | Angle congruence is indepe... |
| cgracgr 28892 | First direction of proposi... |
| cgraid 28893 | Angle congruence is reflex... |
| cgraswap 28894 | Swap rays in a congruence ... |
| cgrcgra 28895 | Triangle congruence implie... |
| cgracom 28896 | Angle congruence commutes.... |
| cgratr 28897 | Angle congruence is transi... |
| flatcgra 28898 | Flat angles are congruent.... |
| cgraswaplr 28899 | Swap both side of angle co... |
| cgrabtwn 28900 | Angle congruence preserves... |
| cgrahl 28901 | Angle congruence preserves... |
| cgracol 28902 | Angle congruence preserves... |
| cgrancol 28903 | Angle congruence preserves... |
| dfcgra2 28904 | This is the full statement... |
| sacgr 28905 | Supplementary angles of co... |
| oacgr 28906 | Vertical angle theorem. V... |
| acopy 28907 | Angle construction. Theor... |
| acopyeu 28908 | Angle construction. Theor... |
| isinag 28912 | Property for point ` X ` t... |
| isinagd 28913 | Sufficient conditions for ... |
| inagflat 28914 | Any point lies in a flat a... |
| inagswap 28915 | Swap the order of the half... |
| inagne1 28916 | Deduce inequality from the... |
| inagne2 28917 | Deduce inequality from the... |
| inagne3 28918 | Deduce inequality from the... |
| inaghl 28919 | The "point lie in angle" r... |
| isleag 28921 | Geometrical "less than" pr... |
| isleagd 28922 | Sufficient condition for "... |
| leagne1 28923 | Deduce inequality from the... |
| leagne2 28924 | Deduce inequality from the... |
| leagne3 28925 | Deduce inequality from the... |
| leagne4 28926 | Deduce inequality from the... |
| cgrg3col4 28927 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28928 | First congruence theorem: ... |
| tgsas 28929 | First congruence theorem: ... |
| tgsas2 28930 | First congruence theorem: ... |
| tgsas3 28931 | First congruence theorem: ... |
| tgasa1 28932 | Second congruence theorem:... |
| tgasa 28933 | Second congruence theorem:... |
| tgsss1 28934 | Third congruence theorem: ... |
| tgsss2 28935 | Third congruence theorem: ... |
| tgsss3 28936 | Third congruence theorem: ... |
| dfcgrg2 28937 | Congruence for two triangl... |
| isoas 28938 | Congruence theorem for iso... |
| iseqlg 28941 | Property of a triangle bei... |
| iseqlgd 28942 | Condition for a triangle t... |
| f1otrgds 28943 | Convenient lemma for ~ f1o... |
| f1otrgitv 28944 | Convenient lemma for ~ f1o... |
| f1otrg 28945 | A bijection between bases ... |
| f1otrge 28946 | A bijection between bases ... |
| ttgval 28949 | Define a function to augme... |
| ttglem 28950 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 28951 | The base set of a subcompl... |
| ttgplusg 28952 | The addition operation of ... |
| ttgsub 28953 | The subtraction operation ... |
| ttgvsca 28954 | The scalar product of a su... |
| ttgds 28955 | The metric of a subcomplex... |
| ttgitvval 28956 | Betweenness for a subcompl... |
| ttgelitv 28957 | Betweenness for a subcompl... |
| ttgbtwnid 28958 | Any subcomplex module equi... |
| ttgcontlem1 28959 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28960 | Any metric space fulfills ... |
| cchhllem 28961 | Lemma for chlbas and chlvs... |
| elee 28968 | Membership in a Euclidean ... |
| mptelee 28969 | A condition for a mapping ... |
| mpteleeOLD 28970 | Obsolete version of ~ mpte... |
| eleenn 28971 | If ` A ` is in ` ( EE `` N... |
| eleei 28972 | The forward direction of ~... |
| eedimeq 28973 | A point belongs to at most... |
| brbtwn 28974 | The binary relation form o... |
| brcgr 28975 | The binary relation form o... |
| fveere 28976 | The function value of a po... |
| fveecn 28977 | The function value of a po... |
| eqeefv 28978 | Two points are equal iff t... |
| eqeelen 28979 | Two points are equal iff t... |
| brbtwn2 28980 | Alternate characterization... |
| colinearalglem1 28981 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28982 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28983 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28984 | Lemma for ~ colinearalg . ... |
| colinearalg 28985 | An algebraic characterizat... |
| eleesub 28986 | Membership of a subtractio... |
| eleesubd 28987 | Membership of a subtractio... |
| axdimuniq 28988 | The unique dimension axiom... |
| axcgrrflx 28989 | ` A ` is as far from ` B `... |
| axcgrtr 28990 | Congruence is transitive. ... |
| axcgrid 28991 | If there is no distance be... |
| axsegconlem1 28992 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28993 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28994 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28995 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28996 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28997 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28998 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28999 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 29000 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 29001 | Lemma for ~ axsegcon . Sh... |
| axsegcon 29002 | Any segment ` A B ` can be... |
| ax5seglem1 29003 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 29004 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 29005 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 29006 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 29007 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 29008 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 29009 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 29010 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 29011 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 29012 | Lemma for ~ ax5seg . Take... |
| ax5seg 29013 | The five segment axiom. T... |
| axbtwnid 29014 | Points are indivisible. T... |
| axpaschlem 29015 | Lemma for ~ axpasch . Set... |
| axpasch 29016 | The inner Pasch axiom. Ta... |
| axlowdimlem1 29017 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 29018 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 29019 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 29020 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 29021 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 29022 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 29023 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 29024 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 29025 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 29026 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 29027 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 29028 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 29029 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 29030 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 29031 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 29032 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 29033 | Lemma for ~ axlowdim . Es... |
| axlowdim1 29034 | The lower dimension axiom ... |
| axlowdim2 29035 | The lower two-dimensional ... |
| axlowdim 29036 | The general lower dimensio... |
| axeuclidlem 29037 | Lemma for ~ axeuclid . Ha... |
| axeuclid 29038 | Euclid's axiom. Take an a... |
| axcontlem1 29039 | Lemma for ~ axcont . Chan... |
| axcontlem2 29040 | Lemma for ~ axcont . The ... |
| axcontlem3 29041 | Lemma for ~ axcont . Give... |
| axcontlem4 29042 | Lemma for ~ axcont . Give... |
| axcontlem5 29043 | Lemma for ~ axcont . Comp... |
| axcontlem6 29044 | Lemma for ~ axcont . Stat... |
| axcontlem7 29045 | Lemma for ~ axcont . Give... |
| axcontlem8 29046 | Lemma for ~ axcont . A po... |
| axcontlem9 29047 | Lemma for ~ axcont . Give... |
| axcontlem10 29048 | Lemma for ~ axcont . Give... |
| axcontlem11 29049 | Lemma for ~ axcont . Elim... |
| axcontlem12 29050 | Lemma for ~ axcont . Elim... |
| axcont 29051 | The axiom of continuity. ... |
| eengv 29054 | The value of the Euclidean... |
| eengstr 29055 | The Euclidean geometry as ... |
| eengbas 29056 | The Base of the Euclidean ... |
| ebtwntg 29057 | The betweenness relation u... |
| ecgrtg 29058 | The congruence relation us... |
| elntg 29059 | The line definition in the... |
| elntg2 29060 | The line definition in the... |
| eengtrkg 29061 | The geometry structure for... |
| eengtrkge 29062 | The geometry structure for... |
| edgfid 29065 | Utility theorem: index-ind... |
| edgfndx 29066 | Index value of the ~ df-ed... |
| edgfndxnn 29067 | The index value of the edg... |
| edgfndxid 29068 | The value of the edge func... |
| basendxltedgfndx 29069 | The index value of the ` B... |
| basendxnedgfndx 29070 | The slots ` Base ` and ` .... |
| vtxval 29075 | The set of vertices of a g... |
| iedgval 29076 | The set of indexed edges o... |
| 1vgrex 29077 | A graph with at least one ... |
| opvtxval 29078 | The set of vertices of a g... |
| opvtxfv 29079 | The set of vertices of a g... |
| opvtxov 29080 | The set of vertices of a g... |
| opiedgval 29081 | The set of indexed edges o... |
| opiedgfv 29082 | The set of indexed edges o... |
| opiedgov 29083 | The set of indexed edges o... |
| opvtxfvi 29084 | The set of vertices of a g... |
| opiedgfvi 29085 | The set of indexed edges o... |
| funvtxdmge2val 29086 | The set of vertices of an ... |
| funiedgdmge2val 29087 | The set of indexed edges o... |
| funvtxdm2val 29088 | The set of vertices of an ... |
| funiedgdm2val 29089 | The set of indexed edges o... |
| funvtxval0 29090 | The set of vertices of an ... |
| basvtxval 29091 | The set of vertices of a g... |
| edgfiedgval 29092 | The set of indexed edges o... |
| funvtxval 29093 | The set of vertices of a g... |
| funiedgval 29094 | The set of indexed edges o... |
| structvtxvallem 29095 | Lemma for ~ structvtxval a... |
| structvtxval 29096 | The set of vertices of an ... |
| structiedg0val 29097 | The set of indexed edges o... |
| structgrssvtxlem 29098 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29099 | The set of vertices of a g... |
| structgrssiedg 29100 | The set of indexed edges o... |
| struct2grstr 29101 | A graph represented as an ... |
| struct2grvtx 29102 | The set of vertices of a g... |
| struct2griedg 29103 | The set of indexed edges o... |
| graop 29104 | Any representation of a gr... |
| grastruct 29105 | Any representation of a gr... |
| gropd 29106 | If any representation of a... |
| grstructd 29107 | If any representation of a... |
| gropeld 29108 | If any representation of a... |
| grstructeld 29109 | If any representation of a... |
| setsvtx 29110 | The vertices of a structur... |
| setsiedg 29111 | The (indexed) edges of a s... |
| snstrvtxval 29112 | The set of vertices of a g... |
| snstriedgval 29113 | The set of indexed edges o... |
| vtxval0 29114 | Degenerated case 1 for ver... |
| iedgval0 29115 | Degenerated case 1 for edg... |
| vtxvalsnop 29116 | Degenerated case 2 for ver... |
| iedgvalsnop 29117 | Degenerated case 2 for edg... |
| vtxval3sn 29118 | Degenerated case 3 for ver... |
| iedgval3sn 29119 | Degenerated case 3 for edg... |
| vtxvalprc 29120 | Degenerated case 4 for ver... |
| iedgvalprc 29121 | Degenerated case 4 for edg... |
| edgval 29124 | The edges of a graph. (Co... |
| iedgedg 29125 | An indexed edge is an edge... |
| edgopval 29126 | The edges of a graph repre... |
| edgov 29127 | The edges of a graph repre... |
| edgstruct 29128 | The edges of a graph repre... |
| edgiedgb 29129 | A set is an edge iff it is... |
| edg0iedg0 29130 | There is no edge in a grap... |
| isuhgr 29135 | The predicate "is an undir... |
| isushgr 29136 | The predicate "is an undir... |
| uhgrf 29137 | The edge function of an un... |
| ushgrf 29138 | The edge function of an un... |
| uhgrss 29139 | An edge is a subset of ver... |
| uhgreq12g 29140 | If two sets have the same ... |
| uhgrfun 29141 | The edge function of an un... |
| uhgrn0 29142 | An edge is a nonempty subs... |
| lpvtx 29143 | The endpoints of a loop (w... |
| ushgruhgr 29144 | An undirected simple hyper... |
| isuhgrop 29145 | The property of being an u... |
| uhgr0e 29146 | The empty graph, with vert... |
| uhgr0vb 29147 | The null graph, with no ve... |
| uhgr0 29148 | The null graph represented... |
| uhgrun 29149 | The union ` U ` of two (un... |
| uhgrunop 29150 | The union of two (undirect... |
| ushgrun 29151 | The union ` U ` of two (un... |
| ushgrunop 29152 | The union of two (undirect... |
| uhgrstrrepe 29153 | Replacing (or adding) the ... |
| incistruhgr 29154 | An _incidence structure_ `... |
| isupgr 29159 | The property of being an u... |
| wrdupgr 29160 | The property of being an u... |
| upgrf 29161 | The edge function of an un... |
| upgrfn 29162 | The edge function of an un... |
| upgrss 29163 | An edge is a subset of ver... |
| upgrn0 29164 | An edge is a nonempty subs... |
| upgrle 29165 | An edge of an undirected p... |
| upgrfi 29166 | An edge is a finite subset... |
| upgrex 29167 | An edge is an unordered pa... |
| upgrbi 29168 | Show that an unordered pai... |
| upgrop 29169 | A pseudograph represented ... |
| isumgr 29170 | The property of being an u... |
| isumgrs 29171 | The simplified property of... |
| wrdumgr 29172 | The property of being an u... |
| umgrf 29173 | The edge function of an un... |
| umgrfn 29174 | The edge function of an un... |
| umgredg2 29175 | An edge of a multigraph ha... |
| umgrbi 29176 | Show that an unordered pai... |
| upgruhgr 29177 | An undirected pseudograph ... |
| umgrupgr 29178 | An undirected multigraph i... |
| umgruhgr 29179 | An undirected multigraph i... |
| upgrle2 29180 | An edge of an undirected p... |
| umgrnloopv 29181 | In a multigraph, there is ... |
| umgredgprv 29182 | In a multigraph, an edge i... |
| umgrnloop 29183 | In a multigraph, there is ... |
| umgrnloop0 29184 | A multigraph has no loops.... |
| umgr0e 29185 | The empty graph, with vert... |
| upgr0e 29186 | The empty graph, with vert... |
| upgr1elem 29187 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29188 | A pseudograph with one edg... |
| upgr0eop 29189 | The empty graph, with vert... |
| upgr1eop 29190 | A pseudograph with one edg... |
| upgr0eopALT 29191 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29192 | Alternate proof of ~ upgr1... |
| upgrun 29193 | The union ` U ` of two pse... |
| upgrunop 29194 | The union of two pseudogra... |
| umgrun 29195 | The union ` U ` of two mul... |
| umgrunop 29196 | The union of two multigrap... |
| umgrislfupgrlem 29197 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29198 | A multigraph is a loop-fre... |
| lfgredgge2 29199 | An edge of a loop-free gra... |
| lfgrnloop 29200 | A loop-free graph has no l... |
| uhgredgiedgb 29201 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29202 | A hypergraph has no edges ... |
| uhgredgn0 29203 | An edge of a hypergraph is... |
| edguhgr 29204 | An edge of a hypergraph is... |
| uhgredgrnv 29205 | An edge of a hypergraph co... |
| uhgredgss 29206 | The set of edges of a hype... |
| upgredgss 29207 | The set of edges of a pseu... |
| umgredgss 29208 | The set of edges of a mult... |
| edgupgr 29209 | Properties of an edge of a... |
| edgumgr 29210 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29211 | In a hypergraph, a vertex ... |
| upgredg 29212 | For each edge in a pseudog... |
| umgredg 29213 | For each edge in a multigr... |
| upgrpredgv 29214 | An edge of a pseudograph a... |
| umgrpredgv 29215 | An edge of a multigraph al... |
| upgredg2vtx 29216 | For a vertex incident to a... |
| upgredgpr 29217 | If a proper pair (of verti... |
| edglnl 29218 | The edges incident with a ... |
| numedglnl 29219 | The number of edges incide... |
| umgredgne 29220 | An edge of a multigraph al... |
| umgrnloop2 29221 | A multigraph has no loops.... |
| umgredgnlp 29222 | An edge of a multigraph is... |
| isuspgr 29227 | The property of being a si... |
| isusgr 29228 | The property of being a si... |
| uspgrf 29229 | The edge function of a sim... |
| usgrf 29230 | The edge function of a sim... |
| isusgrs 29231 | The property of being a si... |
| usgrfs 29232 | The edge function of a sim... |
| usgrfun 29233 | The edge function of a sim... |
| usgredgss 29234 | The set of edges of a simp... |
| edgusgr 29235 | An edge of a simple graph ... |
| isuspgrop 29236 | The property of being an u... |
| isusgrop 29237 | The property of being an u... |
| usgrop 29238 | A simple graph represented... |
| isausgr 29239 | The property of an ordered... |
| ausgrusgrb 29240 | The equivalence of the def... |
| usgrausgri 29241 | A simple graph represented... |
| ausgrumgri 29242 | If an alternatively define... |
| ausgrusgri 29243 | The equivalence of the def... |
| usgrausgrb 29244 | The equivalence of the def... |
| usgredgop 29245 | An edge of a simple graph ... |
| usgrf1o 29246 | The edge function of a sim... |
| usgrf1 29247 | The edge function of a sim... |
| uspgrf1oedg 29248 | The edge function of a sim... |
| usgrss 29249 | An edge is a subset of ver... |
| uspgredgiedg 29250 | In a simple pseudograph, f... |
| uspgriedgedg 29251 | In a simple pseudograph, f... |
| uspgrushgr 29252 | A simple pseudograph is an... |
| uspgrupgr 29253 | A simple pseudograph is an... |
| uspgrupgrushgr 29254 | A graph is a simple pseudo... |
| usgruspgr 29255 | A simple graph is a simple... |
| usgrumgr 29256 | A simple graph is an undir... |
| usgrumgruspgr 29257 | A graph is a simple graph ... |
| usgruspgrb 29258 | A class is a simple graph ... |
| uspgruhgr 29259 | An undirected simple pseud... |
| usgrupgr 29260 | A simple graph is an undir... |
| usgruhgr 29261 | A simple graph is an undir... |
| usgrislfuspgr 29262 | A simple graph is a loop-f... |
| uspgrun 29263 | The union ` U ` of two sim... |
| uspgrunop 29264 | The union of two simple ps... |
| usgrun 29265 | The union ` U ` of two sim... |
| usgrunop 29266 | The union of two simple gr... |
| usgredg2 29267 | The value of the "edge fun... |
| usgredg2ALT 29268 | Alternate proof of ~ usgre... |
| usgredgprv 29269 | In a simple graph, an edge... |
| usgredgprvALT 29270 | Alternate proof of ~ usgre... |
| usgredgppr 29271 | An edge of a simple graph ... |
| usgrpredgv 29272 | An edge of a simple graph ... |
| edgssv2 29273 | An edge of a simple graph ... |
| usgredg 29274 | For each edge in a simple ... |
| usgrnloopv 29275 | In a simple graph, there i... |
| usgrnloopvALT 29276 | Alternate proof of ~ usgrn... |
| usgrnloop 29277 | In a simple graph, there i... |
| usgrnloopALT 29278 | Alternate proof of ~ usgrn... |
| usgrnloop0 29279 | A simple graph has no loop... |
| usgrnloop0ALT 29280 | Alternate proof of ~ usgrn... |
| usgredgne 29281 | An edge of a simple graph ... |
| usgrf1oedg 29282 | The edge function of a sim... |
| uhgr2edg 29283 | If a vertex is adjacent to... |
| umgr2edg 29284 | If a vertex is adjacent to... |
| usgr2edg 29285 | If a vertex is adjacent to... |
| umgr2edg1 29286 | If a vertex is adjacent to... |
| usgr2edg1 29287 | If a vertex is adjacent to... |
| umgrvad2edg 29288 | If a vertex is adjacent to... |
| umgr2edgneu 29289 | If a vertex is adjacent to... |
| usgrsizedg 29290 | In a simple graph, the siz... |
| usgredg3 29291 | The value of the "edge fun... |
| usgredg4 29292 | For a vertex incident to a... |
| usgredgreu 29293 | For a vertex incident to a... |
| usgredg2vtx 29294 | For a vertex incident to a... |
| uspgredg2vtxeu 29295 | For a vertex incident to a... |
| usgredg2vtxeu 29296 | For a vertex incident to a... |
| usgredg2vtxeuALT 29297 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29298 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29299 | In a simple pseudograph, t... |
| usgredg2vlem1 29300 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29301 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29302 | In a simple graph, the map... |
| usgriedgleord 29303 | Alternate version of ~ usg... |
| ushgredgedg 29304 | In a simple hypergraph the... |
| usgredgedg 29305 | In a simple graph there is... |
| ushgredgedgloop 29306 | In a simple hypergraph the... |
| uspgredgleord 29307 | In a simple pseudograph th... |
| usgredgleord 29308 | In a simple graph the numb... |
| usgredgleordALT 29309 | Alternate proof for ~ usgr... |
| usgrstrrepe 29310 | Replacing (or adding) the ... |
| usgr0e 29311 | The empty graph, with vert... |
| usgr0vb 29312 | The null graph, with no ve... |
| uhgr0v0e 29313 | The null graph, with no ve... |
| uhgr0vsize0 29314 | The size of a hypergraph w... |
| uhgr0edgfi 29315 | A graph of order 0 (i.e. w... |
| usgr0v 29316 | The null graph, with no ve... |
| uhgr0vusgr 29317 | The null graph, with no ve... |
| usgr0 29318 | The null graph represented... |
| uspgr1e 29319 | A simple pseudograph with ... |
| usgr1e 29320 | A simple graph with one ed... |
| usgr0eop 29321 | The empty graph, with vert... |
| uspgr1eop 29322 | A simple pseudograph with ... |
| uspgr1ewop 29323 | A simple pseudograph with ... |
| uspgr1v1eop 29324 | A simple pseudograph with ... |
| usgr1eop 29325 | A simple graph with (at le... |
| uspgr2v1e2w 29326 | A simple pseudograph with ... |
| usgr2v1e2w 29327 | A simple graph with two ve... |
| edg0usgr 29328 | A class without edges is a... |
| lfuhgr1v0e 29329 | A loop-free hypergraph wit... |
| usgr1vr 29330 | A simple graph with one ve... |
| usgr1v 29331 | A class with one (or no) v... |
| usgr1v0edg 29332 | A class with one (or no) v... |
| usgrexmpldifpr 29333 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29334 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29335 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29336 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29337 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29338 | ` G ` is a simple graph of... |
| griedg0prc 29339 | The class of empty graphs ... |
| griedg0ssusgr 29340 | The class of all simple gr... |
| usgrprc 29341 | The class of simple graphs... |
| relsubgr 29344 | The class of the subgraph ... |
| subgrv 29345 | If a class is a subgraph o... |
| issubgr 29346 | The property of a set to b... |
| issubgr2 29347 | The property of a set to b... |
| subgrprop 29348 | The properties of a subgra... |
| subgrprop2 29349 | The properties of a subgra... |
| uhgrissubgr 29350 | The property of a hypergra... |
| subgrprop3 29351 | The properties of a subgra... |
| egrsubgr 29352 | An empty graph consisting ... |
| 0grsubgr 29353 | The null graph (represente... |
| 0uhgrsubgr 29354 | The null graph (as hypergr... |
| uhgrsubgrself 29355 | A hypergraph is a subgraph... |
| subgrfun 29356 | The edge function of a sub... |
| subgruhgrfun 29357 | The edge function of a sub... |
| subgreldmiedg 29358 | An element of the domain o... |
| subgruhgredgd 29359 | An edge of a subgraph of a... |
| subumgredg2 29360 | An edge of a subgraph of a... |
| subuhgr 29361 | A subgraph of a hypergraph... |
| subupgr 29362 | A subgraph of a pseudograp... |
| subumgr 29363 | A subgraph of a multigraph... |
| subusgr 29364 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29365 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29366 | A spanning subgraph ` S ` ... |
| uhgrspan 29367 | A spanning subgraph ` S ` ... |
| upgrspan 29368 | A spanning subgraph ` S ` ... |
| umgrspan 29369 | A spanning subgraph ` S ` ... |
| usgrspan 29370 | A spanning subgraph ` S ` ... |
| uhgrspanop 29371 | A spanning subgraph of a h... |
| upgrspanop 29372 | A spanning subgraph of a p... |
| umgrspanop 29373 | A spanning subgraph of a m... |
| usgrspanop 29374 | A spanning subgraph of a s... |
| uhgrspan1lem1 29375 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29376 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29377 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29378 | The induced subgraph ` S `... |
| upgrreslem 29379 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29380 | Lemma for ~ umgrres and ~ ... |
| upgrres 29381 | A subgraph obtained by rem... |
| umgrres 29382 | A subgraph obtained by rem... |
| usgrres 29383 | A subgraph obtained by rem... |
| upgrres1lem1 29384 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29385 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29386 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29387 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29388 | A pseudograph obtained by ... |
| umgrres1 29389 | A multigraph obtained by r... |
| usgrres1 29390 | Restricting a simple graph... |
| isfusgr 29393 | The property of being a fi... |
| fusgrvtxfi 29394 | A finite simple graph has ... |
| isfusgrf1 29395 | The property of being a fi... |
| isfusgrcl 29396 | The property of being a fi... |
| fusgrusgr 29397 | A finite simple graph is a... |
| opfusgr 29398 | A finite simple graph repr... |
| usgredgffibi 29399 | The number of edges in a s... |
| fusgredgfi 29400 | In a finite simple graph t... |
| usgr1v0e 29401 | The size of a (finite) sim... |
| usgrfilem 29402 | In a finite simple graph, ... |
| fusgrfisbase 29403 | Induction base for ~ fusgr... |
| fusgrfisstep 29404 | Induction step in ~ fusgrf... |
| fusgrfis 29405 | A finite simple graph is o... |
| fusgrfupgrfs 29406 | A finite simple graph is a... |
| nbgrprc0 29409 | The set of neighbors is em... |
| nbgrcl 29410 | If a class ` X ` has at le... |
| nbgrval 29411 | The set of neighbors of a ... |
| dfnbgr2 29412 | Alternate definition of th... |
| dfnbgr3 29413 | Alternate definition of th... |
| nbgrnvtx0 29414 | If a class ` X ` is not a ... |
| nbgrel 29415 | Characterization of a neig... |
| nbgrisvtx 29416 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29417 | The neighbors of a vertex ... |
| nbuhgr 29418 | The set of neighbors of a ... |
| nbupgr 29419 | The set of neighbors of a ... |
| nbupgrel 29420 | A neighbor of a vertex in ... |
| nbumgrvtx 29421 | The set of neighbors of a ... |
| nbumgr 29422 | The set of neighbors of an... |
| nbusgrvtx 29423 | The set of neighbors of a ... |
| nbusgr 29424 | The set of neighbors of an... |
| nbgr2vtx1edg 29425 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29426 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29427 | If a hypergraph has two ve... |
| nbusgreledg 29428 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29429 | A vertex which is not endp... |
| nbgr0vtx 29430 | In a null graph (with no v... |
| nbgr0edglem 29431 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29432 | In an empty graph (with no... |
| nbgr1vtx 29433 | In a graph with one vertex... |
| nbgrnself 29434 | A vertex in a graph is not... |
| nbgrnself2 29435 | A class ` X ` is not a nei... |
| nbgrssovtx 29436 | The neighbors of a vertex ... |
| nbgrssvwo2 29437 | The neighbors of a vertex ... |
| nbgrsym 29438 | In a graph, the neighborho... |
| nbupgrres 29439 | The neighborhood of a vert... |
| usgrnbcnvfv 29440 | Applying the edge function... |
| nbusgredgeu 29441 | For each neighbor of a ver... |
| edgnbusgreu 29442 | For each edge incident to ... |
| nbusgredgeu0 29443 | For each neighbor of a ver... |
| nbusgrf1o0 29444 | The mapping of neighbors o... |
| nbusgrf1o1 29445 | The set of neighbors of a ... |
| nbusgrf1o 29446 | The set of neighbors of a ... |
| nbedgusgr 29447 | The number of neighbors of... |
| edgusgrnbfin 29448 | The number of neighbors of... |
| nbusgrfi 29449 | The class of neighbors of ... |
| nbfiusgrfi 29450 | The class of neighbors of ... |
| hashnbusgrnn0 29451 | The number of neighbors of... |
| nbfusgrlevtxm1 29452 | The number of neighbors of... |
| nbfusgrlevtxm2 29453 | If there is a vertex which... |
| nbusgrvtxm1 29454 | If the number of neighbors... |
| nb3grprlem1 29455 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29456 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29457 | The neighbors of a vertex ... |
| nb3grpr2 29458 | The neighbors of a vertex ... |
| nb3gr2nb 29459 | If the neighbors of two ve... |
| uvtxval 29462 | The set of all universal v... |
| uvtxel 29463 | A universal vertex, i.e. a... |
| uvtxisvtx 29464 | A universal vertex is a ve... |
| uvtxssvtx 29465 | The set of the universal v... |
| vtxnbuvtx 29466 | A universal vertex has all... |
| uvtxnbgrss 29467 | A universal vertex has all... |
| uvtxnbgrvtx 29468 | A universal vertex is neig... |
| uvtx0 29469 | There is no universal vert... |
| isuvtx 29470 | The set of all universal v... |
| uvtxel1 29471 | Characterization of a univ... |
| uvtx01vtx 29472 | If a graph/class has no ed... |
| uvtx2vtx1edg 29473 | If a graph has two vertice... |
| uvtx2vtx1edgb 29474 | If a hypergraph has two ve... |
| uvtxnbgr 29475 | A universal vertex has all... |
| uvtxnbgrb 29476 | A vertex is universal iff ... |
| uvtxusgr 29477 | The set of all universal v... |
| uvtxusgrel 29478 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29479 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29480 | If the number of neighbors... |
| uvtxnbvtxm1 29481 | A universal vertex has ` n... |
| nbupgruvtxres 29482 | The neighborhood of a univ... |
| uvtxupgrres 29483 | A universal vertex is univ... |
| cplgruvtxb 29488 | A graph ` G ` is complete ... |
| prcliscplgr 29489 | A proper class (representi... |
| iscplgr 29490 | The property of being a co... |
| iscplgrnb 29491 | A graph is complete iff al... |
| iscplgredg 29492 | A graph ` G ` is complete ... |
| iscusgr 29493 | The property of being a co... |
| cusgrusgr 29494 | A complete simple graph is... |
| cusgrcplgr 29495 | A complete simple graph is... |
| iscusgrvtx 29496 | A simple graph is complete... |
| cusgruvtxb 29497 | A simple graph is complete... |
| iscusgredg 29498 | A simple graph is complete... |
| cusgredg 29499 | In a complete simple graph... |
| cplgr0 29500 | The null graph (with no ve... |
| cusgr0 29501 | The null graph (with no ve... |
| cplgr0v 29502 | A null graph (with no vert... |
| cusgr0v 29503 | A graph with no vertices a... |
| cplgr1vlem 29504 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29505 | A graph with one vertex is... |
| cusgr1v 29506 | A graph with one vertex an... |
| cplgr2v 29507 | An undirected hypergraph w... |
| cplgr2vpr 29508 | An undirected hypergraph w... |
| nbcplgr 29509 | In a complete graph, each ... |
| cplgr3v 29510 | A pseudograph with three (... |
| cusgr3vnbpr 29511 | The neighbors of a vertex ... |
| cplgrop 29512 | A complete graph represent... |
| cusgrop 29513 | A complete simple graph re... |
| cusgrexilem1 29514 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29515 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29516 | An arbitrary set regarded ... |
| cusgrexilem2 29517 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29518 | An arbitrary set ` V ` reg... |
| cusgrexg 29519 | For each set there is a se... |
| structtousgr 29520 | Any (extensible) structure... |
| structtocusgr 29521 | Any (extensible) structure... |
| cffldtocusgr 29522 | The field of complex numbe... |
| cffldtocusgrOLD 29523 | Obsolete version of ~ cffl... |
| cusgrres 29524 | Restricting a complete sim... |
| cusgrsizeindb0 29525 | Base case of the induction... |
| cusgrsizeindb1 29526 | Base case of the induction... |
| cusgrsizeindslem 29527 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29528 | Part 1 of induction step i... |
| cusgrsize2inds 29529 | Induction step in ~ cusgrs... |
| cusgrsize 29530 | The size of a finite compl... |
| cusgrfilem1 29531 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29532 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29533 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29534 | If the size of a complete ... |
| usgredgsscusgredg 29535 | A simple graph is a subgra... |
| usgrsscusgr 29536 | A simple graph is a subgra... |
| sizusglecusglem1 29537 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29538 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29539 | The size of a simple graph... |
| fusgrmaxsize 29540 | The maximum size of a fini... |
| vtxdgfval 29543 | The value of the vertex de... |
| vtxdgval 29544 | The degree of a vertex. (... |
| vtxdgfival 29545 | The degree of a vertex for... |
| vtxdgop 29546 | The vertex degree expresse... |
| vtxdgf 29547 | The vertex degree function... |
| vtxdgelxnn0 29548 | The degree of a vertex is ... |
| vtxdg0v 29549 | The degree of a vertex in ... |
| vtxdg0e 29550 | The degree of a vertex in ... |
| vtxdgfisnn0 29551 | The degree of a vertex in ... |
| vtxdgfisf 29552 | The vertex degree function... |
| vtxdeqd 29553 | Equality theorem for the v... |
| vtxduhgr0e 29554 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29555 | The degree of the vertex i... |
| vdumgr0 29556 | A vertex in a multigraph h... |
| vtxdun 29557 | The degree of a vertex in ... |
| vtxdfiun 29558 | The degree of a vertex in ... |
| vtxduhgrun 29559 | The degree of a vertex in ... |
| vtxduhgrfiun 29560 | The degree of a vertex in ... |
| vtxdlfgrval 29561 | The value of the vertex de... |
| vtxdumgrval 29562 | The value of the vertex de... |
| vtxdusgrval 29563 | The value of the vertex de... |
| vtxd0nedgb 29564 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29565 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29566 | The value of the vertex de... |
| vtxdusgrfvedg 29567 | The value of the vertex de... |
| vtxduhgr0nedg 29568 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29569 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29570 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29571 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29572 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29573 | The vertex degree function... |
| vtxdgfusgr 29574 | In a finite simple graph, ... |
| fusgrn0degnn0 29575 | In a nonempty, finite grap... |
| 1loopgruspgr 29576 | A graph with one edge whic... |
| 1loopgredg 29577 | The set of edges in a grap... |
| 1loopgrnb0 29578 | In a graph (simple pseudog... |
| 1loopgrvd2 29579 | The vertex degree of a one... |
| 1loopgrvd0 29580 | The vertex degree of a one... |
| 1hevtxdg0 29581 | The vertex degree of verte... |
| 1hevtxdg1 29582 | The vertex degree of verte... |
| 1hegrvtxdg1 29583 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29584 | The vertex degree of a gra... |
| 1egrvtxdg1 29585 | The vertex degree of a one... |
| 1egrvtxdg1r 29586 | The vertex degree of a one... |
| 1egrvtxdg0 29587 | The vertex degree of a one... |
| p1evtxdeqlem 29588 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29589 | If an edge ` E ` which doe... |
| p1evtxdp1 29590 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29591 | The set of vertices in a g... |
| uspgrloopvtxel 29592 | A vertex in a graph (simpl... |
| uspgrloopiedg 29593 | The set of edges in a grap... |
| uspgrloopedg 29594 | The set of edges in a grap... |
| uspgrloopnb0 29595 | In a graph (simple pseudog... |
| uspgrloopvd2 29596 | The vertex degree of a one... |
| umgr2v2evtx 29597 | The set of vertices in a m... |
| umgr2v2evtxel 29598 | A vertex in a multigraph w... |
| umgr2v2eiedg 29599 | The edge function in a mul... |
| umgr2v2eedg 29600 | The set of edges in a mult... |
| umgr2v2e 29601 | A multigraph with two edge... |
| umgr2v2enb1 29602 | In a multigraph with two e... |
| umgr2v2evd2 29603 | In a multigraph with two e... |
| hashnbusgrvd 29604 | In a simple graph, the num... |
| usgruvtxvdb 29605 | In a finite simple graph w... |
| vdiscusgrb 29606 | A finite simple graph with... |
| vdiscusgr 29607 | In a finite complete simpl... |
| vtxdusgradjvtx 29608 | The degree of a vertex in ... |
| usgrvd0nedg 29609 | If a vertex in a simple gr... |
| uhgrvd00 29610 | If every vertex in a hyper... |
| usgrvd00 29611 | If every vertex in a simpl... |
| vdegp1ai 29612 | The induction step for a v... |
| vdegp1bi 29613 | The induction step for a v... |
| vdegp1ci 29614 | The induction step for a v... |
| vtxdginducedm1lem1 29615 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29616 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29617 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29618 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29619 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29620 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29621 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29622 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29623 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29624 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29625 | Induction step of ~ finsum... |
| finsumvtxdg2size 29626 | The sum of the degrees of ... |
| fusgr1th 29627 | The sum of the degrees of ... |
| finsumvtxdgeven 29628 | The sum of the degrees of ... |
| vtxdgoddnumeven 29629 | The number of vertices of ... |
| fusgrvtxdgonume 29630 | The number of vertices of ... |
| isrgr 29635 | The property of a class be... |
| rgrprop 29636 | The properties of a k-regu... |
| isrusgr 29637 | The property of being a k-... |
| rusgrprop 29638 | The properties of a k-regu... |
| rusgrrgr 29639 | A k-regular simple graph i... |
| rusgrusgr 29640 | A k-regular simple graph i... |
| finrusgrfusgr 29641 | A finite regular simple gr... |
| isrusgr0 29642 | The property of being a k-... |
| rusgrprop0 29643 | The properties of a k-regu... |
| usgreqdrusgr 29644 | If all vertices in a simpl... |
| fusgrregdegfi 29645 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29646 | If all vertices in a nonem... |
| frusgrnn0 29647 | In a nonempty finite k-reg... |
| 0edg0rgr 29648 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29649 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29650 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29651 | A simple graph is 0-regula... |
| 0vtxrgr 29652 | A null graph (with no vert... |
| 0vtxrusgr 29653 | A graph with no vertices a... |
| 0uhgrrusgr 29654 | The null graph as hypergra... |
| 0grrusgr 29655 | The null graph represented... |
| 0grrgr 29656 | The null graph represented... |
| cusgrrusgr 29657 | A complete simple graph wi... |
| cusgrm1rusgr 29658 | A finite simple graph with... |
| rusgrpropnb 29659 | The properties of a k-regu... |
| rusgrpropedg 29660 | The properties of a k-regu... |
| rusgrpropadjvtx 29661 | The properties of a k-regu... |
| rusgrnumwrdl2 29662 | In a k-regular simple grap... |
| rusgr1vtxlem 29663 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29664 | If a k-regular simple grap... |
| rgrusgrprc 29665 | The class of 0-regular sim... |
| rusgrprc 29666 | The class of 0-regular sim... |
| rgrprc 29667 | The class of 0-regular gra... |
| rgrprcx 29668 | The class of 0-regular gra... |
| rgrx0ndm 29669 | 0 is not in the domain of ... |
| rgrx0nd 29670 | The potentially alternativ... |
| ewlksfval 29677 | The set of s-walks of edge... |
| isewlk 29678 | Conditions for a function ... |
| ewlkprop 29679 | Properties of an s-walk of... |
| ewlkinedg 29680 | The intersection (common v... |
| ewlkle 29681 | An s-walk of edges is also... |
| upgrewlkle2 29682 | In a pseudograph, there is... |
| wkslem1 29683 | Lemma 1 for walks to subst... |
| wkslem2 29684 | Lemma 2 for walks to subst... |
| wksfval 29685 | The set of walks (in an un... |
| iswlk 29686 | Properties of a pair of fu... |
| wlkprop 29687 | Properties of a walk. (Co... |
| wlkv 29688 | The classes involved in a ... |
| iswlkg 29689 | Generalization of ~ iswlk ... |
| wlkf 29690 | The mapping enumerating th... |
| wlkcl 29691 | A walk has length ` # ( F ... |
| wlkp 29692 | The mapping enumerating th... |
| wlkpwrd 29693 | The sequence of vertices o... |
| wlklenvp1 29694 | The number of vertices of ... |
| wksv 29695 | The class of walks is a se... |
| wlkn0 29696 | The sequence of vertices o... |
| wlklenvm1 29697 | The number of edges of a w... |
| ifpsnprss 29698 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29699 | Each pair of adjacent vert... |
| wlkvtxiedg 29700 | The vertices of a walk are... |
| relwlk 29701 | The set ` ( Walks `` G ) `... |
| wlkvv 29702 | If there is at least one w... |
| wlkop 29703 | A walk is an ordered pair.... |
| wlkcpr 29704 | A walk as class with two c... |
| wlk2f 29705 | If there is a walk ` W ` t... |
| wlkcomp 29706 | A walk expressed by proper... |
| wlkcompim 29707 | Implications for the prope... |
| wlkelwrd 29708 | The components of a walk a... |
| wlkeq 29709 | Conditions for two walks (... |
| edginwlk 29710 | The value of the edge func... |
| upgredginwlk 29711 | The value of the edge func... |
| iedginwlk 29712 | The value of the edge func... |
| wlkl1loop 29713 | A walk of length 1 from a ... |
| wlk1walk 29714 | A walk is a 1-walk "on the... |
| wlk1ewlk 29715 | A walk is an s-walk "on th... |
| upgriswlk 29716 | Properties of a pair of fu... |
| upgrwlkedg 29717 | The edges of a walk in a p... |
| upgrwlkcompim 29718 | Implications for the prope... |
| wlkvtxedg 29719 | The vertices of a walk are... |
| upgrwlkvtxedg 29720 | The pairs of connected ver... |
| uspgr2wlkeq 29721 | Conditions for two walks w... |
| uspgr2wlkeq2 29722 | Conditions for two walks w... |
| uspgr2wlkeqi 29723 | Conditions for two walks w... |
| umgrwlknloop 29724 | In a multigraph, each walk... |
| wlkv0 29725 | If there is a walk in the ... |
| g0wlk0 29726 | There is no walk in a null... |
| 0wlk0 29727 | There is no walk for the e... |
| wlk0prc 29728 | There is no walk in a null... |
| wlklenvclwlk 29729 | The number of vertices in ... |
| wlkson 29730 | The set of walks between t... |
| iswlkon 29731 | Properties of a pair of fu... |
| wlkonprop 29732 | Properties of a walk betwe... |
| wlkpvtx 29733 | A walk connects vertices. ... |
| wlkepvtx 29734 | The endpoints of a walk ar... |
| wlkoniswlk 29735 | A walk between two vertice... |
| wlkonwlk 29736 | A walk is a walk between i... |
| wlkonwlk1l 29737 | A walk is a walk from its ... |
| wlksoneq1eq2 29738 | Two walks with identical s... |
| wlkonl1iedg 29739 | If there is a walk between... |
| wlkon2n0 29740 | The length of a walk betwe... |
| 2wlklem 29741 | Lemma for theorems for wal... |
| upgr2wlk 29742 | Properties of a pair of fu... |
| wlkreslem 29743 | Lemma for ~ wlkres . (Con... |
| wlkres 29744 | The restriction ` <. H , Q... |
| redwlklem 29745 | Lemma for ~ redwlk . (Con... |
| redwlk 29746 | A walk ending at the last ... |
| wlkp1lem1 29747 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29748 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29749 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29750 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29751 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29752 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29753 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29754 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29755 | Append one path segment (e... |
| wlkdlem1 29756 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29757 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29758 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29759 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29760 | Two words representing a w... |
| lfgrwlkprop 29761 | Two adjacent vertices in a... |
| lfgriswlk 29762 | Conditions for a pair of f... |
| lfgrwlknloop 29763 | In a loop-free graph, each... |
| reltrls 29768 | The set ` ( Trails `` G ) ... |
| trlsfval 29769 | The set of trails (in an u... |
| istrl 29770 | Conditions for a pair of c... |
| trliswlk 29771 | A trail is a walk. (Contr... |
| trlf1 29772 | The enumeration ` F ` of a... |
| trlreslem 29773 | Lemma for ~ trlres . Form... |
| trlres 29774 | The restriction ` <. H , Q... |
| upgrtrls 29775 | The set of trails in a pse... |
| upgristrl 29776 | Properties of a pair of fu... |
| upgrf1istrl 29777 | Properties of a pair of a ... |
| wksonproplem 29778 | Lemma for theorems for pro... |
| trlsonfval 29779 | The set of trails between ... |
| istrlson 29780 | Properties of a pair of fu... |
| trlsonprop 29781 | Properties of a trail betw... |
| trlsonistrl 29782 | A trail between two vertic... |
| trlsonwlkon 29783 | A trail between two vertic... |
| trlontrl 29784 | A trail is a trail between... |
| relpths 29793 | The set ` ( Paths `` G ) `... |
| pthsfval 29794 | The set of paths (in an un... |
| spthsfval 29795 | The set of simple paths (i... |
| ispth 29796 | Conditions for a pair of c... |
| isspth 29797 | Conditions for a pair of c... |
| pthistrl 29798 | A path is a trail (in an u... |
| spthispth 29799 | A simple path is a path (i... |
| pthiswlk 29800 | A path is a walk (in an un... |
| spthiswlk 29801 | A simple path is a walk (i... |
| pthdivtx 29802 | The inner vertices of a pa... |
| pthdadjvtx 29803 | The adjacent vertices of a... |
| dfpth2 29804 | Alternate definition for a... |
| pthdifv 29805 | The vertices of a path are... |
| 2pthnloop 29806 | A path of length at least ... |
| upgr2pthnlp 29807 | A path of length at least ... |
| spthdifv 29808 | The vertices of a simple p... |
| spthdep 29809 | A simple path (at least of... |
| pthdepisspth 29810 | A path with different star... |
| upgrwlkdvdelem 29811 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29812 | In a pseudograph, all edge... |
| upgrspthswlk 29813 | The set of simple paths in... |
| upgrwlkdvspth 29814 | A walk consisting of diffe... |
| pthsonfval 29815 | The set of paths between t... |
| spthson 29816 | The set of simple paths be... |
| ispthson 29817 | Properties of a pair of fu... |
| isspthson 29818 | Properties of a pair of fu... |
| pthsonprop 29819 | Properties of a path betwe... |
| spthonprop 29820 | Properties of a simple pat... |
| pthonispth 29821 | A path between two vertice... |
| pthontrlon 29822 | A path between two vertice... |
| pthonpth 29823 | A path is a path between i... |
| isspthonpth 29824 | A pair of functions is a s... |
| spthonisspth 29825 | A simple path between to v... |
| spthonpthon 29826 | A simple path between two ... |
| spthonepeq 29827 | The endpoints of a simple ... |
| uhgrwkspthlem1 29828 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29829 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29830 | Any walk of length 1 betwe... |
| usgr2wlkneq 29831 | The vertices and edges are... |
| usgr2wlkspthlem1 29832 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29833 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29834 | In a simple graph, any wal... |
| usgr2trlncl 29835 | In a simple graph, any tra... |
| usgr2trlspth 29836 | In a simple graph, any tra... |
| usgr2pthspth 29837 | In a simple graph, any pat... |
| usgr2pthlem 29838 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29839 | In a simple graph, there i... |
| usgr2pth0 29840 | In a simply graph, there i... |
| pthdlem1 29841 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29842 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29843 | Lemma 2 for ~ pthd . (Con... |
| pthd 29844 | Two words representing a t... |
| clwlks 29847 | The set of closed walks (i... |
| isclwlk 29848 | A pair of functions repres... |
| clwlkiswlk 29849 | A closed walk is a walk (i... |
| clwlkwlk 29850 | Closed walks are walks (in... |
| clwlkswks 29851 | Closed walks are walks (in... |
| isclwlke 29852 | Properties of a pair of fu... |
| isclwlkupgr 29853 | Properties of a pair of fu... |
| clwlkcomp 29854 | A closed walk expressed by... |
| clwlkcompim 29855 | Implications for the prope... |
| upgrclwlkcompim 29856 | Implications for the prope... |
| clwlkcompbp 29857 | Basic properties of the co... |
| clwlkl1loop 29858 | A closed walk of length 1 ... |
| crcts 29863 | The set of circuits (in an... |
| cycls 29864 | The set of cycles (in an u... |
| iscrct 29865 | Sufficient and necessary c... |
| iscycl 29866 | Sufficient and necessary c... |
| crctprop 29867 | The properties of a circui... |
| cyclprop 29868 | The properties of a cycle:... |
| crctisclwlk 29869 | A circuit is a closed walk... |
| crctistrl 29870 | A circuit is a trail. (Co... |
| crctiswlk 29871 | A circuit is a walk. (Con... |
| cyclispth 29872 | A cycle is a path. (Contr... |
| cycliswlk 29873 | A cycle is a walk. (Contr... |
| cycliscrct 29874 | A cycle is a circuit. (Co... |
| cyclnumvtx 29875 | The number of vertices of ... |
| cyclnspth 29876 | A (non-trivial) cycle is n... |
| pthisspthorcycl 29877 | A path is either a simple ... |
| pthspthcyc 29878 | A pair ` <. F , P >. ` rep... |
| cyclispthon 29879 | A cycle is a path starting... |
| lfgrn1cycl 29880 | In a loop-free graph there... |
| usgr2trlncrct 29881 | In a simple graph, any tra... |
| umgrn1cycl 29882 | In a multigraph graph (wit... |
| uspgrn2crct 29883 | In a simple pseudograph th... |
| usgrn2cycl 29884 | In a simple graph there ar... |
| crctcshwlkn0lem1 29885 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29886 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29887 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29888 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29889 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29890 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29891 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29892 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29893 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29894 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29895 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29896 | Cyclically shifting the in... |
| crctcshwlk 29897 | Cyclically shifting the in... |
| crctcshtrl 29898 | Cyclically shifting the in... |
| crctcsh 29899 | Cyclically shifting the in... |
| wwlks 29910 | The set of walks (in an un... |
| iswwlks 29911 | A word over the set of ver... |
| wwlksn 29912 | The set of walks (in an un... |
| iswwlksn 29913 | A word over the set of ver... |
| wwlksnprcl 29914 | Derivation of the length o... |
| iswwlksnx 29915 | Properties of a word to re... |
| wwlkbp 29916 | Basic properties of a walk... |
| wwlknbp 29917 | Basic properties of a walk... |
| wwlknp 29918 | Properties of a set being ... |
| wwlknbp1 29919 | Other basic properties of ... |
| wwlknvtx 29920 | The symbols of a word ` W ... |
| wwlknllvtx 29921 | If a word ` W ` represents... |
| wwlknlsw 29922 | If a word represents a wal... |
| wspthsn 29923 | The set of simple paths of... |
| iswspthn 29924 | An element of the set of s... |
| wspthnp 29925 | Properties of a set being ... |
| wwlksnon 29926 | The set of walks of a fixe... |
| wspthsnon 29927 | The set of simple paths of... |
| iswwlksnon 29928 | The set of walks of a fixe... |
| wwlksnon0 29929 | Sufficient conditions for ... |
| wwlksonvtx 29930 | If a word ` W ` represents... |
| iswspthsnon 29931 | The set of simple paths of... |
| wwlknon 29932 | An element of the set of w... |
| wspthnon 29933 | An element of the set of s... |
| wspthnonp 29934 | Properties of a set being ... |
| wspthneq1eq2 29935 | Two simple paths with iden... |
| wwlksn0s 29936 | The set of all walks as wo... |
| wwlkssswrd 29937 | Walks (represented by word... |
| wwlksn0 29938 | A walk of length 0 is repr... |
| 0enwwlksnge1 29939 | In graphs without edges, t... |
| wwlkswwlksn 29940 | A walk of a fixed length a... |
| wwlkssswwlksn 29941 | The walks of a fixed lengt... |
| wlkiswwlks1 29942 | The sequence of vertices i... |
| wlklnwwlkln1 29943 | The sequence of vertices i... |
| wlkiswwlks2lem1 29944 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29945 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29946 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29947 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29948 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29949 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29950 | A walk as word corresponds... |
| wlkiswwlks 29951 | A walk as word corresponds... |
| wlkiswwlksupgr2 29952 | A walk as word corresponds... |
| wlkiswwlkupgr 29953 | A walk as word corresponds... |
| wlkswwlksf1o 29954 | The mapping of (ordinary) ... |
| wlkswwlksen 29955 | The set of walks as words ... |
| wwlksm1edg 29956 | Removing the trailing edge... |
| wlklnwwlkln2lem 29957 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29958 | A walk of length ` N ` as ... |
| wlklnwwlkn 29959 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29960 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29961 | A walk of length ` N ` as ... |
| wlknewwlksn 29962 | If a walk in a pseudograph... |
| wlknwwlksnbij 29963 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29964 | In a simple pseudograph, t... |
| wlknwwlksneqs 29965 | The set of walks of a fixe... |
| wwlkseq 29966 | Equality of two walks (as ... |
| wwlksnred 29967 | Reduction of a walk (as wo... |
| wwlksnext 29968 | Extension of a walk (as wo... |
| wwlksnextbi 29969 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29970 | For each walk (as word) of... |
| wwlksnredwwlkn0 29971 | For each walk (as word) of... |
| wwlksnextwrd 29972 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29973 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29974 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29975 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29976 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29977 | There is a bijection betwe... |
| wwlksnexthasheq 29978 | The number of the extensio... |
| disjxwwlksn 29979 | Sets of walks (as words) e... |
| wwlksnndef 29980 | Conditions for ` WWalksN `... |
| wwlksnfi 29981 | The number of walks repres... |
| wlksnfi 29982 | The number of walks of fix... |
| wlksnwwlknvbij 29983 | There is a bijection betwe... |
| wwlksnextproplem1 29984 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29985 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29986 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29987 | Adding additional properti... |
| disjxwwlkn 29988 | Sets of walks (as words) e... |
| hashwwlksnext 29989 | Number of walks (as words)... |
| wwlksnwwlksnon 29990 | A walk of fixed length is ... |
| wspthsnwspthsnon 29991 | A simple path of fixed len... |
| wspthsnonn0vne 29992 | If the set of simple paths... |
| wspthsswwlkn 29993 | The set of simple paths of... |
| wspthnfi 29994 | In a finite graph, the set... |
| wwlksnonfi 29995 | In a finite graph, the set... |
| wspthsswwlknon 29996 | The set of simple paths of... |
| wspthnonfi 29997 | In a finite graph, the set... |
| wspniunwspnon 29998 | The set of nonempty simple... |
| wspn0 29999 | If there are no vertices, ... |
| 2wlkdlem1 30000 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 30001 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 30002 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 30003 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 30004 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 30005 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 30006 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 30007 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 30008 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 30009 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 30010 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 30011 | Construction of a walk fro... |
| 2wlkond 30012 | A walk of length 2 from on... |
| 2trld 30013 | Construction of a trail fr... |
| 2trlond 30014 | A trail of length 2 from o... |
| 2pthd 30015 | A path of length 2 from on... |
| 2spthd 30016 | A simple path of length 2 ... |
| 2pthond 30017 | A simple path of length 2 ... |
| 2pthon3v 30018 | For a vertex adjacent to t... |
| umgr2adedgwlklem 30019 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 30020 | In a multigraph, two adjac... |
| umgr2adedgwlkon 30021 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 30022 | Alternate proof for ~ umgr... |
| umgr2adedgspth 30023 | In a multigraph, two adjac... |
| umgr2wlk 30024 | In a multigraph, there is ... |
| umgr2wlkon 30025 | For each pair of adjacent ... |
| elwwlks2s3 30026 | A walk of length 2 as word... |
| midwwlks2s3 30027 | There is a vertex between ... |
| wwlks2onv 30028 | If a length 3 string repre... |
| elwwlks2ons3im 30029 | A walk as word of length 2... |
| elwwlks2ons3 30030 | For each walk of length 2 ... |
| s3wwlks2on 30031 | A length 3 string which re... |
| sps3wwlks2on 30032 | A length 3 string which re... |
| usgrwwlks2on 30033 | A walk of length 2 between... |
| umgrwwlks2on 30034 | A walk of length 2 between... |
| wwlks2onsym 30035 | There is a walk of length ... |
| elwwlks2on 30036 | A walk of length 2 between... |
| elwspths2on 30037 | A simple path of length 2 ... |
| elwspths2onw 30038 | A simple path of length 2 ... |
| wpthswwlks2on 30039 | For two different vertices... |
| 2wspdisj 30040 | All simple paths of length... |
| 2wspiundisj 30041 | All simple paths of length... |
| usgr2wspthons3 30042 | A simple path of length 2 ... |
| usgr2wspthon 30043 | A simple path of length 2 ... |
| elwwlks2 30044 | A walk of length 2 between... |
| elwspths2spth 30045 | A simple path of length 2 ... |
| rusgrnumwwlkl1 30046 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 30047 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 30048 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 30049 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 30050 | Induction base 1 for ~ rus... |
| rusgr0edg 30051 | Special case for graphs wi... |
| rusgrnumwwlks 30052 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30053 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30054 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30055 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30056 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30057 | In a ` K `-regular graph, ... |
| clwwlk 30060 | The set of closed walks (i... |
| isclwwlk 30061 | Properties of a word to re... |
| clwwlkbp 30062 | Basic properties of a clos... |
| clwwlkgt0 30063 | There is no empty closed w... |
| clwwlksswrd 30064 | Closed walks (represented ... |
| clwwlk1loop 30065 | A closed walk of length 1 ... |
| clwwlkccatlem 30066 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30067 | The concatenation of two w... |
| umgrclwwlkge2 30068 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30069 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30070 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30071 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30072 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30073 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30074 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30075 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30076 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30077 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30078 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30079 | A closed walk as word of l... |
| clwlkclwwlk2 30080 | A closed walk corresponds ... |
| clwlkclwwlkflem 30081 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30082 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30083 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30084 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30085 | ` F ` is a function from t... |
| clwlkclwwlkfo 30086 | ` F ` is a function from t... |
| clwlkclwwlkf1 30087 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30088 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30089 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30090 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30091 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30092 | Cyclically shifting a clos... |
| clwwisshclwwsn 30093 | Cyclically shifting a clos... |
| erclwwlkrel 30094 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30095 | Two classes are equivalent... |
| erclwwlkeqlen 30096 | If two classes are equival... |
| erclwwlkref 30097 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30098 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30099 | ` .~ ` is a transitive rel... |
| erclwwlk 30100 | ` .~ ` is an equivalence r... |
| clwwlkn 30103 | The set of closed walks of... |
| isclwwlkn 30104 | A word over the set of ver... |
| clwwlkn0 30105 | There is no closed walk of... |
| clwwlkneq0 30106 | Sufficient conditions for ... |
| clwwlkclwwlkn 30107 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30108 | The closed walks of a fixe... |
| clwwlknlen 30109 | The length of a word repre... |
| clwwlknnn 30110 | The length of a closed wal... |
| clwwlknwrd 30111 | A closed walk of a fixed l... |
| clwwlknbp 30112 | Basic properties of a clos... |
| isclwwlknx 30113 | Characterization of a word... |
| clwwlknp 30114 | Properties of a set being ... |
| clwwlknwwlksn 30115 | A word representing a clos... |
| clwwlknlbonbgr1 30116 | The last but one vertex in... |
| clwwlkinwwlk 30117 | If the initial vertex of a... |
| clwwlkn1 30118 | A closed walk of length 1 ... |
| loopclwwlkn1b 30119 | The singleton word consist... |
| clwwlkn1loopb 30120 | A word represents a closed... |
| clwwlkn2 30121 | A closed walk of length 2 ... |
| clwwlknfi 30122 | If there is only a finite ... |
| clwwlkel 30123 | Obtaining a closed walk (a... |
| clwwlkf 30124 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30125 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30126 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30127 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30128 | F is a 1-1 onto function, ... |
| clwwlken 30129 | The set of closed walks of... |
| clwwlknwwlkncl 30130 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30131 | A nonempty word over verti... |
| clwwlknwwlksnb 30132 | A word over vertices repre... |
| clwwlkext2edg 30133 | If a word concatenated wit... |
| wwlksext2clwwlk 30134 | If a word represents a wal... |
| wwlksubclwwlk 30135 | Any prefix of a word repre... |
| clwwnisshclwwsn 30136 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30137 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30138 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30139 | The set of cyclical shifts... |
| clwwlknccat 30140 | The concatenation of two w... |
| umgr2cwwk2dif 30141 | If a word represents a clo... |
| umgr2cwwkdifex 30142 | If a word represents a clo... |
| erclwwlknrel 30143 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30144 | Two classes are equivalent... |
| erclwwlkneqlen 30145 | If two classes are equival... |
| erclwwlknref 30146 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30147 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30148 | ` .~ ` is a transitive rel... |
| erclwwlkn 30149 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30150 | The quotient set of the se... |
| hashclwwlkn0 30151 | The number of closed walks... |
| eclclwwlkn1 30152 | An equivalence class accor... |
| eleclclwwlkn 30153 | A member of an equivalence... |
| hashecclwwlkn1 30154 | The size of every equivale... |
| umgrhashecclwwlk 30155 | The size of every equivale... |
| fusgrhashclwwlkn 30156 | The size of the set of clo... |
| clwwlkndivn 30157 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30158 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30159 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30160 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30161 | There is a one-to-one onto... |
| clwlkssizeeq 30162 | The size of the set of clo... |
| clwlksndivn 30163 | The size of the set of clo... |
| clwwlknonmpo 30166 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30167 | The set of closed walks on... |
| isclwwlknon 30168 | A word over the set of ver... |
| clwwlk0on0 30169 | There is no word over the ... |
| clwwlknon0 30170 | Sufficient conditions for ... |
| clwwlknonfin 30171 | In a finite graph ` G ` , ... |
| clwwlknonel 30172 | Characterization of a word... |
| clwwlknonccat 30173 | The concatenation of two w... |
| clwwlknon1 30174 | The set of closed walks on... |
| clwwlknon1loop 30175 | If there is a loop at vert... |
| clwwlknon1nloop 30176 | If there is no loop at ver... |
| clwwlknon1sn 30177 | The set of (closed) walks ... |
| clwwlknon1le1 30178 | There is at most one (clos... |
| clwwlknon2 30179 | The set of closed walks on... |
| clwwlknon2x 30180 | The set of closed walks on... |
| s2elclwwlknon2 30181 | Sufficient conditions of a... |
| clwwlknon2num 30182 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30183 | A word over vertices repre... |
| clwwlknonex2lem1 30184 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30185 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30186 | Extending a closed walk ` ... |
| clwwlknonex2e 30187 | Extending a closed walk ` ... |
| clwwlknondisj 30188 | The sets of closed walks o... |
| clwwlknun 30189 | The set of closed walks of... |
| clwwlkvbij 30190 | There is a bijection betwe... |
| 0ewlk 30191 | The empty set (empty seque... |
| 1ewlk 30192 | A sequence of 1 edge is an... |
| 0wlk 30193 | A pair of an empty set (of... |
| is0wlk 30194 | A pair of an empty set (of... |
| 0wlkonlem1 30195 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30196 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30197 | A walk of length 0 from a ... |
| 0wlkons1 30198 | A walk of length 0 from a ... |
| 0trl 30199 | A pair of an empty set (of... |
| is0trl 30200 | A pair of an empty set (of... |
| 0trlon 30201 | A trail of length 0 from a... |
| 0pth 30202 | A pair of an empty set (of... |
| 0spth 30203 | A pair of an empty set (of... |
| 0pthon 30204 | A path of length 0 from a ... |
| 0pthon1 30205 | A path of length 0 from a ... |
| 0pthonv 30206 | For each vertex there is a... |
| 0clwlk 30207 | A pair of an empty set (of... |
| 0clwlkv 30208 | Any vertex (more precisely... |
| 0clwlk0 30209 | There is no closed walk in... |
| 0crct 30210 | A pair of an empty set (of... |
| 0cycl 30211 | A pair of an empty set (of... |
| 1pthdlem1 30212 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30213 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30214 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30215 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30216 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30217 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30218 | In a graph with two vertic... |
| 1trld 30219 | In a graph with two vertic... |
| 1pthd 30220 | In a graph with two vertic... |
| 1pthond 30221 | In a graph with two vertic... |
| upgr1wlkdlem1 30222 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30223 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30224 | In a pseudograph with two ... |
| upgr1trld 30225 | In a pseudograph with two ... |
| upgr1pthd 30226 | In a pseudograph with two ... |
| upgr1pthond 30227 | In a pseudograph with two ... |
| lppthon 30228 | A loop (which is an edge a... |
| lp1cycl 30229 | A loop (which is an edge a... |
| 1pthon2v 30230 | For each pair of adjacent ... |
| 1pthon2ve 30231 | For each pair of adjacent ... |
| wlk2v2elem1 30232 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30233 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30234 | In a graph with two vertic... |
| ntrl2v2e 30235 | A walk which is not a trai... |
| 3wlkdlem1 30236 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30237 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30238 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30239 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30240 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30241 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30242 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30243 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30244 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30245 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30246 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30247 | Construction of a walk fro... |
| 3wlkond 30248 | A walk of length 3 from on... |
| 3trld 30249 | Construction of a trail fr... |
| 3trlond 30250 | A trail of length 3 from o... |
| 3pthd 30251 | A path of length 3 from on... |
| 3pthond 30252 | A path of length 3 from on... |
| 3spthd 30253 | A simple path of length 3 ... |
| 3spthond 30254 | A simple path of length 3 ... |
| 3cycld 30255 | Construction of a 3-cycle ... |
| 3cyclpd 30256 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30257 | If there is a cycle of len... |
| uhgr3cyclexlem 30258 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30259 | If there are three differe... |
| umgr3cyclex 30260 | If there are three (differ... |
| umgr3v3e3cycl 30261 | If and only if there is a ... |
| upgr4cycl4dv4e 30262 | If there is a cycle of len... |
| dfconngr1 30265 | Alternative definition of ... |
| isconngr 30266 | The property of being a co... |
| isconngr1 30267 | The property of being a co... |
| cusconngr 30268 | A complete hypergraph is c... |
| 0conngr 30269 | A graph without vertices i... |
| 0vconngr 30270 | A graph without vertices i... |
| 1conngr 30271 | A graph with (at most) one... |
| conngrv2edg 30272 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30273 | A vertex in a connected mu... |
| releupth 30276 | The set ` ( EulerPaths `` ... |
| eupths 30277 | The Eulerian paths on the ... |
| iseupth 30278 | The property " ` <. F , P ... |
| iseupthf1o 30279 | The property " ` <. F , P ... |
| eupthi 30280 | Properties of an Eulerian ... |
| eupthf1o 30281 | The ` F ` function in an E... |
| eupthfi 30282 | Any graph with an Eulerian... |
| eupthseg 30283 | The ` N ` -th edge in an e... |
| upgriseupth 30284 | The property " ` <. F , P ... |
| upgreupthi 30285 | Properties of an Eulerian ... |
| upgreupthseg 30286 | The ` N ` -th edge in an e... |
| eupthcl 30287 | An Eulerian path has lengt... |
| eupthistrl 30288 | An Eulerian path is a trai... |
| eupthiswlk 30289 | An Eulerian path is a walk... |
| eupthpf 30290 | The ` P ` function in an E... |
| eupth0 30291 | There is an Eulerian path ... |
| eupthres 30292 | The restriction ` <. H , Q... |
| eupthp1 30293 | Append one path segment to... |
| eupth2eucrct 30294 | Append one path segment to... |
| eupth2lem1 30295 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30296 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30297 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30298 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30299 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30300 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30301 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30302 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30303 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30304 | Formerly part of proof of ... |
| eupth2lem3lem1 30305 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30306 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30307 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30308 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30309 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30310 | Formerly part of proof of ... |
| eupth2lem3lem7 30311 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30312 | Formerly part of proof of ... |
| eupth2lem3 30313 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30314 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30315 | Lemma for ~ eupth2 (induct... |
| eupth2 30316 | The only vertices of odd d... |
| eulerpathpr 30317 | A graph with an Eulerian p... |
| eulerpath 30318 | A pseudograph with an Eule... |
| eulercrct 30319 | A pseudograph with an Eule... |
| eucrctshift 30320 | Cyclically shifting the in... |
| eucrct2eupth1 30321 | Removing one edge ` ( I ``... |
| eucrct2eupth 30322 | Removing one edge ` ( I ``... |
| konigsbergvtx 30323 | The set of vertices of the... |
| konigsbergiedg 30324 | The indexed edges of the K... |
| konigsbergiedgw 30325 | The indexed edges of the K... |
| konigsbergssiedgwpr 30326 | Each subset of the indexed... |
| konigsbergssiedgw 30327 | Each subset of the indexed... |
| konigsbergumgr 30328 | The Königsberg graph ... |
| konigsberglem1 30329 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30330 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30331 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30332 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30333 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30334 | The Königsberg Bridge... |
| isfrgr 30337 | The property of being a fr... |
| frgrusgr 30338 | A friendship graph is a si... |
| frgr0v 30339 | Any null graph (set with n... |
| frgr0vb 30340 | Any null graph (without ve... |
| frgruhgr0v 30341 | Any null graph (without ve... |
| frgr0 30342 | The null graph (graph with... |
| frcond1 30343 | The friendship condition: ... |
| frcond2 30344 | The friendship condition: ... |
| frgreu 30345 | Variant of ~ frcond2 : An... |
| frcond3 30346 | The friendship condition, ... |
| frcond4 30347 | The friendship condition, ... |
| frgr1v 30348 | Any graph with (at most) o... |
| nfrgr2v 30349 | Any graph with two (differ... |
| frgr3vlem1 30350 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30351 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30352 | Any graph with three verti... |
| 1vwmgr 30353 | Every graph with one verte... |
| 3vfriswmgrlem 30354 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30355 | Every friendship graph wit... |
| 1to2vfriswmgr 30356 | Every friendship graph wit... |
| 1to3vfriswmgr 30357 | Every friendship graph wit... |
| 1to3vfriendship 30358 | The friendship theorem for... |
| 2pthfrgrrn 30359 | Between any two (different... |
| 2pthfrgrrn2 30360 | Between any two (different... |
| 2pthfrgr 30361 | Between any two (different... |
| 3cyclfrgrrn1 30362 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30363 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30364 | Every vertex in a friendsh... |
| 3cyclfrgr 30365 | Every vertex in a friendsh... |
| 4cycl2v2nb 30366 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30367 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30368 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30369 | A graph with a 4-cycle is ... |
| frgrnbnb 30370 | If two neighbors ` U ` and... |
| frgrconngr 30371 | A friendship graph is conn... |
| vdgn0frgrv2 30372 | A vertex in a friendship g... |
| vdgn1frgrv2 30373 | Any vertex in a friendship... |
| vdgn1frgrv3 30374 | Any vertex in a friendship... |
| vdgfrgrgt2 30375 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30376 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30377 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30378 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30379 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30380 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30381 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30382 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30383 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30384 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30385 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30386 | In a friendship graph, two... |
| frgrwopreglem4a 30387 | In a friendship graph any ... |
| frgrwopreglem5a 30388 | If a friendship graph has ... |
| frgrwopreglem1 30389 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30390 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30391 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30392 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30393 | According to statement 5 i... |
| frgrwopregbsn 30394 | According to statement 5 i... |
| frgrwopreg1 30395 | According to statement 5 i... |
| frgrwopreg2 30396 | According to statement 5 i... |
| frgrwopreglem5lem 30397 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30398 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30399 | Alternate direct proof of ... |
| frgrwopreg 30400 | In a friendship graph ther... |
| frgrregorufr0 30401 | In a friendship graph ther... |
| frgrregorufr 30402 | If there is a vertex havin... |
| frgrregorufrg 30403 | If there is a vertex havin... |
| frgr2wwlkeu 30404 | For two different vertices... |
| frgr2wwlkn0 30405 | In a friendship graph, the... |
| frgr2wwlk1 30406 | In a friendship graph, the... |
| frgr2wsp1 30407 | In a friendship graph, the... |
| frgr2wwlkeqm 30408 | If there is a (simple) pat... |
| frgrhash2wsp 30409 | The number of simple paths... |
| fusgreg2wsplem 30410 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30411 | The set of paths of length... |
| fusgreghash2wspv 30412 | According to statement 7 i... |
| fusgreg2wsp 30413 | In a finite simple graph, ... |
| 2wspmdisj 30414 | The sets of paths of lengt... |
| fusgreghash2wsp 30415 | In a finite k-regular grap... |
| frrusgrord0lem 30416 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30417 | If a nonempty finite frien... |
| frrusgrord 30418 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30419 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30420 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30421 | If the initial vertex of a... |
| clwwnonrepclwwnon 30422 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30423 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30424 | Value of operation ` C ` ,... |
| 2clwwlk2 30425 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30426 | Characterization of an ele... |
| 2clwwlk2clwwlk 30427 | An element of the value of... |
| numclwwlk1lem2foalem 30428 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30429 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30430 | Characterization of an ele... |
| numclwwlk1lem2foa 30431 | Going forth and back from ... |
| numclwwlk1lem2f 30432 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30433 | Value of the function ` T ... |
| numclwwlk1lem2f1 30434 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30435 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30436 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30437 | The set of double loops of... |
| numclwwlk1 30438 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30439 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30440 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30441 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30442 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30443 | The sets of the two repres... |
| wlkl0 30444 | There is exactly one walk ... |
| clwlknon2num 30445 | There are k walks of lengt... |
| numclwlk1lem1 30446 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30447 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30448 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30449 | Value of operation ` H ` ,... |
| numclwwlkovh 30450 | Value of operation ` H ` ,... |
| numclwwlkovq 30451 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30452 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30453 | In a friendship graph, for... |
| numclwlk2lem2f 30454 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30455 | Value of the function ` R ... |
| numclwlk2lem2f1o 30456 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30457 | In a friendship graph, the... |
| numclwwlk2 30458 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30459 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30460 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30461 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30462 | Statement 12 in [Huneke] p... |
| numclwwlk4 30463 | The total number of closed... |
| numclwwlk5lem 30464 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30465 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30466 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30467 | For a prime divisor ` P ` ... |
| numclwwlk7 30468 | Statement 14 in [Huneke] p... |
| numclwwlk8 30469 | The size of the set of clo... |
| frgrreggt1 30470 | If a finite nonempty frien... |
| frgrreg 30471 | If a finite nonempty frien... |
| frgrregord013 30472 | If a finite friendship gra... |
| frgrregord13 30473 | If a nonempty finite frien... |
| frgrogt3nreg 30474 | If a finite friendship gra... |
| friendshipgt3 30475 | The friendship theorem for... |
| friendship 30476 | The friendship theorem: I... |
| conventions 30477 |
H... |
| conventions-labels 30478 |
... |
| conventions-comments 30479 |
... |
| natded 30480 | Here are typical n... |
| ex-natded5.2 30481 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30482 | A more efficient proof of ... |
| ex-natded5.2i 30483 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30484 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30485 | A more efficient proof of ... |
| ex-natded5.3i 30486 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30487 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30488 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30489 | A more efficient proof of ... |
| ex-natded5.8 30490 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30491 | A more efficient proof of ... |
| ex-natded5.13 30492 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30493 | A more efficient proof of ... |
| ex-natded9.20 30494 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30495 | A more efficient proof of ... |
| ex-natded9.26 30496 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30497 | A more efficient proof of ... |
| ex-or 30498 | Example for ~ df-or . Exa... |
| ex-an 30499 | Example for ~ df-an . Exa... |
| ex-dif 30500 | Example for ~ df-dif . Ex... |
| ex-un 30501 | Example for ~ df-un . Exa... |
| ex-in 30502 | Example for ~ df-in . Exa... |
| ex-uni 30503 | Example for ~ df-uni . Ex... |
| ex-ss 30504 | Example for ~ df-ss . Exa... |
| ex-pss 30505 | Example for ~ df-pss . Ex... |
| ex-pw 30506 | Example for ~ df-pw . Exa... |
| ex-pr 30507 | Example for ~ df-pr . (Co... |
| ex-br 30508 | Example for ~ df-br . Exa... |
| ex-opab 30509 | Example for ~ df-opab . E... |
| ex-eprel 30510 | Example for ~ df-eprel . ... |
| ex-id 30511 | Example for ~ df-id . Exa... |
| ex-po 30512 | Example for ~ df-po . Exa... |
| ex-xp 30513 | Example for ~ df-xp . Exa... |
| ex-cnv 30514 | Example for ~ df-cnv . Ex... |
| ex-co 30515 | Example for ~ df-co . Exa... |
| ex-dm 30516 | Example for ~ df-dm . Exa... |
| ex-rn 30517 | Example for ~ df-rn . Exa... |
| ex-res 30518 | Example for ~ df-res . Ex... |
| ex-ima 30519 | Example for ~ df-ima . Ex... |
| ex-fv 30520 | Example for ~ df-fv . Exa... |
| ex-1st 30521 | Example for ~ df-1st . Ex... |
| ex-2nd 30522 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30523 | Example for ~ df-dec , 100... |
| ex-fl 30524 | Example for ~ df-fl . Exa... |
| ex-ceil 30525 | Example for ~ df-ceil . (... |
| ex-mod 30526 | Example for ~ df-mod . (C... |
| ex-exp 30527 | Example for ~ df-exp . (C... |
| ex-fac 30528 | Example for ~ df-fac . (C... |
| ex-bc 30529 | Example for ~ df-bc . (Co... |
| ex-hash 30530 | Example for ~ df-hash . (... |
| ex-sqrt 30531 | Example for ~ df-sqrt . (... |
| ex-abs 30532 | Example for ~ df-abs . (C... |
| ex-dvds 30533 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30534 | Example for ~ df-gcd . (C... |
| ex-lcm 30535 | Example for ~ df-lcm . (C... |
| ex-prmo 30536 | Example for ~ df-prmo : ` ... |
| aevdemo 30537 | Proof illustrating the com... |
| ex-ind-dvds 30538 | Example of a proof by indu... |
| ex-fpar 30539 | Formalized example provide... |
| avril1 30540 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30541 | The law of excluded middle... |
| helloworld 30542 | The classic "Hello world" ... |
| 1p1e2apr1 30543 | One plus one equals two. ... |
| eqid1 30544 | Law of identity (reflexivi... |
| 1div0apr 30545 | Division by zero is forbid... |
| topnfbey 30546 | Nothing seems to be imposs... |
| 9p10ne21 30547 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30548 | 9 + 10 equals 21. This as... |
| nrt2irr 30550 | The ` N ` -th root of 2 is... |
| isplig 30553 | The predicate "is a planar... |
| ispligb 30554 | The predicate "is a planar... |
| tncp 30555 | In any planar incidence ge... |
| l2p 30556 | For any line in a planar i... |
| lpni 30557 | For any line in a planar i... |
| nsnlplig 30558 | There is no "one-point lin... |
| nsnlpligALT 30559 | Alternate version of ~ nsn... |
| n0lplig 30560 | There is no "empty line" i... |
| n0lpligALT 30561 | Alternate version of ~ n0l... |
| eulplig 30562 | Through two distinct point... |
| pliguhgr 30563 | Any planar incidence geome... |
| dummylink 30564 | Alias for ~ a1ii that may ... |
| id1 30565 | Alias for ~ idALT that may... |
| isgrpo 30574 | The predicate "is a group ... |
| isgrpoi 30575 | Properties that determine ... |
| grpofo 30576 | A group operation maps ont... |
| grpocl 30577 | Closure law for a group op... |
| grpolidinv 30578 | A group has a left identit... |
| grpon0 30579 | The base set of a group is... |
| grpoass 30580 | A group operation is assoc... |
| grpoidinvlem1 30581 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30582 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30583 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30584 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30585 | A group has a left and rig... |
| grpoideu 30586 | The left identity element ... |
| grporndm 30587 | A group's range in terms o... |
| 0ngrp 30588 | The empty set is not a gro... |
| gidval 30589 | The value of the identity ... |
| grpoidval 30590 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30591 | The identity element of a ... |
| grpoidinv2 30592 | A group's properties using... |
| grpolid 30593 | The identity element of a ... |
| grporid 30594 | The identity element of a ... |
| grporcan 30595 | Right cancellation law for... |
| grpoinveu 30596 | The left inverse element o... |
| grpoid 30597 | Two ways of saying that an... |
| grporn 30598 | The range of a group opera... |
| grpoinvfval 30599 | The inverse function of a ... |
| grpoinvval 30600 | The inverse of a group ele... |
| grpoinvcl 30601 | A group element's inverse ... |
| grpoinv 30602 | The properties of a group ... |
| grpolinv 30603 | The left inverse of a grou... |
| grporinv 30604 | The right inverse of a gro... |
| grpoinvid1 30605 | The inverse of a group ele... |
| grpoinvid2 30606 | The inverse of a group ele... |
| grpolcan 30607 | Left cancellation law for ... |
| grpo2inv 30608 | Double inverse law for gro... |
| grpoinvf 30609 | Mapping of the inverse fun... |
| grpoinvop 30610 | The inverse of the group o... |
| grpodivfval 30611 | Group division (or subtrac... |
| grpodivval 30612 | Group division (or subtrac... |
| grpodivinv 30613 | Group division by an inver... |
| grpoinvdiv 30614 | Inverse of a group divisio... |
| grpodivf 30615 | Mapping for group division... |
| grpodivcl 30616 | Closure of group division ... |
| grpodivdiv 30617 | Double group division. (C... |
| grpomuldivass 30618 | Associative-type law for m... |
| grpodivid 30619 | Division of a group member... |
| grponpcan 30620 | Cancellation law for group... |
| isablo 30623 | The predicate "is an Abeli... |
| ablogrpo 30624 | An Abelian group operation... |
| ablocom 30625 | An Abelian group operation... |
| ablo32 30626 | Commutative/associative la... |
| ablo4 30627 | Commutative/associative la... |
| isabloi 30628 | Properties that determine ... |
| ablomuldiv 30629 | Law for group multiplicati... |
| ablodivdiv 30630 | Law for double group divis... |
| ablodivdiv4 30631 | Law for double group divis... |
| ablodiv32 30632 | Swap the second and third ... |
| ablonncan 30633 | Cancellation law for group... |
| ablonnncan1 30634 | Cancellation law for group... |
| vcrel 30637 | The class of all complex v... |
| vciOLD 30638 | Obsolete version of ~ cvsi... |
| vcsm 30639 | Functionality of th scalar... |
| vccl 30640 | Closure of the scalar prod... |
| vcidOLD 30641 | Identity element for the s... |
| vcdi 30642 | Distributive law for the s... |
| vcdir 30643 | Distributive law for the s... |
| vcass 30644 | Associative law for the sc... |
| vc2OLD 30645 | A vector plus itself is tw... |
| vcablo 30646 | Vector addition is an Abel... |
| vcgrp 30647 | Vector addition is a group... |
| vclcan 30648 | Left cancellation law for ... |
| vczcl 30649 | The zero vector is a vecto... |
| vc0rid 30650 | The zero vector is a right... |
| vc0 30651 | Zero times a vector is the... |
| vcz 30652 | Anything times the zero ve... |
| vcm 30653 | Minus 1 times a vector is ... |
| isvclem 30654 | Lemma for ~ isvcOLD . (Co... |
| vcex 30655 | The components of a comple... |
| isvcOLD 30656 | The predicate "is a comple... |
| isvciOLD 30657 | Properties that determine ... |
| cnaddabloOLD 30658 | Obsolete version of ~ cnad... |
| cnidOLD 30659 | Obsolete version of ~ cnad... |
| cncvcOLD 30660 | Obsolete version of ~ cncv... |
| nvss 30670 | Structure of the class of ... |
| nvvcop 30671 | A normed complex vector sp... |
| nvrel 30679 | The class of all normed co... |
| vafval 30680 | Value of the function for ... |
| bafval 30681 | Value of the function for ... |
| smfval 30682 | Value of the function for ... |
| 0vfval 30683 | Value of the function for ... |
| nmcvfval 30684 | Value of the norm function... |
| nvop2 30685 | A normed complex vector sp... |
| nvvop 30686 | The vector space component... |
| isnvlem 30687 | Lemma for ~ isnv . (Contr... |
| nvex 30688 | The components of a normed... |
| isnv 30689 | The predicate "is a normed... |
| isnvi 30690 | Properties that determine ... |
| nvi 30691 | The properties of a normed... |
| nvvc 30692 | The vector space component... |
| nvablo 30693 | The vector addition operat... |
| nvgrp 30694 | The vector addition operat... |
| nvgf 30695 | Mapping for the vector add... |
| nvsf 30696 | Mapping for the scalar mul... |
| nvgcl 30697 | Closure law for the vector... |
| nvcom 30698 | The vector addition (group... |
| nvass 30699 | The vector addition (group... |
| nvadd32 30700 | Commutative/associative la... |
| nvrcan 30701 | Right cancellation law for... |
| nvadd4 30702 | Rearrangement of 4 terms i... |
| nvscl 30703 | Closure law for the scalar... |
| nvsid 30704 | Identity element for the s... |
| nvsass 30705 | Associative law for the sc... |
| nvscom 30706 | Commutative law for the sc... |
| nvdi 30707 | Distributive law for the s... |
| nvdir 30708 | Distributive law for the s... |
| nv2 30709 | A vector plus itself is tw... |
| vsfval 30710 | Value of the function for ... |
| nvzcl 30711 | Closure law for the zero v... |
| nv0rid 30712 | The zero vector is a right... |
| nv0lid 30713 | The zero vector is a left ... |
| nv0 30714 | Zero times a vector is the... |
| nvsz 30715 | Anything times the zero ve... |
| nvinv 30716 | Minus 1 times a vector is ... |
| nvinvfval 30717 | Function for the negative ... |
| nvm 30718 | Vector subtraction in term... |
| nvmval 30719 | Value of vector subtractio... |
| nvmval2 30720 | Value of vector subtractio... |
| nvmfval 30721 | Value of the function for ... |
| nvmf 30722 | Mapping for the vector sub... |
| nvmcl 30723 | Closure law for the vector... |
| nvnnncan1 30724 | Cancellation law for vecto... |
| nvmdi 30725 | Distributive law for scala... |
| nvnegneg 30726 | Double negative of a vecto... |
| nvmul0or 30727 | If a scalar product is zer... |
| nvrinv 30728 | A vector minus itself. (C... |
| nvlinv 30729 | Minus a vector plus itself... |
| nvpncan2 30730 | Cancellation law for vecto... |
| nvpncan 30731 | Cancellation law for vecto... |
| nvaddsub 30732 | Commutative/associative la... |
| nvnpcan 30733 | Cancellation law for a nor... |
| nvaddsub4 30734 | Rearrangement of 4 terms i... |
| nvmeq0 30735 | The difference between two... |
| nvmid 30736 | A vector minus itself is t... |
| nvf 30737 | Mapping for the norm funct... |
| nvcl 30738 | The norm of a normed compl... |
| nvcli 30739 | The norm of a normed compl... |
| nvs 30740 | Proportionality property o... |
| nvsge0 30741 | The norm of a scalar produ... |
| nvm1 30742 | The norm of the negative o... |
| nvdif 30743 | The norm of the difference... |
| nvpi 30744 | The norm of a vector plus ... |
| nvz0 30745 | The norm of a zero vector ... |
| nvz 30746 | The norm of a vector is ze... |
| nvtri 30747 | Triangle inequality for th... |
| nvmtri 30748 | Triangle inequality for th... |
| nvabs 30749 | Norm difference property o... |
| nvge0 30750 | The norm of a normed compl... |
| nvgt0 30751 | A nonzero norm is positive... |
| nv1 30752 | From any nonzero vector, c... |
| nvop 30753 | A complex inner product sp... |
| cnnv 30754 | The set of complex numbers... |
| cnnvg 30755 | The vector addition (group... |
| cnnvba 30756 | The base set of the normed... |
| cnnvs 30757 | The scalar product operati... |
| cnnvnm 30758 | The norm operation of the ... |
| cnnvm 30759 | The vector subtraction ope... |
| elimnv 30760 | Hypothesis elimination lem... |
| elimnvu 30761 | Hypothesis elimination lem... |
| imsval 30762 | Value of the induced metri... |
| imsdval 30763 | Value of the induced metri... |
| imsdval2 30764 | Value of the distance func... |
| nvnd 30765 | The norm of a normed compl... |
| imsdf 30766 | Mapping for the induced me... |
| imsmetlem 30767 | Lemma for ~ imsmet . (Con... |
| imsmet 30768 | The induced metric of a no... |
| imsxmet 30769 | The induced metric of a no... |
| cnims 30770 | The metric induced on the ... |
| vacn 30771 | Vector addition is jointly... |
| nmcvcn 30772 | The norm of a normed compl... |
| nmcnc 30773 | The norm of a normed compl... |
| smcnlem 30774 | Lemma for ~ smcn . (Contr... |
| smcn 30775 | Scalar multiplication is j... |
| vmcn 30776 | Vector subtraction is join... |
| dipfval 30779 | The inner product function... |
| ipval 30780 | Value of the inner product... |
| ipval2lem2 30781 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30782 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30783 | Lemma for ~ ipval3 . (Con... |
| ipval2 30784 | Expansion of the inner pro... |
| 4ipval2 30785 | Four times the inner produ... |
| ipval3 30786 | Expansion of the inner pro... |
| ipidsq 30787 | The inner product of a vec... |
| ipnm 30788 | Norm expressed in terms of... |
| dipcl 30789 | An inner product is a comp... |
| ipf 30790 | Mapping for the inner prod... |
| dipcj 30791 | The complex conjugate of a... |
| ipipcj 30792 | An inner product times its... |
| diporthcom 30793 | Orthogonality (meaning inn... |
| dip0r 30794 | Inner product with a zero ... |
| dip0l 30795 | Inner product with a zero ... |
| ipz 30796 | The inner product of a vec... |
| dipcn 30797 | Inner product is jointly c... |
| sspval 30800 | The set of all subspaces o... |
| isssp 30801 | The predicate "is a subspa... |
| sspid 30802 | A normed complex vector sp... |
| sspnv 30803 | A subspace is a normed com... |
| sspba 30804 | The base set of a subspace... |
| sspg 30805 | Vector addition on a subsp... |
| sspgval 30806 | Vector addition on a subsp... |
| ssps 30807 | Scalar multiplication on a... |
| sspsval 30808 | Scalar multiplication on a... |
| sspmlem 30809 | Lemma for ~ sspm and other... |
| sspmval 30810 | Vector addition on a subsp... |
| sspm 30811 | Vector subtraction on a su... |
| sspz 30812 | The zero vector of a subsp... |
| sspn 30813 | The norm on a subspace is ... |
| sspnval 30814 | The norm on a subspace in ... |
| sspimsval 30815 | The induced metric on a su... |
| sspims 30816 | The induced metric on a su... |
| lnoval 30829 | The set of linear operator... |
| islno 30830 | The predicate "is a linear... |
| lnolin 30831 | Basic linearity property o... |
| lnof 30832 | A linear operator is a map... |
| lno0 30833 | The value of a linear oper... |
| lnocoi 30834 | The composition of two lin... |
| lnoadd 30835 | Addition property of a lin... |
| lnosub 30836 | Subtraction property of a ... |
| lnomul 30837 | Scalar multiplication prop... |
| nvo00 30838 | Two ways to express a zero... |
| nmoofval 30839 | The operator norm function... |
| nmooval 30840 | The operator norm function... |
| nmosetre 30841 | The set in the supremum of... |
| nmosetn0 30842 | The set in the supremum of... |
| nmoxr 30843 | The norm of an operator is... |
| nmooge0 30844 | The norm of an operator is... |
| nmorepnf 30845 | The norm of an operator is... |
| nmoreltpnf 30846 | The norm of any operator i... |
| nmogtmnf 30847 | The norm of an operator is... |
| nmoolb 30848 | A lower bound for an opera... |
| nmoubi 30849 | An upper bound for an oper... |
| nmoub3i 30850 | An upper bound for an oper... |
| nmoub2i 30851 | An upper bound for an oper... |
| nmobndi 30852 | Two ways to express that a... |
| nmounbi 30853 | Two ways two express that ... |
| nmounbseqi 30854 | An unbounded operator dete... |
| nmounbseqiALT 30855 | Alternate shorter proof of... |
| nmobndseqi 30856 | A bounded sequence determi... |
| nmobndseqiALT 30857 | Alternate shorter proof of... |
| bloval 30858 | The class of bounded linea... |
| isblo 30859 | The predicate "is a bounde... |
| isblo2 30860 | The predicate "is a bounde... |
| bloln 30861 | A bounded operator is a li... |
| blof 30862 | A bounded operator is an o... |
| nmblore 30863 | The norm of a bounded oper... |
| 0ofval 30864 | The zero operator between ... |
| 0oval 30865 | Value of the zero operator... |
| 0oo 30866 | The zero operator is an op... |
| 0lno 30867 | The zero operator is linea... |
| nmoo0 30868 | The operator norm of the z... |
| 0blo 30869 | The zero operator is a bou... |
| nmlno0lem 30870 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30871 | The norm of a linear opera... |
| nmlno0 30872 | The norm of a linear opera... |
| nmlnoubi 30873 | An upper bound for the ope... |
| nmlnogt0 30874 | The norm of a nonzero line... |
| lnon0 30875 | The domain of a nonzero li... |
| nmblolbii 30876 | A lower bound for the norm... |
| nmblolbi 30877 | A lower bound for the norm... |
| isblo3i 30878 | The predicate "is a bounde... |
| blo3i 30879 | Properties that determine ... |
| blometi 30880 | Upper bound for the distan... |
| blocnilem 30881 | Lemma for ~ blocni and ~ l... |
| blocni 30882 | A linear operator is conti... |
| lnocni 30883 | If a linear operator is co... |
| blocn 30884 | A linear operator is conti... |
| blocn2 30885 | A bounded linear operator ... |
| ajfval 30886 | The adjoint function. (Co... |
| hmoval 30887 | The set of Hermitian (self... |
| ishmo 30888 | The predicate "is a hermit... |
| phnv 30891 | Every complex inner produc... |
| phrel 30892 | The class of all complex i... |
| phnvi 30893 | Every complex inner produc... |
| isphg 30894 | The predicate "is a comple... |
| phop 30895 | A complex inner product sp... |
| cncph 30896 | The set of complex numbers... |
| elimph 30897 | Hypothesis elimination lem... |
| elimphu 30898 | Hypothesis elimination lem... |
| isph 30899 | The predicate "is an inner... |
| phpar2 30900 | The parallelogram law for ... |
| phpar 30901 | The parallelogram law for ... |
| ip0i 30902 | A slight variant of Equati... |
| ip1ilem 30903 | Lemma for ~ ip1i . (Contr... |
| ip1i 30904 | Equation 6.47 of [Ponnusam... |
| ip2i 30905 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30906 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30907 | Distributive law for inner... |
| ipasslem1 30908 | Lemma for ~ ipassi . Show... |
| ipasslem2 30909 | Lemma for ~ ipassi . Show... |
| ipasslem3 30910 | Lemma for ~ ipassi . Show... |
| ipasslem4 30911 | Lemma for ~ ipassi . Show... |
| ipasslem5 30912 | Lemma for ~ ipassi . Show... |
| ipasslem7 30913 | Lemma for ~ ipassi . Show... |
| ipasslem8 30914 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30915 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30916 | Lemma for ~ ipassi . Show... |
| ipasslem11 30917 | Lemma for ~ ipassi . Show... |
| ipassi 30918 | Associative law for inner ... |
| dipdir 30919 | Distributive law for inner... |
| dipdi 30920 | Distributive law for inner... |
| ip2dii 30921 | Inner product of two sums.... |
| dipass 30922 | Associative law for inner ... |
| dipassr 30923 | "Associative" law for seco... |
| dipassr2 30924 | "Associative" law for inne... |
| dipsubdir 30925 | Distributive law for inner... |
| dipsubdi 30926 | Distributive law for inner... |
| pythi 30927 | The Pythagorean theorem fo... |
| siilem1 30928 | Lemma for ~ sii . (Contri... |
| siilem2 30929 | Lemma for ~ sii . (Contri... |
| siii 30930 | Inference from ~ sii . (C... |
| sii 30931 | Obsolete version of ~ ipca... |
| ipblnfi 30932 | A function ` F ` generated... |
| ip2eqi 30933 | Two vectors are equal iff ... |
| phoeqi 30934 | A condition implying that ... |
| ajmoi 30935 | Every operator has at most... |
| ajfuni 30936 | The adjoint function is a ... |
| ajfun 30937 | The adjoint function is a ... |
| ajval 30938 | Value of the adjoint funct... |
| iscbn 30941 | A complex Banach space is ... |
| cbncms 30942 | The induced metric on comp... |
| bnnv 30943 | Every complex Banach space... |
| bnrel 30944 | The class of all complex B... |
| bnsscmcl 30945 | A subspace of a Banach spa... |
| cnbn 30946 | The set of complex numbers... |
| ubthlem1 30947 | Lemma for ~ ubth . The fu... |
| ubthlem2 30948 | Lemma for ~ ubth . Given ... |
| ubthlem3 30949 | Lemma for ~ ubth . Prove ... |
| ubth 30950 | Uniform Boundedness Theore... |
| minvecolem1 30951 | Lemma for ~ minveco . The... |
| minvecolem2 30952 | Lemma for ~ minveco . Any... |
| minvecolem3 30953 | Lemma for ~ minveco . The... |
| minvecolem4a 30954 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30955 | Lemma for ~ minveco . The... |
| minvecolem4c 30956 | Lemma for ~ minveco . The... |
| minvecolem4 30957 | Lemma for ~ minveco . The... |
| minvecolem5 30958 | Lemma for ~ minveco . Dis... |
| minvecolem6 30959 | Lemma for ~ minveco . Any... |
| minvecolem7 30960 | Lemma for ~ minveco . Sin... |
| minveco 30961 | Minimizing vector theorem,... |
| ishlo 30964 | The predicate "is a comple... |
| hlobn 30965 | Every complex Hilbert spac... |
| hlph 30966 | Every complex Hilbert spac... |
| hlrel 30967 | The class of all complex H... |
| hlnv 30968 | Every complex Hilbert spac... |
| hlnvi 30969 | Every complex Hilbert spac... |
| hlvc 30970 | Every complex Hilbert spac... |
| hlcmet 30971 | The induced metric on a co... |
| hlmet 30972 | The induced metric on a co... |
| hlpar2 30973 | The parallelogram law sati... |
| hlpar 30974 | The parallelogram law sati... |
| hlex 30975 | The base set of a Hilbert ... |
| hladdf 30976 | Mapping for Hilbert space ... |
| hlcom 30977 | Hilbert space vector addit... |
| hlass 30978 | Hilbert space vector addit... |
| hl0cl 30979 | The Hilbert space zero vec... |
| hladdid 30980 | Hilbert space addition wit... |
| hlmulf 30981 | Mapping for Hilbert space ... |
| hlmulid 30982 | Hilbert space scalar multi... |
| hlmulass 30983 | Hilbert space scalar multi... |
| hldi 30984 | Hilbert space scalar multi... |
| hldir 30985 | Hilbert space scalar multi... |
| hlmul0 30986 | Hilbert space scalar multi... |
| hlipf 30987 | Mapping for Hilbert space ... |
| hlipcj 30988 | Conjugate law for Hilbert ... |
| hlipdir 30989 | Distributive law for Hilbe... |
| hlipass 30990 | Associative law for Hilber... |
| hlipgt0 30991 | The inner product of a Hil... |
| hlcompl 30992 | Completeness of a Hilbert ... |
| cnchl 30993 | The set of complex numbers... |
| htthlem 30994 | Lemma for ~ htth . The co... |
| htth 30995 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 31051 | The group (addition) opera... |
| h2hsm 31052 | The scalar product operati... |
| h2hnm 31053 | The norm function of Hilbe... |
| h2hvs 31054 | The vector subtraction ope... |
| h2hmetdval 31055 | Value of the distance func... |
| h2hcau 31056 | The Cauchy sequences of Hi... |
| h2hlm 31057 | The limit sequences of Hil... |
| axhilex-zf 31058 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31059 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31060 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31061 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31062 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31063 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31064 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31065 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31066 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31067 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31068 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31069 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31070 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31071 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31072 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31073 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31074 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31075 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31088 | The Hilbert space scalar p... |
| hvaddcl 31089 | Closure of vector addition... |
| hvmulcl 31090 | Closure of scalar multipli... |
| hvmulcli 31091 | Closure inference for scal... |
| hvsubf 31092 | Mapping domain and codomai... |
| hvsubval 31093 | Value of vector subtractio... |
| hvsubcl 31094 | Closure of vector subtract... |
| hvaddcli 31095 | Closure of vector addition... |
| hvcomi 31096 | Commutation of vector addi... |
| hvsubvali 31097 | Value of vector subtractio... |
| hvsubcli 31098 | Closure of vector subtract... |
| ifhvhv0 31099 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31100 | Addition with the zero vec... |
| hvmul0 31101 | Scalar multiplication with... |
| hvmul0or 31102 | If a scalar product is zer... |
| hvsubid 31103 | Subtraction of a vector fr... |
| hvnegid 31104 | Addition of negative of a ... |
| hv2neg 31105 | Two ways to express the ne... |
| hvaddlidi 31106 | Addition with the zero vec... |
| hvnegidi 31107 | Addition of negative of a ... |
| hv2negi 31108 | Two ways to express the ne... |
| hvm1neg 31109 | Convert minus one times a ... |
| hvaddsubval 31110 | Value of vector addition i... |
| hvadd32 31111 | Commutative/associative la... |
| hvadd12 31112 | Commutative/associative la... |
| hvadd4 31113 | Hilbert vector space addit... |
| hvsub4 31114 | Hilbert vector space addit... |
| hvaddsub12 31115 | Commutative/associative la... |
| hvpncan 31116 | Addition/subtraction cance... |
| hvpncan2 31117 | Addition/subtraction cance... |
| hvaddsubass 31118 | Associativity of sum and d... |
| hvpncan3 31119 | Subtraction and addition o... |
| hvmulcom 31120 | Scalar multiplication comm... |
| hvsubass 31121 | Hilbert vector space assoc... |
| hvsub32 31122 | Hilbert vector space commu... |
| hvmulassi 31123 | Scalar multiplication asso... |
| hvmulcomi 31124 | Scalar multiplication comm... |
| hvmul2negi 31125 | Double negative in scalar ... |
| hvsubdistr1 31126 | Scalar multiplication dist... |
| hvsubdistr2 31127 | Scalar multiplication dist... |
| hvdistr1i 31128 | Scalar multiplication dist... |
| hvsubdistr1i 31129 | Scalar multiplication dist... |
| hvassi 31130 | Hilbert vector space assoc... |
| hvadd32i 31131 | Hilbert vector space commu... |
| hvsubassi 31132 | Hilbert vector space assoc... |
| hvsub32i 31133 | Hilbert vector space commu... |
| hvadd12i 31134 | Hilbert vector space commu... |
| hvadd4i 31135 | Hilbert vector space addit... |
| hvsubsub4i 31136 | Hilbert vector space addit... |
| hvsubsub4 31137 | Hilbert vector space addit... |
| hv2times 31138 | Two times a vector. (Cont... |
| hvnegdii 31139 | Distribution of negative o... |
| hvsubeq0i 31140 | If the difference between ... |
| hvsubcan2i 31141 | Vector cancellation law. ... |
| hvaddcani 31142 | Cancellation law for vecto... |
| hvsubaddi 31143 | Relationship between vecto... |
| hvnegdi 31144 | Distribution of negative o... |
| hvsubeq0 31145 | If the difference between ... |
| hvaddeq0 31146 | If the sum of two vectors ... |
| hvaddcan 31147 | Cancellation law for vecto... |
| hvaddcan2 31148 | Cancellation law for vecto... |
| hvmulcan 31149 | Cancellation law for scala... |
| hvmulcan2 31150 | Cancellation law for scala... |
| hvsubcan 31151 | Cancellation law for vecto... |
| hvsubcan2 31152 | Cancellation law for vecto... |
| hvsub0 31153 | Subtraction of a zero vect... |
| hvsubadd 31154 | Relationship between vecto... |
| hvaddsub4 31155 | Hilbert vector space addit... |
| hicl 31157 | Closure of inner product. ... |
| hicli 31158 | Closure inference for inne... |
| his5 31163 | Associative law for inner ... |
| his52 31164 | Associative law for inner ... |
| his35 31165 | Move scalar multiplication... |
| his35i 31166 | Move scalar multiplication... |
| his7 31167 | Distributive law for inner... |
| hiassdi 31168 | Distributive/associative l... |
| his2sub 31169 | Distributive law for inner... |
| his2sub2 31170 | Distributive law for inner... |
| hire 31171 | A necessary and sufficient... |
| hiidrcl 31172 | Real closure of inner prod... |
| hi01 31173 | Inner product with the 0 v... |
| hi02 31174 | Inner product with the 0 v... |
| hiidge0 31175 | Inner product with self is... |
| his6 31176 | Zero inner product with se... |
| his1i 31177 | Conjugate law for inner pr... |
| abshicom 31178 | Commuted inner products ha... |
| hial0 31179 | A vector whose inner produ... |
| hial02 31180 | A vector whose inner produ... |
| hisubcomi 31181 | Two vector subtractions si... |
| hi2eq 31182 | Lemma used to prove equali... |
| hial2eq 31183 | Two vectors whose inner pr... |
| hial2eq2 31184 | Two vectors whose inner pr... |
| orthcom 31185 | Orthogonality commutes. (... |
| normlem0 31186 | Lemma used to derive prope... |
| normlem1 31187 | Lemma used to derive prope... |
| normlem2 31188 | Lemma used to derive prope... |
| normlem3 31189 | Lemma used to derive prope... |
| normlem4 31190 | Lemma used to derive prope... |
| normlem5 31191 | Lemma used to derive prope... |
| normlem6 31192 | Lemma used to derive prope... |
| normlem7 31193 | Lemma used to derive prope... |
| normlem8 31194 | Lemma used to derive prope... |
| normlem9 31195 | Lemma used to derive prope... |
| normlem7tALT 31196 | Lemma used to derive prope... |
| bcseqi 31197 | Equality case of Bunjakova... |
| normlem9at 31198 | Lemma used to derive prope... |
| dfhnorm2 31199 | Alternate definition of th... |
| normf 31200 | The norm function maps fro... |
| normval 31201 | The value of the norm of a... |
| normcl 31202 | Real closure of the norm o... |
| normge0 31203 | The norm of a vector is no... |
| normgt0 31204 | The norm of nonzero vector... |
| norm0 31205 | The norm of a zero vector.... |
| norm-i 31206 | Theorem 3.3(i) of [Beran] ... |
| normne0 31207 | A norm is nonzero iff its ... |
| normcli 31208 | Real closure of the norm o... |
| normsqi 31209 | The square of a norm. (Co... |
| norm-i-i 31210 | Theorem 3.3(i) of [Beran] ... |
| normsq 31211 | The square of a norm. (Co... |
| normsub0i 31212 | Two vectors are equal iff ... |
| normsub0 31213 | Two vectors are equal iff ... |
| norm-ii-i 31214 | Triangle inequality for no... |
| norm-ii 31215 | Triangle inequality for no... |
| norm-iii-i 31216 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31217 | Theorem 3.3(iii) of [Beran... |
| normsubi 31218 | Negative doesn't change th... |
| normpythi 31219 | Analogy to Pythagorean the... |
| normsub 31220 | Swapping order of subtract... |
| normneg 31221 | The norm of a vector equal... |
| normpyth 31222 | Analogy to Pythagorean the... |
| normpyc 31223 | Corollary to Pythagorean t... |
| norm3difi 31224 | Norm of differences around... |
| norm3adifii 31225 | Norm of differences around... |
| norm3lem 31226 | Lemma involving norm of di... |
| norm3dif 31227 | Norm of differences around... |
| norm3dif2 31228 | Norm of differences around... |
| norm3lemt 31229 | Lemma involving norm of di... |
| norm3adifi 31230 | Norm of differences around... |
| normpari 31231 | Parallelogram law for norm... |
| normpar 31232 | Parallelogram law for norm... |
| normpar2i 31233 | Corollary of parallelogram... |
| polid2i 31234 | Generalized polarization i... |
| polidi 31235 | Polarization identity. Re... |
| polid 31236 | Polarization identity. Re... |
| hilablo 31237 | Hilbert space vector addit... |
| hilid 31238 | The group identity element... |
| hilvc 31239 | Hilbert space is a complex... |
| hilnormi 31240 | Hilbert space norm in term... |
| hilhhi 31241 | Deduce the structure of Hi... |
| hhnv 31242 | Hilbert space is a normed ... |
| hhva 31243 | The group (addition) opera... |
| hhba 31244 | The base set of Hilbert sp... |
| hh0v 31245 | The zero vector of Hilbert... |
| hhsm 31246 | The scalar product operati... |
| hhvs 31247 | The vector subtraction ope... |
| hhnm 31248 | The norm function of Hilbe... |
| hhims 31249 | The induced metric of Hilb... |
| hhims2 31250 | Hilbert space distance met... |
| hhmet 31251 | The induced metric of Hilb... |
| hhxmet 31252 | The induced metric of Hilb... |
| hhmetdval 31253 | Value of the distance func... |
| hhip 31254 | The inner product operatio... |
| hhph 31255 | The Hilbert space of the H... |
| bcsiALT 31256 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31257 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31258 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31259 | Corollary of the Bunjakova... |
| bcs3 31260 | Corollary of the Bunjakova... |
| hcau 31261 | Member of the set of Cauch... |
| hcauseq 31262 | A Cauchy sequences on a Hi... |
| hcaucvg 31263 | A Cauchy sequence on a Hil... |
| seq1hcau 31264 | A sequence on a Hilbert sp... |
| hlimi 31265 | Express the predicate: Th... |
| hlimseqi 31266 | A sequence with a limit on... |
| hlimveci 31267 | Closure of the limit of a ... |
| hlimconvi 31268 | Convergence of a sequence ... |
| hlim2 31269 | The limit of a sequence on... |
| hlimadd 31270 | Limit of the sum of two se... |
| hilmet 31271 | The Hilbert space norm det... |
| hilxmet 31272 | The Hilbert space norm det... |
| hilmetdval 31273 | Value of the distance func... |
| hilims 31274 | Hilbert space distance met... |
| hhcau 31275 | The Cauchy sequences of Hi... |
| hhlm 31276 | The limit sequences of Hil... |
| hhcmpl 31277 | Lemma used for derivation ... |
| hilcompl 31278 | Lemma used for derivation ... |
| hhcms 31280 | The Hilbert space induced ... |
| hhhl 31281 | The Hilbert space structur... |
| hilcms 31282 | The Hilbert space norm det... |
| hilhl 31283 | The Hilbert space of the H... |
| issh 31285 | Subspace ` H ` of a Hilber... |
| issh2 31286 | Subspace ` H ` of a Hilber... |
| shss 31287 | A subspace is a subset of ... |
| shel 31288 | A member of a subspace of ... |
| shex 31289 | The set of subspaces of a ... |
| shssii 31290 | A closed subspace of a Hil... |
| sheli 31291 | A member of a subspace of ... |
| shelii 31292 | A member of a subspace of ... |
| sh0 31293 | The zero vector belongs to... |
| shaddcl 31294 | Closure of vector addition... |
| shmulcl 31295 | Closure of vector scalar m... |
| issh3 31296 | Subspace ` H ` of a Hilber... |
| shsubcl 31297 | Closure of vector subtract... |
| isch 31299 | Closed subspace ` H ` of a... |
| isch2 31300 | Closed subspace ` H ` of a... |
| chsh 31301 | A closed subspace is a sub... |
| chsssh 31302 | Closed subspaces are subsp... |
| chex 31303 | The set of closed subspace... |
| chshii 31304 | A closed subspace is a sub... |
| ch0 31305 | The zero vector belongs to... |
| chss 31306 | A closed subspace of a Hil... |
| chel 31307 | A member of a closed subsp... |
| chssii 31308 | A closed subspace of a Hil... |
| cheli 31309 | A member of a closed subsp... |
| chelii 31310 | A member of a closed subsp... |
| chlimi 31311 | The limit property of a cl... |
| hlim0 31312 | The zero sequence in Hilbe... |
| hlimcaui 31313 | If a sequence in Hilbert s... |
| hlimf 31314 | Function-like behavior of ... |
| hlimuni 31315 | A Hilbert space sequence c... |
| hlimreui 31316 | The limit of a Hilbert spa... |
| hlimeui 31317 | The limit of a Hilbert spa... |
| isch3 31318 | A Hilbert subspace is clos... |
| chcompl 31319 | Completeness of a closed s... |
| helch 31320 | The Hilbert lattice one (w... |
| ifchhv 31321 | Prove ` if ( A e. CH , A ,... |
| helsh 31322 | Hilbert space is a subspac... |
| shsspwh 31323 | Subspaces are subsets of H... |
| chsspwh 31324 | Closed subspaces are subse... |
| hsn0elch 31325 | The zero subspace belongs ... |
| norm1 31326 | From any nonzero Hilbert s... |
| norm1exi 31327 | A normalized vector exists... |
| norm1hex 31328 | A normalized vector can ex... |
| elch0 31331 | Membership in zero for clo... |
| h0elch 31332 | The zero subspace is a clo... |
| h0elsh 31333 | The zero subspace is a sub... |
| hhssva 31334 | The vector addition operat... |
| hhsssm 31335 | The scalar multiplication ... |
| hhssnm 31336 | The norm operation on a su... |
| issubgoilem 31337 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31338 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31339 | Abelian group property of ... |
| hhssablo 31340 | Abelian group property of ... |
| hhssnv 31341 | Normed complex vector spac... |
| hhssnvt 31342 | Normed complex vector spac... |
| hhsst 31343 | A member of ` SH ` is a su... |
| hhshsslem1 31344 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31345 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31346 | The predicate " ` H ` is a... |
| hhsssh2 31347 | The predicate " ` H ` is a... |
| hhssba 31348 | The base set of a subspace... |
| hhssvs 31349 | The vector subtraction ope... |
| hhssvsf 31350 | Mapping of the vector subt... |
| hhssims 31351 | Induced metric of a subspa... |
| hhssims2 31352 | Induced metric of a subspa... |
| hhssmet 31353 | Induced metric of a subspa... |
| hhssmetdval 31354 | Value of the distance func... |
| hhsscms 31355 | The induced metric of a cl... |
| hhssbnOLD 31356 | Obsolete version of ~ cssb... |
| ocval 31357 | Value of orthogonal comple... |
| ocel 31358 | Membership in orthogonal c... |
| shocel 31359 | Membership in orthogonal c... |
| ocsh 31360 | The orthogonal complement ... |
| shocsh 31361 | The orthogonal complement ... |
| ocss 31362 | An orthogonal complement i... |
| shocss 31363 | An orthogonal complement i... |
| occon 31364 | Contraposition law for ort... |
| occon2 31365 | Double contraposition for ... |
| occon2i 31366 | Double contraposition for ... |
| oc0 31367 | The zero vector belongs to... |
| ocorth 31368 | Members of a subset and it... |
| shocorth 31369 | Members of a subspace and ... |
| ococss 31370 | Inclusion in complement of... |
| shococss 31371 | Inclusion in complement of... |
| shorth 31372 | Members of orthogonal subs... |
| ocin 31373 | Intersection of a Hilbert ... |
| occon3 31374 | Hilbert lattice contraposi... |
| ocnel 31375 | A nonzero vector in the co... |
| chocvali 31376 | Value of the orthogonal co... |
| shuni 31377 | Two subspaces with trivial... |
| chocunii 31378 | Lemma for uniqueness part ... |
| pjhthmo 31379 | Projection Theorem, unique... |
| occllem 31380 | Lemma for ~ occl . (Contr... |
| occl 31381 | Closure of complement of H... |
| shoccl 31382 | Closure of complement of H... |
| choccl 31383 | Closure of complement of H... |
| choccli 31384 | Closure of ` CH ` orthocom... |
| shsval 31389 | Value of subspace sum of t... |
| shsss 31390 | The subspace sum is a subs... |
| shsel 31391 | Membership in the subspace... |
| shsel3 31392 | Membership in the subspace... |
| shseli 31393 | Membership in subspace sum... |
| shscli 31394 | Closure of subspace sum. ... |
| shscl 31395 | Closure of subspace sum. ... |
| shscom 31396 | Commutative law for subspa... |
| shsva 31397 | Vector sum belongs to subs... |
| shsel1 31398 | A subspace sum contains a ... |
| shsel2 31399 | A subspace sum contains a ... |
| shsvs 31400 | Vector subtraction belongs... |
| shsub1 31401 | Subspace sum is an upper b... |
| shsub2 31402 | Subspace sum is an upper b... |
| choc0 31403 | The orthocomplement of the... |
| choc1 31404 | The orthocomplement of the... |
| chocnul 31405 | Orthogonal complement of t... |
| shintcli 31406 | Closure of intersection of... |
| shintcl 31407 | The intersection of a none... |
| chintcli 31408 | The intersection of a none... |
| chintcl 31409 | The intersection (infimum)... |
| spanval 31410 | Value of the linear span o... |
| hsupval 31411 | Value of supremum of set o... |
| chsupval 31412 | The value of the supremum ... |
| spancl 31413 | The span of a subset of Hi... |
| elspancl 31414 | A member of a span is a ve... |
| shsupcl 31415 | Closure of the subspace su... |
| hsupcl 31416 | Closure of supremum of set... |
| chsupcl 31417 | Closure of supremum of sub... |
| hsupss 31418 | Subset relation for suprem... |
| chsupss 31419 | Subset relation for suprem... |
| hsupunss 31420 | The union of a set of Hilb... |
| chsupunss 31421 | The union of a set of clos... |
| spanss2 31422 | A subset of Hilbert space ... |
| shsupunss 31423 | The union of a set of subs... |
| spanid 31424 | A subspace of Hilbert spac... |
| spanss 31425 | Ordering relationship for ... |
| spanssoc 31426 | The span of a subset of Hi... |
| sshjval 31427 | Value of join for subsets ... |
| shjval 31428 | Value of join in ` SH ` . ... |
| chjval 31429 | Value of join in ` CH ` . ... |
| chjvali 31430 | Value of join in ` CH ` . ... |
| sshjval3 31431 | Value of join for subsets ... |
| sshjcl 31432 | Closure of join for subset... |
| shjcl 31433 | Closure of join in ` SH ` ... |
| chjcl 31434 | Closure of join in ` CH ` ... |
| shjcom 31435 | Commutative law for Hilber... |
| shless 31436 | Subset implies subset of s... |
| shlej1 31437 | Add disjunct to both sides... |
| shlej2 31438 | Add disjunct to both sides... |
| shincli 31439 | Closure of intersection of... |
| shscomi 31440 | Commutative law for subspa... |
| shsvai 31441 | Vector sum belongs to subs... |
| shsel1i 31442 | A subspace sum contains a ... |
| shsel2i 31443 | A subspace sum contains a ... |
| shsvsi 31444 | Vector subtraction belongs... |
| shunssi 31445 | Union is smaller than subs... |
| shunssji 31446 | Union is smaller than Hilb... |
| shsleji 31447 | Subspace sum is smaller th... |
| shjcomi 31448 | Commutative law for join i... |
| shsub1i 31449 | Subspace sum is an upper b... |
| shsub2i 31450 | Subspace sum is an upper b... |
| shub1i 31451 | Hilbert lattice join is an... |
| shjcli 31452 | Closure of ` CH ` join. (... |
| shjshcli 31453 | ` SH ` closure of join. (... |
| shlessi 31454 | Subset implies subset of s... |
| shlej1i 31455 | Add disjunct to both sides... |
| shlej2i 31456 | Add disjunct to both sides... |
| shslej 31457 | Subspace sum is smaller th... |
| shincl 31458 | Closure of intersection of... |
| shub1 31459 | Hilbert lattice join is an... |
| shub2 31460 | A subspace is a subset of ... |
| shsidmi 31461 | Idempotent law for Hilbert... |
| shslubi 31462 | The least upper bound law ... |
| shlesb1i 31463 | Hilbert lattice ordering i... |
| shsval2i 31464 | An alternate way to expres... |
| shsval3i 31465 | An alternate way to expres... |
| shmodsi 31466 | The modular law holds for ... |
| shmodi 31467 | The modular law is implied... |
| pjhthlem1 31468 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31469 | Lemma for ~ pjhth . (Cont... |
| pjhth 31470 | Projection Theorem: Any H... |
| pjhtheu 31471 | Projection Theorem: Any H... |
| pjhfval 31473 | The value of the projectio... |
| pjhval 31474 | Value of a projection. (C... |
| pjpreeq 31475 | Equality with a projection... |
| pjeq 31476 | Equality with a projection... |
| axpjcl 31477 | Closure of a projection in... |
| pjhcl 31478 | Closure of a projection in... |
| omlsilem 31479 | Lemma for orthomodular law... |
| omlsii 31480 | Subspace inference form of... |
| omlsi 31481 | Subspace form of orthomodu... |
| ococi 31482 | Complement of complement o... |
| ococ 31483 | Complement of complement o... |
| dfch2 31484 | Alternate definition of th... |
| ococin 31485 | The double complement is t... |
| hsupval2 31486 | Alternate definition of su... |
| chsupval2 31487 | The value of the supremum ... |
| sshjval2 31488 | Value of join in the set o... |
| chsupid 31489 | A subspace is the supremum... |
| chsupsn 31490 | Value of supremum of subse... |
| shlub 31491 | Hilbert lattice join is th... |
| shlubi 31492 | Hilbert lattice join is th... |
| pjhtheu2 31493 | Uniqueness of ` y ` for th... |
| pjcli 31494 | Closure of a projection in... |
| pjhcli 31495 | Closure of a projection in... |
| pjpjpre 31496 | Decomposition of a vector ... |
| axpjpj 31497 | Decomposition of a vector ... |
| pjclii 31498 | Closure of a projection in... |
| pjhclii 31499 | Closure of a projection in... |
| pjpj0i 31500 | Decomposition of a vector ... |
| pjpji 31501 | Decomposition of a vector ... |
| pjpjhth 31502 | Projection Theorem: Any H... |
| pjpjhthi 31503 | Projection Theorem: Any H... |
| pjop 31504 | Orthocomplement projection... |
| pjpo 31505 | Projection in terms of ort... |
| pjopi 31506 | Orthocomplement projection... |
| pjpoi 31507 | Projection in terms of ort... |
| pjoc1i 31508 | Projection of a vector in ... |
| pjchi 31509 | Projection of a vector in ... |
| pjoccl 31510 | The part of a vector that ... |
| pjoc1 31511 | Projection of a vector in ... |
| pjomli 31512 | Subspace form of orthomodu... |
| pjoml 31513 | Subspace form of orthomodu... |
| pjococi 31514 | Proof of orthocomplement t... |
| pjoc2i 31515 | Projection of a vector in ... |
| pjoc2 31516 | Projection of a vector in ... |
| sh0le 31517 | The zero subspace is the s... |
| ch0le 31518 | The zero subspace is the s... |
| shle0 31519 | No subspace is smaller tha... |
| chle0 31520 | No Hilbert lattice element... |
| chnlen0 31521 | A Hilbert lattice element ... |
| ch0pss 31522 | The zero subspace is a pro... |
| orthin 31523 | The intersection of orthog... |
| ssjo 31524 | The lattice join of a subs... |
| shne0i 31525 | A nonzero subspace has a n... |
| shs0i 31526 | Hilbert subspace sum with ... |
| shs00i 31527 | Two subspaces are zero iff... |
| ch0lei 31528 | The closed subspace zero i... |
| chle0i 31529 | No Hilbert closed subspace... |
| chne0i 31530 | A nonzero closed subspace ... |
| chocini 31531 | Intersection of a closed s... |
| chj0i 31532 | Join with lattice zero in ... |
| chm1i 31533 | Meet with lattice one in `... |
| chjcli 31534 | Closure of ` CH ` join. (... |
| chsleji 31535 | Subspace sum is smaller th... |
| chseli 31536 | Membership in subspace sum... |
| chincli 31537 | Closure of Hilbert lattice... |
| chsscon3i 31538 | Hilbert lattice contraposi... |
| chsscon1i 31539 | Hilbert lattice contraposi... |
| chsscon2i 31540 | Hilbert lattice contraposi... |
| chcon2i 31541 | Hilbert lattice contraposi... |
| chcon1i 31542 | Hilbert lattice contraposi... |
| chcon3i 31543 | Hilbert lattice contraposi... |
| chunssji 31544 | Union is smaller than ` CH... |
| chjcomi 31545 | Commutative law for join i... |
| chub1i 31546 | ` CH ` join is an upper bo... |
| chub2i 31547 | ` CH ` join is an upper bo... |
| chlubi 31548 | Hilbert lattice join is th... |
| chlubii 31549 | Hilbert lattice join is th... |
| chlej1i 31550 | Add join to both sides of ... |
| chlej2i 31551 | Add join to both sides of ... |
| chlej12i 31552 | Add join to both sides of ... |
| chlejb1i 31553 | Hilbert lattice ordering i... |
| chdmm1i 31554 | De Morgan's law for meet i... |
| chdmm2i 31555 | De Morgan's law for meet i... |
| chdmm3i 31556 | De Morgan's law for meet i... |
| chdmm4i 31557 | De Morgan's law for meet i... |
| chdmj1i 31558 | De Morgan's law for join i... |
| chdmj2i 31559 | De Morgan's law for join i... |
| chdmj3i 31560 | De Morgan's law for join i... |
| chdmj4i 31561 | De Morgan's law for join i... |
| chnlei 31562 | Equivalent expressions for... |
| chjassi 31563 | Associative law for Hilber... |
| chj00i 31564 | Two Hilbert lattice elemen... |
| chjoi 31565 | The join of a closed subsp... |
| chj1i 31566 | Join with Hilbert lattice ... |
| chm0i 31567 | Meet with Hilbert lattice ... |
| chm0 31568 | Meet with Hilbert lattice ... |
| shjshsi 31569 | Hilbert lattice join equal... |
| shjshseli 31570 | A closed subspace sum equa... |
| chne0 31571 | A nonzero closed subspace ... |
| chocin 31572 | Intersection of a closed s... |
| chssoc 31573 | A closed subspace less tha... |
| chj0 31574 | Join with Hilbert lattice ... |
| chslej 31575 | Subspace sum is smaller th... |
| chincl 31576 | Closure of Hilbert lattice... |
| chsscon3 31577 | Hilbert lattice contraposi... |
| chsscon1 31578 | Hilbert lattice contraposi... |
| chsscon2 31579 | Hilbert lattice contraposi... |
| chpsscon3 31580 | Hilbert lattice contraposi... |
| chpsscon1 31581 | Hilbert lattice contraposi... |
| chpsscon2 31582 | Hilbert lattice contraposi... |
| chjcom 31583 | Commutative law for Hilber... |
| chub1 31584 | Hilbert lattice join is gr... |
| chub2 31585 | Hilbert lattice join is gr... |
| chlub 31586 | Hilbert lattice join is th... |
| chlej1 31587 | Add join to both sides of ... |
| chlej2 31588 | Add join to both sides of ... |
| chlejb1 31589 | Hilbert lattice ordering i... |
| chlejb2 31590 | Hilbert lattice ordering i... |
| chnle 31591 | Equivalent expressions for... |
| chjo 31592 | The join of a closed subsp... |
| chabs1 31593 | Hilbert lattice absorption... |
| chabs2 31594 | Hilbert lattice absorption... |
| chabs1i 31595 | Hilbert lattice absorption... |
| chabs2i 31596 | Hilbert lattice absorption... |
| chjidm 31597 | Idempotent law for Hilbert... |
| chjidmi 31598 | Idempotent law for Hilbert... |
| chj12i 31599 | A rearrangement of Hilbert... |
| chj4i 31600 | Rearrangement of the join ... |
| chjjdiri 31601 | Hilbert lattice join distr... |
| chdmm1 31602 | De Morgan's law for meet i... |
| chdmm2 31603 | De Morgan's law for meet i... |
| chdmm3 31604 | De Morgan's law for meet i... |
| chdmm4 31605 | De Morgan's law for meet i... |
| chdmj1 31606 | De Morgan's law for join i... |
| chdmj2 31607 | De Morgan's law for join i... |
| chdmj3 31608 | De Morgan's law for join i... |
| chdmj4 31609 | De Morgan's law for join i... |
| chjass 31610 | Associative law for Hilber... |
| chj12 31611 | A rearrangement of Hilbert... |
| chj4 31612 | Rearrangement of the join ... |
| ledii 31613 | An ortholattice is distrib... |
| lediri 31614 | An ortholattice is distrib... |
| lejdii 31615 | An ortholattice is distrib... |
| lejdiri 31616 | An ortholattice is distrib... |
| ledi 31617 | An ortholattice is distrib... |
| spansn0 31618 | The span of the singleton ... |
| span0 31619 | The span of the empty set ... |
| elspani 31620 | Membership in the span of ... |
| spanuni 31621 | The span of a union is the... |
| spanun 31622 | The span of a union is the... |
| sshhococi 31623 | The join of two Hilbert sp... |
| hne0 31624 | Hilbert space has a nonzer... |
| chsup0 31625 | The supremum of the empty ... |
| h1deoi 31626 | Membership in orthocomplem... |
| h1dei 31627 | Membership in 1-dimensiona... |
| h1did 31628 | A generating vector belong... |
| h1dn0 31629 | A nonzero vector generates... |
| h1de2i 31630 | Membership in 1-dimensiona... |
| h1de2bi 31631 | Membership in 1-dimensiona... |
| h1de2ctlem 31632 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31633 | Membership in 1-dimensiona... |
| spansni 31634 | The span of a singleton in... |
| elspansni 31635 | Membership in the span of ... |
| spansn 31636 | The span of a singleton in... |
| spansnch 31637 | The span of a Hilbert spac... |
| spansnsh 31638 | The span of a Hilbert spac... |
| spansnchi 31639 | The span of a singleton in... |
| spansnid 31640 | A vector belongs to the sp... |
| spansnmul 31641 | A scalar product with a ve... |
| elspansncl 31642 | A member of a span of a si... |
| elspansn 31643 | Membership in the span of ... |
| elspansn2 31644 | Membership in the span of ... |
| spansncol 31645 | The singletons of collinea... |
| spansneleqi 31646 | Membership relation implie... |
| spansneleq 31647 | Membership relation that i... |
| spansnss 31648 | The span of the singleton ... |
| elspansn3 31649 | A member of the span of th... |
| elspansn4 31650 | A span membership conditio... |
| elspansn5 31651 | A vector belonging to both... |
| spansnss2 31652 | The span of the singleton ... |
| normcan 31653 | Cancellation-type law that... |
| pjspansn 31654 | A projection on the span o... |
| spansnpji 31655 | A subset of Hilbert space ... |
| spanunsni 31656 | The span of the union of a... |
| spanpr 31657 | The span of a pair of vect... |
| h1datomi 31658 | A 1-dimensional subspace i... |
| h1datom 31659 | A 1-dimensional subspace i... |
| cmbr 31661 | Binary relation expressing... |
| pjoml2i 31662 | Variation of orthomodular ... |
| pjoml3i 31663 | Variation of orthomodular ... |
| pjoml4i 31664 | Variation of orthomodular ... |
| pjoml5i 31665 | The orthomodular law. Rem... |
| pjoml6i 31666 | An equivalent of the ortho... |
| cmbri 31667 | Binary relation expressing... |
| cmcmlem 31668 | Commutation is symmetric. ... |
| cmcmi 31669 | Commutation is symmetric. ... |
| cmcm2i 31670 | Commutation with orthocomp... |
| cmcm3i 31671 | Commutation with orthocomp... |
| cmcm4i 31672 | Commutation with orthocomp... |
| cmbr2i 31673 | Alternate definition of th... |
| cmcmii 31674 | Commutation is symmetric. ... |
| cmcm2ii 31675 | Commutation with orthocomp... |
| cmcm3ii 31676 | Commutation with orthocomp... |
| cmbr3i 31677 | Alternate definition for t... |
| cmbr4i 31678 | Alternate definition for t... |
| lecmi 31679 | Comparable Hilbert lattice... |
| lecmii 31680 | Comparable Hilbert lattice... |
| cmj1i 31681 | A Hilbert lattice element ... |
| cmj2i 31682 | A Hilbert lattice element ... |
| cmm1i 31683 | A Hilbert lattice element ... |
| cmm2i 31684 | A Hilbert lattice element ... |
| cmbr3 31685 | Alternate definition for t... |
| cm0 31686 | The zero Hilbert lattice e... |
| cmidi 31687 | The commutes relation is r... |
| pjoml2 31688 | Variation of orthomodular ... |
| pjoml3 31689 | Variation of orthomodular ... |
| pjoml5 31690 | The orthomodular law. Rem... |
| cmcm 31691 | Commutation is symmetric. ... |
| cmcm3 31692 | Commutation with orthocomp... |
| cmcm2 31693 | Commutation with orthocomp... |
| lecm 31694 | Comparable Hilbert lattice... |
| fh1 31695 | Foulis-Holland Theorem. I... |
| fh2 31696 | Foulis-Holland Theorem. I... |
| cm2j 31697 | A lattice element that com... |
| fh1i 31698 | Foulis-Holland Theorem. I... |
| fh2i 31699 | Foulis-Holland Theorem. I... |
| fh3i 31700 | Variation of the Foulis-Ho... |
| fh4i 31701 | Variation of the Foulis-Ho... |
| cm2ji 31702 | A lattice element that com... |
| cm2mi 31703 | A lattice element that com... |
| qlax1i 31704 | One of the equations showi... |
| qlax2i 31705 | One of the equations showi... |
| qlax3i 31706 | One of the equations showi... |
| qlax4i 31707 | One of the equations showi... |
| qlax5i 31708 | One of the equations showi... |
| qlaxr1i 31709 | One of the conditions show... |
| qlaxr2i 31710 | One of the conditions show... |
| qlaxr4i 31711 | One of the conditions show... |
| qlaxr5i 31712 | One of the conditions show... |
| qlaxr3i 31713 | A variation of the orthomo... |
| chscllem1 31714 | Lemma for ~ chscl . (Cont... |
| chscllem2 31715 | Lemma for ~ chscl . (Cont... |
| chscllem3 31716 | Lemma for ~ chscl . (Cont... |
| chscllem4 31717 | Lemma for ~ chscl . (Cont... |
| chscl 31718 | The subspace sum of two cl... |
| osumi 31719 | If two closed subspaces of... |
| osumcori 31720 | Corollary of ~ osumi . (C... |
| osumcor2i 31721 | Corollary of ~ osumi , sho... |
| osum 31722 | If two closed subspaces of... |
| spansnji 31723 | The subspace sum of a clos... |
| spansnj 31724 | The subspace sum of a clos... |
| spansnscl 31725 | The subspace sum of a clos... |
| sumspansn 31726 | The sum of two vectors bel... |
| spansnm0i 31727 | The meet of different one-... |
| nonbooli 31728 | A Hilbert lattice with two... |
| spansncvi 31729 | Hilbert space has the cove... |
| spansncv 31730 | Hilbert space has the cove... |
| 5oalem1 31731 | Lemma for orthoarguesian l... |
| 5oalem2 31732 | Lemma for orthoarguesian l... |
| 5oalem3 31733 | Lemma for orthoarguesian l... |
| 5oalem4 31734 | Lemma for orthoarguesian l... |
| 5oalem5 31735 | Lemma for orthoarguesian l... |
| 5oalem6 31736 | Lemma for orthoarguesian l... |
| 5oalem7 31737 | Lemma for orthoarguesian l... |
| 5oai 31738 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31739 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31740 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31741 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31742 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31743 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31744 | Lemma for 3OA (weak) ortho... |
| 3oai 31745 | 3OA (weak) orthoarguesian ... |
| pjorthi 31746 | Projection components on o... |
| pjch1 31747 | Property of identity proje... |
| pjo 31748 | The orthogonal projection.... |
| pjcompi 31749 | Component of a projection.... |
| pjidmi 31750 | A projection is idempotent... |
| pjadjii 31751 | A projection is self-adjoi... |
| pjaddii 31752 | Projection of vector sum i... |
| pjinormii 31753 | The inner product of a pro... |
| pjmulii 31754 | Projection of (scalar) pro... |
| pjsubii 31755 | Projection of vector diffe... |
| pjsslem 31756 | Lemma for subset relations... |
| pjss2i 31757 | Subset relationship for pr... |
| pjssmii 31758 | Projection meet property. ... |
| pjssge0ii 31759 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31760 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31761 | The projection on a subspa... |
| pjadji 31762 | A projection is self-adjoi... |
| pjaddi 31763 | Projection of vector sum i... |
| pjinormi 31764 | The inner product of a pro... |
| pjsubi 31765 | Projection of vector diffe... |
| pjmuli 31766 | Projection of scalar produ... |
| pjige0i 31767 | The inner product of a pro... |
| pjige0 31768 | The inner product of a pro... |
| pjcjt2 31769 | The projection on a subspa... |
| pj0i 31770 | The projection of the zero... |
| pjch 31771 | Projection of a vector in ... |
| pjid 31772 | The projection of a vector... |
| pjvec 31773 | The set of vectors belongi... |
| pjocvec 31774 | The set of vectors belongi... |
| pjocini 31775 | Membership of projection i... |
| pjini 31776 | Membership of projection i... |
| pjjsi 31777 | A sufficient condition for... |
| pjfni 31778 | Functionality of a project... |
| pjrni 31779 | The range of a projection.... |
| pjfoi 31780 | A projection maps onto its... |
| pjfi 31781 | The mapping of a projectio... |
| pjvi 31782 | The value of a projection ... |
| pjhfo 31783 | A projection maps onto its... |
| pjrn 31784 | The range of a projection.... |
| pjhf 31785 | The mapping of a projectio... |
| pjfn 31786 | Functionality of a project... |
| pjsumi 31787 | The projection on a subspa... |
| pj11i 31788 | One-to-one correspondence ... |
| pjdsi 31789 | Vector decomposition into ... |
| pjds3i 31790 | Vector decomposition into ... |
| pj11 31791 | One-to-one correspondence ... |
| pjmfn 31792 | Functionality of the proje... |
| pjmf1 31793 | The projector function map... |
| pjoi0 31794 | The inner product of proje... |
| pjoi0i 31795 | The inner product of proje... |
| pjopythi 31796 | Pythagorean theorem for pr... |
| pjopyth 31797 | Pythagorean theorem for pr... |
| pjnormi 31798 | The norm of the projection... |
| pjpythi 31799 | Pythagorean theorem for pr... |
| pjneli 31800 | If a vector does not belon... |
| pjnorm 31801 | The norm of the projection... |
| pjpyth 31802 | Pythagorean theorem for pr... |
| pjnel 31803 | If a vector does not belon... |
| pjnorm2 31804 | A vector belongs to the su... |
| mayete3i 31805 | Mayet's equation E_3. Par... |
| mayetes3i 31806 | Mayet's equation E^*_3, de... |
| hosmval 31812 | Value of the sum of two Hi... |
| hommval 31813 | Value of the scalar produc... |
| hodmval 31814 | Value of the difference of... |
| hfsmval 31815 | Value of the sum of two Hi... |
| hfmmval 31816 | Value of the scalar produc... |
| hosval 31817 | Value of the sum of two Hi... |
| homval 31818 | Value of the scalar produc... |
| hodval 31819 | Value of the difference of... |
| hfsval 31820 | Value of the sum of two Hi... |
| hfmval 31821 | Value of the scalar produc... |
| hoscl 31822 | Closure of the sum of two ... |
| homcl 31823 | Closure of the scalar prod... |
| hodcl 31824 | Closure of the difference ... |
| ho0val 31827 | Value of the zero Hilbert ... |
| ho0f 31828 | Functionality of the zero ... |
| df0op2 31829 | Alternate definition of Hi... |
| dfiop2 31830 | Alternate definition of Hi... |
| hoif 31831 | Functionality of the Hilbe... |
| hoival 31832 | The value of the Hilbert s... |
| hoico1 31833 | Composition with the Hilbe... |
| hoico2 31834 | Composition with the Hilbe... |
| hoaddcl 31835 | The sum of Hilbert space o... |
| homulcl 31836 | The scalar product of a Hi... |
| hoeq 31837 | Equality of Hilbert space ... |
| hoeqi 31838 | Equality of Hilbert space ... |
| hoscli 31839 | Closure of Hilbert space o... |
| hodcli 31840 | Closure of Hilbert space o... |
| hocoi 31841 | Composition of Hilbert spa... |
| hococli 31842 | Closure of composition of ... |
| hocofi 31843 | Mapping of composition of ... |
| hocofni 31844 | Functionality of compositi... |
| hoaddcli 31845 | Mapping of sum of Hilbert ... |
| hosubcli 31846 | Mapping of difference of H... |
| hoaddfni 31847 | Functionality of sum of Hi... |
| hosubfni 31848 | Functionality of differenc... |
| hoaddcomi 31849 | Commutativity of sum of Hi... |
| hosubcl 31850 | Mapping of difference of H... |
| hoaddcom 31851 | Commutativity of sum of Hi... |
| hodsi 31852 | Relationship between Hilbe... |
| hoaddassi 31853 | Associativity of sum of Hi... |
| hoadd12i 31854 | Commutative/associative la... |
| hoadd32i 31855 | Commutative/associative la... |
| hocadddiri 31856 | Distributive law for Hilbe... |
| hocsubdiri 31857 | Distributive law for Hilbe... |
| ho2coi 31858 | Double composition of Hilb... |
| hoaddass 31859 | Associativity of sum of Hi... |
| hoadd32 31860 | Commutative/associative la... |
| hoadd4 31861 | Rearrangement of 4 terms i... |
| hocsubdir 31862 | Distributive law for Hilbe... |
| hoaddridi 31863 | Sum of a Hilbert space ope... |
| hodidi 31864 | Difference of a Hilbert sp... |
| ho0coi 31865 | Composition of the zero op... |
| hoid1i 31866 | Composition of Hilbert spa... |
| hoid1ri 31867 | Composition of Hilbert spa... |
| hoaddrid 31868 | Sum of a Hilbert space ope... |
| hodid 31869 | Difference of a Hilbert sp... |
| hon0 31870 | A Hilbert space operator i... |
| hodseqi 31871 | Subtraction and addition o... |
| ho0subi 31872 | Subtraction of Hilbert spa... |
| honegsubi 31873 | Relationship between Hilbe... |
| ho0sub 31874 | Subtraction of Hilbert spa... |
| hosubid1 31875 | The zero operator subtract... |
| honegsub 31876 | Relationship between Hilbe... |
| homullid 31877 | An operator equals its sca... |
| homco1 31878 | Associative law for scalar... |
| homulass 31879 | Scalar product associative... |
| hoadddi 31880 | Scalar product distributiv... |
| hoadddir 31881 | Scalar product reverse dis... |
| homul12 31882 | Swap first and second fact... |
| honegneg 31883 | Double negative of a Hilbe... |
| hosubneg 31884 | Relationship between opera... |
| hosubdi 31885 | Scalar product distributiv... |
| honegdi 31886 | Distribution of negative o... |
| honegsubdi 31887 | Distribution of negative o... |
| honegsubdi2 31888 | Distribution of negative o... |
| hosubsub2 31889 | Law for double subtraction... |
| hosub4 31890 | Rearrangement of 4 terms i... |
| hosubadd4 31891 | Rearrangement of 4 terms i... |
| hoaddsubass 31892 | Associative-type law for a... |
| hoaddsub 31893 | Law for operator addition ... |
| hosubsub 31894 | Law for double subtraction... |
| hosubsub4 31895 | Law for double subtraction... |
| ho2times 31896 | Two times a Hilbert space ... |
| hoaddsubassi 31897 | Associativity of sum and d... |
| hoaddsubi 31898 | Law for sum and difference... |
| hosd1i 31899 | Hilbert space operator sum... |
| hosd2i 31900 | Hilbert space operator sum... |
| hopncani 31901 | Hilbert space operator can... |
| honpcani 31902 | Hilbert space operator can... |
| hosubeq0i 31903 | If the difference between ... |
| honpncani 31904 | Hilbert space operator can... |
| ho01i 31905 | A condition implying that ... |
| ho02i 31906 | A condition implying that ... |
| hoeq1 31907 | A condition implying that ... |
| hoeq2 31908 | A condition implying that ... |
| adjmo 31909 | Every Hilbert space operat... |
| adjsym 31910 | Symmetry property of an ad... |
| eigrei 31911 | A necessary and sufficient... |
| eigre 31912 | A necessary and sufficient... |
| eigposi 31913 | A sufficient condition (fi... |
| eigorthi 31914 | A necessary and sufficient... |
| eigorth 31915 | A necessary and sufficient... |
| nmopval 31933 | Value of the norm of a Hil... |
| elcnop 31934 | Property defining a contin... |
| ellnop 31935 | Property defining a linear... |
| lnopf 31936 | A linear Hilbert space ope... |
| elbdop 31937 | Property defining a bounde... |
| bdopln 31938 | A bounded linear Hilbert s... |
| bdopf 31939 | A bounded linear Hilbert s... |
| nmopsetretALT 31940 | The set in the supremum of... |
| nmopsetretHIL 31941 | The set in the supremum of... |
| nmopsetn0 31942 | The set in the supremum of... |
| nmopxr 31943 | The norm of a Hilbert spac... |
| nmoprepnf 31944 | The norm of a Hilbert spac... |
| nmopgtmnf 31945 | The norm of a Hilbert spac... |
| nmopreltpnf 31946 | The norm of a Hilbert spac... |
| nmopre 31947 | The norm of a bounded oper... |
| elbdop2 31948 | Property defining a bounde... |
| elunop 31949 | Property defining a unitar... |
| elhmop 31950 | Property defining a Hermit... |
| hmopf 31951 | A Hermitian operator is a ... |
| hmopex 31952 | The class of Hermitian ope... |
| nmfnval 31953 | Value of the norm of a Hil... |
| nmfnsetre 31954 | The set in the supremum of... |
| nmfnsetn0 31955 | The set in the supremum of... |
| nmfnxr 31956 | The norm of any Hilbert sp... |
| nmfnrepnf 31957 | The norm of a Hilbert spac... |
| nlfnval 31958 | Value of the null space of... |
| elcnfn 31959 | Property defining a contin... |
| ellnfn 31960 | Property defining a linear... |
| lnfnf 31961 | A linear Hilbert space fun... |
| dfadj2 31962 | Alternate definition of th... |
| funadj 31963 | Functionality of the adjoi... |
| dmadjss 31964 | The domain of the adjoint ... |
| dmadjop 31965 | A member of the domain of ... |
| adjeu 31966 | Elementhood in the domain ... |
| adjval 31967 | Value of the adjoint funct... |
| adjval2 31968 | Value of the adjoint funct... |
| cnvadj 31969 | The adjoint function equal... |
| funcnvadj 31970 | The converse of the adjoin... |
| adj1o 31971 | The adjoint function maps ... |
| dmadjrn 31972 | The adjoint of an operator... |
| eigvecval 31973 | The set of eigenvectors of... |
| eigvalfval 31974 | The eigenvalues of eigenve... |
| specval 31975 | The value of the spectrum ... |
| speccl 31976 | The spectrum of an operato... |
| hhlnoi 31977 | The linear operators of Hi... |
| hhnmoi 31978 | The norm of an operator in... |
| hhbloi 31979 | A bounded linear operator ... |
| hh0oi 31980 | The zero operator in Hilbe... |
| hhcno 31981 | The continuous operators o... |
| hhcnf 31982 | The continuous functionals... |
| dmadjrnb 31983 | The adjoint of an operator... |
| nmoplb 31984 | A lower bound for an opera... |
| nmopub 31985 | An upper bound for an oper... |
| nmopub2tALT 31986 | An upper bound for an oper... |
| nmopub2tHIL 31987 | An upper bound for an oper... |
| nmopge0 31988 | The norm of any Hilbert sp... |
| nmopgt0 31989 | A linear Hilbert space ope... |
| cnopc 31990 | Basic continuity property ... |
| lnopl 31991 | Basic linearity property o... |
| unop 31992 | Basic inner product proper... |
| unopf1o 31993 | A unitary operator in Hilb... |
| unopnorm 31994 | A unitary operator is idem... |
| cnvunop 31995 | The inverse (converse) of ... |
| unopadj 31996 | The inverse (converse) of ... |
| unoplin 31997 | A unitary operator is line... |
| counop 31998 | The composition of two uni... |
| hmop 31999 | Basic inner product proper... |
| hmopre 32000 | The inner product of the v... |
| nmfnlb 32001 | A lower bound for a functi... |
| nmfnleub 32002 | An upper bound for the nor... |
| nmfnleub2 32003 | An upper bound for the nor... |
| nmfnge0 32004 | The norm of any Hilbert sp... |
| elnlfn 32005 | Membership in the null spa... |
| elnlfn2 32006 | Membership in the null spa... |
| cnfnc 32007 | Basic continuity property ... |
| lnfnl 32008 | Basic linearity property o... |
| adjcl 32009 | Closure of the adjoint of ... |
| adj1 32010 | Property of an adjoint Hil... |
| adj2 32011 | Property of an adjoint Hil... |
| adjeq 32012 | A property that determines... |
| adjadj 32013 | Double adjoint. Theorem 3... |
| adjvalval 32014 | Value of the value of the ... |
| unopadj2 32015 | The adjoint of a unitary o... |
| hmopadj 32016 | A Hermitian operator is se... |
| hmdmadj 32017 | Every Hermitian operator h... |
| hmopadj2 32018 | An operator is Hermitian i... |
| hmoplin 32019 | A Hermitian operator is li... |
| brafval 32020 | The bra of a vector, expre... |
| braval 32021 | A bra-ket juxtaposition, e... |
| braadd 32022 | Linearity property of bra ... |
| bramul 32023 | Linearity property of bra ... |
| brafn 32024 | The bra function is a func... |
| bralnfn 32025 | The Dirac bra function is ... |
| bracl 32026 | Closure of the bra functio... |
| bra0 32027 | The Dirac bra of the zero ... |
| brafnmul 32028 | Anti-linearity property of... |
| kbfval 32029 | The outer product of two v... |
| kbop 32030 | The outer product of two v... |
| kbval 32031 | The value of the operator ... |
| kbmul 32032 | Multiplication property of... |
| kbpj 32033 | If a vector ` A ` has norm... |
| eleigvec 32034 | Membership in the set of e... |
| eleigvec2 32035 | Membership in the set of e... |
| eleigveccl 32036 | Closure of an eigenvector ... |
| eigvalval 32037 | The eigenvalue of an eigen... |
| eigvalcl 32038 | An eigenvalue is a complex... |
| eigvec1 32039 | Property of an eigenvector... |
| eighmre 32040 | The eigenvalues of a Hermi... |
| eighmorth 32041 | Eigenvectors of a Hermitia... |
| nmopnegi 32042 | Value of the norm of the n... |
| lnop0 32043 | The value of a linear Hilb... |
| lnopmul 32044 | Multiplicative property of... |
| lnopli 32045 | Basic scalar product prope... |
| lnopfi 32046 | A linear Hilbert space ope... |
| lnop0i 32047 | The value of a linear Hilb... |
| lnopaddi 32048 | Additive property of a lin... |
| lnopmuli 32049 | Multiplicative property of... |
| lnopaddmuli 32050 | Sum/product property of a ... |
| lnopsubi 32051 | Subtraction property for a... |
| lnopsubmuli 32052 | Subtraction/product proper... |
| lnopmulsubi 32053 | Product/subtraction proper... |
| homco2 32054 | Move a scalar product out ... |
| idunop 32055 | The identity function (res... |
| 0cnop 32056 | The identically zero funct... |
| 0cnfn 32057 | The identically zero funct... |
| idcnop 32058 | The identity function (res... |
| idhmop 32059 | The Hilbert space identity... |
| 0hmop 32060 | The identically zero funct... |
| 0lnop 32061 | The identically zero funct... |
| 0lnfn 32062 | The identically zero funct... |
| nmop0 32063 | The norm of the zero opera... |
| nmfn0 32064 | The norm of the identicall... |
| hmopbdoptHIL 32065 | A Hermitian operator is a ... |
| hoddii 32066 | Distributive law for Hilbe... |
| hoddi 32067 | Distributive law for Hilbe... |
| nmop0h 32068 | The norm of any operator o... |
| idlnop 32069 | The identity function (res... |
| 0bdop 32070 | The identically zero opera... |
| adj0 32071 | Adjoint of the zero operat... |
| nmlnop0iALT 32072 | A linear operator with a z... |
| nmlnop0iHIL 32073 | A linear operator with a z... |
| nmlnopgt0i 32074 | A linear Hilbert space ope... |
| nmlnop0 32075 | A linear operator with a z... |
| nmlnopne0 32076 | A linear operator with a n... |
| lnopmi 32077 | The scalar product of a li... |
| lnophsi 32078 | The sum of two linear oper... |
| lnophdi 32079 | The difference of two line... |
| lnopcoi 32080 | The composition of two lin... |
| lnopco0i 32081 | The composition of a linea... |
| lnopeq0lem1 32082 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 32083 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 32084 | A condition implying that ... |
| lnopeqi 32085 | Two linear Hilbert space o... |
| lnopeq 32086 | Two linear Hilbert space o... |
| lnopunilem1 32087 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 32088 | Lemma for ~ lnopunii . (C... |
| lnopunii 32089 | If a linear operator (whos... |
| elunop2 32090 | An operator is unitary iff... |
| nmopun 32091 | Norm of a unitary Hilbert ... |
| unopbd 32092 | A unitary operator is a bo... |
| lnophmlem1 32093 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 32094 | Lemma for ~ lnophmi . (Co... |
| lnophmi 32095 | A linear operator is Hermi... |
| lnophm 32096 | A linear operator is Hermi... |
| hmops 32097 | The sum of two Hermitian o... |
| hmopm 32098 | The scalar product of a He... |
| hmopd 32099 | The difference of two Herm... |
| hmopco 32100 | The composition of two com... |
| nmbdoplbi 32101 | A lower bound for the norm... |
| nmbdoplb 32102 | A lower bound for the norm... |
| nmcexi 32103 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 32104 | The norm of a continuous l... |
| nmcoplbi 32105 | A lower bound for the norm... |
| nmcopex 32106 | The norm of a continuous l... |
| nmcoplb 32107 | A lower bound for the norm... |
| nmophmi 32108 | The norm of the scalar pro... |
| bdophmi 32109 | The scalar product of a bo... |
| lnconi 32110 | Lemma for ~ lnopconi and ~... |
| lnopconi 32111 | A condition equivalent to ... |
| lnopcon 32112 | A condition equivalent to ... |
| lnopcnbd 32113 | A linear operator is conti... |
| lncnopbd 32114 | A continuous linear operat... |
| lncnbd 32115 | A continuous linear operat... |
| lnopcnre 32116 | A linear operator is conti... |
| lnfnli 32117 | Basic property of a linear... |
| lnfnfi 32118 | A linear Hilbert space fun... |
| lnfn0i 32119 | The value of a linear Hilb... |
| lnfnaddi 32120 | Additive property of a lin... |
| lnfnmuli 32121 | Multiplicative property of... |
| lnfnaddmuli 32122 | Sum/product property of a ... |
| lnfnsubi 32123 | Subtraction property for a... |
| lnfn0 32124 | The value of a linear Hilb... |
| lnfnmul 32125 | Multiplicative property of... |
| nmbdfnlbi 32126 | A lower bound for the norm... |
| nmbdfnlb 32127 | A lower bound for the norm... |
| nmcfnexi 32128 | The norm of a continuous l... |
| nmcfnlbi 32129 | A lower bound for the norm... |
| nmcfnex 32130 | The norm of a continuous l... |
| nmcfnlb 32131 | A lower bound of the norm ... |
| lnfnconi 32132 | A condition equivalent to ... |
| lnfncon 32133 | A condition equivalent to ... |
| lnfncnbd 32134 | A linear functional is con... |
| imaelshi 32135 | The image of a subspace un... |
| rnelshi 32136 | The range of a linear oper... |
| nlelshi 32137 | The null space of a linear... |
| nlelchi 32138 | The null space of a contin... |
| riesz3i 32139 | A continuous linear functi... |
| riesz4i 32140 | A continuous linear functi... |
| riesz4 32141 | A continuous linear functi... |
| riesz1 32142 | Part 1 of the Riesz repres... |
| riesz2 32143 | Part 2 of the Riesz repres... |
| cnlnadjlem1 32144 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 32145 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 32146 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 32147 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32148 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32149 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32150 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32151 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32152 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32153 | Every continuous linear op... |
| cnlnadjeui 32154 | Every continuous linear op... |
| cnlnadjeu 32155 | Every continuous linear op... |
| cnlnadj 32156 | Every continuous linear op... |
| cnlnssadj 32157 | Every continuous linear Hi... |
| bdopssadj 32158 | Every bounded linear Hilbe... |
| bdopadj 32159 | Every bounded linear Hilbe... |
| adjbdln 32160 | The adjoint of a bounded l... |
| adjbdlnb 32161 | An operator is bounded and... |
| adjbd1o 32162 | The mapping of adjoints of... |
| adjlnop 32163 | The adjoint of an operator... |
| adjsslnop 32164 | Every operator with an adj... |
| nmopadjlei 32165 | Property of the norm of an... |
| nmopadjlem 32166 | Lemma for ~ nmopadji . (C... |
| nmopadji 32167 | Property of the norm of an... |
| adjeq0 32168 | An operator is zero iff it... |
| adjmul 32169 | The adjoint of the scalar ... |
| adjadd 32170 | The adjoint of the sum of ... |
| nmoptrii 32171 | Triangle inequality for th... |
| nmopcoi 32172 | Upper bound for the norm o... |
| bdophsi 32173 | The sum of two bounded lin... |
| bdophdi 32174 | The difference between two... |
| bdopcoi 32175 | The composition of two bou... |
| nmoptri2i 32176 | Triangle-type inequality f... |
| adjcoi 32177 | The adjoint of a compositi... |
| nmopcoadji 32178 | The norm of an operator co... |
| nmopcoadj2i 32179 | The norm of an operator co... |
| nmopcoadj0i 32180 | An operator composed with ... |
| unierri 32181 | If we approximate a chain ... |
| branmfn 32182 | The norm of the bra functi... |
| brabn 32183 | The bra of a vector is a b... |
| rnbra 32184 | The set of bras equals the... |
| bra11 32185 | The bra function maps vect... |
| bracnln 32186 | A bra is a continuous line... |
| cnvbraval 32187 | Value of the converse of t... |
| cnvbracl 32188 | Closure of the converse of... |
| cnvbrabra 32189 | The converse bra of the br... |
| bracnvbra 32190 | The bra of the converse br... |
| bracnlnval 32191 | The vector that a continuo... |
| cnvbramul 32192 | Multiplication property of... |
| kbass1 32193 | Dirac bra-ket associative ... |
| kbass2 32194 | Dirac bra-ket associative ... |
| kbass3 32195 | Dirac bra-ket associative ... |
| kbass4 32196 | Dirac bra-ket associative ... |
| kbass5 32197 | Dirac bra-ket associative ... |
| kbass6 32198 | Dirac bra-ket associative ... |
| leopg 32199 | Ordering relation for posi... |
| leop 32200 | Ordering relation for oper... |
| leop2 32201 | Ordering relation for oper... |
| leop3 32202 | Operator ordering in terms... |
| leoppos 32203 | Binary relation defining a... |
| leoprf2 32204 | The ordering relation for ... |
| leoprf 32205 | The ordering relation for ... |
| leopsq 32206 | The square of a Hermitian ... |
| 0leop 32207 | The zero operator is a pos... |
| idleop 32208 | The identity operator is a... |
| leopadd 32209 | The sum of two positive op... |
| leopmuli 32210 | The scalar product of a no... |
| leopmul 32211 | The scalar product of a po... |
| leopmul2i 32212 | Scalar product applied to ... |
| leoptri 32213 | The positive operator orde... |
| leoptr 32214 | The positive operator orde... |
| leopnmid 32215 | A bounded Hermitian operat... |
| nmopleid 32216 | A nonzero, bounded Hermiti... |
| opsqrlem1 32217 | Lemma for opsqri . (Contr... |
| opsqrlem2 32218 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32219 | Lemma for opsqri . (Contr... |
| opsqrlem4 32220 | Lemma for opsqri . (Contr... |
| opsqrlem5 32221 | Lemma for opsqri . (Contr... |
| opsqrlem6 32222 | Lemma for opsqri . (Contr... |
| pjhmopi 32223 | A projector is a Hermitian... |
| pjlnopi 32224 | A projector is a linear op... |
| pjnmopi 32225 | The operator norm of a pro... |
| pjbdlni 32226 | A projector is a bounded l... |
| pjhmop 32227 | A projection is a Hermitia... |
| hmopidmchi 32228 | An idempotent Hermitian op... |
| hmopidmpji 32229 | An idempotent Hermitian op... |
| hmopidmch 32230 | An idempotent Hermitian op... |
| hmopidmpj 32231 | An idempotent Hermitian op... |
| pjsdii 32232 | Distributive law for Hilbe... |
| pjddii 32233 | Distributive law for Hilbe... |
| pjsdi2i 32234 | Chained distributive law f... |
| pjcoi 32235 | Composition of projections... |
| pjcocli 32236 | Closure of composition of ... |
| pjcohcli 32237 | Closure of composition of ... |
| pjadjcoi 32238 | Adjoint of composition of ... |
| pjcofni 32239 | Functionality of compositi... |
| pjss1coi 32240 | Subset relationship for pr... |
| pjss2coi 32241 | Subset relationship for pr... |
| pjssmi 32242 | Projection meet property. ... |
| pjssge0i 32243 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32244 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32245 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32246 | Composition of projections... |
| pjscji 32247 | The projection of orthogon... |
| pjssumi 32248 | The projection on a subspa... |
| pjssposi 32249 | Projector ordering can be ... |
| pjordi 32250 | The definition of projecto... |
| pjssdif2i 32251 | The projection subspace of... |
| pjssdif1i 32252 | A necessary and sufficient... |
| pjimai 32253 | The image of a projection.... |
| pjidmcoi 32254 | A projection is idempotent... |
| pjoccoi 32255 | Composition of projections... |
| pjtoi 32256 | Subspace sum of projection... |
| pjoci 32257 | Projection of orthocomplem... |
| pjidmco 32258 | A projection operator is i... |
| dfpjop 32259 | Definition of projection o... |
| pjhmopidm 32260 | Two ways to express the se... |
| elpjidm 32261 | A projection operator is i... |
| elpjhmop 32262 | A projection operator is H... |
| 0leopj 32263 | A projector is a positive ... |
| pjadj2 32264 | A projector is self-adjoin... |
| pjadj3 32265 | A projector is self-adjoin... |
| elpjch 32266 | Reconstruction of the subs... |
| elpjrn 32267 | Reconstruction of the subs... |
| pjinvari 32268 | A closed subspace ` H ` wi... |
| pjin1i 32269 | Lemma for Theorem 1.22 of ... |
| pjin2i 32270 | Lemma for Theorem 1.22 of ... |
| pjin3i 32271 | Lemma for Theorem 1.22 of ... |
| pjclem1 32272 | Lemma for projection commu... |
| pjclem2 32273 | Lemma for projection commu... |
| pjclem3 32274 | Lemma for projection commu... |
| pjclem4a 32275 | Lemma for projection commu... |
| pjclem4 32276 | Lemma for projection commu... |
| pjci 32277 | Two subspaces commute iff ... |
| pjcmul1i 32278 | A necessary and sufficient... |
| pjcmul2i 32279 | The projection subspace of... |
| pjcohocli 32280 | Closure of composition of ... |
| pjadj2coi 32281 | Adjoint of double composit... |
| pj2cocli 32282 | Closure of double composit... |
| pj3lem1 32283 | Lemma for projection tripl... |
| pj3si 32284 | Stronger projection triple... |
| pj3i 32285 | Projection triplet theorem... |
| pj3cor1i 32286 | Projection triplet corolla... |
| pjs14i 32287 | Theorem S-14 of Watanabe, ... |
| isst 32290 | Property of a state. (Con... |
| ishst 32291 | Property of a complex Hilb... |
| sticl 32292 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32293 | Real closure of the value ... |
| hstcl 32294 | Closure of the value of a ... |
| hst1a 32295 | Unit value of a Hilbert-sp... |
| hstel2 32296 | Properties of a Hilbert-sp... |
| hstorth 32297 | Orthogonality property of ... |
| hstosum 32298 | Orthogonal sum property of... |
| hstoc 32299 | Sum of a Hilbert-space-val... |
| hstnmoc 32300 | Sum of norms of a Hilbert-... |
| stge0 32301 | The value of a state is no... |
| stle1 32302 | The value of a state is le... |
| hstle1 32303 | The norm of the value of a... |
| hst1h 32304 | The norm of a Hilbert-spac... |
| hst0h 32305 | The norm of a Hilbert-spac... |
| hstpyth 32306 | Pythagorean property of a ... |
| hstle 32307 | Ordering property of a Hil... |
| hstles 32308 | Ordering property of a Hil... |
| hstoh 32309 | A Hilbert-space-valued sta... |
| hst0 32310 | A Hilbert-space-valued sta... |
| sthil 32311 | The value of a state at th... |
| stj 32312 | The value of a state on a ... |
| sto1i 32313 | The state of a subspace pl... |
| sto2i 32314 | The state of the orthocomp... |
| stge1i 32315 | If a state is greater than... |
| stle0i 32316 | If a state is less than or... |
| stlei 32317 | Ordering law for states. ... |
| stlesi 32318 | Ordering law for states. ... |
| stji1i 32319 | Join of components of Sasa... |
| stm1i 32320 | State of component of unit... |
| stm1ri 32321 | State of component of unit... |
| stm1addi 32322 | Sum of states whose meet i... |
| staddi 32323 | If the sum of 2 states is ... |
| stm1add3i 32324 | Sum of states whose meet i... |
| stadd3i 32325 | If the sum of 3 states is ... |
| st0 32326 | The state of the zero subs... |
| strlem1 32327 | Lemma for strong state the... |
| strlem2 32328 | Lemma for strong state the... |
| strlem3a 32329 | Lemma for strong state the... |
| strlem3 32330 | Lemma for strong state the... |
| strlem4 32331 | Lemma for strong state the... |
| strlem5 32332 | Lemma for strong state the... |
| strlem6 32333 | Lemma for strong state the... |
| stri 32334 | Strong state theorem. The... |
| strb 32335 | Strong state theorem (bidi... |
| hstrlem2 32336 | Lemma for strong set of CH... |
| hstrlem3a 32337 | Lemma for strong set of CH... |
| hstrlem3 32338 | Lemma for strong set of CH... |
| hstrlem4 32339 | Lemma for strong set of CH... |
| hstrlem5 32340 | Lemma for strong set of CH... |
| hstrlem6 32341 | Lemma for strong set of CH... |
| hstri 32342 | Hilbert space admits a str... |
| hstrbi 32343 | Strong CH-state theorem (b... |
| largei 32344 | A Hilbert lattice admits a... |
| jplem1 32345 | Lemma for Jauch-Piron theo... |
| jplem2 32346 | Lemma for Jauch-Piron theo... |
| jpi 32347 | The function ` S ` , that ... |
| golem1 32348 | Lemma for Godowski's equat... |
| golem2 32349 | Lemma for Godowski's equat... |
| goeqi 32350 | Godowski's equation, shown... |
| stcltr1i 32351 | Property of a strong class... |
| stcltr2i 32352 | Property of a strong class... |
| stcltrlem1 32353 | Lemma for strong classical... |
| stcltrlem2 32354 | Lemma for strong classical... |
| stcltrthi 32355 | Theorem for classically st... |
| cvbr 32359 | Binary relation expressing... |
| cvbr2 32360 | Binary relation expressing... |
| cvcon3 32361 | Contraposition law for the... |
| cvpss 32362 | The covers relation implie... |
| cvnbtwn 32363 | The covers relation implie... |
| cvnbtwn2 32364 | The covers relation implie... |
| cvnbtwn3 32365 | The covers relation implie... |
| cvnbtwn4 32366 | The covers relation implie... |
| cvnsym 32367 | The covers relation is not... |
| cvnref 32368 | The covers relation is not... |
| cvntr 32369 | The covers relation is not... |
| spansncv2 32370 | Hilbert space has the cove... |
| mdbr 32371 | Binary relation expressing... |
| mdi 32372 | Consequence of the modular... |
| mdbr2 32373 | Binary relation expressing... |
| mdbr3 32374 | Binary relation expressing... |
| mdbr4 32375 | Binary relation expressing... |
| dmdbr 32376 | Binary relation expressing... |
| dmdmd 32377 | The dual modular pair prop... |
| mddmd 32378 | The modular pair property ... |
| dmdi 32379 | Consequence of the dual mo... |
| dmdbr2 32380 | Binary relation expressing... |
| dmdi2 32381 | Consequence of the dual mo... |
| dmdbr3 32382 | Binary relation expressing... |
| dmdbr4 32383 | Binary relation expressing... |
| dmdi4 32384 | Consequence of the dual mo... |
| dmdbr5 32385 | Binary relation expressing... |
| mddmd2 32386 | Relationship between modul... |
| mdsl0 32387 | A sublattice condition tha... |
| ssmd1 32388 | Ordering implies the modul... |
| ssmd2 32389 | Ordering implies the modul... |
| ssdmd1 32390 | Ordering implies the dual ... |
| ssdmd2 32391 | Ordering implies the dual ... |
| dmdsl3 32392 | Sublattice mapping for a d... |
| mdsl3 32393 | Sublattice mapping for a m... |
| mdslle1i 32394 | Order preservation of the ... |
| mdslle2i 32395 | Order preservation of the ... |
| mdslj1i 32396 | Join preservation of the o... |
| mdslj2i 32397 | Meet preservation of the r... |
| mdsl1i 32398 | If the modular pair proper... |
| mdsl2i 32399 | If the modular pair proper... |
| mdsl2bi 32400 | If the modular pair proper... |
| cvmdi 32401 | The covering property impl... |
| mdslmd1lem1 32402 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32403 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32404 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32405 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32406 | Preservation of the modula... |
| mdslmd2i 32407 | Preservation of the modula... |
| mdsldmd1i 32408 | Preservation of the dual m... |
| mdslmd3i 32409 | Modular pair conditions th... |
| mdslmd4i 32410 | Modular pair condition tha... |
| csmdsymi 32411 | Cross-symmetry implies M-s... |
| mdexchi 32412 | An exchange lemma for modu... |
| cvmd 32413 | The covering property impl... |
| cvdmd 32414 | The covering property impl... |
| ela 32416 | Atoms in a Hilbert lattice... |
| elat2 32417 | Expanded membership relati... |
| elatcv0 32418 | A Hilbert lattice element ... |
| atcv0 32419 | An atom covers the zero su... |
| atssch 32420 | Atoms are a subset of the ... |
| atelch 32421 | An atom is a Hilbert latti... |
| atne0 32422 | An atom is not the Hilbert... |
| atss 32423 | A lattice element smaller ... |
| atsseq 32424 | Two atoms in a subset rela... |
| atcveq0 32425 | A Hilbert lattice element ... |
| h1da 32426 | A 1-dimensional subspace i... |
| spansna 32427 | The span of the singleton ... |
| sh1dle 32428 | A 1-dimensional subspace i... |
| ch1dle 32429 | A 1-dimensional subspace i... |
| atom1d 32430 | The 1-dimensional subspace... |
| superpos 32431 | Superposition Principle. ... |
| chcv1 32432 | The Hilbert lattice has th... |
| chcv2 32433 | The Hilbert lattice has th... |
| chjatom 32434 | The join of a closed subsp... |
| shatomici 32435 | The lattice of Hilbert sub... |
| hatomici 32436 | The Hilbert lattice is ato... |
| hatomic 32437 | A Hilbert lattice is atomi... |
| shatomistici 32438 | The lattice of Hilbert sub... |
| hatomistici 32439 | ` CH ` is atomistic, i.e. ... |
| chpssati 32440 | Two Hilbert lattice elemen... |
| chrelati 32441 | The Hilbert lattice is rel... |
| chrelat2i 32442 | A consequence of relative ... |
| cvati 32443 | If a Hilbert lattice eleme... |
| cvbr4i 32444 | An alternate way to expres... |
| cvexchlem 32445 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32446 | The Hilbert lattice satisf... |
| chrelat2 32447 | A consequence of relative ... |
| chrelat3 32448 | A consequence of relative ... |
| chrelat3i 32449 | A consequence of the relat... |
| chrelat4i 32450 | A consequence of relative ... |
| cvexch 32451 | The Hilbert lattice satisf... |
| cvp 32452 | The Hilbert lattice satisf... |
| atnssm0 32453 | The meet of a Hilbert latt... |
| atnemeq0 32454 | The meet of distinct atoms... |
| atssma 32455 | The meet with an atom's su... |
| atcv0eq 32456 | Two atoms covering the zer... |
| atcv1 32457 | Two atoms covering the zer... |
| atexch 32458 | The Hilbert lattice satisf... |
| atomli 32459 | An assertion holding in at... |
| atoml2i 32460 | An assertion holding in at... |
| atordi 32461 | An ordering law for a Hilb... |
| atcvatlem 32462 | Lemma for ~ atcvati . (Co... |
| atcvati 32463 | A nonzero Hilbert lattice ... |
| atcvat2i 32464 | A Hilbert lattice element ... |
| atord 32465 | An ordering law for a Hilb... |
| atcvat2 32466 | A Hilbert lattice element ... |
| chirredlem1 32467 | Lemma for ~ chirredi . (C... |
| chirredlem2 32468 | Lemma for ~ chirredi . (C... |
| chirredlem3 32469 | Lemma for ~ chirredi . (C... |
| chirredlem4 32470 | Lemma for ~ chirredi . (C... |
| chirredi 32471 | The Hilbert lattice is irr... |
| chirred 32472 | The Hilbert lattice is irr... |
| atcvat3i 32473 | A condition implying that ... |
| atcvat4i 32474 | A condition implying exist... |
| atdmd 32475 | Two Hilbert lattice elemen... |
| atmd 32476 | Two Hilbert lattice elemen... |
| atmd2 32477 | Two Hilbert lattice elemen... |
| atabsi 32478 | Absorption of an incompara... |
| atabs2i 32479 | Absorption of an incompara... |
| mdsymlem1 32480 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32481 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32482 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32483 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32484 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32485 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32486 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32487 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32488 | M-symmetry of the Hilbert ... |
| mdsym 32489 | M-symmetry of the Hilbert ... |
| dmdsym 32490 | Dual M-symmetry of the Hil... |
| atdmd2 32491 | Two Hilbert lattice elemen... |
| sumdmdii 32492 | If the subspace sum of two... |
| cmmdi 32493 | Commuting subspaces form a... |
| cmdmdi 32494 | Commuting subspaces form a... |
| sumdmdlem 32495 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32496 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32497 | The subspace sum of two Hi... |
| dmdbr4ati 32498 | Dual modular pair property... |
| dmdbr5ati 32499 | Dual modular pair property... |
| dmdbr6ati 32500 | Dual modular pair property... |
| dmdbr7ati 32501 | Dual modular pair property... |
| mdoc1i 32502 | Orthocomplements form a mo... |
| mdoc2i 32503 | Orthocomplements form a mo... |
| dmdoc1i 32504 | Orthocomplements form a du... |
| dmdoc2i 32505 | Orthocomplements form a du... |
| mdcompli 32506 | A condition equivalent to ... |
| dmdcompli 32507 | A condition equivalent to ... |
| mddmdin0i 32508 | If dual modular implies mo... |
| cdjreui 32509 | A member of the sum of dis... |
| cdj1i 32510 | Two ways to express " ` A ... |
| cdj3lem1 32511 | A property of " ` A ` and ... |
| cdj3lem2 32512 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32513 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32514 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32515 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32516 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32517 | Lemma for ~ cdj3i . The s... |
| cdj3i 32518 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32519 | (_This theorem is a dummy ... |
| sa-abvi 32520 | A theorem about the univer... |
| xfree 32521 | A partial converse to ~ 19... |
| xfree2 32522 | A partial converse to ~ 19... |
| addltmulALT 32523 | A proof readability experi... |
| ad11antr 32524 | Deduction adding 11 conjun... |
| simp-12l 32525 | Simplification of a conjun... |
| simp-12r 32526 | Simplification of a conjun... |
| an52ds 32527 | Inference exchanging the l... |
| an62ds 32528 | Inference exchanging the l... |
| an72ds 32529 | Inference exchanging the l... |
| an82ds 32530 | Inference exchanging the l... |
| syl22anbrc 32531 | Syllogism inference. (Con... |
| bian1d 32532 | Adding a superfluous conju... |
| bian1dOLD 32533 | Obsolete version of ~ bian... |
| orim12da 32534 | Deduce a disjunction from ... |
| or3di 32535 | Distributive law for disju... |
| or3dir 32536 | Distributive law for disju... |
| 3o1cs 32537 | Deduction eliminating disj... |
| 3o2cs 32538 | Deduction eliminating disj... |
| 3o3cs 32539 | Deduction eliminating disj... |
| 13an22anass 32540 | Associative law for four c... |
| sbc2iedf 32541 | Conversion of implicit sub... |
| rspc2daf 32542 | Double restricted speciali... |
| ralcom4f 32543 | Commutation of restricted ... |
| rexcom4f 32544 | Commutation of restricted ... |
| 19.9d2rf 32545 | A deduction version of one... |
| 19.9d2r 32546 | A deduction version of one... |
| r19.29ffa 32547 | A commonly used pattern ba... |
| n0limd 32548 | Deduction rule for nonempt... |
| reu6dv 32549 | A condition which implies ... |
| eqtrb 32550 | A transposition of equalit... |
| eqelbid 32551 | A variable elimination law... |
| opsbc2ie 32552 | Conversion of implicit sub... |
| opreu2reuALT 32553 | Correspondence between uni... |
| 2reucom 32556 | Double restricted existent... |
| 2reu2rex1 32557 | Double restricted existent... |
| 2reureurex 32558 | Double restricted existent... |
| 2reu2reu2 32559 | Double restricted existent... |
| opreu2reu1 32560 | Equivalent definition of t... |
| sq2reunnltb 32561 | There exists a unique deco... |
| addsqnot2reu 32562 | For each complex number ` ... |
| sbceqbidf 32563 | Equality theorem for class... |
| sbcies 32564 | A special version of class... |
| mo5f 32565 | Alternate definition of "a... |
| nmo 32566 | Negation of "at most one".... |
| reuxfrdf 32567 | Transfer existential uniqu... |
| rexunirn 32568 | Restricted existential qua... |
| rmoxfrd 32569 | Transfer "at most one" res... |
| rmoun 32570 | "At most one" restricted e... |
| rmounid 32571 | A case where an "at most o... |
| riotaeqbidva 32572 | Equivalent wff's yield equ... |
| dmrab 32573 | Domain of a restricted cla... |
| difrab2 32574 | Difference of two restrict... |
| elrabrd 32575 | Deduction version of ~ elr... |
| rabexgfGS 32576 | Separation Scheme in terms... |
| rabsnel 32577 | Truth implied by equality ... |
| rabsspr 32578 | Conditions for a restricte... |
| rabsstp 32579 | Conditions for a restricte... |
| 3unrab 32580 | Union of three restricted ... |
| foresf1o 32581 | From a surjective function... |
| rabfodom 32582 | Domination relation for re... |
| rabrexfi 32583 | Conditions for a class abs... |
| abrexdomjm 32584 | An indexed set is dominate... |
| abrexdom2jm 32585 | An indexed set is dominate... |
| abrexexd 32586 | Existence of a class abstr... |
| elabreximd 32587 | Class substitution in an i... |
| elabreximdv 32588 | Class substitution in an i... |
| abrexss 32589 | A necessary condition for ... |
| nelun 32590 | Negated membership for a u... |
| snsssng 32591 | If a singleton is a subset... |
| n0nsnel 32592 | If a class with one elemen... |
| inin 32593 | Intersection with an inter... |
| difininv 32594 | Condition for the intersec... |
| difeq 32595 | Rewriting an equation with... |
| eqdif 32596 | If both set differences of... |
| indifbi 32597 | Two ways to express equali... |
| diffib 32598 | Case where ~ diffi is a bi... |
| difxp1ss 32599 | Difference law for Cartesi... |
| difxp2ss 32600 | Difference law for Cartesi... |
| indifundif 32601 | A remarkable equation with... |
| elpwincl1 32602 | Closure of intersection wi... |
| elpwdifcl 32603 | Closure of class differenc... |
| elpwiuncl 32604 | Closure of indexed union w... |
| elpreq 32605 | Equality wihin a pair. (C... |
| prssad 32606 | If a pair is a subset of a... |
| prssbd 32607 | If a pair is a subset of a... |
| nelpr 32608 | A set ` A ` not in a pair ... |
| inpr0 32609 | Rewrite an empty intersect... |
| neldifpr1 32610 | The first element of a pai... |
| neldifpr2 32611 | The second element of a pa... |
| unidifsnel 32612 | The other element of a pai... |
| unidifsnne 32613 | The other element of a pai... |
| tpssg 32614 | An unordered triple of ele... |
| tpssd 32615 | Deduction version of tpssi... |
| tpssad 32616 | If an ordered triple is a ... |
| tpssbd 32617 | If an ordered triple is a ... |
| tpsscd 32618 | If an ordered triple is a ... |
| ifeqeqx 32619 | An equality theorem tailor... |
| elimifd 32620 | Elimination of a condition... |
| elim2if 32621 | Elimination of two conditi... |
| elim2ifim 32622 | Elimination of two conditi... |
| ifeq3da 32623 | Given an expression ` C ` ... |
| ifnetrue 32624 | Deduce truth from a condit... |
| ifnefals 32625 | Deduce falsehood from a co... |
| ifnebib 32626 | The converse of ~ ifbi hol... |
| uniinn0 32627 | Sufficient and necessary c... |
| uniin1 32628 | Union of intersection. Ge... |
| uniin2 32629 | Union of intersection. Ge... |
| difuncomp 32630 | Express a class difference... |
| elpwunicl 32631 | Closure of a set union wit... |
| cbviunf 32632 | Rule used to change the bo... |
| iuneq12daf 32633 | Equality deduction for ind... |
| iunin1f 32634 | Indexed union of intersect... |
| ssiun3 32635 | Subset equivalence for an ... |
| ssiun2sf 32636 | Subset relationship for an... |
| iuninc 32637 | The union of an increasing... |
| iundifdifd 32638 | The intersection of a set ... |
| iundifdif 32639 | The intersection of a set ... |
| iunrdx 32640 | Re-index an indexed union.... |
| iunpreima 32641 | Preimage of an indexed uni... |
| iunrnmptss 32642 | A subset relation for an i... |
| iunxunsn 32643 | Appending a set to an inde... |
| iunxunpr 32644 | Appending two sets to an i... |
| iunxpssiun1 32645 | Provide an upper bound for... |
| iinabrex 32646 | Rewriting an indexed inter... |
| disjnf 32647 | In case ` x ` is not free ... |
| cbvdisjf 32648 | Change bound variables in ... |
| disjss1f 32649 | A subset of a disjoint col... |
| disjeq1f 32650 | Equality theorem for disjo... |
| disjxun0 32651 | Simplify a disjoint union.... |
| disjdifprg 32652 | A trivial partition into a... |
| disjdifprg2 32653 | A trivial partition of a s... |
| disji2f 32654 | Property of a disjoint col... |
| disjif 32655 | Property of a disjoint col... |
| disjorf 32656 | Two ways to say that a col... |
| disjorsf 32657 | Two ways to say that a col... |
| disjif2 32658 | Property of a disjoint col... |
| disjabrex 32659 | Rewriting a disjoint colle... |
| disjabrexf 32660 | Rewriting a disjoint colle... |
| disjpreima 32661 | A preimage of a disjoint s... |
| disjrnmpt 32662 | Rewriting a disjoint colle... |
| disjin 32663 | If a collection is disjoin... |
| disjin2 32664 | If a collection is disjoin... |
| disjxpin 32665 | Derive a disjunction over ... |
| iundisjf 32666 | Rewrite a countable union ... |
| iundisj2f 32667 | A disjoint union is disjoi... |
| disjrdx 32668 | Re-index a disjunct collec... |
| disjex 32669 | Two ways to say that two c... |
| disjexc 32670 | A variant of ~ disjex , ap... |
| disjunsn 32671 | Append an element to a dis... |
| disjun0 32672 | Adding the empty element p... |
| disjiunel 32673 | A set of elements B of a d... |
| disjuniel 32674 | A set of elements B of a d... |
| xpdisjres 32675 | Restriction of a constant ... |
| opeldifid 32676 | Ordered pair elementhood o... |
| difres 32677 | Case when class difference... |
| imadifxp 32678 | Image of the difference wi... |
| relfi 32679 | A relation (set) is finite... |
| 0res 32680 | Restriction of the empty f... |
| fcoinver 32681 | Build an equivalence relat... |
| fcoinvbr 32682 | Binary relation for the eq... |
| breq1dd 32683 | Equality deduction for a b... |
| breq2dd 32684 | Equality deduction for a b... |
| brab2d 32685 | Expressing that two sets a... |
| brabgaf 32686 | The law of concretion for ... |
| brelg 32687 | Two things in a binary rel... |
| br8d 32688 | Substitution for an eight-... |
| fnfvor 32689 | Relation between two funct... |
| ofrco 32690 | Function relation between ... |
| opabdm 32691 | Domain of an ordered-pair ... |
| opabrn 32692 | Range of an ordered-pair c... |
| opabssi 32693 | Sufficient condition for a... |
| opabid2ss 32694 | One direction of ~ opabid2... |
| ssrelf 32695 | A subclass relationship de... |
| eqrelrd2 32696 | A version of ~ eqrelrdv2 w... |
| erbr3b 32697 | Biconditional for equivale... |
| iunsnima 32698 | Image of a singleton by an... |
| iunsnima2 32699 | Version of ~ iunsnima with... |
| fconst7v 32700 | An alternative way to expr... |
| constcof 32701 | Composition with a constan... |
| ac6sf2 32702 | Alternate version of ~ ac6... |
| ac6mapd 32703 | Axiom of choice equivalent... |
| fnresin 32704 | Restriction of a function ... |
| fresunsn 32705 | Recover the original funct... |
| f1o3d 32706 | Describe an implicit one-t... |
| eldmne0 32707 | A function of nonempty dom... |
| f1rnen 32708 | Equinumerosity of the rang... |
| f1oeq3dd 32709 | Equality deduction for one... |
| rinvf1o 32710 | Sufficient conditions for ... |
| fresf1o 32711 | Conditions for a restricti... |
| nfpconfp 32712 | The set of fixed points of... |
| fmptco1f1o 32713 | The action of composing (t... |
| cofmpt2 32714 | Express composition of a m... |
| f1mptrn 32715 | Express injection for a ma... |
| dfimafnf 32716 | Alternate definition of th... |
| funimass4f 32717 | Membership relation for th... |
| suppss2f 32718 | Show that the support of a... |
| ofrn 32719 | The range of the function ... |
| ofrn2 32720 | The range of the function ... |
| off2 32721 | The function operation pro... |
| ofresid 32722 | Applying an operation rest... |
| unipreima 32723 | Preimage of a class union.... |
| opfv 32724 | Value of a function produc... |
| xppreima 32725 | The preimage of a Cartesia... |
| 2ndimaxp 32726 | Image of a cartesian produ... |
| dmdju 32727 | Domain of a disjoint union... |
| djussxp2 32728 | Stronger version of ~ djus... |
| 2ndresdju 32729 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32730 | The ` 2nd ` function restr... |
| xppreima2 32731 | The preimage of a Cartesia... |
| abfmpunirn 32732 | Membership in a union of a... |
| rabfmpunirn 32733 | Membership in a union of a... |
| abfmpeld 32734 | Membership in an element o... |
| abfmpel 32735 | Membership in an element o... |
| fmptdF 32736 | Domain and codomain of the... |
| fmptcof2 32737 | Composition of two functio... |
| fcomptf 32738 | Express composition of two... |
| acunirnmpt 32739 | Axiom of choice for the un... |
| acunirnmpt2 32740 | Axiom of choice for the un... |
| acunirnmpt2f 32741 | Axiom of choice for the un... |
| aciunf1lem 32742 | Choice in an index union. ... |
| aciunf1 32743 | Choice in an index union. ... |
| ofoprabco 32744 | Function operation as a co... |
| ofpreima 32745 | Express the preimage of a ... |
| ofpreima2 32746 | Express the preimage of a ... |
| funcnvmpt 32747 | Condition for a function i... |
| funcnv5mpt 32748 | Two ways to say that a fun... |
| funcnv4mpt 32749 | Two ways to say that a fun... |
| preimane 32750 | Different elements have di... |
| fnpreimac 32751 | Choose a set ` x ` contain... |
| fgreu 32752 | Exactly one point of a fun... |
| fcnvgreu 32753 | If the converse of a relat... |
| rnmposs 32754 | The range of an operation ... |
| mptssALT 32755 | Deduce subset relation of ... |
| dfcnv2 32756 | Alternative definition of ... |
| partfun2 32757 | Rewrite a function defined... |
| rnressnsn 32758 | The range of a restriction... |
| mpomptxf 32759 | Express a two-argument fun... |
| of0r 32760 | Function operation with th... |
| elmaprd 32761 | Deduction associated with ... |
| suppovss 32762 | A bound for the support of... |
| elsuppfnd 32763 | Deduce membership in the s... |
| fisuppov1 32764 | Formula building theorem f... |
| suppun2 32765 | The support of a union is ... |
| fdifsupp 32766 | Express the support of a f... |
| suppiniseg 32767 | Relation between the suppo... |
| fsuppinisegfi 32768 | The initial segment ` ( ``... |
| fressupp 32769 | The restriction of a funct... |
| fdifsuppconst 32770 | A function is a zero const... |
| ressupprn 32771 | The range of a function re... |
| supppreima 32772 | Express the support of a f... |
| fsupprnfi 32773 | Finite support implies fin... |
| mptiffisupp 32774 | Conditions for a mapping f... |
| cosnopne 32775 | Composition of two ordered... |
| cosnop 32776 | Composition of two ordered... |
| cnvprop 32777 | Converse of a pair of orde... |
| brprop 32778 | Binary relation for a pair... |
| mptprop 32779 | Rewrite pairs of ordered p... |
| coprprop 32780 | Composition of two pairs o... |
| fmptunsnop 32781 | Two ways to express a func... |
| gtiso 32782 | Two ways to write a strict... |
| isoun 32783 | Infer an isomorphism from ... |
| disjdsct 32784 | A disjoint collection is d... |
| df1stres 32785 | Definition for a restricti... |
| df2ndres 32786 | Definition for a restricti... |
| 1stpreimas 32787 | The preimage of a singleto... |
| 1stpreima 32788 | The preimage by ` 1st ` is... |
| 2ndpreima 32789 | The preimage by ` 2nd ` is... |
| curry2ima 32790 | The image of a curried fun... |
| preiman0 32791 | The preimage of a nonempty... |
| intimafv 32792 | The intersection of an ima... |
| snct 32793 | A singleton is countable. ... |
| prct 32794 | An unordered pair is count... |
| mpocti 32795 | An operation is countable ... |
| abrexct 32796 | An image set of a countabl... |
| mptctf 32797 | A countable mapping set is... |
| abrexctf 32798 | An image set of a countabl... |
| padct 32799 | Index a countable set with... |
| f1od2 32800 | Sufficient condition for a... |
| fcobij 32801 | Composing functions with a... |
| fcobijfs 32802 | Composing finitely support... |
| fcobijfs2 32803 | Composing finitely support... |
| suppss3 32804 | Deduce a function's suppor... |
| fsuppcurry1 32805 | Finite support of a currie... |
| fsuppcurry2 32806 | Finite support of a currie... |
| offinsupp1 32807 | Finite support for a funct... |
| ffs2 32808 | Rewrite a function's suppo... |
| ffsrn 32809 | The range of a finitely su... |
| cocnvf1o 32810 | Composing with the inverse... |
| resf1o 32811 | Restriction of functions t... |
| maprnin 32812 | Restricting the range of t... |
| fpwrelmapffslem 32813 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32814 | Define a canonical mapping... |
| fpwrelmapffs 32815 | Define a canonical mapping... |
| sgnval2 32816 | Value of the signum of a r... |
| creq0 32817 | The real representation of... |
| 1nei 32818 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32819 | An integer unit times itse... |
| nnmulge 32820 | Multiplying by a positive ... |
| submuladdd 32821 | The product of a differenc... |
| muldivdid 32822 | Distribution of division o... |
| binom2subadd 32823 | The difference of the squa... |
| cjsubd 32824 | Complex conjugate distribu... |
| re0cj 32825 | The conjugate of a pure im... |
| receqid 32826 | Real numbers equal to thei... |
| pythagreim 32827 | A simplified version of th... |
| efiargd 32828 | The exponential of the "ar... |
| arginv 32829 | The argument of the invers... |
| argcj 32830 | The argument of the conjug... |
| quad3d 32831 | Variant of quadratic equat... |
| lt2addrd 32832 | If the right-hand side of ... |
| nn0mnfxrd 32833 | Nonnegative integers or mi... |
| xrlelttric 32834 | Trichotomy law for extende... |
| xaddeq0 32835 | Two extended reals which a... |
| rexmul2 32836 | If the result ` A ` of an ... |
| xrinfm 32837 | The extended real numbers ... |
| le2halvesd 32838 | A sum is less than the who... |
| xraddge02 32839 | A number is less than or e... |
| xrge0addge 32840 | A number is less than or e... |
| xlt2addrd 32841 | If the right-hand side of ... |
| xrge0infss 32842 | Any subset of nonnegative ... |
| xrge0infssd 32843 | Inequality deduction for i... |
| xrge0addcld 32844 | Nonnegative extended reals... |
| xrge0subcld 32845 | Condition for closure of n... |
| infxrge0lb 32846 | A member of a set of nonne... |
| infxrge0glb 32847 | The infimum of a set of no... |
| infxrge0gelb 32848 | The infimum of a set of no... |
| xrofsup 32849 | The supremum is preserved ... |
| supxrnemnf 32850 | The supremum of a nonempty... |
| xnn0gt0 32851 | Nonzero extended nonnegati... |
| xnn01gt 32852 | An extended nonnegative in... |
| nn0xmulclb 32853 | Finite multiplication in t... |
| xnn0nn0d 32854 | Conditions for an extended... |
| xnn0nnd 32855 | Conditions for an extended... |
| joiniooico 32856 | Disjoint joining an open i... |
| ubico 32857 | A right-open interval does... |
| xeqlelt 32858 | Equality in terms of 'less... |
| eliccelico 32859 | Relate elementhood to a cl... |
| elicoelioo 32860 | Relate elementhood to a cl... |
| iocinioc2 32861 | Intersection between two o... |
| xrdifh 32862 | Class difference of a half... |
| iocinif 32863 | Relate intersection of two... |
| difioo 32864 | The difference between two... |
| difico 32865 | The difference between two... |
| uzssico 32866 | Upper integer sets are a s... |
| fz2ssnn0 32867 | A finite set of sequential... |
| nndiffz1 32868 | Upper set of the positive ... |
| ssnnssfz 32869 | For any finite subset of `... |
| fzm1ne1 32870 | Elementhood of an integer ... |
| fzspl 32871 | Split the last element of ... |
| fzdif2 32872 | Split the last element of ... |
| fzodif2 32873 | Split the last element of ... |
| fzodif1 32874 | Set difference of two half... |
| fzsplit3 32875 | Split a finite interval of... |
| nn0diffz0 32876 | Upper set of the nonnegati... |
| bcm1n 32877 | The proportion of one bino... |
| iundisjfi 32878 | Rewrite a countable union ... |
| iundisj2fi 32879 | A disjoint union is disjoi... |
| iundisjcnt 32880 | Rewrite a countable union ... |
| iundisj2cnt 32881 | A countable disjoint union... |
| f1ocnt 32882 | Given a countable set ` A ... |
| fz1nnct 32883 | NN and integer ranges star... |
| fz1nntr 32884 | NN and integer ranges star... |
| fzo0opth 32885 | Equality for a half open i... |
| nn0difffzod 32886 | A nonnegative integer that... |
| suppssnn0 32887 | Show that the support of a... |
| hashunif 32888 | The cardinality of a disjo... |
| hashxpe 32889 | The size of the Cartesian ... |
| hashgt1 32890 | Restate "set contains at l... |
| hashpss 32891 | The size of a proper subse... |
| hashne0 32892 | Deduce that the size of a ... |
| hashimaf1 32893 | Taking the image of a set ... |
| elq2 32894 | Elementhood in the rationa... |
| znumd 32895 | Numerator of an integer. ... |
| zdend 32896 | Denominator of an integer.... |
| numdenneg 32897 | Numerator and denominator ... |
| divnumden2 32898 | Calculate the reduced form... |
| expgt0b 32899 | A real number ` A ` raised... |
| nn0split01 32900 | Split 0 and 1 from the non... |
| nn0disj01 32901 | The pair ` { 0 , 1 } ` doe... |
| nnindf 32902 | Principle of Mathematical ... |
| nn0min 32903 | Extracting the minimum pos... |
| subne0nn 32904 | A nonnegative difference i... |
| ltesubnnd 32905 | Subtracting an integer num... |
| fprodeq02 32906 | If one of the factors is z... |
| pr01ssre 32907 | The range of the indicator... |
| fprodex01 32908 | A product of factors equal... |
| prodpr 32909 | A product over a pair is t... |
| prodtp 32910 | A product over a triple is... |
| fsumub 32911 | An upper bound for a term ... |
| fsumiunle 32912 | Upper bound for a sum of n... |
| dfdec100 32913 | Split the hundreds from a ... |
| sgncl 32914 | Closure of the signum. (C... |
| sgnclre 32915 | Closure of the signum. (C... |
| sgnneg 32916 | Negation of the signum. (... |
| sgn3da 32917 | A conditional containing a... |
| sgnmul 32918 | Signum of a product. (Con... |
| sgnmulrp2 32919 | Multiplication by a positi... |
| sgnsub 32920 | Subtraction of a number of... |
| sgnnbi 32921 | Negative signum. (Contrib... |
| sgnpbi 32922 | Positive signum. (Contrib... |
| sgn0bi 32923 | Zero signum. (Contributed... |
| sgnsgn 32924 | Signum is idempotent. (Co... |
| sgnmulsgn 32925 | If two real numbers are of... |
| sgnmulsgp 32926 | If two real numbers are of... |
| nexple 32927 | A lower bound for an expon... |
| 2exple2exp 32928 | If a nonnegative integer `... |
| expevenpos 32929 | Even powers are positive. ... |
| oexpled 32930 | Odd power monomials are mo... |
| indv 32933 | Value of the indicator fun... |
| indval 32934 | Value of the indicator fun... |
| indval2 32935 | Alternate value of the ind... |
| indf 32936 | An indicator function as a... |
| indfval 32937 | Value of the indicator fun... |
| ind1 32938 | Value of the indicator fun... |
| ind0 32939 | Value of the indicator fun... |
| ind1a 32940 | Value of the indicator fun... |
| indconst0 32941 | Indicator of the empty set... |
| indconst1 32942 | Indicator of the whole set... |
| indpi1 32943 | Preimage of the singleton ... |
| indsum 32944 | Finite sum of a product wi... |
| indsumin 32945 | Finite sum of a product wi... |
| prodindf 32946 | The product of indicators ... |
| indsn 32947 | The indicator function of ... |
| indf1o 32948 | The bijection between a po... |
| indpreima 32949 | A function with range ` { ... |
| indf1ofs 32950 | The bijection between fini... |
| indsupp 32951 | The support of the indicat... |
| indfsd 32952 | The indicator function of ... |
| indfsid 32953 | Conditions for a function ... |
| dp2eq1 32956 | Equality theorem for the d... |
| dp2eq2 32957 | Equality theorem for the d... |
| dp2eq1i 32958 | Equality theorem for the d... |
| dp2eq2i 32959 | Equality theorem for the d... |
| dp2eq12i 32960 | Equality theorem for the d... |
| dp20u 32961 | Add a zero in the tenths (... |
| dp20h 32962 | Add a zero in the unit pla... |
| dp2cl 32963 | Closure for the decimal fr... |
| dp2clq 32964 | Closure for a decimal frac... |
| rpdp2cl 32965 | Closure for a decimal frac... |
| rpdp2cl2 32966 | Closure for a decimal frac... |
| dp2lt10 32967 | Decimal fraction builds re... |
| dp2lt 32968 | Comparing two decimal frac... |
| dp2ltsuc 32969 | Comparing a decimal fracti... |
| dp2ltc 32970 | Comparing two decimal expa... |
| dpval 32973 | Define the value of the de... |
| dpcl 32974 | Prove that the closure of ... |
| dpfrac1 32975 | Prove a simple equivalence... |
| dpval2 32976 | Value of the decimal point... |
| dpval3 32977 | Value of the decimal point... |
| dpmul10 32978 | Multiply by 10 a decimal e... |
| decdiv10 32979 | Divide a decimal number by... |
| dpmul100 32980 | Multiply by 100 a decimal ... |
| dp3mul10 32981 | Multiply by 10 a decimal e... |
| dpmul1000 32982 | Multiply by 1000 a decimal... |
| dpval3rp 32983 | Value of the decimal point... |
| dp0u 32984 | Add a zero in the tenths p... |
| dp0h 32985 | Remove a zero in the units... |
| rpdpcl 32986 | Closure of the decimal poi... |
| dplt 32987 | Comparing two decimal expa... |
| dplti 32988 | Comparing a decimal expans... |
| dpgti 32989 | Comparing a decimal expans... |
| dpltc 32990 | Comparing two decimal inte... |
| dpexpp1 32991 | Add one zero to the mantis... |
| 0dp2dp 32992 | Multiply by 10 a decimal e... |
| dpadd2 32993 | Addition with one decimal,... |
| dpadd 32994 | Addition with one decimal.... |
| dpadd3 32995 | Addition with two decimals... |
| dpmul 32996 | Multiplication with one de... |
| dpmul4 32997 | An upper bound to multipli... |
| threehalves 32998 | Example theorem demonstrat... |
| 1mhdrd 32999 | Example theorem demonstrat... |
| xdivval 33002 | Value of division: the (un... |
| xrecex 33003 | Existence of reciprocal of... |
| xmulcand 33004 | Cancellation law for exten... |
| xreceu 33005 | Existential uniqueness of ... |
| xdivcld 33006 | Closure law for the extend... |
| xdivcl 33007 | Closure law for the extend... |
| xdivmul 33008 | Relationship between divis... |
| rexdiv 33009 | The extended real division... |
| xdivrec 33010 | Relationship between divis... |
| xdivid 33011 | A number divided by itself... |
| xdiv0 33012 | Division into zero is zero... |
| xdiv0rp 33013 | Division into zero is zero... |
| eliccioo 33014 | Membership in a closed int... |
| elxrge02 33015 | Elementhood in the set of ... |
| xdivpnfrp 33016 | Plus infinity divided by a... |
| rpxdivcld 33017 | Closure law for extended d... |
| xrpxdivcld 33018 | Closure law for extended d... |
| wrdres 33019 | Condition for the restrict... |
| wrdsplex 33020 | Existence of a split of a ... |
| wrdfsupp 33021 | A word has finite support.... |
| wrdpmcl 33022 | Closure of a word with per... |
| pfx1s2 33023 | The prefix of length 1 of ... |
| pfxrn2 33024 | The range of a prefix of a... |
| pfxrn3 33025 | Express the range of a pre... |
| pfxf1 33026 | Condition for a prefix to ... |
| s1f1 33027 | Conditions for a length 1 ... |
| s2rnOLD 33028 | Obsolete version of ~ s2rn... |
| s2f1 33029 | Conditions for a length 2 ... |
| s3rnOLD 33030 | Obsolete version of ~ s2rn... |
| s3f1 33031 | Conditions for a length 3 ... |
| s3clhash 33032 | Closure of the words of le... |
| ccatf1 33033 | Conditions for a concatena... |
| pfxlsw2ccat 33034 | Reconstruct a word from it... |
| ccatws1f1o 33035 | Conditions for the concate... |
| ccatws1f1olast 33036 | Two ways to reorder symbol... |
| wrdt2ind 33037 | Perform an induction over ... |
| swrdrn2 33038 | The range of a subword is ... |
| swrdrn3 33039 | Express the range of a sub... |
| swrdf1 33040 | Condition for a subword to... |
| swrdrndisj 33041 | Condition for the range of... |
| splfv3 33042 | Symbols to the right of a ... |
| 1cshid 33043 | Cyclically shifting a sing... |
| cshw1s2 33044 | Cyclically shifting a leng... |
| cshwrnid 33045 | Cyclically shifting a word... |
| cshf1o 33046 | Condition for the cyclic s... |
| ressplusf 33047 | The group operation functi... |
| ressnm 33048 | The norm in a restricted s... |
| abvpropd2 33049 | Weaker version of ~ abvpro... |
| ressprs 33050 | The restriction of a prose... |
| posrasymb 33051 | A poset ordering is asymet... |
| odutos 33052 | Being a toset is a self-du... |
| tlt2 33053 | In a Toset, two elements m... |
| tlt3 33054 | In a Toset, two elements m... |
| trleile 33055 | In a Toset, two elements m... |
| toslublem 33056 | Lemma for ~ toslub and ~ x... |
| toslub 33057 | In a toset, the lowest upp... |
| tosglblem 33058 | Lemma for ~ tosglb and ~ x... |
| tosglb 33059 | Same theorem as ~ toslub ,... |
| clatp0cl 33060 | The poset zero of a comple... |
| clatp1cl 33061 | The poset one of a complet... |
| mntoval 33066 | Operation value of the mon... |
| ismnt 33067 | Express the statement " ` ... |
| ismntd 33068 | Property of being a monoto... |
| mntf 33069 | A monotone function is a f... |
| mgcoval 33070 | Operation value of the mon... |
| mgcval 33071 | Monotone Galois connection... |
| mgcf1 33072 | The lower adjoint ` F ` of... |
| mgcf2 33073 | The upper adjoint ` G ` of... |
| mgccole1 33074 | An inequality for the kern... |
| mgccole2 33075 | Inequality for the closure... |
| mgcmnt1 33076 | The lower adjoint ` F ` of... |
| mgcmnt2 33077 | The upper adjoint ` G ` of... |
| mgcmntco 33078 | A Galois connection like s... |
| dfmgc2lem 33079 | Lemma for dfmgc2, backward... |
| dfmgc2 33080 | Alternate definition of th... |
| mgcmnt1d 33081 | Galois connection implies ... |
| mgcmnt2d 33082 | Galois connection implies ... |
| mgccnv 33083 | The inverse Galois connect... |
| pwrssmgc 33084 | Given a function ` F ` , e... |
| mgcf1olem1 33085 | Property of a Galois conne... |
| mgcf1olem2 33086 | Property of a Galois conne... |
| mgcf1o 33087 | Given a Galois connection,... |
| xrs0 33090 | The zero of the extended r... |
| xrslt 33091 | The "strictly less than" r... |
| xrsinvgval 33092 | The inversion operation in... |
| xrsmulgzz 33093 | The "multiple" function in... |
| xrstos 33094 | The extended real numbers ... |
| xrsclat 33095 | The extended real numbers ... |
| xrsp0 33096 | The poset 0 of the extende... |
| xrsp1 33097 | The poset 1 of the extende... |
| xrge00 33098 | The zero of the extended n... |
| xrge0mulgnn0 33099 | The group multiple functio... |
| xrge0addass 33100 | Associativity of extended ... |
| xrge0addgt0 33101 | The sum of nonnegative and... |
| xrge0adddir 33102 | Right-distributivity of ex... |
| xrge0adddi 33103 | Left-distributivity of ext... |
| xrge0npcan 33104 | Extended nonnegative real ... |
| fsumrp0cl 33105 | Closure of a finite sum of... |
| mndcld 33106 | Closure of the operation o... |
| mndassd 33107 | A monoid operation is asso... |
| mndlrinv 33108 | In a monoid, if an element... |
| mndlrinvb 33109 | In a monoid, if an element... |
| mndlactf1 33110 | If an element ` X ` of a m... |
| mndlactfo 33111 | An element ` X ` of a mono... |
| mndractf1 33112 | If an element ` X ` of a m... |
| mndractfo 33113 | An element ` X ` of a mono... |
| mndlactf1o 33114 | An element ` X ` of a mono... |
| mndractf1o 33115 | An element ` X ` of a mono... |
| cmn4d 33116 | Commutative/associative la... |
| cmn246135 33117 | Rearrange terms in a commu... |
| cmn145236 33118 | Rearrange terms in a commu... |
| submcld 33119 | Submonoids are closed unde... |
| abliso 33120 | The image of an Abelian gr... |
| lmhmghmd 33121 | A module homomorphism is a... |
| mhmimasplusg 33122 | Value of the operation of ... |
| lmhmimasvsca 33123 | Value of the scalar produc... |
| grpinvinvd 33124 | Double inverse law for gro... |
| grpsubcld 33125 | Closure of group subtracti... |
| subgcld 33126 | A subgroup is closed under... |
| subgsubcld 33127 | A subgroup is closed under... |
| subgmulgcld 33128 | Closure of the group multi... |
| ressmulgnn0d 33129 | Values for the group multi... |
| ablcomd 33130 | An abelian group operation... |
| gsumsubg 33131 | The group sum in a subgrou... |
| gsumsra 33132 | The group sum in a subring... |
| gsummpt2co 33133 | Split a finite sum into a ... |
| gsummpt2d 33134 | Express a finite sum over ... |
| lmodvslmhm 33135 | Scalar multiplication in a... |
| gsumvsmul1 33136 | Pull a scalar multiplicati... |
| gsummptres 33137 | Extend a finite group sum ... |
| gsummptres2 33138 | Extend a finite group sum ... |
| gsummptfsres 33139 | Extend a finitely supporte... |
| gsummptf1od 33140 | Re-index a finite group su... |
| gsummptrev 33141 | Revert ordering in a group... |
| gsummptp1 33142 | Reindex a zero-based sum a... |
| gsummptfzsplitra 33143 | Split a group sum expresse... |
| gsummptfzsplitla 33144 | Split a group sum expresse... |
| gsummptfsf1o 33145 | Re-index a finite group su... |
| gsumfs2d 33146 | Express a finite sum over ... |
| gsumzresunsn 33147 | Append an element to a fin... |
| gsumpart 33148 | Express a group sum as a d... |
| gsumtp 33149 | Group sum of an unordered ... |
| gsumzrsum 33150 | Relate a group sum on ` ZZ... |
| gsummulgc2 33151 | A finite group sum multipl... |
| gsumhashmul 33152 | Express a group sum by gro... |
| gsummulsubdishift1 33153 | Distribute a subtraction o... |
| gsummulsubdishift2 33154 | Distribute a subtraction o... |
| gsummulsubdishift1s 33155 | Distribute a subtraction o... |
| gsummulsubdishift2s 33156 | Distribute a subtraction o... |
| xrge0tsmsd 33157 | Any finite or infinite sum... |
| xrge0tsmsbi 33158 | Any limit of a finite or i... |
| xrge0tsmseq 33159 | Any limit of a finite or i... |
| gsumwun 33160 | In a commutative ring, a g... |
| gsumwrd2dccatlem 33161 | Lemma for ~ gsumwrd2dccat ... |
| gsumwrd2dccat 33162 | Rewrite a sum ranging over... |
| cntzun 33163 | The centralizer of a union... |
| cntzsnid 33164 | The centralizer of the ide... |
| cntrcrng 33165 | The center of a ring is a ... |
| symgfcoeu 33166 | Uniqueness property of per... |
| symgcom 33167 | Two permutations ` X ` and... |
| symgcom2 33168 | Two permutations ` X ` and... |
| symgcntz 33169 | All elements of a (finite)... |
| odpmco 33170 | The composition of two odd... |
| symgsubg 33171 | The value of the group sub... |
| pmtrprfv2 33172 | In a transposition of two ... |
| pmtrcnel 33173 | Composing a permutation ` ... |
| pmtrcnel2 33174 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33175 | Composing a permutation ` ... |
| fzo0pmtrlast 33176 | Reorder a half-open intege... |
| wrdpmtrlast 33177 | Reorder a word, so that th... |
| pmtridf1o 33178 | Transpositions of ` X ` an... |
| pmtridfv1 33179 | Value at X of the transpos... |
| pmtridfv2 33180 | Value at Y of the transpos... |
| psgnid 33181 | Permutation sign of the id... |
| psgndmfi 33182 | For a finite base set, the... |
| pmtrto1cl 33183 | Useful lemma for the follo... |
| psgnfzto1stlem 33184 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33185 | Value of our permutation `... |
| fzto1st1 33186 | Special case where the per... |
| fzto1st 33187 | The function moving one el... |
| fzto1stinvn 33188 | Value of the inverse of ou... |
| psgnfzto1st 33189 | The permutation sign for m... |
| tocycval 33192 | Value of the cycle builder... |
| tocycfv 33193 | Function value of a permut... |
| tocycfvres1 33194 | A cyclic permutation is a ... |
| tocycfvres2 33195 | A cyclic permutation is th... |
| cycpmfvlem 33196 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33197 | Value of a cycle function ... |
| cycpmfv2 33198 | Value of a cycle function ... |
| cycpmfv3 33199 | Values outside of the orbi... |
| cycpmcl 33200 | Cyclic permutations are pe... |
| tocycf 33201 | The permutation cycle buil... |
| tocyc01 33202 | Permutation cycles built f... |
| cycpm2tr 33203 | A cyclic permutation of 2 ... |
| cycpm2cl 33204 | Closure for the 2-cycles. ... |
| cyc2fv1 33205 | Function value of a 2-cycl... |
| cyc2fv2 33206 | Function value of a 2-cycl... |
| trsp2cyc 33207 | Exhibit the word a transpo... |
| cycpmco2f1 33208 | The word U used in ~ cycpm... |
| cycpmco2rn 33209 | The orbit of the compositi... |
| cycpmco2lem1 33210 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33211 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33212 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33213 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33214 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33215 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33216 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33217 | The composition of a cycli... |
| cyc2fvx 33218 | Function value of a 2-cycl... |
| cycpm3cl 33219 | Closure of the 3-cycles in... |
| cycpm3cl2 33220 | Closure of the 3-cycles in... |
| cyc3fv1 33221 | Function value of a 3-cycl... |
| cyc3fv2 33222 | Function value of a 3-cycl... |
| cyc3fv3 33223 | Function value of a 3-cycl... |
| cyc3co2 33224 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33225 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33226 | A formula for computing co... |
| cycpmrn 33227 | The range of the word used... |
| tocyccntz 33228 | All elements of a (finite)... |
| evpmval 33229 | Value of the set of even p... |
| cnmsgn0g 33230 | The neutral element of the... |
| evpmsubg 33231 | The alternating group is a... |
| evpmid 33232 | The identity is an even pe... |
| altgnsg 33233 | The alternating group ` ( ... |
| cyc3evpm 33234 | 3-Cycles are even permutat... |
| cyc3genpmlem 33235 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33236 | The alternating group ` A ... |
| cycpmgcl 33237 | Cyclic permutations are pe... |
| cycpmconjslem1 33238 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33239 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33240 | All cycles of the same len... |
| cyc3conja 33241 | All 3-cycles are conjugate... |
| sgnsv 33244 | The sign mapping. (Contri... |
| sgnsval 33245 | The sign value. (Contribu... |
| sgnsf 33246 | The sign function. (Contr... |
| fxpval 33249 | Value of the set of fixed ... |
| fxpss 33250 | The set of fixed points is... |
| fxpgaval 33251 | Value of the set of fixed ... |
| isfxp 33252 | Property of being a fixed ... |
| fxpgaeq 33253 | A fixed point ` X ` is inv... |
| conjga 33254 | Group conjugation induces ... |
| cntrval2 33255 | Express the center ` Z ` o... |
| fxpsubm 33256 | Provided the group action ... |
| fxpsubg 33257 | The fixed points of a grou... |
| fxpsubrg 33258 | The fixed points of a grou... |
| fxpsdrg 33259 | The fixed points of a grou... |
| inftmrel 33264 | The infinitesimal relation... |
| isinftm 33265 | Express ` x ` is infinites... |
| isarchi 33266 | Express the predicate " ` ... |
| pnfinf 33267 | Plus infinity is an infini... |
| xrnarchi 33268 | The completed real line is... |
| isarchi2 33269 | Alternative way to express... |
| submarchi 33270 | A submonoid is archimedean... |
| isarchi3 33271 | This is the usual definiti... |
| archirng 33272 | Property of Archimedean or... |
| archirngz 33273 | Property of Archimedean le... |
| archiexdiv 33274 | In an Archimedean group, g... |
| archiabllem1a 33275 | Lemma for ~ archiabl : In... |
| archiabllem1b 33276 | Lemma for ~ archiabl . (C... |
| archiabllem1 33277 | Archimedean ordered groups... |
| archiabllem2a 33278 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33279 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33280 | Lemma for ~ archiabl . (C... |
| archiabllem2 33281 | Archimedean ordered groups... |
| archiabl 33282 | Archimedean left- and righ... |
| isarchiofld 33283 | Axiom of Archimedes : a ch... |
| isslmd 33286 | The predicate "is a semimo... |
| slmdlema 33287 | Lemma for properties of a ... |
| lmodslmd 33288 | Left semimodules generaliz... |
| slmdcmn 33289 | A semimodule is a commutat... |
| slmdmnd 33290 | A semimodule is a monoid. ... |
| slmdsrg 33291 | The scalar component of a ... |
| slmdbn0 33292 | The base set of a semimodu... |
| slmdacl 33293 | Closure of ring addition f... |
| slmdmcl 33294 | Closure of ring multiplica... |
| slmdsn0 33295 | The set of scalars in a se... |
| slmdvacl 33296 | Closure of vector addition... |
| slmdass 33297 | Semiring left module vecto... |
| slmdvscl 33298 | Closure of scalar product ... |
| slmdvsdi 33299 | Distributive law for scala... |
| slmdvsdir 33300 | Distributive law for scala... |
| slmdvsass 33301 | Associative law for scalar... |
| slmd0cl 33302 | The ring zero in a semimod... |
| slmd1cl 33303 | The ring unity in a semiri... |
| slmdvs1 33304 | Scalar product with ring u... |
| slmd0vcl 33305 | The zero vector is a vecto... |
| slmd0vlid 33306 | Left identity law for the ... |
| slmd0vrid 33307 | Right identity law for the... |
| slmd0vs 33308 | Zero times a vector is the... |
| slmdvs0 33309 | Anything times the zero ve... |
| gsumvsca1 33310 | Scalar product of a finite... |
| gsumvsca2 33311 | Scalar product of a finite... |
| prmsimpcyc 33312 | A group of prime order is ... |
| ringrngd 33313 | A unital ring is a non-uni... |
| ringdi22 33314 | Expand the product of two ... |
| urpropd 33315 | Sufficient condition for r... |
| subrgmcld 33316 | A subring is closed under ... |
| ress1r 33317 | ` 1r ` is unaffected by re... |
| ringm1expp1 33318 | Ring exponentiation of min... |
| ringinvval 33319 | The ring inverse expressed... |
| dvrcan5 33320 | Cancellation law for commo... |
| subrgchr 33321 | If ` A ` is a subring of `... |
| rmfsupp2 33322 | A mapping of a multiplicat... |
| unitnz 33323 | In a nonzero ring, a unit ... |
| isunit2 33324 | Alternate definition of be... |
| isunit3 33325 | Alternate definition of be... |
| elrgspnlem1 33326 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem2 33327 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem3 33328 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem4 33329 | Lemma for ~ elrgspn . (Co... |
| elrgspn 33330 | Membership in the subring ... |
| elrgspnsubrunlem1 33331 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrunlem2 33332 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrun 33333 | Membership in the ring spa... |
| irrednzr 33334 | A ring with an irreducible... |
| 0ringsubrg 33335 | A subring of a zero ring i... |
| 0ringcring 33336 | The zero ring is commutati... |
| reldmrloc 33341 | Ring localization is a pro... |
| erlval 33342 | Value of the ring localiza... |
| rlocval 33343 | Expand the value of the ri... |
| erlcl1 33344 | Closure for the ring local... |
| erlcl2 33345 | Closure for the ring local... |
| erldi 33346 | Main property of the ring ... |
| erlbrd 33347 | Deduce the ring localizati... |
| erlbr2d 33348 | Deduce the ring localizati... |
| erler 33349 | The relation used to build... |
| elrlocbasi 33350 | Membership in the basis of... |
| rlocbas 33351 | The base set of a ring loc... |
| rlocaddval 33352 | Value of the addition in t... |
| rlocmulval 33353 | Value of the addition in t... |
| rloccring 33354 | The ring localization ` L ... |
| rloc0g 33355 | The zero of a ring localiz... |
| rloc1r 33356 | The multiplicative identit... |
| rlocf1 33357 | The embedding ` F ` of a r... |
| domnmuln0rd 33358 | In a domain, factors of a ... |
| domnprodn0 33359 | In a domain, a finite prod... |
| domnprodeq0 33360 | A product over a domain is... |
| domnpropd 33361 | If two structures have the... |
| idompropd 33362 | If two structures have the... |
| idomrcan 33363 | Right-cancellation law for... |
| domnlcanOLD 33364 | Obsolete version of ~ domn... |
| domnlcanbOLD 33365 | Obsolete version of ~ domn... |
| idomrcanOLD 33366 | Obsolete version of ~ idom... |
| 1rrg 33367 | The multiplicative identit... |
| rrgsubm 33368 | The left regular elements ... |
| subrdom 33369 | A subring of a domain is a... |
| subridom 33370 | A subring of an integral d... |
| subrfld 33371 | A subring of a field is an... |
| eufndx 33374 | Index value of the Euclide... |
| eufid 33375 | Utility theorem: index-ind... |
| ringinveu 33378 | If a ring unit element ` X... |
| isdrng4 33379 | A division ring is a ring ... |
| rndrhmcl 33380 | The image of a division ri... |
| qfld 33381 | The field of rational numb... |
| subsdrg 33382 | A subring of a sub-divisio... |
| sdrgdvcl 33383 | A sub-division-ring is clo... |
| sdrginvcl 33384 | A sub-division-ring is clo... |
| primefldchr 33385 | The characteristic of a pr... |
| fracval 33388 | Value of the field of frac... |
| fracbas 33389 | The base of the field of f... |
| fracerl 33390 | Rewrite the ring localizat... |
| fracf1 33391 | The embedding of a commuta... |
| fracfld 33392 | The field of fractions of ... |
| idomsubr 33393 | Every integral domain is i... |
| fldgenval 33396 | Value of the field generat... |
| fldgenssid 33397 | The field generated by a s... |
| fldgensdrg 33398 | A generated subfield is a ... |
| fldgenssv 33399 | A generated subfield is a ... |
| fldgenss 33400 | Generated subfields preser... |
| fldgenidfld 33401 | The subfield generated by ... |
| fldgenssp 33402 | The field generated by a s... |
| fldgenid 33403 | The subfield of a field ` ... |
| fldgenfld 33404 | A generated subfield is a ... |
| primefldgen1 33405 | The prime field of a divis... |
| 1fldgenq 33406 | The field of rational numb... |
| rhmdvd 33407 | A ring homomorphism preser... |
| kerunit 33408 | If a unit element lies in ... |
| reldmresv 33411 | The scalar restriction is ... |
| resvval 33412 | Value of structure restric... |
| resvid2 33413 | General behavior of trivia... |
| resvval2 33414 | Value of nontrivial struct... |
| resvsca 33415 | Base set of a structure re... |
| resvlem 33416 | Other elements of a scalar... |
| resvbas 33417 | ` Base ` is unaffected by ... |
| resvplusg 33418 | ` +g ` is unaffected by sc... |
| resvvsca 33419 | ` .s ` is unaffected by sc... |
| resvmulr 33420 | ` .r ` is unaffected by sc... |
| resv0g 33421 | ` 0g ` is unaffected by sc... |
| resv1r 33422 | ` 1r ` is unaffected by sc... |
| resvcmn 33423 | Scalar restriction preserv... |
| gzcrng 33424 | The gaussian integers form... |
| cnfldfld 33425 | The complex numbers form a... |
| reofld 33426 | The real numbers form an o... |
| nn0omnd 33427 | The nonnegative integers f... |
| gsumind 33428 | The group sum of an indica... |
| rearchi 33429 | The field of the real numb... |
| nn0archi 33430 | The monoid of the nonnegat... |
| xrge0slmod 33431 | The extended nonnegative r... |
| qusker 33432 | The kernel of a quotient m... |
| eqgvscpbl 33433 | The left coset equivalence... |
| qusvscpbl 33434 | The quotient map distribut... |
| qusvsval 33435 | Value of the scalar multip... |
| imaslmod 33436 | The image structure of a l... |
| imasmhm 33437 | Given a function ` F ` wit... |
| imasghm 33438 | Given a function ` F ` wit... |
| imasrhm 33439 | Given a function ` F ` wit... |
| imaslmhm 33440 | Given a function ` F ` wit... |
| quslmod 33441 | If ` G ` is a submodule in... |
| quslmhm 33442 | If ` G ` is a submodule of... |
| quslvec 33443 | If ` S ` is a vector subsp... |
| ecxpid 33444 | The equivalence class of a... |
| qsxpid 33445 | The quotient set of a cart... |
| qusxpid 33446 | The Group quotient equival... |
| qustriv 33447 | The quotient of a group ` ... |
| qustrivr 33448 | Converse of ~ qustriv . (... |
| znfermltl 33449 | Fermat's little theorem in... |
| islinds5 33450 | A set is linearly independ... |
| ellspds 33451 | Variation on ~ ellspd . (... |
| 0ellsp 33452 | Zero is in all spans. (Co... |
| 0nellinds 33453 | The group identity cannot ... |
| rspsnid 33454 | A principal ideal contains... |
| elrsp 33455 | Write the elements of a ri... |
| ellpi 33456 | Elementhood in a left prin... |
| lpirlidllpi 33457 | In a principal ideal ring,... |
| rspidlid 33458 | The ideal span of an ideal... |
| pidlnz 33459 | A principal ideal generate... |
| lbslsp 33460 | Any element of a left modu... |
| lindssn 33461 | Any singleton of a nonzero... |
| lindflbs 33462 | Conditions for an independ... |
| islbs5 33463 | An equivalent formulation ... |
| linds2eq 33464 | Deduce equality of element... |
| lindfpropd 33465 | Property deduction for lin... |
| lindspropd 33466 | Property deduction for lin... |
| dvdsruassoi 33467 | If two elements ` X ` and ... |
| dvdsruasso 33468 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33469 | A reformulation of ~ dvdsr... |
| dvdsrspss 33470 | In a ring, an element ` X ... |
| rspsnasso 33471 | Two elements ` X ` and ` Y... |
| unitprodclb 33472 | A finite product is a unit... |
| elgrplsmsn 33473 | Membership in a sumset wit... |
| lsmsnorb 33474 | The sumset of a group with... |
| lsmsnorb2 33475 | The sumset of a single ele... |
| elringlsm 33476 | Membership in a product of... |
| elringlsmd 33477 | Membership in a product of... |
| ringlsmss 33478 | Closure of the product of ... |
| ringlsmss1 33479 | The product of an ideal ` ... |
| ringlsmss2 33480 | The product with an ideal ... |
| lsmsnpridl 33481 | The product of the ring wi... |
| lsmsnidl 33482 | The product of the ring wi... |
| lsmidllsp 33483 | The sum of two ideals is t... |
| lsmidl 33484 | The sum of two ideals is a... |
| lsmssass 33485 | Group sum is associative, ... |
| grplsm0l 33486 | Sumset with the identity s... |
| grplsmid 33487 | The direct sum of an eleme... |
| quslsm 33488 | Express the image by the q... |
| qusbas2 33489 | Alternate definition of th... |
| qus0g 33490 | The identity element of a ... |
| qusima 33491 | The image of a subgroup by... |
| qusrn 33492 | The natural map from eleme... |
| nsgqus0 33493 | A normal subgroup ` N ` is... |
| nsgmgclem 33494 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33495 | There is a monotone Galois... |
| nsgqusf1olem1 33496 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33497 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33498 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33499 | The canonical projection h... |
| lmhmqusker 33500 | A surjective module homomo... |
| lmicqusker 33501 | The image ` H ` of a modul... |
| lidlmcld 33502 | An ideal is closed under l... |
| intlidl 33503 | The intersection of a none... |
| 0ringidl 33504 | The zero ideal is the only... |
| pidlnzb 33505 | A principal ideal is nonze... |
| lidlunitel 33506 | If an ideal ` I ` contains... |
| unitpidl1 33507 | The ideal ` I ` generated ... |
| rhmquskerlem 33508 | The mapping ` J ` induced ... |
| rhmqusker 33509 | A surjective ring homomorp... |
| ricqusker 33510 | The image ` H ` of a ring ... |
| elrspunidl 33511 | Elementhood in the span of... |
| elrspunsn 33512 | Membership to the span of ... |
| lidlincl 33513 | Ideals are closed under in... |
| idlinsubrg 33514 | The intersection between a... |
| rhmimaidl 33515 | The image of an ideal ` I ... |
| drngidl 33516 | A nonzero ring is a divisi... |
| drngidlhash 33517 | A ring is a division ring ... |
| prmidlval 33520 | The class of prime ideals ... |
| isprmidl 33521 | The predicate "is a prime ... |
| prmidlnr 33522 | A prime ideal is a proper ... |
| prmidl 33523 | The main property of a pri... |
| prmidl2 33524 | A condition that shows an ... |
| idlmulssprm 33525 | Let ` P ` be a prime ideal... |
| pridln1 33526 | A proper ideal cannot cont... |
| prmidlidl 33527 | A prime ideal is an ideal.... |
| prmidlssidl 33528 | Prime ideals as a subset o... |
| cringm4 33529 | Commutative/associative la... |
| isprmidlc 33530 | The predicate "is prime id... |
| prmidlc 33531 | Property of a prime ideal ... |
| 0ringprmidl 33532 | The trivial ring does not ... |
| prmidl0 33533 | The zero ideal of a commut... |
| rhmpreimaprmidl 33534 | The preimage of a prime id... |
| qsidomlem1 33535 | If the quotient ring of a ... |
| qsidomlem2 33536 | A quotient by a prime idea... |
| qsidom 33537 | An ideal ` I ` in the comm... |
| qsnzr 33538 | A quotient of a non-zero r... |
| ssdifidllem 33539 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 33540 | Let ` R ` be a ring, and l... |
| ssdifidlprm 33541 | If the set ` S ` of ~ ssdi... |
| mxidlval 33544 | The set of maximal ideals ... |
| ismxidl 33545 | The predicate "is a maxima... |
| mxidlidl 33546 | A maximal ideal is an idea... |
| mxidlnr 33547 | A maximal ideal is proper.... |
| mxidlmax 33548 | A maximal ideal is a maxim... |
| mxidln1 33549 | One is not contained in an... |
| mxidlnzr 33550 | A ring with a maximal idea... |
| mxidlmaxv 33551 | An ideal ` I ` strictly co... |
| crngmxidl 33552 | In a commutative ring, max... |
| mxidlprm 33553 | Every maximal ideal is pri... |
| mxidlirredi 33554 | In an integral domain, the... |
| mxidlirred 33555 | In a principal ideal domai... |
| ssmxidllem 33556 | The set ` P ` used in the ... |
| ssmxidl 33557 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 33558 | In a nonzero ring, the zer... |
| drng0mxidl 33559 | In a division ring, the ze... |
| drngmxidl 33560 | The zero ideal is the only... |
| drngmxidlr 33561 | If a ring's only maximal i... |
| krull 33562 | Krull's theorem: Any nonz... |
| mxidlnzrb 33563 | A ring is nonzero if and o... |
| krullndrng 33564 | Krull's theorem for non-di... |
| opprabs 33565 | The opposite ring of the o... |
| oppreqg 33566 | Group coset equivalence re... |
| opprnsg 33567 | Normal subgroups of the op... |
| opprlidlabs 33568 | The ideals of the opposite... |
| oppr2idl 33569 | Two sided ideal of the opp... |
| opprmxidlabs 33570 | The maximal ideal of the o... |
| opprqusbas 33571 | The base of the quotient o... |
| opprqusplusg 33572 | The group operation of the... |
| opprqus0g 33573 | The group identity element... |
| opprqusmulr 33574 | The multiplication operati... |
| opprqus1r 33575 | The ring unity of the quot... |
| opprqusdrng 33576 | The quotient of the opposi... |
| qsdrngilem 33577 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33578 | A quotient by a maximal le... |
| qsdrnglem2 33579 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33580 | An ideal ` M ` is both lef... |
| qsfld 33581 | An ideal ` M ` in the comm... |
| mxidlprmALT 33582 | Every maximal ideal is pri... |
| idlsrgstr 33585 | A constructed semiring of ... |
| idlsrgval 33586 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33587 | Base of the ideals of a ri... |
| idlsrgplusg 33588 | Additive operation of the ... |
| idlsrg0g 33589 | The zero ideal is the addi... |
| idlsrgmulr 33590 | Multiplicative operation o... |
| idlsrgtset 33591 | Topology component of the ... |
| idlsrgmulrval 33592 | Value of the ring multipli... |
| idlsrgmulrcl 33593 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33594 | In a commutative ring, the... |
| idlsrgmulrss2 33595 | The product of two ideals ... |
| idlsrgmulrssin 33596 | In a commutative ring, the... |
| idlsrgmnd 33597 | The ideals of a ring form ... |
| idlsrgcmnd 33598 | The ideals of a ring form ... |
| rprmval 33599 | The prime elements of a ri... |
| isrprm 33600 | Property for ` P ` to be a... |
| rprmcl 33601 | A ring prime is an element... |
| rprmdvds 33602 | If a ring prime ` Q ` divi... |
| rprmnz 33603 | A ring prime is nonzero. ... |
| rprmnunit 33604 | A ring prime is not a unit... |
| rsprprmprmidl 33605 | In a commutative ring, ide... |
| rsprprmprmidlb 33606 | In an integral domain, an ... |
| rprmndvdsr1 33607 | A ring prime element does ... |
| rprmasso 33608 | In an integral domain, the... |
| rprmasso2 33609 | In an integral domain, if ... |
| rprmasso3 33610 | In an integral domain, if ... |
| unitmulrprm 33611 | A ring unit multiplied by ... |
| rprmndvdsru 33612 | A ring prime element does ... |
| rprmirredlem 33613 | Lemma for ~ rprmirred . (... |
| rprmirred 33614 | In an integral domain, rin... |
| rprmirredb 33615 | In a principal ideal domai... |
| rprmdvdspow 33616 | If a prime element divides... |
| rprmdvdsprod 33617 | If a prime element ` Q ` d... |
| 1arithidomlem1 33618 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 33619 | Lemma for ~ 1arithidom : i... |
| 1arithidom 33620 | Uniqueness of prime factor... |
| isufd 33623 | The property of being a Un... |
| ufdprmidl 33624 | In a unique factorization ... |
| ufdidom 33625 | A nonzero unique factoriza... |
| pidufd 33626 | Every principal ideal doma... |
| 1arithufdlem1 33627 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 33628 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 33629 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 33630 | Lemma for ~ 1arithufd . N... |
| 1arithufd 33631 | Existence of a factorizati... |
| dfufd2lem 33632 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 33633 | Alternative definition of ... |
| zringidom 33634 | The ring of integers is an... |
| zringpid 33635 | The ring of integers is a ... |
| dfprm3 33636 | The (positive) prime eleme... |
| zringfrac 33637 | The field of fractions of ... |
| assaassd 33638 | Left-associative property ... |
| assaassrd 33639 | Right-associative property... |
| 0ringmon1p 33640 | There are no monic polynom... |
| fply1 33641 | Conditions for a function ... |
| ply1lvec 33642 | In a division ring, the un... |
| evls1fn 33643 | Functionality of the subri... |
| evls1dm 33644 | The domain of the subring ... |
| evls1fvf 33645 | The subring evaluation fun... |
| evl1fvf 33646 | The univariate polynomial ... |
| evl1fpws 33647 | Evaluation of a univariate... |
| ressply1evls1 33648 | Subring evaluation of a un... |
| ressdeg1 33649 | The degree of a univariate... |
| ressply10g 33650 | A restricted polynomial al... |
| ressply1mon1p 33651 | The monic polynomials of a... |
| ressply1invg 33652 | An element of a restricted... |
| ressply1sub 33653 | A restricted polynomial al... |
| ressasclcl 33654 | Closure of the univariate ... |
| evls1subd 33655 | Univariate polynomial eval... |
| deg1le0eq0 33656 | A polynomial with nonposit... |
| ply1asclunit 33657 | A non-zero scalar polynomi... |
| ply1unit 33658 | In a field ` F ` , a polyn... |
| evl1deg1 33659 | Evaluation of a univariate... |
| evl1deg2 33660 | Evaluation of a univariate... |
| evl1deg3 33661 | Evaluation of a univariate... |
| evls1monply1 33662 | Subring evaluation of a sc... |
| ply1dg1rt 33663 | Express the root ` - B / A... |
| ply1dg1rtn0 33664 | Polynomials of degree 1 ov... |
| ply1mulrtss 33665 | The roots of a factor ` F ... |
| deg1prod 33666 | Degree of a product of pol... |
| ply1dg3rt0irred 33667 | If a cubic polynomial over... |
| m1pmeq 33668 | If two monic polynomials `... |
| ply1fermltl 33669 | Fermat's little theorem fo... |
| coe1mon 33670 | Coefficient vector of a mo... |
| ply1moneq 33671 | Two monomials are equal if... |
| ply1coedeg 33672 | Decompose a univariate pol... |
| coe1zfv 33673 | The coefficients of the ze... |
| coe1vr1 33674 | Polynomial coefficient of ... |
| deg1vr 33675 | The degree of the variable... |
| vr1nz 33676 | A univariate polynomial va... |
| ply1degltel 33677 | Characterize elementhood i... |
| ply1degleel 33678 | Characterize elementhood i... |
| ply1degltlss 33679 | The space ` S ` of the uni... |
| gsummoncoe1fzo 33680 | A coefficient of the polyn... |
| gsummoncoe1fz 33681 | A coefficient of the polyn... |
| ply1gsumz 33682 | If a polynomial given as a... |
| deg1addlt 33683 | If both factors have degre... |
| ig1pnunit 33684 | The polynomial ideal gener... |
| ig1pmindeg 33685 | The polynomial ideal gener... |
| q1pdir 33686 | Distribution of univariate... |
| q1pvsca 33687 | Scalar multiplication prop... |
| r1pvsca 33688 | Scalar multiplication prop... |
| r1p0 33689 | Polynomial remainder opera... |
| r1pcyc 33690 | The polynomial remainder o... |
| r1padd1 33691 | Addition property of the p... |
| r1pid2OLD 33692 | Obsolete version of ~ r1pi... |
| r1plmhm 33693 | The univariate polynomial ... |
| r1pquslmic 33694 | The univariate polynomial ... |
| psrbasfsupp 33695 | Rewrite a finite support f... |
| extvval 33698 | Value of the "variable ext... |
| extvfval 33699 | The "variable extension" f... |
| extvfv 33700 | The "variable extension" f... |
| extvfvv 33701 | The "variable extension" f... |
| extvfvvcl 33702 | Closure for the "variable ... |
| extvfvcl 33703 | Closure for the "variable ... |
| extvfvalf 33704 | The "variable extension" f... |
| mvrvalind 33705 | Value of the generating el... |
| mplmulmvr 33706 | Multiply a polynomial ` F ... |
| evlscaval 33707 | Polynomial evaluation for ... |
| evlvarval 33708 | Polynomial evaluation buil... |
| evlextv 33709 | Evaluating a variable-exte... |
| mplvrpmlem 33710 | Lemma for ~ mplvrpmga and ... |
| mplvrpmfgalem 33711 | Permuting variables in a m... |
| mplvrpmga 33712 | The action of permuting va... |
| mplvrpmmhm 33713 | The action of permuting va... |
| mplvrpmrhm 33714 | The action of permuting va... |
| splyval 33719 | The symmetric polynomials ... |
| splysubrg 33720 | The symmetric polynomials ... |
| issply 33721 | Conditions for being a sym... |
| esplyval 33722 | The elementary polynomials... |
| esplyfval 33723 | The ` K ` -th elementary p... |
| esplyfval0 33724 | The ` 0 ` -th elementary s... |
| esplyfval2 33725 | When ` K ` is out-of-bound... |
| esplylem 33726 | Lemma for ~ esplyfv and ot... |
| esplympl 33727 | Elementary symmetric polyn... |
| esplymhp 33728 | The ` K ` -th elementary s... |
| esplyfv1 33729 | Coefficient for the ` K ` ... |
| esplyfv 33730 | Coefficient for the ` K ` ... |
| esplysply 33731 | The ` K ` -th elementary s... |
| esplyfval3 33732 | Alternate expression for t... |
| esplyind 33733 | A recursive formula for th... |
| esplyindfv 33734 | A recursive formula for th... |
| esplyfvn 33735 | Express the last elementar... |
| vietadeg1 33736 | The degree of a product of... |
| vietalem 33737 | Lemma for ~ vieta : induct... |
| vieta 33738 | Vieta's Formulas: Coeffic... |
| sra1r 33739 | The unity element of a sub... |
| sradrng 33740 | Condition for a subring al... |
| sraidom 33741 | Condition for a subring al... |
| srasubrg 33742 | A subring of the original ... |
| sralvec 33743 | Given a sub division ring ... |
| srafldlvec 33744 | Given a subfield ` F ` of ... |
| resssra 33745 | The subring algebra of a r... |
| lsssra 33746 | A subring is a subspace of... |
| srapwov 33747 | The "power" operation on a... |
| drgext0g 33748 | The additive neutral eleme... |
| drgextvsca 33749 | The scalar multiplication ... |
| drgext0gsca 33750 | The additive neutral eleme... |
| drgextsubrg 33751 | The scalar field is a subr... |
| drgextlsp 33752 | The scalar field is a subs... |
| drgextgsum 33753 | Group sum in a division ri... |
| lvecdimfi 33754 | Finite version of ~ lvecdi... |
| exsslsb 33755 | Any finite generating set ... |
| lbslelsp 33756 | The size of a basis ` X ` ... |
| dimval 33759 | The dimension of a vector ... |
| dimvalfi 33760 | The dimension of a vector ... |
| dimcl 33761 | Closure of the vector spac... |
| lmimdim 33762 | Module isomorphisms preser... |
| lmicdim 33763 | Module isomorphisms preser... |
| lvecdim0i 33764 | A vector space of dimensio... |
| lvecdim0 33765 | A vector space of dimensio... |
| lssdimle 33766 | The dimension of a linear ... |
| dimpropd 33767 | If two structures have the... |
| rlmdim 33768 | The left vector space indu... |
| rgmoddimOLD 33769 | Obsolete version of ~ rlmd... |
| frlmdim 33770 | Dimension of a free left m... |
| tnglvec 33771 | Augmenting a structure wit... |
| tngdim 33772 | Dimension of a left vector... |
| rrxdim 33773 | Dimension of the generaliz... |
| matdim 33774 | Dimension of the space of ... |
| lbslsat 33775 | A nonzero vector ` X ` is ... |
| lsatdim 33776 | A line, spanned by a nonze... |
| drngdimgt0 33777 | The dimension of a vector ... |
| lmhmlvec2 33778 | A homomorphism of left vec... |
| kerlmhm 33779 | The kernel of a vector spa... |
| imlmhm 33780 | The image of a vector spac... |
| ply1degltdimlem 33781 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 33782 | The space ` S ` of the uni... |
| lindsunlem 33783 | Lemma for ~ lindsun . (Co... |
| lindsun 33784 | Condition for the union of... |
| lbsdiflsp0 33785 | The linear spans of two di... |
| dimkerim 33786 | Given a linear map ` F ` b... |
| qusdimsum 33787 | Let ` W ` be a vector spac... |
| fedgmullem1 33788 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33789 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33790 | The multiplicativity formu... |
| dimlssid 33791 | If the dimension of a line... |
| lvecendof1f1o 33792 | If an endomorphism ` U ` o... |
| lactlmhm 33793 | In an associative algebra ... |
| assalactf1o 33794 | In an associative algebra ... |
| assarrginv 33795 | If an element ` X ` of an ... |
| assafld 33796 | If an algebra ` A ` of fin... |
| relfldext 33803 | The field extension is a r... |
| brfldext 33804 | The field extension relati... |
| ccfldextrr 33805 | The field of the complex n... |
| fldextfld1 33806 | A field extension is only ... |
| fldextfld2 33807 | A field extension is only ... |
| fldextsubrg 33808 | Field extension implies a ... |
| sdrgfldext 33809 | A field ` E ` and any sub-... |
| fldextress 33810 | Field extension implies a ... |
| brfinext 33811 | The finite field extension... |
| extdgval 33812 | Value of the field extensi... |
| fldextsdrg 33813 | Deduce sub-division-ring f... |
| fldextsralvec 33814 | The subring algebra associ... |
| extdgcl 33815 | Closure of the field exten... |
| extdggt0 33816 | Degrees of field extension... |
| fldexttr 33817 | Field extension is a trans... |
| fldextid 33818 | The field extension relati... |
| extdgid 33819 | A trivial field extension ... |
| fldsdrgfldext 33820 | A sub-division-ring of a f... |
| fldsdrgfldext2 33821 | A sub-sub-division-ring of... |
| extdgmul 33822 | The multiplicativity formu... |
| finextfldext 33823 | A finite field extension i... |
| finexttrb 33824 | The extension ` E ` of ` K... |
| extdg1id 33825 | If the degree of the exten... |
| extdg1b 33826 | The degree of the extensio... |
| fldgenfldext 33827 | A subfield ` F ` extended ... |
| fldextchr 33828 | The characteristic of a su... |
| evls1fldgencl 33829 | Closure of the subring pol... |
| ccfldsrarelvec 33830 | The subring algebra of the... |
| ccfldextdgrr 33831 | The degree of the field ex... |
| fldextrspunlsplem 33832 | Lemma for ~ fldextrspunlsp... |
| fldextrspunlsp 33833 | Lemma for ~ fldextrspunfld... |
| fldextrspunlem1 33834 | Lemma for ~ fldextrspunfld... |
| fldextrspunfld 33835 | The ring generated by the ... |
| fldextrspunlem2 33836 | Part of the proof of Propo... |
| fldextrspundgle 33837 | Inequality involving the d... |
| fldextrspundglemul 33838 | Given two field extensions... |
| fldextrspundgdvdslem 33839 | Lemma for ~ fldextrspundgd... |
| fldextrspundgdvds 33840 | Given two finite extension... |
| fldext2rspun 33841 | Given two field extensions... |
| irngval 33844 | The elements of a field ` ... |
| elirng 33845 | Property for an element ` ... |
| irngss 33846 | All elements of a subring ... |
| irngssv 33847 | An integral element is an ... |
| 0ringirng 33848 | A zero ring ` R ` has no i... |
| irngnzply1lem 33849 | In the case of a field ` E... |
| irngnzply1 33850 | In the case of a field ` E... |
| extdgfialglem1 33851 | Lemma for ~ extdgfialg . ... |
| extdgfialglem2 33852 | Lemma for ~ extdgfialg . ... |
| extdgfialg 33853 | A finite field extension `... |
| bralgext 33856 | Express the fact that a fi... |
| finextalg 33857 | A finite field extension i... |
| ply1annidllem 33860 | Write the set ` Q ` of pol... |
| ply1annidl 33861 | The set ` Q ` of polynomia... |
| ply1annnr 33862 | The set ` Q ` of polynomia... |
| ply1annig1p 33863 | The ideal ` Q ` of polynom... |
| minplyval 33864 | Expand the value of the mi... |
| minplycl 33865 | The minimal polynomial is ... |
| ply1annprmidl 33866 | The set ` Q ` of polynomia... |
| minplymindeg 33867 | The minimal polynomial of ... |
| minplyann 33868 | The minimal polynomial for... |
| minplyirredlem 33869 | Lemma for ~ minplyirred . ... |
| minplyirred 33870 | A nonzero minimal polynomi... |
| irngnminplynz 33871 | Integral elements have non... |
| minplym1p 33872 | A minimal polynomial is mo... |
| minplynzm1p 33873 | If a minimal polynomial is... |
| minplyelirng 33874 | If the minimial polynomial... |
| irredminply 33875 | An irreducible, monic, ann... |
| algextdeglem1 33876 | Lemma for ~ algextdeg . (... |
| algextdeglem2 33877 | Lemma for ~ algextdeg . B... |
| algextdeglem3 33878 | Lemma for ~ algextdeg . T... |
| algextdeglem4 33879 | Lemma for ~ algextdeg . B... |
| algextdeglem5 33880 | Lemma for ~ algextdeg . T... |
| algextdeglem6 33881 | Lemma for ~ algextdeg . B... |
| algextdeglem7 33882 | Lemma for ~ algextdeg . T... |
| algextdeglem8 33883 | Lemma for ~ algextdeg . T... |
| algextdeg 33884 | The degree of an algebraic... |
| rtelextdg2lem 33885 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 33886 | If an element ` X ` is a s... |
| fldext2chn 33887 | In a non-empty chain ` T `... |
| constrrtll 33890 | In the construction of con... |
| constrrtlc1 33891 | In the construction of con... |
| constrrtlc2 33892 | In the construction of con... |
| constrrtcclem 33893 | In the construction of con... |
| constrrtcc 33894 | In the construction of con... |
| isconstr 33895 | Property of being a constr... |
| constr0 33896 | The first step of the cons... |
| constrsuc 33897 | Membership in the successo... |
| constrlim 33898 | Limit step of the construc... |
| constrsscn 33899 | Closure of the constructib... |
| constrsslem 33900 | Lemma for ~ constrss . Th... |
| constr01 33901 | ` 0 ` and ` 1 ` are in all... |
| constrss 33902 | Constructed points are in ... |
| constrmon 33903 | The construction of constr... |
| constrconj 33904 | If a point ` X ` of the co... |
| constrfin 33905 | Each step of the construct... |
| constrelextdg2 33906 | If the ` N ` -th step ` ( ... |
| constrextdg2lem 33907 | Lemma for ~ constrextdg2 (... |
| constrextdg2 33908 | Any step ` ( C `` N ) ` of... |
| constrext2chnlem 33909 | Lemma for ~ constrext2chn ... |
| constrfiss 33910 | For any finite set ` A ` o... |
| constrllcllem 33911 | Constructible numbers are ... |
| constrlccllem 33912 | Constructible numbers are ... |
| constrcccllem 33913 | Constructible numbers are ... |
| constrcbvlem 33914 | Technical lemma for elimin... |
| constrllcl 33915 | Constructible numbers are ... |
| constrlccl 33916 | Constructible numbers are ... |
| constrcccl 33917 | Constructible numbers are ... |
| constrext2chn 33918 | If a constructible number ... |
| constrcn 33919 | Constructible numbers are ... |
| nn0constr 33920 | Nonnegative integers are c... |
| constraddcl 33921 | Constructive numbers are c... |
| constrnegcl 33922 | Constructible numbers are ... |
| zconstr 33923 | Integers are constructible... |
| constrdircl 33924 | Constructible numbers are ... |
| iconstr 33925 | The imaginary unit ` _i ` ... |
| constrremulcl 33926 | If two real numbers ` X ` ... |
| constrcjcl 33927 | Constructible numbers are ... |
| constrrecl 33928 | Constructible numbers are ... |
| constrimcl 33929 | Constructible numbers are ... |
| constrmulcl 33930 | Constructible numbers are ... |
| constrreinvcl 33931 | If a real number ` X ` is ... |
| constrinvcl 33932 | Constructible numbers are ... |
| constrcon 33933 | Contradiction of construct... |
| constrsdrg 33934 | Constructible numbers form... |
| constrfld 33935 | The constructible numbers ... |
| constrresqrtcl 33936 | If a positive real number ... |
| constrabscl 33937 | Constructible numbers are ... |
| constrsqrtcl 33938 | Constructible numbers are ... |
| 2sqr3minply 33939 | The polynomial ` ( ( X ^ 3... |
| 2sqr3nconstr 33940 | Doubling the cube is an im... |
| cos9thpiminplylem1 33941 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem2 33942 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem3 33943 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem4 33944 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem5 33945 | The constructed complex nu... |
| cos9thpiminplylem6 33946 | Evaluation of the polynomi... |
| cos9thpiminply 33947 | The polynomial ` ( ( X ^ 3... |
| cos9thpinconstrlem1 33948 | The complex number ` O ` ,... |
| cos9thpinconstrlem2 33949 | The complex number ` A ` i... |
| cos9thpinconstr 33950 | Trisecting an angle is an ... |
| trisecnconstr 33951 | Not all angles can be tris... |
| smatfval 33954 | Value of the submatrix. (... |
| smatrcl 33955 | Closure of the rectangular... |
| smatlem 33956 | Lemma for the next theorem... |
| smattl 33957 | Entries of a submatrix, to... |
| smattr 33958 | Entries of a submatrix, to... |
| smatbl 33959 | Entries of a submatrix, bo... |
| smatbr 33960 | Entries of a submatrix, bo... |
| smatcl 33961 | Closure of the square subm... |
| matmpo 33962 | Write a square matrix as a... |
| 1smat1 33963 | The submatrix of the ident... |
| submat1n 33964 | One case where the submatr... |
| submatres 33965 | Special case where the sub... |
| submateqlem1 33966 | Lemma for ~ submateq . (C... |
| submateqlem2 33967 | Lemma for ~ submateq . (C... |
| submateq 33968 | Sufficient condition for t... |
| submatminr1 33969 | If we take a submatrix by ... |
| lmatval 33972 | Value of the literal matri... |
| lmatfval 33973 | Entries of a literal matri... |
| lmatfvlem 33974 | Useful lemma to extract li... |
| lmatcl 33975 | Closure of the literal mat... |
| lmat22lem 33976 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 33977 | Entry of a 2x2 literal mat... |
| lmat22e12 33978 | Entry of a 2x2 literal mat... |
| lmat22e21 33979 | Entry of a 2x2 literal mat... |
| lmat22e22 33980 | Entry of a 2x2 literal mat... |
| lmat22det 33981 | The determinant of a liter... |
| mdetpmtr1 33982 | The determinant of a matri... |
| mdetpmtr2 33983 | The determinant of a matri... |
| mdetpmtr12 33984 | The determinant of a matri... |
| mdetlap1 33985 | A Laplace expansion of the... |
| madjusmdetlem1 33986 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 33987 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 33988 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 33989 | Lemma for ~ madjusmdet . ... |
| madjusmdet 33990 | Express the cofactor of th... |
| mdetlap 33991 | Laplace expansion of the d... |
| ist0cld 33992 | The predicate "is a T_0 sp... |
| txomap 33993 | Given two open maps ` F ` ... |
| qtopt1 33994 | If every equivalence class... |
| qtophaus 33995 | If an open map's graph in ... |
| circtopn 33996 | The topology of the unit c... |
| circcn 33997 | The function gluing the re... |
| reff 33998 | For any cover refinement, ... |
| locfinreflem 33999 | A locally finite refinemen... |
| locfinref 34000 | A locally finite refinemen... |
| iscref 34003 | The property that every op... |
| crefeq 34004 | Equality theorem for the "... |
| creftop 34005 | A space where every open c... |
| crefi 34006 | The property that every op... |
| crefdf 34007 | A formulation of ~ crefi e... |
| crefss 34008 | The "every open cover has ... |
| cmpcref 34009 | Equivalent definition of c... |
| cmpfiref 34010 | Every open cover of a Comp... |
| ldlfcntref 34013 | Every open cover of a Lind... |
| ispcmp 34016 | The predicate "is a paraco... |
| cmppcmp 34017 | Every compact space is par... |
| dispcmp 34018 | Every discrete space is pa... |
| pcmplfin 34019 | Given a paracompact topolo... |
| pcmplfinf 34020 | Given a paracompact topolo... |
| rspecval 34023 | Value of the spectrum of t... |
| rspecbas 34024 | The prime ideals form the ... |
| rspectset 34025 | Topology component of the ... |
| rspectopn 34026 | The topology component of ... |
| zarcls0 34027 | The closure of the identit... |
| zarcls1 34028 | The unit ideal ` B ` is th... |
| zarclsun 34029 | The union of two closed se... |
| zarclsiin 34030 | In a Zariski topology, the... |
| zarclsint 34031 | The intersection of a fami... |
| zarclssn 34032 | The closed points of Zaris... |
| zarcls 34033 | The open sets of the Zaris... |
| zartopn 34034 | The Zariski topology is a ... |
| zartop 34035 | The Zariski topology is a ... |
| zartopon 34036 | The points of the Zariski ... |
| zar0ring 34037 | The Zariski Topology of th... |
| zart0 34038 | The Zariski topology is T_... |
| zarmxt1 34039 | The Zariski topology restr... |
| zarcmplem 34040 | Lemma for ~ zarcmp . (Con... |
| zarcmp 34041 | The Zariski topology is co... |
| rspectps 34042 | The spectrum of a ring ` R... |
| rhmpreimacnlem 34043 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 34044 | The function mapping a pri... |
| metidval 34049 | Value of the metric identi... |
| metidss 34050 | As a relation, the metric ... |
| metidv 34051 | ` A ` and ` B ` identify b... |
| metideq 34052 | Basic property of the metr... |
| metider 34053 | The metric identification ... |
| pstmval 34054 | Value of the metric induce... |
| pstmfval 34055 | Function value of the metr... |
| pstmxmet 34056 | The metric induced by a ps... |
| hauseqcn 34057 | In a Hausdorff topology, t... |
| elunitge0 34058 | An element of the closed u... |
| unitssxrge0 34059 | The closed unit interval i... |
| unitdivcld 34060 | Necessary conditions for a... |
| iistmd 34061 | The closed unit interval f... |
| unicls 34062 | The union of the closed se... |
| tpr2tp 34063 | The usual topology on ` ( ... |
| tpr2uni 34064 | The usual topology on ` ( ... |
| xpinpreima 34065 | Rewrite the cartesian prod... |
| xpinpreima2 34066 | Rewrite the cartesian prod... |
| sqsscirc1 34067 | The complex square of side... |
| sqsscirc2 34068 | The complex square of side... |
| cnre2csqlem 34069 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 34070 | Image of a centered square... |
| tpr2rico 34071 | For any point of an open s... |
| cnvordtrestixx 34072 | The restriction of the 'gr... |
| prsdm 34073 | Domain of the relation of ... |
| prsrn 34074 | Range of the relation of a... |
| prsss 34075 | Relation of a subproset. ... |
| prsssdm 34076 | Domain of a subproset rela... |
| ordtprsval 34077 | Value of the order topolog... |
| ordtprsuni 34078 | Value of the order topolog... |
| ordtcnvNEW 34079 | The order dual generates t... |
| ordtrestNEW 34080 | The subspace topology of a... |
| ordtrest2NEWlem 34081 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 34082 | An interval-closed set ` A... |
| ordtconnlem1 34083 | Connectedness in the order... |
| ordtconn 34084 | Connectedness in the order... |
| mndpluscn 34085 | A mapping that is both a h... |
| mhmhmeotmd 34086 | Deduce a Topological Monoi... |
| rmulccn 34087 | Multiplication by a real c... |
| raddcn 34088 | Addition in the real numbe... |
| xrmulc1cn 34089 | The operation multiplying ... |
| fmcncfil 34090 | The image of a Cauchy filt... |
| xrge0hmph 34091 | The extended nonnegative r... |
| xrge0iifcnv 34092 | Define a bijection from ` ... |
| xrge0iifcv 34093 | The defined function's val... |
| xrge0iifiso 34094 | The defined bijection from... |
| xrge0iifhmeo 34095 | Expose a homeomorphism fro... |
| xrge0iifhom 34096 | The defined function from ... |
| xrge0iif1 34097 | Condition for the defined ... |
| xrge0iifmhm 34098 | The defined function from ... |
| xrge0pluscn 34099 | The addition operation of ... |
| xrge0mulc1cn 34100 | The operation multiplying ... |
| xrge0tps 34101 | The extended nonnegative r... |
| xrge0topn 34102 | The topology of the extend... |
| xrge0haus 34103 | The topology of the extend... |
| xrge0tmd 34104 | The extended nonnegative r... |
| xrge0tmdALT 34105 | Alternate proof of ~ xrge0... |
| lmlim 34106 | Relate a limit in a given ... |
| lmlimxrge0 34107 | Relate a limit in the nonn... |
| rge0scvg 34108 | Implication of convergence... |
| fsumcvg4 34109 | A serie with finite suppor... |
| pnfneige0 34110 | A neighborhood of ` +oo ` ... |
| lmxrge0 34111 | Express "sequence ` F ` co... |
| lmdvg 34112 | If a monotonic sequence of... |
| lmdvglim 34113 | If a monotonic real number... |
| pl1cn 34114 | A univariate polynomial is... |
| zringnm 34117 | The norm (function) for a ... |
| zzsnm 34118 | The norm of the ring of th... |
| zlm0 34119 | Zero of a ` ZZ ` -module. ... |
| zlm1 34120 | Unity element of a ` ZZ ` ... |
| zlmds 34121 | Distance in a ` ZZ ` -modu... |
| zlmtset 34122 | Topology in a ` ZZ ` -modu... |
| zlmnm 34123 | Norm of a ` ZZ ` -module (... |
| zhmnrg 34124 | The ` ZZ ` -module built f... |
| nmmulg 34125 | The norm of a group produc... |
| zrhnm 34126 | The norm of the image by `... |
| cnzh 34127 | The ` ZZ ` -module of ` CC... |
| rezh 34128 | The ` ZZ ` -module of ` RR... |
| qqhval 34131 | Value of the canonical hom... |
| zrhf1ker 34132 | The kernel of the homomorp... |
| zrhchr 34133 | The kernel of the homomorp... |
| zrhker 34134 | The kernel of the homomorp... |
| zrhunitpreima 34135 | The preimage by ` ZRHom ` ... |
| elzrhunit 34136 | Condition for the image by... |
| zrhneg 34137 | The canonical homomorphism... |
| zrhcntr 34138 | The canonical representati... |
| elzdif0 34139 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 34140 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 34141 | Value of the canonical hom... |
| qqhvval 34142 | Value of the canonical hom... |
| qqh0 34143 | The image of ` 0 ` by the ... |
| qqh1 34144 | The image of ` 1 ` by the ... |
| qqhf 34145 | ` QQHom ` as a function. ... |
| qqhvq 34146 | The image of a quotient by... |
| qqhghm 34147 | The ` QQHom ` homomorphism... |
| qqhrhm 34148 | The ` QQHom ` homomorphism... |
| qqhnm 34149 | The norm of the image by `... |
| qqhcn 34150 | The ` QQHom ` homomorphism... |
| qqhucn 34151 | The ` QQHom ` homomorphism... |
| rrhval 34155 | Value of the canonical hom... |
| rrhcn 34156 | If the topology of ` R ` i... |
| rrhf 34157 | If the topology of ` R ` i... |
| isrrext 34159 | Express the property " ` R... |
| rrextnrg 34160 | An extension of ` RR ` is ... |
| rrextdrg 34161 | An extension of ` RR ` is ... |
| rrextnlm 34162 | The norm of an extension o... |
| rrextchr 34163 | The ring characteristic of... |
| rrextcusp 34164 | An extension of ` RR ` is ... |
| rrexttps 34165 | An extension of ` RR ` is ... |
| rrexthaus 34166 | The topology of an extensi... |
| rrextust 34167 | The uniformity of an exten... |
| rerrext 34168 | The field of the real numb... |
| cnrrext 34169 | The field of the complex n... |
| qqtopn 34170 | The topology of the field ... |
| rrhfe 34171 | If ` R ` is an extension o... |
| rrhcne 34172 | If ` R ` is an extension o... |
| rrhqima 34173 | The ` RRHom ` homomorphism... |
| rrh0 34174 | The image of ` 0 ` by the ... |
| xrhval 34177 | The value of the embedding... |
| zrhre 34178 | The ` ZRHom ` homomorphism... |
| qqhre 34179 | The ` QQHom ` homomorphism... |
| rrhre 34180 | The ` RRHom ` homomorphism... |
| relmntop 34183 | Manifold is a relation. (... |
| ismntoplly 34184 | Property of being a manifo... |
| ismntop 34185 | Property of being a manifo... |
| esumex 34188 | An extended sum is a set b... |
| esumcl 34189 | Closure for extended sum i... |
| esumeq12dvaf 34190 | Equality deduction for ext... |
| esumeq12dva 34191 | Equality deduction for ext... |
| esumeq12d 34192 | Equality deduction for ext... |
| esumeq1 34193 | Equality theorem for an ex... |
| esumeq1d 34194 | Equality theorem for an ex... |
| esumeq2 34195 | Equality theorem for exten... |
| esumeq2d 34196 | Equality deduction for ext... |
| esumeq2dv 34197 | Equality deduction for ext... |
| esumeq2sdv 34198 | Equality deduction for ext... |
| nfesum1 34199 | Bound-variable hypothesis ... |
| nfesum2 34200 | Bound-variable hypothesis ... |
| cbvesum 34201 | Change bound variable in a... |
| cbvesumv 34202 | Change bound variable in a... |
| esumid 34203 | Identify the extended sum ... |
| esumgsum 34204 | A finite extended sum is t... |
| esumval 34205 | Develop the value of the e... |
| esumel 34206 | The extended sum is a limi... |
| esumnul 34207 | Extended sum over the empt... |
| esum0 34208 | Extended sum of zero. (Co... |
| esumf1o 34209 | Re-index an extended sum u... |
| esumc 34210 | Convert from the collectio... |
| esumrnmpt 34211 | Rewrite an extended sum in... |
| esumsplit 34212 | Split an extended sum into... |
| esummono 34213 | Extended sum is monotonic.... |
| esumpad 34214 | Extend an extended sum by ... |
| esumpad2 34215 | Remove zeroes from an exte... |
| esumadd 34216 | Addition of infinite sums.... |
| esumle 34217 | If all of the terms of an ... |
| gsumesum 34218 | Relate a group sum on ` ( ... |
| esumlub 34219 | The extended sum is the lo... |
| esumaddf 34220 | Addition of infinite sums.... |
| esumlef 34221 | If all of the terms of an ... |
| esumcst 34222 | The extended sum of a cons... |
| esumsnf 34223 | The extended sum of a sing... |
| esumsn 34224 | The extended sum of a sing... |
| esumpr 34225 | Extended sum over a pair. ... |
| esumpr2 34226 | Extended sum over a pair, ... |
| esumrnmpt2 34227 | Rewrite an extended sum in... |
| esumfzf 34228 | Formulating a partial exte... |
| esumfsup 34229 | Formulating an extended su... |
| esumfsupre 34230 | Formulating an extended su... |
| esumss 34231 | Change the index set to a ... |
| esumpinfval 34232 | The value of the extended ... |
| esumpfinvallem 34233 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34234 | The value of the extended ... |
| esumpfinvalf 34235 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34236 | The value of the extended ... |
| esumpcvgval 34237 | The value of the extended ... |
| esumpmono 34238 | The partial sums in an ext... |
| esumcocn 34239 | Lemma for ~ esummulc2 and ... |
| esummulc1 34240 | An extended sum multiplied... |
| esummulc2 34241 | An extended sum multiplied... |
| esumdivc 34242 | An extended sum divided by... |
| hashf2 34243 | Lemma for ~ hasheuni . (C... |
| hasheuni 34244 | The cardinality of a disjo... |
| esumcvg 34245 | The sequence of partial su... |
| esumcvg2 34246 | Simpler version of ~ esumc... |
| esumcvgsum 34247 | The value of the extended ... |
| esumsup 34248 | Express an extended sum as... |
| esumgect 34249 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34250 | All terms of a converging ... |
| esum2dlem 34251 | Lemma for ~ esum2d (finite... |
| esum2d 34252 | Write a double extended su... |
| esumiun 34253 | Sum over a nonnecessarily ... |
| ofceq 34256 | Equality theorem for funct... |
| ofcfval 34257 | Value of an operation appl... |
| ofcval 34258 | Evaluate a function/consta... |
| ofcfn 34259 | The function operation pro... |
| ofcfeqd2 34260 | Equality theorem for funct... |
| ofcfval3 34261 | General value of ` ( F oFC... |
| ofcf 34262 | The function/constant oper... |
| ofcfval2 34263 | The function operation exp... |
| ofcfval4 34264 | The function/constant oper... |
| ofcc 34265 | Left operation by a consta... |
| ofcof 34266 | Relate function operation ... |
| sigaex 34269 | Lemma for ~ issiga and ~ i... |
| sigaval 34270 | The set of sigma-algebra w... |
| issiga 34271 | An alternative definition ... |
| isrnsiga 34272 | The property of being a si... |
| 0elsiga 34273 | A sigma-algebra contains t... |
| baselsiga 34274 | A sigma-algebra contains i... |
| sigasspw 34275 | A sigma-algebra is a set o... |
| sigaclcu 34276 | A sigma-algebra is closed ... |
| sigaclcuni 34277 | A sigma-algebra is closed ... |
| sigaclfu 34278 | A sigma-algebra is closed ... |
| sigaclcu2 34279 | A sigma-algebra is closed ... |
| sigaclfu2 34280 | A sigma-algebra is closed ... |
| sigaclcu3 34281 | A sigma-algebra is closed ... |
| issgon 34282 | Property of being a sigma-... |
| sgon 34283 | A sigma-algebra is a sigma... |
| elsigass 34284 | An element of a sigma-alge... |
| elrnsiga 34285 | Dropping the base informat... |
| isrnsigau 34286 | The property of being a si... |
| unielsiga 34287 | A sigma-algebra contains i... |
| dmvlsiga 34288 | Lebesgue-measurable subset... |
| pwsiga 34289 | Any power set forms a sigm... |
| prsiga 34290 | The smallest possible sigm... |
| sigaclci 34291 | A sigma-algebra is closed ... |
| difelsiga 34292 | A sigma-algebra is closed ... |
| unelsiga 34293 | A sigma-algebra is closed ... |
| inelsiga 34294 | A sigma-algebra is closed ... |
| sigainb 34295 | Building a sigma-algebra f... |
| insiga 34296 | The intersection of a coll... |
| sigagenval 34299 | Value of the generated sig... |
| sigagensiga 34300 | A generated sigma-algebra ... |
| sgsiga 34301 | A generated sigma-algebra ... |
| unisg 34302 | The sigma-algebra generate... |
| dmsigagen 34303 | A sigma-algebra can be gen... |
| sssigagen 34304 | A set is a subset of the s... |
| sssigagen2 34305 | A subset of the generating... |
| elsigagen 34306 | Any element of a set is al... |
| elsigagen2 34307 | Any countable union of ele... |
| sigagenss 34308 | The generated sigma-algebr... |
| sigagenss2 34309 | Sufficient condition for i... |
| sigagenid 34310 | The sigma-algebra generate... |
| ispisys 34311 | The property of being a pi... |
| ispisys2 34312 | The property of being a pi... |
| inelpisys 34313 | Pi-systems are closed unde... |
| sigapisys 34314 | All sigma-algebras are pi-... |
| isldsys 34315 | The property of being a la... |
| pwldsys 34316 | The power set of the unive... |
| unelldsys 34317 | Lambda-systems are closed ... |
| sigaldsys 34318 | All sigma-algebras are lam... |
| ldsysgenld 34319 | The intersection of all la... |
| sigapildsyslem 34320 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34321 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34322 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34323 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34324 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34325 | The lambda system ` E ` ge... |
| dynkin 34326 | Dynkin's lambda-pi theorem... |
| isros 34327 | The property of being a ri... |
| rossspw 34328 | A ring of sets is a collec... |
| 0elros 34329 | A ring of sets contains th... |
| unelros 34330 | A ring of sets is closed u... |
| difelros 34331 | A ring of sets is closed u... |
| inelros 34332 | A ring of sets is closed u... |
| fiunelros 34333 | A ring of sets is closed u... |
| issros 34334 | The property of being a se... |
| srossspw 34335 | A semiring of sets is a co... |
| 0elsros 34336 | A semiring of sets contain... |
| inelsros 34337 | A semiring of sets is clos... |
| diffiunisros 34338 | In semiring of sets, compl... |
| rossros 34339 | Rings of sets are semiring... |
| brsiga 34342 | The Borel Algebra on real ... |
| brsigarn 34343 | The Borel Algebra is a sig... |
| brsigasspwrn 34344 | The Borel Algebra is a set... |
| unibrsiga 34345 | The union of the Borel Alg... |
| cldssbrsiga 34346 | A Borel Algebra contains a... |
| sxval 34349 | Value of the product sigma... |
| sxsiga 34350 | A product sigma-algebra is... |
| sxsigon 34351 | A product sigma-algebra is... |
| sxuni 34352 | The base set of a product ... |
| elsx 34353 | The cartesian product of t... |
| measbase 34356 | The base set of a measure ... |
| measval 34357 | The value of the ` measure... |
| ismeas 34358 | The property of being a me... |
| isrnmeas 34359 | The property of being a me... |
| dmmeas 34360 | The domain of a measure is... |
| measbasedom 34361 | The base set of a measure ... |
| measfrge0 34362 | A measure is a function ov... |
| measfn 34363 | A measure is a function on... |
| measvxrge0 34364 | The values of a measure ar... |
| measvnul 34365 | The measure of the empty s... |
| measge0 34366 | A measure is nonnegative. ... |
| measle0 34367 | If the measure of a given ... |
| measvun 34368 | The measure of a countable... |
| measxun2 34369 | The measure the union of t... |
| measun 34370 | The measure the union of t... |
| measvunilem 34371 | Lemma for ~ measvuni . (C... |
| measvunilem0 34372 | Lemma for ~ measvuni . (C... |
| measvuni 34373 | The measure of a countable... |
| measssd 34374 | A measure is monotone with... |
| measunl 34375 | A measure is sub-additive ... |
| measiuns 34376 | The measure of the union o... |
| measiun 34377 | A measure is sub-additive.... |
| meascnbl 34378 | A measure is continuous fr... |
| measinblem 34379 | Lemma for ~ measinb . (Co... |
| measinb 34380 | Building a measure restric... |
| measres 34381 | Building a measure restric... |
| measinb2 34382 | Building a measure restric... |
| measdivcst 34383 | Division of a measure by a... |
| measdivcstALTV 34384 | Alternate version of ~ mea... |
| cntmeas 34385 | The Counting measure is a ... |
| pwcntmeas 34386 | The counting measure is a ... |
| cntnevol 34387 | Counting and Lebesgue meas... |
| voliune 34388 | The Lebesgue measure funct... |
| volfiniune 34389 | The Lebesgue measure funct... |
| volmeas 34390 | The Lebesgue measure is a ... |
| ddeval1 34393 | Value of the delta measure... |
| ddeval0 34394 | Value of the delta measure... |
| ddemeas 34395 | The Dirac delta measure is... |
| relae 34399 | 'almost everywhere' is a r... |
| brae 34400 | 'almost everywhere' relati... |
| braew 34401 | 'almost everywhere' relati... |
| truae 34402 | A truth holds almost every... |
| aean 34403 | A conjunction holds almost... |
| faeval 34405 | Value of the 'almost every... |
| relfae 34406 | The 'almost everywhere' bu... |
| brfae 34407 | 'almost everywhere' relati... |
| ismbfm 34410 | The predicate " ` F ` is a... |
| elunirnmbfm 34411 | The property of being a me... |
| mbfmfun 34412 | A measurable function is a... |
| mbfmf 34413 | A measurable function as a... |
| mbfmcnvima 34414 | The preimage by a measurab... |
| isanmbfm 34415 | The predicate to be a meas... |
| mbfmbfmOLD 34416 | A measurable function to a... |
| mbfmbfm 34417 | A measurable function to a... |
| mbfmcst 34418 | A constant function is mea... |
| 1stmbfm 34419 | The first projection map i... |
| 2ndmbfm 34420 | The second projection map ... |
| imambfm 34421 | If the sigma-algebra in th... |
| cnmbfm 34422 | A continuous function is m... |
| mbfmco 34423 | The composition of two mea... |
| mbfmco2 34424 | The pair building of two m... |
| mbfmvolf 34425 | Measurable functions with ... |
| elmbfmvol2 34426 | Measurable functions with ... |
| mbfmcnt 34427 | All functions are measurab... |
| br2base 34428 | The base set for the gener... |
| dya2ub 34429 | An upper bound for a dyadi... |
| sxbrsigalem0 34430 | The closed half-spaces of ... |
| sxbrsigalem3 34431 | The sigma-algebra generate... |
| dya2iocival 34432 | The function ` I ` returns... |
| dya2iocress 34433 | Dyadic intervals are subse... |
| dya2iocbrsiga 34434 | Dyadic intervals are Borel... |
| dya2icobrsiga 34435 | Dyadic intervals are Borel... |
| dya2icoseg 34436 | For any point and any clos... |
| dya2icoseg2 34437 | For any point and any open... |
| dya2iocrfn 34438 | The function returning dya... |
| dya2iocct 34439 | The dyadic rectangle set i... |
| dya2iocnrect 34440 | For any point of an open r... |
| dya2iocnei 34441 | For any point of an open s... |
| dya2iocuni 34442 | Every open set of ` ( RR X... |
| dya2iocucvr 34443 | The dyadic rectangular set... |
| sxbrsigalem1 34444 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34445 | The sigma-algebra generate... |
| sxbrsigalem4 34446 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34447 | First direction for ~ sxbr... |
| sxbrsigalem6 34448 | First direction for ~ sxbr... |
| sxbrsiga 34449 | The product sigma-algebra ... |
| omsval 34452 | Value of the function mapp... |
| omsfval 34453 | Value of the outer measure... |
| omscl 34454 | A closure lemma for the co... |
| omsf 34455 | A constructed outer measur... |
| oms0 34456 | A constructed outer measur... |
| omsmon 34457 | A constructed outer measur... |
| omssubaddlem 34458 | For any small margin ` E `... |
| omssubadd 34459 | A constructed outer measur... |
| carsgval 34462 | Value of the Caratheodory ... |
| carsgcl 34463 | Closure of the Caratheodor... |
| elcarsg 34464 | Property of being a Carath... |
| baselcarsg 34465 | The universe set, ` O ` , ... |
| 0elcarsg 34466 | The empty set is Caratheod... |
| carsguni 34467 | The union of all Caratheod... |
| elcarsgss 34468 | Caratheodory measurable se... |
| difelcarsg 34469 | The Caratheodory measurabl... |
| inelcarsg 34470 | The Caratheodory measurabl... |
| unelcarsg 34471 | The Caratheodory-measurabl... |
| difelcarsg2 34472 | The Caratheodory-measurabl... |
| carsgmon 34473 | Utility lemma: Apply mono... |
| carsgsigalem 34474 | Lemma for the following th... |
| fiunelcarsg 34475 | The Caratheodory measurabl... |
| carsgclctunlem1 34476 | Lemma for ~ carsgclctun . ... |
| carsggect 34477 | The outer measure is count... |
| carsgclctunlem2 34478 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34479 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34480 | The Caratheodory measurabl... |
| carsgsiga 34481 | The Caratheodory measurabl... |
| omsmeas 34482 | The restriction of a const... |
| pmeasmono 34483 | This theorem's hypotheses ... |
| pmeasadd 34484 | A premeasure on a ring of ... |
| itgeq12dv 34485 | Equality theorem for an in... |
| sitgval 34491 | Value of the simple functi... |
| issibf 34492 | The predicate " ` F ` is a... |
| sibf0 34493 | The constant zero function... |
| sibfmbl 34494 | A simple function is measu... |
| sibff 34495 | A simple function is a fun... |
| sibfrn 34496 | A simple function has fini... |
| sibfima 34497 | Any preimage of a singleto... |
| sibfinima 34498 | The measure of the interse... |
| sibfof 34499 | Applying function operatio... |
| sitgfval 34500 | Value of the Bochner integ... |
| sitgclg 34501 | Closure of the Bochner int... |
| sitgclbn 34502 | Closure of the Bochner int... |
| sitgclcn 34503 | Closure of the Bochner int... |
| sitgclre 34504 | Closure of the Bochner int... |
| sitg0 34505 | The integral of the consta... |
| sitgf 34506 | The integral for simple fu... |
| sitgaddlemb 34507 | Lemma for * sitgadd . (Co... |
| sitmval 34508 | Value of the simple functi... |
| sitmfval 34509 | Value of the integral dist... |
| sitmcl 34510 | Closure of the integral di... |
| sitmf 34511 | The integral metric as a f... |
| oddpwdc 34513 | Lemma for ~ eulerpart . T... |
| oddpwdcv 34514 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 34515 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 34516 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 34517 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 34518 | Lemma for ~ eulerpart . (... |
| eulerpartlems 34519 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 34520 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 34521 | Lemma for ~ eulerpart . (... |
| eulerpartleme 34522 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 34523 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 34524 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 34525 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 34526 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 34527 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 34528 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 34529 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 34530 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 34531 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 34532 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 34533 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 34534 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 34535 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 34536 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 34537 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 34538 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 34539 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 34540 | Lemma for ~ eulerpart . (... |
| eulerpart 34541 | Euler's theorem on partiti... |
| subiwrd 34544 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 34545 | Length of a subword of an ... |
| iwrdsplit 34546 | Lemma for ~ sseqp1 . (Con... |
| sseqval 34547 | Value of the strong sequen... |
| sseqfv1 34548 | Value of the strong sequen... |
| sseqfn 34549 | A strong recursive sequenc... |
| sseqmw 34550 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 34551 | A strong recursive sequenc... |
| sseqfres 34552 | The first elements in the ... |
| sseqfv2 34553 | Value of the strong sequen... |
| sseqp1 34554 | Value of the strong sequen... |
| fiblem 34557 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 34558 | Value of the Fibonacci seq... |
| fib1 34559 | Value of the Fibonacci seq... |
| fibp1 34560 | Value of the Fibonacci seq... |
| fib2 34561 | Value of the Fibonacci seq... |
| fib3 34562 | Value of the Fibonacci seq... |
| fib4 34563 | Value of the Fibonacci seq... |
| fib5 34564 | Value of the Fibonacci seq... |
| fib6 34565 | Value of the Fibonacci seq... |
| elprob 34568 | The property of being a pr... |
| domprobmeas 34569 | A probability measure is a... |
| domprobsiga 34570 | The domain of a probabilit... |
| probtot 34571 | The probability of the uni... |
| prob01 34572 | A probability is an elemen... |
| probnul 34573 | The probability of the emp... |
| unveldomd 34574 | The universe is an element... |
| unveldom 34575 | The universe is an element... |
| nuleldmp 34576 | The empty set is an elemen... |
| probcun 34577 | The probability of the uni... |
| probun 34578 | The probability of the uni... |
| probdif 34579 | The probability of the dif... |
| probinc 34580 | A probability law is incre... |
| probdsb 34581 | The probability of the com... |
| probmeasd 34582 | A probability measure is a... |
| probvalrnd 34583 | The value of a probability... |
| probtotrnd 34584 | The probability of the uni... |
| totprobd 34585 | Law of total probability, ... |
| totprob 34586 | Law of total probability. ... |
| probfinmeasb 34587 | Build a probability measur... |
| probfinmeasbALTV 34588 | Alternate version of ~ pro... |
| probmeasb 34589 | Build a probability from a... |
| cndprobval 34592 | The value of the condition... |
| cndprobin 34593 | An identity linking condit... |
| cndprob01 34594 | The conditional probabilit... |
| cndprobtot 34595 | The conditional probabilit... |
| cndprobnul 34596 | The conditional probabilit... |
| cndprobprob 34597 | The conditional probabilit... |
| bayesth 34598 | Bayes Theorem. (Contribut... |
| rrvmbfm 34601 | A real-valued random varia... |
| isrrvv 34602 | Elementhood to the set of ... |
| rrvvf 34603 | A real-valued random varia... |
| rrvfn 34604 | A real-valued random varia... |
| rrvdm 34605 | The domain of a random var... |
| rrvrnss 34606 | The range of a random vari... |
| rrvf2 34607 | A real-valued random varia... |
| rrvdmss 34608 | The domain of a random var... |
| rrvfinvima 34609 | For a real-value random va... |
| 0rrv 34610 | The constant function equa... |
| rrvadd 34611 | The sum of two random vari... |
| rrvmulc 34612 | A random variable multipli... |
| rrvsum 34613 | An indexed sum of random v... |
| boolesineq 34614 | Boole's inequality (union ... |
| orvcval 34617 | Value of the preimage mapp... |
| orvcval2 34618 | Another way to express the... |
| elorvc 34619 | Elementhood of a preimage.... |
| orvcval4 34620 | The value of the preimage ... |
| orvcoel 34621 | If the relation produces o... |
| orvccel 34622 | If the relation produces c... |
| elorrvc 34623 | Elementhood of a preimage ... |
| orrvcval4 34624 | The value of the preimage ... |
| orrvcoel 34625 | If the relation produces o... |
| orrvccel 34626 | If the relation produces c... |
| orvcgteel 34627 | Preimage maps produced by ... |
| orvcelval 34628 | Preimage maps produced by ... |
| orvcelel 34629 | Preimage maps produced by ... |
| dstrvval 34630 | The value of the distribut... |
| dstrvprob 34631 | The distribution of a rand... |
| orvclteel 34632 | Preimage maps produced by ... |
| dstfrvel 34633 | Elementhood of preimage ma... |
| dstfrvunirn 34634 | The limit of all preimage ... |
| orvclteinc 34635 | Preimage maps produced by ... |
| dstfrvinc 34636 | A cumulative distribution ... |
| dstfrvclim1 34637 | The limit of the cumulativ... |
| coinfliplem 34638 | Division in the extended r... |
| coinflipprob 34639 | The ` P ` we defined for c... |
| coinflipspace 34640 | The space of our coin-flip... |
| coinflipuniv 34641 | The universe of our coin-f... |
| coinfliprv 34642 | The ` X ` we defined for c... |
| coinflippv 34643 | The probability of heads i... |
| coinflippvt 34644 | The probability of tails i... |
| ballotlemoex 34645 | ` O ` is a set. (Contribu... |
| ballotlem1 34646 | The size of the universe i... |
| ballotlemelo 34647 | Elementhood in ` O ` . (C... |
| ballotlem2 34648 | The probability that the f... |
| ballotlemfval 34649 | The value of ` F ` . (Con... |
| ballotlemfelz 34650 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 34651 | If the ` J ` th ballot is ... |
| ballotlemfc0 34652 | ` F ` takes value 0 betwee... |
| ballotlemfcc 34653 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 34654 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 34655 | ` ( F `` C ) ` always star... |
| ballotleme 34656 | Elements of ` E ` . (Cont... |
| ballotlemodife 34657 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 34658 | If the first pick is a vot... |
| ballotlem5 34659 | If A is not ahead througho... |
| ballotlemi 34660 | Value of ` I ` for a given... |
| ballotlemiex 34661 | Properties of ` ( I `` C )... |
| ballotlemi1 34662 | The first tie cannot be re... |
| ballotlemii 34663 | The first tie cannot be re... |
| ballotlemsup 34664 | The set of zeroes of ` F `... |
| ballotlemimin 34665 | ` ( I `` C ) ` is the firs... |
| ballotlemic 34666 | If the first vote is for B... |
| ballotlem1c 34667 | If the first vote is for A... |
| ballotlemsval 34668 | Value of ` S ` . (Contrib... |
| ballotlemsv 34669 | Value of ` S ` evaluated a... |
| ballotlemsgt1 34670 | ` S ` maps values less tha... |
| ballotlemsdom 34671 | Domain of ` S ` for a give... |
| ballotlemsel1i 34672 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 34673 | The defined ` S ` is a bij... |
| ballotlemsi 34674 | The image by ` S ` of the ... |
| ballotlemsima 34675 | The image by ` S ` of an i... |
| ballotlemieq 34676 | If two countings share the... |
| ballotlemrval 34677 | Value of ` R ` . (Contrib... |
| ballotlemscr 34678 | The image of ` ( R `` C ) ... |
| ballotlemrv 34679 | Value of ` R ` evaluated a... |
| ballotlemrv1 34680 | Value of ` R ` before the ... |
| ballotlemrv2 34681 | Value of ` R ` after the t... |
| ballotlemro 34682 | Range of ` R ` is included... |
| ballotlemgval 34683 | Expand the value of ` .^ `... |
| ballotlemgun 34684 | A property of the defined ... |
| ballotlemfg 34685 | Express the value of ` ( F... |
| ballotlemfrc 34686 | Express the value of ` ( F... |
| ballotlemfrci 34687 | Reverse counting preserves... |
| ballotlemfrceq 34688 | Value of ` F ` for a rever... |
| ballotlemfrcn0 34689 | Value of ` F ` for a rever... |
| ballotlemrc 34690 | Range of ` R ` . (Contrib... |
| ballotlemirc 34691 | Applying ` R ` does not ch... |
| ballotlemrinv0 34692 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 34693 | ` R ` is its own inverse :... |
| ballotlem1ri 34694 | When the vote on the first... |
| ballotlem7 34695 | ` R ` is a bijection betwe... |
| ballotlem8 34696 | There are as many counting... |
| ballotth 34697 | Bertrand's ballot problem ... |
| fzssfzo 34698 | Condition for an integer i... |
| gsumncl 34699 | Closure of a group sum in ... |
| gsumnunsn 34700 | Closure of a group sum in ... |
| ccatmulgnn0dir 34701 | Concatenation of words fol... |
| ofcccat 34702 | Letterwise operations on w... |
| ofcs1 34703 | Letterwise operations on a... |
| ofcs2 34704 | Letterwise operations on a... |
| plymul02 34705 | Product of a polynomial wi... |
| plymulx0 34706 | Coefficients of a polynomi... |
| plymulx 34707 | Coefficients of a polynomi... |
| plyrecld 34708 | Closure of a polynomial wi... |
| signsplypnf 34709 | The quotient of a polynomi... |
| signsply0 34710 | Lemma for the rule of sign... |
| signspval 34711 | The value of the skipping ... |
| signsw0glem 34712 | Neutral element property o... |
| signswbase 34713 | The base of ` W ` is the u... |
| signswplusg 34714 | The operation of ` W ` . ... |
| signsw0g 34715 | The neutral element of ` W... |
| signswmnd 34716 | ` W ` is a monoid structur... |
| signswrid 34717 | The zero-skipping operatio... |
| signswlid 34718 | The zero-skipping operatio... |
| signswn0 34719 | The zero-skipping operatio... |
| signswch 34720 | The zero-skipping operatio... |
| signslema 34721 | Computational part of ~~? ... |
| signstfv 34722 | Value of the zero-skipping... |
| signstfval 34723 | Value of the zero-skipping... |
| signstcl 34724 | Closure of the zero skippi... |
| signstf 34725 | The zero skipping sign wor... |
| signstlen 34726 | Length of the zero skippin... |
| signstf0 34727 | Sign of a single letter wo... |
| signstfvn 34728 | Zero-skipping sign in a wo... |
| signsvtn0 34729 | If the last letter is nonz... |
| signstfvp 34730 | Zero-skipping sign in a wo... |
| signstfvneq0 34731 | In case the first letter i... |
| signstfvcl 34732 | Closure of the zero skippi... |
| signstfvc 34733 | Zero-skipping sign in a wo... |
| signstres 34734 | Restriction of a zero skip... |
| signstfveq0a 34735 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 34736 | In case the last letter is... |
| signsvvfval 34737 | The value of ` V ` , which... |
| signsvvf 34738 | ` V ` is a function. (Con... |
| signsvf0 34739 | There is no change of sign... |
| signsvf1 34740 | In a single-letter word, w... |
| signsvfn 34741 | Number of changes in a wor... |
| signsvtp 34742 | Adding a letter of the sam... |
| signsvtn 34743 | Adding a letter of a diffe... |
| signsvfpn 34744 | Adding a letter of the sam... |
| signsvfnn 34745 | Adding a letter of a diffe... |
| signlem0 34746 | Adding a zero as the highe... |
| signshf 34747 | ` H ` , corresponding to t... |
| signshwrd 34748 | ` H ` , corresponding to t... |
| signshlen 34749 | Length of ` H ` , correspo... |
| signshnz 34750 | ` H ` is not the empty wor... |
| iblidicc 34751 | The identity function is i... |
| rpsqrtcn 34752 | Continuity of the real pos... |
| divsqrtid 34753 | A real number divided by i... |
| cxpcncf1 34754 | The power function on comp... |
| efmul2picn 34755 | Multiplying by ` ( _i x. (... |
| fct2relem 34756 | Lemma for ~ ftc2re . (Con... |
| ftc2re 34757 | The Fundamental Theorem of... |
| fdvposlt 34758 | Functions with a positive ... |
| fdvneggt 34759 | Functions with a negative ... |
| fdvposle 34760 | Functions with a nonnegati... |
| fdvnegge 34761 | Functions with a nonpositi... |
| prodfzo03 34762 | A product of three factors... |
| actfunsnf1o 34763 | The action ` F ` of extend... |
| actfunsnrndisj 34764 | The action ` F ` of extend... |
| itgexpif 34765 | The basis for the circle m... |
| fsum2dsub 34766 | Lemma for ~ breprexp - Re-... |
| reprval 34769 | Value of the representatio... |
| repr0 34770 | There is exactly one repre... |
| reprf 34771 | Members of the representat... |
| reprsum 34772 | Sums of values of the memb... |
| reprle 34773 | Upper bound to the terms i... |
| reprsuc 34774 | Express the representation... |
| reprfi 34775 | Bounded representations ar... |
| reprss 34776 | Representations with terms... |
| reprinrn 34777 | Representations with term ... |
| reprlt 34778 | There are no representatio... |
| hashreprin 34779 | Express a sum of represent... |
| reprgt 34780 | There are no representatio... |
| reprinfz1 34781 | For the representation of ... |
| reprfi2 34782 | Corollary of ~ reprinfz1 .... |
| reprfz1 34783 | Corollary of ~ reprinfz1 .... |
| hashrepr 34784 | Develop the number of repr... |
| reprpmtf1o 34785 | Transposing ` 0 ` and ` X ... |
| reprdifc 34786 | Express the representation... |
| chpvalz 34787 | Value of the second Chebys... |
| chtvalz 34788 | Value of the Chebyshev fun... |
| breprexplema 34789 | Lemma for ~ breprexp (indu... |
| breprexplemb 34790 | Lemma for ~ breprexp (clos... |
| breprexplemc 34791 | Lemma for ~ breprexp (indu... |
| breprexp 34792 | Express the ` S ` th power... |
| breprexpnat 34793 | Express the ` S ` th power... |
| vtsval 34796 | Value of the Vinogradov tr... |
| vtscl 34797 | Closure of the Vinogradov ... |
| vtsprod 34798 | Express the Vinogradov tri... |
| circlemeth 34799 | The Hardy, Littlewood and ... |
| circlemethnat 34800 | The Hardy, Littlewood and ... |
| circlevma 34801 | The Circle Method, where t... |
| circlemethhgt 34802 | The circle method, where t... |
| hgt750lemc 34806 | An upper bound to the summ... |
| hgt750lemd 34807 | An upper bound to the summ... |
| hgt749d 34808 | A deduction version of ~ a... |
| logdivsqrle 34809 | Conditions for ` ( ( log `... |
| hgt750lem 34810 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 34811 | Decimal multiplication gal... |
| hgt750lemf 34812 | Lemma for the statement 7.... |
| hgt750lemg 34813 | Lemma for the statement 7.... |
| oddprm2 34814 | Two ways to write the set ... |
| hgt750lemb 34815 | An upper bound on the cont... |
| hgt750lema 34816 | An upper bound on the cont... |
| hgt750leme 34817 | An upper bound on the cont... |
| tgoldbachgnn 34818 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 34819 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 34820 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 34821 | Odd integers greater than ... |
| tgoldbachgt 34822 | Odd integers greater than ... |
| istrkg2d 34825 | Property of fulfilling dim... |
| axtglowdim2ALTV 34826 | Alternate version of ~ axt... |
| axtgupdim2ALTV 34827 | Alternate version of ~ axt... |
| afsval 34830 | Value of the AFS relation ... |
| brafs 34831 | Binary relation form of th... |
| tg5segofs 34832 | Rephrase ~ axtg5seg using ... |
| lpadval 34835 | Value of the ` leftpad ` f... |
| lpadlem1 34836 | Lemma for the ` leftpad ` ... |
| lpadlem3 34837 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 34838 | Length of a left-padded wo... |
| lpadlem2 34839 | Lemma for the ` leftpad ` ... |
| lpadlen2 34840 | Length of a left-padded wo... |
| lpadmax 34841 | Length of a left-padded wo... |
| lpadleft 34842 | The contents of prefix of ... |
| lpadright 34843 | The suffix of a left-padde... |
| bnj170 34856 | ` /\ ` -manipulation. (Co... |
| bnj240 34857 | ` /\ ` -manipulation. (Co... |
| bnj248 34858 | ` /\ ` -manipulation. (Co... |
| bnj250 34859 | ` /\ ` -manipulation. (Co... |
| bnj251 34860 | ` /\ ` -manipulation. (Co... |
| bnj252 34861 | ` /\ ` -manipulation. (Co... |
| bnj253 34862 | ` /\ ` -manipulation. (Co... |
| bnj255 34863 | ` /\ ` -manipulation. (Co... |
| bnj256 34864 | ` /\ ` -manipulation. (Co... |
| bnj257 34865 | ` /\ ` -manipulation. (Co... |
| bnj258 34866 | ` /\ ` -manipulation. (Co... |
| bnj268 34867 | ` /\ ` -manipulation. (Co... |
| bnj290 34868 | ` /\ ` -manipulation. (Co... |
| bnj291 34869 | ` /\ ` -manipulation. (Co... |
| bnj312 34870 | ` /\ ` -manipulation. (Co... |
| bnj334 34871 | ` /\ ` -manipulation. (Co... |
| bnj345 34872 | ` /\ ` -manipulation. (Co... |
| bnj422 34873 | ` /\ ` -manipulation. (Co... |
| bnj432 34874 | ` /\ ` -manipulation. (Co... |
| bnj446 34875 | ` /\ ` -manipulation. (Co... |
| bnj23 34876 | First-order logic and set ... |
| bnj31 34877 | First-order logic and set ... |
| bnj62 34878 | First-order logic and set ... |
| bnj89 34879 | First-order logic and set ... |
| bnj90 34880 | First-order logic and set ... |
| bnj101 34881 | First-order logic and set ... |
| bnj105 34882 | First-order logic and set ... |
| bnj115 34883 | First-order logic and set ... |
| bnj132 34884 | First-order logic and set ... |
| bnj133 34885 | First-order logic and set ... |
| bnj156 34886 | First-order logic and set ... |
| bnj158 34887 | First-order logic and set ... |
| bnj168 34888 | First-order logic and set ... |
| bnj206 34889 | First-order logic and set ... |
| bnj216 34890 | First-order logic and set ... |
| bnj219 34891 | First-order logic and set ... |
| bnj226 34892 | First-order logic and set ... |
| bnj228 34893 | First-order logic and set ... |
| bnj519 34894 | First-order logic and set ... |
| bnj524 34895 | First-order logic and set ... |
| bnj525 34896 | First-order logic and set ... |
| bnj534 34897 | First-order logic and set ... |
| bnj538 34898 | First-order logic and set ... |
| bnj529 34899 | First-order logic and set ... |
| bnj551 34900 | First-order logic and set ... |
| bnj563 34901 | First-order logic and set ... |
| bnj564 34902 | First-order logic and set ... |
| bnj593 34903 | First-order logic and set ... |
| bnj596 34904 | First-order logic and set ... |
| bnj610 34905 | Pass from equality ( ` x =... |
| bnj642 34906 | ` /\ ` -manipulation. (Co... |
| bnj643 34907 | ` /\ ` -manipulation. (Co... |
| bnj645 34908 | ` /\ ` -manipulation. (Co... |
| bnj658 34909 | ` /\ ` -manipulation. (Co... |
| bnj667 34910 | ` /\ ` -manipulation. (Co... |
| bnj705 34911 | ` /\ ` -manipulation. (Co... |
| bnj706 34912 | ` /\ ` -manipulation. (Co... |
| bnj707 34913 | ` /\ ` -manipulation. (Co... |
| bnj708 34914 | ` /\ ` -manipulation. (Co... |
| bnj721 34915 | ` /\ ` -manipulation. (Co... |
| bnj832 34916 | ` /\ ` -manipulation. (Co... |
| bnj835 34917 | ` /\ ` -manipulation. (Co... |
| bnj836 34918 | ` /\ ` -manipulation. (Co... |
| bnj837 34919 | ` /\ ` -manipulation. (Co... |
| bnj769 34920 | ` /\ ` -manipulation. (Co... |
| bnj770 34921 | ` /\ ` -manipulation. (Co... |
| bnj771 34922 | ` /\ ` -manipulation. (Co... |
| bnj887 34923 | ` /\ ` -manipulation. (Co... |
| bnj918 34924 | First-order logic and set ... |
| bnj919 34925 | First-order logic and set ... |
| bnj923 34926 | First-order logic and set ... |
| bnj927 34927 | First-order logic and set ... |
| bnj931 34928 | First-order logic and set ... |
| bnj937 34929 | First-order logic and set ... |
| bnj941 34930 | First-order logic and set ... |
| bnj945 34931 | Technical lemma for ~ bnj6... |
| bnj946 34932 | First-order logic and set ... |
| bnj951 34933 | ` /\ ` -manipulation. (Co... |
| bnj956 34934 | First-order logic and set ... |
| bnj976 34935 | First-order logic and set ... |
| bnj982 34936 | First-order logic and set ... |
| bnj1019 34937 | First-order logic and set ... |
| bnj1023 34938 | First-order logic and set ... |
| bnj1095 34939 | First-order logic and set ... |
| bnj1096 34940 | First-order logic and set ... |
| bnj1098 34941 | First-order logic and set ... |
| bnj1101 34942 | First-order logic and set ... |
| bnj1113 34943 | First-order logic and set ... |
| bnj1109 34944 | First-order logic and set ... |
| bnj1131 34945 | First-order logic and set ... |
| bnj1138 34946 | First-order logic and set ... |
| bnj1142 34947 | First-order logic and set ... |
| bnj1143 34948 | First-order logic and set ... |
| bnj1146 34949 | First-order logic and set ... |
| bnj1149 34950 | First-order logic and set ... |
| bnj1185 34951 | First-order logic and set ... |
| bnj1196 34952 | First-order logic and set ... |
| bnj1198 34953 | First-order logic and set ... |
| bnj1209 34954 | First-order logic and set ... |
| bnj1211 34955 | First-order logic and set ... |
| bnj1213 34956 | First-order logic and set ... |
| bnj1212 34957 | First-order logic and set ... |
| bnj1219 34958 | First-order logic and set ... |
| bnj1224 34959 | First-order logic and set ... |
| bnj1230 34960 | First-order logic and set ... |
| bnj1232 34961 | First-order logic and set ... |
| bnj1235 34962 | First-order logic and set ... |
| bnj1239 34963 | First-order logic and set ... |
| bnj1238 34964 | First-order logic and set ... |
| bnj1241 34965 | First-order logic and set ... |
| bnj1247 34966 | First-order logic and set ... |
| bnj1254 34967 | First-order logic and set ... |
| bnj1262 34968 | First-order logic and set ... |
| bnj1266 34969 | First-order logic and set ... |
| bnj1265 34970 | First-order logic and set ... |
| bnj1275 34971 | First-order logic and set ... |
| bnj1276 34972 | First-order logic and set ... |
| bnj1292 34973 | First-order logic and set ... |
| bnj1293 34974 | First-order logic and set ... |
| bnj1294 34975 | First-order logic and set ... |
| bnj1299 34976 | First-order logic and set ... |
| bnj1304 34977 | First-order logic and set ... |
| bnj1316 34978 | First-order logic and set ... |
| bnj1317 34979 | First-order logic and set ... |
| bnj1322 34980 | First-order logic and set ... |
| bnj1340 34981 | First-order logic and set ... |
| bnj1345 34982 | First-order logic and set ... |
| bnj1350 34983 | First-order logic and set ... |
| bnj1351 34984 | First-order logic and set ... |
| bnj1352 34985 | First-order logic and set ... |
| bnj1361 34986 | First-order logic and set ... |
| bnj1366 34987 | First-order logic and set ... |
| bnj1379 34988 | First-order logic and set ... |
| bnj1383 34989 | First-order logic and set ... |
| bnj1385 34990 | First-order logic and set ... |
| bnj1386 34991 | First-order logic and set ... |
| bnj1397 34992 | First-order logic and set ... |
| bnj1400 34993 | First-order logic and set ... |
| bnj1405 34994 | First-order logic and set ... |
| bnj1422 34995 | First-order logic and set ... |
| bnj1424 34996 | First-order logic and set ... |
| bnj1436 34997 | First-order logic and set ... |
| bnj1441 34998 | First-order logic and set ... |
| bnj1441g 34999 | First-order logic and set ... |
| bnj1454 35000 | First-order logic and set ... |
| bnj1459 35001 | First-order logic and set ... |
| bnj1464 35002 | Conversion of implicit sub... |
| bnj1465 35003 | First-order logic and set ... |
| bnj1468 35004 | Conversion of implicit sub... |
| bnj1476 35005 | First-order logic and set ... |
| bnj1502 35006 | First-order logic and set ... |
| bnj1503 35007 | First-order logic and set ... |
| bnj1517 35008 | First-order logic and set ... |
| bnj1521 35009 | First-order logic and set ... |
| bnj1533 35010 | First-order logic and set ... |
| bnj1534 35011 | First-order logic and set ... |
| bnj1536 35012 | First-order logic and set ... |
| bnj1538 35013 | First-order logic and set ... |
| bnj1541 35014 | First-order logic and set ... |
| bnj1542 35015 | First-order logic and set ... |
| bnj110 35016 | Well-founded induction res... |
| bnj157 35017 | Well-founded induction res... |
| bnj66 35018 | Technical lemma for ~ bnj6... |
| bnj91 35019 | First-order logic and set ... |
| bnj92 35020 | First-order logic and set ... |
| bnj93 35021 | Technical lemma for ~ bnj9... |
| bnj95 35022 | Technical lemma for ~ bnj1... |
| bnj96 35023 | Technical lemma for ~ bnj1... |
| bnj97 35024 | Technical lemma for ~ bnj1... |
| bnj98 35025 | Technical lemma for ~ bnj1... |
| bnj106 35026 | First-order logic and set ... |
| bnj118 35027 | First-order logic and set ... |
| bnj121 35028 | First-order logic and set ... |
| bnj124 35029 | Technical lemma for ~ bnj1... |
| bnj125 35030 | Technical lemma for ~ bnj1... |
| bnj126 35031 | Technical lemma for ~ bnj1... |
| bnj130 35032 | Technical lemma for ~ bnj1... |
| bnj149 35033 | Technical lemma for ~ bnj1... |
| bnj150 35034 | Technical lemma for ~ bnj1... |
| bnj151 35035 | Technical lemma for ~ bnj1... |
| bnj154 35036 | Technical lemma for ~ bnj1... |
| bnj155 35037 | Technical lemma for ~ bnj1... |
| bnj153 35038 | Technical lemma for ~ bnj8... |
| bnj207 35039 | Technical lemma for ~ bnj8... |
| bnj213 35040 | First-order logic and set ... |
| bnj222 35041 | Technical lemma for ~ bnj2... |
| bnj229 35042 | Technical lemma for ~ bnj5... |
| bnj517 35043 | Technical lemma for ~ bnj5... |
| bnj518 35044 | Technical lemma for ~ bnj8... |
| bnj523 35045 | Technical lemma for ~ bnj8... |
| bnj526 35046 | Technical lemma for ~ bnj8... |
| bnj528 35047 | Technical lemma for ~ bnj8... |
| bnj535 35048 | Technical lemma for ~ bnj8... |
| bnj539 35049 | Technical lemma for ~ bnj8... |
| bnj540 35050 | Technical lemma for ~ bnj8... |
| bnj543 35051 | Technical lemma for ~ bnj8... |
| bnj544 35052 | Technical lemma for ~ bnj8... |
| bnj545 35053 | Technical lemma for ~ bnj8... |
| bnj546 35054 | Technical lemma for ~ bnj8... |
| bnj548 35055 | Technical lemma for ~ bnj8... |
| bnj553 35056 | Technical lemma for ~ bnj8... |
| bnj554 35057 | Technical lemma for ~ bnj8... |
| bnj556 35058 | Technical lemma for ~ bnj8... |
| bnj557 35059 | Technical lemma for ~ bnj8... |
| bnj558 35060 | Technical lemma for ~ bnj8... |
| bnj561 35061 | Technical lemma for ~ bnj8... |
| bnj562 35062 | Technical lemma for ~ bnj8... |
| bnj570 35063 | Technical lemma for ~ bnj8... |
| bnj571 35064 | Technical lemma for ~ bnj8... |
| bnj605 35065 | Technical lemma. This lem... |
| bnj581 35066 | Technical lemma for ~ bnj5... |
| bnj589 35067 | Technical lemma for ~ bnj8... |
| bnj590 35068 | Technical lemma for ~ bnj8... |
| bnj591 35069 | Technical lemma for ~ bnj8... |
| bnj594 35070 | Technical lemma for ~ bnj8... |
| bnj580 35071 | Technical lemma for ~ bnj5... |
| bnj579 35072 | Technical lemma for ~ bnj8... |
| bnj602 35073 | Equality theorem for the `... |
| bnj607 35074 | Technical lemma for ~ bnj8... |
| bnj609 35075 | Technical lemma for ~ bnj8... |
| bnj611 35076 | Technical lemma for ~ bnj8... |
| bnj600 35077 | Technical lemma for ~ bnj8... |
| bnj601 35078 | Technical lemma for ~ bnj8... |
| bnj852 35079 | Technical lemma for ~ bnj6... |
| bnj864 35080 | Technical lemma for ~ bnj6... |
| bnj865 35081 | Technical lemma for ~ bnj6... |
| bnj873 35082 | Technical lemma for ~ bnj6... |
| bnj849 35083 | Technical lemma for ~ bnj6... |
| bnj882 35084 | Definition (using hypothes... |
| bnj18eq1 35085 | Equality theorem for trans... |
| bnj893 35086 | Property of ` _trCl ` . U... |
| bnj900 35087 | Technical lemma for ~ bnj6... |
| bnj906 35088 | Property of ` _trCl ` . (... |
| bnj908 35089 | Technical lemma for ~ bnj6... |
| bnj911 35090 | Technical lemma for ~ bnj6... |
| bnj916 35091 | Technical lemma for ~ bnj6... |
| bnj917 35092 | Technical lemma for ~ bnj6... |
| bnj934 35093 | Technical lemma for ~ bnj6... |
| bnj929 35094 | Technical lemma for ~ bnj6... |
| bnj938 35095 | Technical lemma for ~ bnj6... |
| bnj944 35096 | Technical lemma for ~ bnj6... |
| bnj953 35097 | Technical lemma for ~ bnj6... |
| bnj958 35098 | Technical lemma for ~ bnj6... |
| bnj1000 35099 | Technical lemma for ~ bnj8... |
| bnj965 35100 | Technical lemma for ~ bnj8... |
| bnj964 35101 | Technical lemma for ~ bnj6... |
| bnj966 35102 | Technical lemma for ~ bnj6... |
| bnj967 35103 | Technical lemma for ~ bnj6... |
| bnj969 35104 | Technical lemma for ~ bnj6... |
| bnj970 35105 | Technical lemma for ~ bnj6... |
| bnj910 35106 | Technical lemma for ~ bnj6... |
| bnj978 35107 | Technical lemma for ~ bnj6... |
| bnj981 35108 | Technical lemma for ~ bnj6... |
| bnj983 35109 | Technical lemma for ~ bnj6... |
| bnj984 35110 | Technical lemma for ~ bnj6... |
| bnj985v 35111 | Version of ~ bnj985 with a... |
| bnj985 35112 | Technical lemma for ~ bnj6... |
| bnj986 35113 | Technical lemma for ~ bnj6... |
| bnj996 35114 | Technical lemma for ~ bnj6... |
| bnj998 35115 | Technical lemma for ~ bnj6... |
| bnj999 35116 | Technical lemma for ~ bnj6... |
| bnj1001 35117 | Technical lemma for ~ bnj6... |
| bnj1006 35118 | Technical lemma for ~ bnj6... |
| bnj1014 35119 | Technical lemma for ~ bnj6... |
| bnj1015 35120 | Technical lemma for ~ bnj6... |
| bnj1018g 35121 | Version of ~ bnj1018 with ... |
| bnj1018 35122 | Technical lemma for ~ bnj6... |
| bnj1020 35123 | Technical lemma for ~ bnj6... |
| bnj1021 35124 | Technical lemma for ~ bnj6... |
| bnj907 35125 | Technical lemma for ~ bnj6... |
| bnj1029 35126 | Property of ` _trCl ` . (... |
| bnj1033 35127 | Technical lemma for ~ bnj6... |
| bnj1034 35128 | Technical lemma for ~ bnj6... |
| bnj1039 35129 | Technical lemma for ~ bnj6... |
| bnj1040 35130 | Technical lemma for ~ bnj6... |
| bnj1047 35131 | Technical lemma for ~ bnj6... |
| bnj1049 35132 | Technical lemma for ~ bnj6... |
| bnj1052 35133 | Technical lemma for ~ bnj6... |
| bnj1053 35134 | Technical lemma for ~ bnj6... |
| bnj1071 35135 | Technical lemma for ~ bnj6... |
| bnj1083 35136 | Technical lemma for ~ bnj6... |
| bnj1090 35137 | Technical lemma for ~ bnj6... |
| bnj1093 35138 | Technical lemma for ~ bnj6... |
| bnj1097 35139 | Technical lemma for ~ bnj6... |
| bnj1110 35140 | Technical lemma for ~ bnj6... |
| bnj1112 35141 | Technical lemma for ~ bnj6... |
| bnj1118 35142 | Technical lemma for ~ bnj6... |
| bnj1121 35143 | Technical lemma for ~ bnj6... |
| bnj1123 35144 | Technical lemma for ~ bnj6... |
| bnj1030 35145 | Technical lemma for ~ bnj6... |
| bnj1124 35146 | Property of ` _trCl ` . (... |
| bnj1133 35147 | Technical lemma for ~ bnj6... |
| bnj1128 35148 | Technical lemma for ~ bnj6... |
| bnj1127 35149 | Property of ` _trCl ` . (... |
| bnj1125 35150 | Property of ` _trCl ` . (... |
| bnj1145 35151 | Technical lemma for ~ bnj6... |
| bnj1147 35152 | Property of ` _trCl ` . (... |
| bnj1137 35153 | Property of ` _trCl ` . (... |
| bnj1148 35154 | Property of ` _pred ` . (... |
| bnj1136 35155 | Technical lemma for ~ bnj6... |
| bnj1152 35156 | Technical lemma for ~ bnj6... |
| bnj1154 35157 | Property of ` Fr ` . (Con... |
| bnj1171 35158 | Technical lemma for ~ bnj6... |
| bnj1172 35159 | Technical lemma for ~ bnj6... |
| bnj1173 35160 | Technical lemma for ~ bnj6... |
| bnj1174 35161 | Technical lemma for ~ bnj6... |
| bnj1175 35162 | Technical lemma for ~ bnj6... |
| bnj1176 35163 | Technical lemma for ~ bnj6... |
| bnj1177 35164 | Technical lemma for ~ bnj6... |
| bnj1186 35165 | Technical lemma for ~ bnj6... |
| bnj1190 35166 | Technical lemma for ~ bnj6... |
| bnj1189 35167 | Technical lemma for ~ bnj6... |
| bnj69 35168 | Existence of a minimal ele... |
| bnj1228 35169 | Existence of a minimal ele... |
| bnj1204 35170 | Well-founded induction. T... |
| bnj1234 35171 | Technical lemma for ~ bnj6... |
| bnj1245 35172 | Technical lemma for ~ bnj6... |
| bnj1256 35173 | Technical lemma for ~ bnj6... |
| bnj1259 35174 | Technical lemma for ~ bnj6... |
| bnj1253 35175 | Technical lemma for ~ bnj6... |
| bnj1279 35176 | Technical lemma for ~ bnj6... |
| bnj1286 35177 | Technical lemma for ~ bnj6... |
| bnj1280 35178 | Technical lemma for ~ bnj6... |
| bnj1296 35179 | Technical lemma for ~ bnj6... |
| bnj1309 35180 | Technical lemma for ~ bnj6... |
| bnj1307 35181 | Technical lemma for ~ bnj6... |
| bnj1311 35182 | Technical lemma for ~ bnj6... |
| bnj1318 35183 | Technical lemma for ~ bnj6... |
| bnj1326 35184 | Technical lemma for ~ bnj6... |
| bnj1321 35185 | Technical lemma for ~ bnj6... |
| bnj1364 35186 | Property of ` _FrSe ` . (... |
| bnj1371 35187 | Technical lemma for ~ bnj6... |
| bnj1373 35188 | Technical lemma for ~ bnj6... |
| bnj1374 35189 | Technical lemma for ~ bnj6... |
| bnj1384 35190 | Technical lemma for ~ bnj6... |
| bnj1388 35191 | Technical lemma for ~ bnj6... |
| bnj1398 35192 | Technical lemma for ~ bnj6... |
| bnj1413 35193 | Property of ` _trCl ` . (... |
| bnj1408 35194 | Technical lemma for ~ bnj1... |
| bnj1414 35195 | Property of ` _trCl ` . (... |
| bnj1415 35196 | Technical lemma for ~ bnj6... |
| bnj1416 35197 | Technical lemma for ~ bnj6... |
| bnj1418 35198 | Property of ` _pred ` . (... |
| bnj1417 35199 | Technical lemma for ~ bnj6... |
| bnj1421 35200 | Technical lemma for ~ bnj6... |
| bnj1444 35201 | Technical lemma for ~ bnj6... |
| bnj1445 35202 | Technical lemma for ~ bnj6... |
| bnj1446 35203 | Technical lemma for ~ bnj6... |
| bnj1447 35204 | Technical lemma for ~ bnj6... |
| bnj1448 35205 | Technical lemma for ~ bnj6... |
| bnj1449 35206 | Technical lemma for ~ bnj6... |
| bnj1442 35207 | Technical lemma for ~ bnj6... |
| bnj1450 35208 | Technical lemma for ~ bnj6... |
| bnj1423 35209 | Technical lemma for ~ bnj6... |
| bnj1452 35210 | Technical lemma for ~ bnj6... |
| bnj1466 35211 | Technical lemma for ~ bnj6... |
| bnj1467 35212 | Technical lemma for ~ bnj6... |
| bnj1463 35213 | Technical lemma for ~ bnj6... |
| bnj1489 35214 | Technical lemma for ~ bnj6... |
| bnj1491 35215 | Technical lemma for ~ bnj6... |
| bnj1312 35216 | Technical lemma for ~ bnj6... |
| bnj1493 35217 | Technical lemma for ~ bnj6... |
| bnj1497 35218 | Technical lemma for ~ bnj6... |
| bnj1498 35219 | Technical lemma for ~ bnj6... |
| bnj60 35220 | Well-founded recursion, pa... |
| bnj1514 35221 | Technical lemma for ~ bnj1... |
| bnj1518 35222 | Technical lemma for ~ bnj1... |
| bnj1519 35223 | Technical lemma for ~ bnj1... |
| bnj1520 35224 | Technical lemma for ~ bnj1... |
| bnj1501 35225 | Technical lemma for ~ bnj1... |
| bnj1500 35226 | Well-founded recursion, pa... |
| bnj1525 35227 | Technical lemma for ~ bnj1... |
| bnj1529 35228 | Technical lemma for ~ bnj1... |
| bnj1523 35229 | Technical lemma for ~ bnj1... |
| bnj1522 35230 | Well-founded recursion, pa... |
| nfan1c 35231 | Variant of ~ nfan and comm... |
| cbvex1v 35232 | Rule used to change bound ... |
| dvelimalcased 35233 | Eliminate a disjoint varia... |
| dvelimalcasei 35234 | Eliminate a disjoint varia... |
| dvelimexcased 35235 | Eliminate a disjoint varia... |
| dvelimexcasei 35236 | Eliminate a disjoint varia... |
| exdifsn 35237 | There exists an element in... |
| srcmpltd 35238 | If a statement is true for... |
| prsrcmpltd 35239 | If a statement is true for... |
| axsepg2 35240 | A generalization of ~ ax-s... |
| axsepg2ALT 35241 | Alternate proof of ~ axsep... |
| dff15 35242 | A one-to-one function in t... |
| f1resveqaeq 35243 | If a function restricted t... |
| f1resrcmplf1dlem 35244 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 35245 | If a function's restrictio... |
| funen1cnv 35246 | If a function is equinumer... |
| xoromon 35247 | ` _om ` is either an ordin... |
| fissorduni 35248 | The union (supremum) of a ... |
| fnrelpredd 35249 | A function that preserves ... |
| cardpred 35250 | The cardinality function p... |
| nummin 35251 | Every nonempty class of nu... |
| r11 35252 | Value of the cumulative hi... |
| r12 35253 | Value of the cumulative hi... |
| r1wf 35254 | Each stage in the cumulati... |
| elwf 35255 | An element of a well-found... |
| r1elcl 35256 | Each set of the cumulative... |
| rankval2b 35257 | Value of an alternate defi... |
| rankval4b 35258 | The rank of a set is the s... |
| rankfilimbi 35259 | If all elements in a finit... |
| rankfilimb 35260 | The rank of a finite well-... |
| r1filimi 35261 | If all elements in a finit... |
| r1filim 35262 | A finite set appears in th... |
| r1omfi 35263 | Hereditarily finite sets a... |
| r1omhf 35264 | A set is hereditarily fini... |
| r1ssel 35265 | A set is a subset of the v... |
| axnulg 35266 | A generalization of ~ ax-n... |
| axnulALT2 35267 | Alternate proof of ~ axnul... |
| r1omfv 35268 | Value of the cumulative hi... |
| trssfir1om 35269 | If every element in a tran... |
| r1omhfb 35270 | The class of all hereditar... |
| prcinf 35271 | Any proper class is litera... |
| fineqvrep 35272 | If all sets are finite, th... |
| fineqvpow 35273 | If all sets are finite, th... |
| fineqvac 35274 | If all sets are finite, th... |
| fineqvacALT 35275 | Shorter proof of ~ fineqva... |
| fineqvomon 35276 | If all sets are finite, th... |
| fineqvomonb 35277 | All sets are finite iff al... |
| omprcomonb 35278 | The class of all finite or... |
| fineqvnttrclselem1 35279 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem2 35280 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem3 35281 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclse 35282 | A counterexample demonstra... |
| fineqvinfep 35283 | A counterexample demonstra... |
| axreg 35285 | Derivation of ~ ax-reg fro... |
| axregscl 35286 | A version of ~ ax-regs wit... |
| axregszf 35287 | Derivation of ~ zfregs usi... |
| setindregs 35288 | Set (epsilon) induction. ... |
| setinds2regs 35289 | Principle of set induction... |
| noinfepfnregs 35290 | There are no infinite desc... |
| noinfepregs 35291 | There are no infinite desc... |
| tz9.1regs 35292 | Every set has a transitive... |
| unir1regs 35293 | The cumulative hierarchy o... |
| trssfir1omregs 35294 | If every element in a tran... |
| r1omhfbregs 35295 | The class of all hereditar... |
| fineqvr1ombregs 35296 | All sets are finite iff al... |
| axregs 35297 | Derivation of ~ ax-regs fr... |
| gblacfnacd 35298 | If ` G ` is a global choic... |
| onvf1odlem1 35299 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem2 35300 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem3 35301 | Lemma for ~ onvf1od . The... |
| onvf1odlem4 35302 | Lemma for ~ onvf1od . If ... |
| onvf1od 35303 | If ` G ` is a global choic... |
| vonf1owev 35304 | If ` F ` is a bijection fr... |
| wevgblacfn 35305 | If ` R ` is a well-orderin... |
| zltp1ne 35306 | Integer ordering relation.... |
| nnltp1ne 35307 | Positive integer ordering ... |
| nn0ltp1ne 35308 | Nonnegative integer orderi... |
| 0nn0m1nnn0 35309 | A number is zero if and on... |
| f1resfz0f1d 35310 | If a function with a seque... |
| fisshasheq 35311 | A finite set is equal to i... |
| revpfxsfxrev 35312 | The reverse of a prefix of... |
| swrdrevpfx 35313 | A subword expressed in ter... |
| lfuhgr 35314 | A hypergraph is loop-free ... |
| lfuhgr2 35315 | A hypergraph is loop-free ... |
| lfuhgr3 35316 | A hypergraph is loop-free ... |
| cplgredgex 35317 | Any two (distinct) vertice... |
| cusgredgex 35318 | Any two (distinct) vertice... |
| cusgredgex2 35319 | Any two distinct vertices ... |
| pfxwlk 35320 | A prefix of a walk is a wa... |
| revwlk 35321 | The reverse of a walk is a... |
| revwlkb 35322 | Two words represent a walk... |
| swrdwlk 35323 | Two matching subwords of a... |
| pthhashvtx 35324 | A graph containing a path ... |
| spthcycl 35325 | A walk is a trivial path i... |
| usgrgt2cycl 35326 | A non-trivial cycle in a s... |
| usgrcyclgt2v 35327 | A simple graph with a non-... |
| subgrwlk 35328 | If a walk exists in a subg... |
| subgrtrl 35329 | If a trail exists in a sub... |
| subgrpth 35330 | If a path exists in a subg... |
| subgrcycl 35331 | If a cycle exists in a sub... |
| cusgr3cyclex 35332 | Every complete simple grap... |
| loop1cycl 35333 | A hypergraph has a cycle o... |
| 2cycld 35334 | Construction of a 2-cycle ... |
| 2cycl2d 35335 | Construction of a 2-cycle ... |
| umgr2cycllem 35336 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 35337 | A multigraph with two dist... |
| dfacycgr1 35340 | An alternate definition of... |
| isacycgr 35341 | The property of being an a... |
| isacycgr1 35342 | The property of being an a... |
| acycgrcycl 35343 | Any cycle in an acyclic gr... |
| acycgr0v 35344 | A null graph (with no vert... |
| acycgr1v 35345 | A multigraph with one vert... |
| acycgr2v 35346 | A simple graph with two ve... |
| prclisacycgr 35347 | A proper class (representi... |
| acycgrislfgr 35348 | An acyclic hypergraph is a... |
| upgracycumgr 35349 | An acyclic pseudograph is ... |
| umgracycusgr 35350 | An acyclic multigraph is a... |
| upgracycusgr 35351 | An acyclic pseudograph is ... |
| cusgracyclt3v 35352 | A complete simple graph is... |
| pthacycspth 35353 | A path in an acyclic graph... |
| acycgrsubgr 35354 | The subgraph of an acyclic... |
| quartfull 35361 | The quartic equation, writ... |
| deranglem 35362 | Lemma for derangements. (... |
| derangval 35363 | Define the derangement fun... |
| derangf 35364 | The derangement number is ... |
| derang0 35365 | The derangement number of ... |
| derangsn 35366 | The derangement number of ... |
| derangenlem 35367 | One half of ~ derangen . ... |
| derangen 35368 | The derangement number is ... |
| subfacval 35369 | The subfactorial is define... |
| derangen2 35370 | Write the derangement numb... |
| subfacf 35371 | The subfactorial is a func... |
| subfaclefac 35372 | The subfactorial is less t... |
| subfac0 35373 | The subfactorial at zero. ... |
| subfac1 35374 | The subfactorial at one. ... |
| subfacp1lem1 35375 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 35376 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 35377 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 35378 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 35379 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 35380 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 35381 | Lemma for ~ subfacp1 . By... |
| subfacp1 35382 | A two-term recurrence for ... |
| subfacval2 35383 | A closed-form expression f... |
| subfaclim 35384 | The subfactorial converges... |
| subfacval3 35385 | Another closed form expres... |
| derangfmla 35386 | The derangements formula, ... |
| erdszelem1 35387 | Lemma for ~ erdsze . (Con... |
| erdszelem2 35388 | Lemma for ~ erdsze . (Con... |
| erdszelem3 35389 | Lemma for ~ erdsze . (Con... |
| erdszelem4 35390 | Lemma for ~ erdsze . (Con... |
| erdszelem5 35391 | Lemma for ~ erdsze . (Con... |
| erdszelem6 35392 | Lemma for ~ erdsze . (Con... |
| erdszelem7 35393 | Lemma for ~ erdsze . (Con... |
| erdszelem8 35394 | Lemma for ~ erdsze . (Con... |
| erdszelem9 35395 | Lemma for ~ erdsze . (Con... |
| erdszelem10 35396 | Lemma for ~ erdsze . (Con... |
| erdszelem11 35397 | Lemma for ~ erdsze . (Con... |
| erdsze 35398 | The Erdős-Szekeres th... |
| erdsze2lem1 35399 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 35400 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 35401 | Generalize the statement o... |
| kur14lem1 35402 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 35403 | Lemma for ~ kur14 . Write... |
| kur14lem3 35404 | Lemma for ~ kur14 . A clo... |
| kur14lem4 35405 | Lemma for ~ kur14 . Compl... |
| kur14lem5 35406 | Lemma for ~ kur14 . Closu... |
| kur14lem6 35407 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 35408 | Lemma for ~ kur14 : main p... |
| kur14lem8 35409 | Lemma for ~ kur14 . Show ... |
| kur14lem9 35410 | Lemma for ~ kur14 . Since... |
| kur14lem10 35411 | Lemma for ~ kur14 . Disch... |
| kur14 35412 | Kuratowski's closure-compl... |
| ispconn 35419 | The property of being a pa... |
| pconncn 35420 | The property of being a pa... |
| pconntop 35421 | A simply connected space i... |
| issconn 35422 | The property of being a si... |
| sconnpconn 35423 | A simply connected space i... |
| sconntop 35424 | A simply connected space i... |
| sconnpht 35425 | A closed path in a simply ... |
| cnpconn 35426 | An image of a path-connect... |
| pconnconn 35427 | A path-connected space is ... |
| txpconn 35428 | The topological product of... |
| ptpconn 35429 | The topological product of... |
| indispconn 35430 | The indiscrete topology (o... |
| connpconn 35431 | A connected and locally pa... |
| qtoppconn 35432 | A quotient of a path-conne... |
| pconnpi1 35433 | All fundamental groups in ... |
| sconnpht2 35434 | Any two paths in a simply ... |
| sconnpi1 35435 | A path-connected topologic... |
| txsconnlem 35436 | Lemma for ~ txsconn . (Co... |
| txsconn 35437 | The topological product of... |
| cvxpconn 35438 | A convex subset of the com... |
| cvxsconn 35439 | A convex subset of the com... |
| blsconn 35440 | An open ball in the comple... |
| cnllysconn 35441 | The topology of the comple... |
| resconn 35442 | A subset of ` RR ` is simp... |
| ioosconn 35443 | An open interval is simply... |
| iccsconn 35444 | A closed interval is simpl... |
| retopsconn 35445 | The real numbers are simpl... |
| iccllysconn 35446 | A closed interval is local... |
| rellysconn 35447 | The real numbers are local... |
| iisconn 35448 | The unit interval is simpl... |
| iillysconn 35449 | The unit interval is local... |
| iinllyconn 35450 | The unit interval is local... |
| fncvm 35453 | Lemma for covering maps. ... |
| cvmscbv 35454 | Change bound variables in ... |
| iscvm 35455 | The property of being a co... |
| cvmtop1 35456 | Reverse closure for a cove... |
| cvmtop2 35457 | Reverse closure for a cove... |
| cvmcn 35458 | A covering map is a contin... |
| cvmcov 35459 | Property of a covering map... |
| cvmsrcl 35460 | Reverse closure for an eve... |
| cvmsi 35461 | One direction of ~ cvmsval... |
| cvmsval 35462 | Elementhood in the set ` S... |
| cvmsss 35463 | An even covering is a subs... |
| cvmsn0 35464 | An even covering is nonemp... |
| cvmsuni 35465 | An even covering of ` U ` ... |
| cvmsdisj 35466 | An even covering of ` U ` ... |
| cvmshmeo 35467 | Every element of an even c... |
| cvmsf1o 35468 | ` F ` , localized to an el... |
| cvmscld 35469 | The sets of an even coveri... |
| cvmsss2 35470 | An open subset of an evenl... |
| cvmcov2 35471 | The covering map property ... |
| cvmseu 35472 | Every element in ` U. T ` ... |
| cvmsiota 35473 | Identify the unique elemen... |
| cvmopnlem 35474 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 35475 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 35476 | A covering map is an open ... |
| cvmliftmolem1 35477 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 35478 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 35479 | A lift of a continuous fun... |
| cvmliftmo 35480 | A lift of a continuous fun... |
| cvmliftlem1 35481 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 35482 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 35483 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 35484 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 35485 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 35486 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 35487 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 35488 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 35489 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 35490 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 35491 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 35492 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 35493 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 35494 | Lemma for ~ cvmlift . Dis... |
| cvmlift 35495 | One of the important prope... |
| cvmfo 35496 | A covering map is an onto ... |
| cvmliftiota 35497 | Write out a function ` H `... |
| cvmlift2lem1 35498 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 35499 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 35500 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 35501 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 35502 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 35503 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 35504 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 35505 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 35506 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 35507 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 35508 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 35509 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 35510 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 35511 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 35512 | A two-dimensional version ... |
| cvmliftphtlem 35513 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 35514 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 35515 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 35516 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 35517 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 35518 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 35519 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 35520 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 35521 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 35522 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 35523 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 35524 | A general version of ~ cvm... |
| snmlff 35525 | The function ` F ` from ~ ... |
| snmlfval 35526 | The function ` F ` from ~ ... |
| snmlval 35527 | The property " ` A ` is si... |
| snmlflim 35528 | If ` A ` is simply normal,... |
| goel 35543 | A "Godel-set of membership... |
| goelel3xp 35544 | A "Godel-set of membership... |
| goeleq12bg 35545 | Two "Godel-set of membersh... |
| gonafv 35546 | The "Godel-set for the She... |
| goaleq12d 35547 | Equality of the "Godel-set... |
| gonanegoal 35548 | The Godel-set for the Shef... |
| satf 35549 | The satisfaction predicate... |
| satfsucom 35550 | The satisfaction predicate... |
| satfn 35551 | The satisfaction predicate... |
| satom 35552 | The satisfaction predicate... |
| satfvsucom 35553 | The satisfaction predicate... |
| satfv0 35554 | The value of the satisfact... |
| satfvsuclem1 35555 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 35556 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 35557 | The value of the satisfact... |
| satfv1lem 35558 | Lemma for ~ satfv1 . (Con... |
| satfv1 35559 | The value of the satisfact... |
| satfsschain 35560 | The binary relation of a s... |
| satfvsucsuc 35561 | The satisfaction predicate... |
| satfbrsuc 35562 | The binary relation of a s... |
| satfrel 35563 | The value of the satisfact... |
| satfdmlem 35564 | Lemma for ~ satfdm . (Con... |
| satfdm 35565 | The domain of the satisfac... |
| satfrnmapom 35566 | The range of the satisfact... |
| satfv0fun 35567 | The value of the satisfact... |
| satf0 35568 | The satisfaction predicate... |
| satf0sucom 35569 | The satisfaction predicate... |
| satf00 35570 | The value of the satisfact... |
| satf0suclem 35571 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 35572 | The value of the satisfact... |
| satf0op 35573 | An element of a value of t... |
| satf0n0 35574 | The value of the satisfact... |
| sat1el2xp 35575 | The first component of an ... |
| fmlafv 35576 | The valid Godel formulas o... |
| fmla 35577 | The set of all valid Godel... |
| fmla0 35578 | The valid Godel formulas o... |
| fmla0xp 35579 | The valid Godel formulas o... |
| fmlasuc0 35580 | The valid Godel formulas o... |
| fmlafvel 35581 | A class is a valid Godel f... |
| fmlasuc 35582 | The valid Godel formulas o... |
| fmla1 35583 | The valid Godel formulas o... |
| isfmlasuc 35584 | The characterization of a ... |
| fmlasssuc 35585 | The Godel formulas of heig... |
| fmlaomn0 35586 | The empty set is not a God... |
| fmlan0 35587 | The empty set is not a God... |
| gonan0 35588 | The "Godel-set of NAND" is... |
| goaln0 35589 | The "Godel-set of universa... |
| gonarlem 35590 | Lemma for ~ gonar (inducti... |
| gonar 35591 | If the "Godel-set of NAND"... |
| goalrlem 35592 | Lemma for ~ goalr (inducti... |
| goalr 35593 | If the "Godel-set of unive... |
| fmla0disjsuc 35594 | The set of valid Godel for... |
| fmlasucdisj 35595 | The valid Godel formulas o... |
| satfdmfmla 35596 | The domain of the satisfac... |
| satffunlem 35597 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 35598 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 35599 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 35600 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 35601 | The domain of an ordered p... |
| satffunlem2lem2 35602 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 35603 | Lemma 1 for ~ satffun : in... |
| satffunlem2 35604 | Lemma 2 for ~ satffun : in... |
| satffun 35605 | The value of the satisfact... |
| satff 35606 | The satisfaction predicate... |
| satfun 35607 | The satisfaction predicate... |
| satfvel 35608 | An element of the value of... |
| satfv0fvfmla0 35609 | The value of the satisfact... |
| satefv 35610 | The simplified satisfactio... |
| sate0 35611 | The simplified satisfactio... |
| satef 35612 | The simplified satisfactio... |
| sate0fv0 35613 | A simplified satisfaction ... |
| satefvfmla0 35614 | The simplified satisfactio... |
| sategoelfvb 35615 | Characterization of a valu... |
| sategoelfv 35616 | Condition of a valuation `... |
| ex-sategoelel 35617 | Example of a valuation of ... |
| ex-sategoel 35618 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 35619 | The value of the satisfact... |
| 2goelgoanfmla1 35620 | Two Godel-sets of membersh... |
| satefvfmla1 35621 | The simplified satisfactio... |
| ex-sategoelelomsuc 35622 | Example of a valuation of ... |
| ex-sategoelel12 35623 | Example of a valuation of ... |
| prv 35624 | The "proves" relation on a... |
| elnanelprv 35625 | The wff ` ( A e. B -/\ B e... |
| prv0 35626 | Every wff encoded as ` U `... |
| prv1n 35627 | No wff encoded as a Godel-... |
| mvtval 35696 | The set of variable typeco... |
| mrexval 35697 | The set of "raw expression... |
| mexval 35698 | The set of expressions, wh... |
| mexval2 35699 | The set of expressions, wh... |
| mdvval 35700 | The set of disjoint variab... |
| mvrsval 35701 | The set of variables in an... |
| mvrsfpw 35702 | The set of variables in an... |
| mrsubffval 35703 | The substitution of some v... |
| mrsubfval 35704 | The substitution of some v... |
| mrsubval 35705 | The substitution of some v... |
| mrsubcv 35706 | The value of a substituted... |
| mrsubvr 35707 | The value of a substituted... |
| mrsubff 35708 | A substitution is a functi... |
| mrsubrn 35709 | Although it is defined for... |
| mrsubff1 35710 | When restricted to complet... |
| mrsubff1o 35711 | When restricted to complet... |
| mrsub0 35712 | The value of the substitut... |
| mrsubf 35713 | A substitution is a functi... |
| mrsubccat 35714 | Substitution distributes o... |
| mrsubcn 35715 | A substitution does not ch... |
| elmrsubrn 35716 | Characterization of the su... |
| mrsubco 35717 | The composition of two sub... |
| mrsubvrs 35718 | The set of variables in a ... |
| msubffval 35719 | A substitution applied to ... |
| msubfval 35720 | A substitution applied to ... |
| msubval 35721 | A substitution applied to ... |
| msubrsub 35722 | A substitution applied to ... |
| msubty 35723 | The type of a substituted ... |
| elmsubrn 35724 | Characterization of substi... |
| msubrn 35725 | Although it is defined for... |
| msubff 35726 | A substitution is a functi... |
| msubco 35727 | The composition of two sub... |
| msubf 35728 | A substitution is a functi... |
| mvhfval 35729 | Value of the function mapp... |
| mvhval 35730 | Value of the function mapp... |
| mpstval 35731 | A pre-statement is an orde... |
| elmpst 35732 | Property of being a pre-st... |
| msrfval 35733 | Value of the reduct of a p... |
| msrval 35734 | Value of the reduct of a p... |
| mpstssv 35735 | A pre-statement is an orde... |
| mpst123 35736 | Decompose a pre-statement ... |
| mpstrcl 35737 | The elements of a pre-stat... |
| msrf 35738 | The reduct of a pre-statem... |
| msrrcl 35739 | If ` X ` and ` Y ` have th... |
| mstaval 35740 | Value of the set of statem... |
| msrid 35741 | The reduct of a statement ... |
| msrfo 35742 | The reduct of a pre-statem... |
| mstapst 35743 | A statement is a pre-state... |
| elmsta 35744 | Property of being a statem... |
| ismfs 35745 | A formal system is a tuple... |
| mfsdisj 35746 | The constants and variable... |
| mtyf2 35747 | The type function maps var... |
| mtyf 35748 | The type function maps var... |
| mvtss 35749 | The set of variable typeco... |
| maxsta 35750 | An axiom is a statement. ... |
| mvtinf 35751 | Each variable typecode has... |
| msubff1 35752 | When restricted to complet... |
| msubff1o 35753 | When restricted to complet... |
| mvhf 35754 | The function mapping varia... |
| mvhf1 35755 | The function mapping varia... |
| msubvrs 35756 | The set of variables in a ... |
| mclsrcl 35757 | Reverse closure for the cl... |
| mclsssvlem 35758 | Lemma for ~ mclsssv . (Co... |
| mclsval 35759 | The function mapping varia... |
| mclsssv 35760 | The closure of a set of ex... |
| ssmclslem 35761 | Lemma for ~ ssmcls . (Con... |
| vhmcls 35762 | All variable hypotheses ar... |
| ssmcls 35763 | The original expressions a... |
| ss2mcls 35764 | The closure is monotonic u... |
| mclsax 35765 | The closure is closed unde... |
| mclsind 35766 | Induction theorem for clos... |
| mppspstlem 35767 | Lemma for ~ mppspst . (Co... |
| mppsval 35768 | Definition of a provable p... |
| elmpps 35769 | Definition of a provable p... |
| mppspst 35770 | A provable pre-statement i... |
| mthmval 35771 | A theorem is a pre-stateme... |
| elmthm 35772 | A theorem is a pre-stateme... |
| mthmi 35773 | A statement whose reduct i... |
| mthmsta 35774 | A theorem is a pre-stateme... |
| mppsthm 35775 | A provable pre-statement i... |
| mthmblem 35776 | Lemma for ~ mthmb . (Cont... |
| mthmb 35777 | If two statements have the... |
| mthmpps 35778 | Given a theorem, there is ... |
| mclsppslem 35779 | The closure is closed unde... |
| mclspps 35780 | The closure is closed unde... |
| rexxfr3d 35834 | Transfer existential quant... |
| rexxfr3dALT 35835 | Longer proof of ~ rexxfr3d... |
| rspssbasd 35836 | The span of a set of ring ... |
| ellcsrspsn 35837 | Membership in a left coset... |
| ply1divalg3 35838 | Uniqueness of polynomial r... |
| r1peuqusdeg1 35839 | Uniqueness of polynomial r... |
| problem1 35861 | Practice problem 1. Clues... |
| problem2 35862 | Practice problem 2. Clues... |
| problem3 35863 | Practice problem 3. Clues... |
| problem4 35864 | Practice problem 4. Clues... |
| problem5 35865 | Practice problem 5. Clues... |
| quad3 35866 | Variant of quadratic equat... |
| climuzcnv 35867 | Utility lemma to convert b... |
| sinccvglem 35868 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 35869 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 35870 | The circumference of a cir... |
| elfzm12 35871 | Membership in a curtailed ... |
| nn0seqcvg 35872 | A strictly-decreasing nonn... |
| lediv2aALT 35873 | Division of both sides of ... |
| abs2sqlei 35874 | The absolute values of two... |
| abs2sqlti 35875 | The absolute values of two... |
| abs2sqle 35876 | The absolute values of two... |
| abs2sqlt 35877 | The absolute values of two... |
| abs2difi 35878 | Difference of absolute val... |
| abs2difabsi 35879 | Absolute value of differen... |
| 2thALT 35880 | Alternate proof of ~ 2th .... |
| orbi2iALT 35881 | Alternate proof of ~ orbi2... |
| pm3.48ALT 35882 | Alternate proof of ~ pm3.4... |
| 3jcadALT 35883 | Alternate proof of ~ 3jcad... |
| currybi 35884 | Biconditional version of C... |
| antnest 35885 | Suppose ` ph ` , ` ps ` ar... |
| antnestlaw3lem 35886 | Lemma for ~ antnestlaw3 . ... |
| antnestlaw1 35887 | A law of nested antecedent... |
| antnestlaw2 35888 | A law of nested antecedent... |
| antnestlaw3 35889 | A law of nested antecedent... |
| antnestALT 35890 | Alternative proof of ~ ant... |
| axextprim 35897 | ~ ax-ext without distinct ... |
| axrepprim 35898 | ~ ax-rep without distinct ... |
| axunprim 35899 | ~ ax-un without distinct v... |
| axpowprim 35900 | ~ ax-pow without distinct ... |
| axregprim 35901 | ~ ax-reg without distinct ... |
| axinfprim 35902 | ~ ax-inf without distinct ... |
| axacprim 35903 | ~ ax-ac without distinct v... |
| untelirr 35904 | We call a class "untanged"... |
| untuni 35905 | The union of a class is un... |
| untsucf 35906 | If a class is untangled, t... |
| unt0 35907 | The null set is untangled.... |
| untint 35908 | If there is an untangled e... |
| efrunt 35909 | If ` A ` is well-founded b... |
| untangtr 35910 | A transitive class is unta... |
| 3jaodd 35911 | Double deduction form of ~... |
| 3orit 35912 | Closed form of ~ 3ori . (... |
| biimpexp 35913 | A biconditional in the ant... |
| nepss 35914 | Two classes are unequal if... |
| 3ccased 35915 | Triple disjunction form of... |
| dfso3 35916 | Expansion of the definitio... |
| brtpid1 35917 | A binary relation involvin... |
| brtpid2 35918 | A binary relation involvin... |
| brtpid3 35919 | A binary relation involvin... |
| iota5f 35920 | A method for computing iot... |
| jath 35921 | Closed form of ~ ja . Pro... |
| xpab 35922 | Cartesian product of two c... |
| nnuni 35923 | The union of a finite ordi... |
| sqdivzi 35924 | Distribution of square ove... |
| supfz 35925 | The supremum of a finite s... |
| inffz 35926 | The infimum of a finite se... |
| fz0n 35927 | The sequence ` ( 0 ... ( N... |
| shftvalg 35928 | Value of a sequence shifte... |
| divcnvlin 35929 | Limit of the ratio of two ... |
| climlec3 35930 | Comparison of a constant t... |
| iexpire 35931 | ` _i ` raised to itself is... |
| bcneg1 35932 | The binomial coefficient o... |
| bcm1nt 35933 | The proportion of one bino... |
| bcprod 35934 | A product identity for bin... |
| bccolsum 35935 | A column-sum rule for bino... |
| iprodefisumlem 35936 | Lemma for ~ iprodefisum . ... |
| iprodefisum 35937 | Applying the exponential f... |
| iprodgam 35938 | An infinite product versio... |
| faclimlem1 35939 | Lemma for ~ faclim . Clos... |
| faclimlem2 35940 | Lemma for ~ faclim . Show... |
| faclimlem3 35941 | Lemma for ~ faclim . Alge... |
| faclim 35942 | An infinite product expres... |
| iprodfac 35943 | An infinite product expres... |
| faclim2 35944 | Another factorial limit du... |
| gcd32 35945 | Swap the second and third ... |
| gcdabsorb 35946 | Absorption law for gcd. (... |
| dftr6 35947 | A potential definition of ... |
| coep 35948 | Composition with the membe... |
| coepr 35949 | Composition with the conve... |
| dffr5 35950 | A quantifier-free definiti... |
| dfso2 35951 | Quantifier-free definition... |
| br8 35952 | Substitution for an eight-... |
| br6 35953 | Substitution for a six-pla... |
| br4 35954 | Substitution for a four-pl... |
| cnvco1 35955 | Another distributive law o... |
| cnvco2 35956 | Another distributive law o... |
| eldm3 35957 | Quantifier-free definition... |
| elrn3 35958 | Quantifier-free definition... |
| pocnv 35959 | The converse of a partial ... |
| socnv 35960 | The converse of a strict o... |
| elintfv 35961 | Membership in an intersect... |
| funpsstri 35962 | A condition for subset tri... |
| fundmpss 35963 | If a class ` F ` is a prop... |
| funsseq 35964 | Given two functions with e... |
| fununiq 35965 | The uniqueness condition o... |
| funbreq 35966 | An equality condition for ... |
| br1steq 35967 | Uniqueness condition for t... |
| br2ndeq 35968 | Uniqueness condition for t... |
| dfdm5 35969 | Definition of domain in te... |
| dfrn5 35970 | Definition of range in ter... |
| opelco3 35971 | Alternate way of saying th... |
| elima4 35972 | Quantifier-free expression... |
| fv1stcnv 35973 | The value of the converse ... |
| fv2ndcnv 35974 | The value of the converse ... |
| elpotr 35975 | A class of transitive sets... |
| dford5reg 35976 | Given ~ ax-reg , an ordina... |
| dfon2lem1 35977 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 35978 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 35979 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 35980 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 35981 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 35982 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 35983 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 35984 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 35985 | Lemma for ~ dfon2 . A cla... |
| dfon2 35986 | ` On ` consists of all set... |
| rdgprc0 35987 | The value of the recursive... |
| rdgprc 35988 | The value of the recursive... |
| dfrdg2 35989 | Alternate definition of th... |
| dfrdg3 35990 | Generalization of ~ dfrdg2... |
| axextdfeq 35991 | A version of ~ ax-ext for ... |
| ax8dfeq 35992 | A version of ~ ax-8 for us... |
| axextdist 35993 | ~ ax-ext with distinctors ... |
| axextbdist 35994 | ~ axextb with distinctors ... |
| 19.12b 35995 | Version of ~ 19.12vv with ... |
| exnel 35996 | There is always a set not ... |
| distel 35997 | Distinctors in terms of me... |
| axextndbi 35998 | ~ axextnd as a bicondition... |
| hbntg 35999 | A more general form of ~ h... |
| hbimtg 36000 | A more general and closed ... |
| hbaltg 36001 | A more general and closed ... |
| hbng 36002 | A more general form of ~ h... |
| hbimg 36003 | A more general form of ~ h... |
| wsuceq123 36008 | Equality theorem for well-... |
| wsuceq1 36009 | Equality theorem for well-... |
| wsuceq2 36010 | Equality theorem for well-... |
| wsuceq3 36011 | Equality theorem for well-... |
| nfwsuc 36012 | Bound-variable hypothesis ... |
| wlimeq12 36013 | Equality theorem for the l... |
| wlimeq1 36014 | Equality theorem for the l... |
| wlimeq2 36015 | Equality theorem for the l... |
| nfwlim 36016 | Bound-variable hypothesis ... |
| elwlim 36017 | Membership in the limit cl... |
| wzel 36018 | The zero of a well-founded... |
| wsuclem 36019 | Lemma for the supremum pro... |
| wsucex 36020 | Existence theorem for well... |
| wsuccl 36021 | If ` X ` is a set with an ... |
| wsuclb 36022 | A well-founded successor i... |
| wlimss 36023 | The class of limit points ... |
| txpss3v 36072 | A tail Cartesian product i... |
| txprel 36073 | A tail Cartesian product i... |
| brtxp 36074 | Characterize a ternary rel... |
| brtxp2 36075 | The binary relation over a... |
| dfpprod2 36076 | Expanded definition of par... |
| pprodcnveq 36077 | A converse law for paralle... |
| pprodss4v 36078 | The parallel product is a ... |
| brpprod 36079 | Characterize a quaternary ... |
| brpprod3a 36080 | Condition for parallel pro... |
| brpprod3b 36081 | Condition for parallel pro... |
| relsset 36082 | The subset class is a bina... |
| brsset 36083 | For sets, the ` SSet ` bin... |
| idsset 36084 | ` _I ` is equal to the int... |
| eltrans 36085 | Membership in the class of... |
| dfon3 36086 | A quantifier-free definiti... |
| dfon4 36087 | Another quantifier-free de... |
| brtxpsd 36088 | Expansion of a common form... |
| brtxpsd2 36089 | Another common abbreviatio... |
| brtxpsd3 36090 | A third common abbreviatio... |
| relbigcup 36091 | The ` Bigcup ` relationshi... |
| brbigcup 36092 | Binary relation over ` Big... |
| dfbigcup2 36093 | ` Bigcup ` using maps-to n... |
| fobigcup 36094 | ` Bigcup ` maps the univer... |
| fnbigcup 36095 | ` Bigcup ` is a function o... |
| fvbigcup 36096 | For sets, ` Bigcup ` yield... |
| elfix 36097 | Membership in the fixpoint... |
| elfix2 36098 | Alternative membership in ... |
| dffix2 36099 | The fixpoints of a class i... |
| fixssdm 36100 | The fixpoints of a class a... |
| fixssrn 36101 | The fixpoints of a class a... |
| fixcnv 36102 | The fixpoints of a class a... |
| fixun 36103 | The fixpoint operator dist... |
| ellimits 36104 | Membership in the class of... |
| limitssson 36105 | The class of all limit ord... |
| dfom5b 36106 | A quantifier-free definiti... |
| sscoid 36107 | A condition for subset and... |
| dffun10 36108 | Another potential definiti... |
| elfuns 36109 | Membership in the class of... |
| elfunsg 36110 | Closed form of ~ elfuns . ... |
| brsingle 36111 | The binary relation form o... |
| elsingles 36112 | Membership in the class of... |
| fnsingle 36113 | The singleton relationship... |
| fvsingle 36114 | The value of the singleton... |
| dfsingles2 36115 | Alternate definition of th... |
| snelsingles 36116 | A singleton is a member of... |
| dfiota3 36117 | A definition of iota using... |
| dffv5 36118 | Another quantifier-free de... |
| unisnif 36119 | Express union of singleton... |
| brimage 36120 | Binary relation form of th... |
| brimageg 36121 | Closed form of ~ brimage .... |
| funimage 36122 | ` Image A ` is a function.... |
| fnimage 36123 | ` Image R ` is a function ... |
| imageval 36124 | The image functor in maps-... |
| fvimage 36125 | Value of the image functor... |
| brcart 36126 | Binary relation form of th... |
| brdomain 36127 | Binary relation form of th... |
| brrange 36128 | Binary relation form of th... |
| brdomaing 36129 | Closed form of ~ brdomain ... |
| brrangeg 36130 | Closed form of ~ brrange .... |
| brimg 36131 | Binary relation form of th... |
| brapply 36132 | Binary relation form of th... |
| brcup 36133 | Binary relation form of th... |
| brcap 36134 | Binary relation form of th... |
| lemsuccf 36135 | Lemma for unfolding differ... |
| brsuccf 36136 | Binary relation form of th... |
| dfsuccf2 36137 | Alternate definition of Sc... |
| funpartlem 36138 | Lemma for ~ funpartfun . ... |
| funpartfun 36139 | The functional part of ` F... |
| funpartss 36140 | The functional part of ` F... |
| funpartfv 36141 | The function value of the ... |
| fullfunfnv 36142 | The full functional part o... |
| fullfunfv 36143 | The function value of the ... |
| brfullfun 36144 | A binary relation form con... |
| brrestrict 36145 | Binary relation form of th... |
| dfrecs2 36146 | A quantifier-free definiti... |
| dfrdg4 36147 | A quantifier-free definiti... |
| dfint3 36148 | Quantifier-free definition... |
| imagesset 36149 | The Image functor applied ... |
| brub 36150 | Binary relation form of th... |
| brlb 36151 | Binary relation form of th... |
| altopex 36156 | Alternative ordered pairs ... |
| altopthsn 36157 | Two alternate ordered pair... |
| altopeq12 36158 | Equality for alternate ord... |
| altopeq1 36159 | Equality for alternate ord... |
| altopeq2 36160 | Equality for alternate ord... |
| altopth1 36161 | Equality of the first memb... |
| altopth2 36162 | Equality of the second mem... |
| altopthg 36163 | Alternate ordered pair the... |
| altopthbg 36164 | Alternate ordered pair the... |
| altopth 36165 | The alternate ordered pair... |
| altopthb 36166 | Alternate ordered pair the... |
| altopthc 36167 | Alternate ordered pair the... |
| altopthd 36168 | Alternate ordered pair the... |
| altxpeq1 36169 | Equality for alternate Car... |
| altxpeq2 36170 | Equality for alternate Car... |
| elaltxp 36171 | Membership in alternate Ca... |
| altopelaltxp 36172 | Alternate ordered pair mem... |
| altxpsspw 36173 | An inclusion rule for alte... |
| altxpexg 36174 | The alternate Cartesian pr... |
| rankaltopb 36175 | Compute the rank of an alt... |
| nfaltop 36176 | Bound-variable hypothesis ... |
| sbcaltop 36177 | Distribution of class subs... |
| cgrrflx2d 36180 | Deduction form of ~ axcgrr... |
| cgrtr4d 36181 | Deduction form of ~ axcgrt... |
| cgrtr4and 36182 | Deduction form of ~ axcgrt... |
| cgrrflx 36183 | Reflexivity law for congru... |
| cgrrflxd 36184 | Deduction form of ~ cgrrfl... |
| cgrcomim 36185 | Congruence commutes on the... |
| cgrcom 36186 | Congruence commutes betwee... |
| cgrcomand 36187 | Deduction form of ~ cgrcom... |
| cgrtr 36188 | Transitivity law for congr... |
| cgrtrand 36189 | Deduction form of ~ cgrtr ... |
| cgrtr3 36190 | Transitivity law for congr... |
| cgrtr3and 36191 | Deduction form of ~ cgrtr3... |
| cgrcoml 36192 | Congruence commutes on the... |
| cgrcomr 36193 | Congruence commutes on the... |
| cgrcomlr 36194 | Congruence commutes on bot... |
| cgrcomland 36195 | Deduction form of ~ cgrcom... |
| cgrcomrand 36196 | Deduction form of ~ cgrcom... |
| cgrcomlrand 36197 | Deduction form of ~ cgrcom... |
| cgrtriv 36198 | Degenerate segments are co... |
| cgrid2 36199 | Identity law for congruenc... |
| cgrdegen 36200 | Two congruent segments are... |
| brofs 36201 | Binary relation form of th... |
| 5segofs 36202 | Rephrase ~ ax5seg using th... |
| ofscom 36203 | The outer five segment pre... |
| cgrextend 36204 | Link congruence over a pai... |
| cgrextendand 36205 | Deduction form of ~ cgrext... |
| segconeq 36206 | Two points that satisfy th... |
| segconeu 36207 | Existential uniqueness ver... |
| btwntriv2 36208 | Betweenness always holds f... |
| btwncomim 36209 | Betweenness commutes. Imp... |
| btwncom 36210 | Betweenness commutes. (Co... |
| btwncomand 36211 | Deduction form of ~ btwnco... |
| btwntriv1 36212 | Betweenness always holds f... |
| btwnswapid 36213 | If you can swap the first ... |
| btwnswapid2 36214 | If you can swap arguments ... |
| btwnintr 36215 | Inner transitivity law for... |
| btwnexch3 36216 | Exchange the first endpoin... |
| btwnexch3and 36217 | Deduction form of ~ btwnex... |
| btwnouttr2 36218 | Outer transitivity law for... |
| btwnexch2 36219 | Exchange the outer point o... |
| btwnouttr 36220 | Outer transitivity law for... |
| btwnexch 36221 | Outer transitivity law for... |
| btwnexchand 36222 | Deduction form of ~ btwnex... |
| btwndiff 36223 | There is always a ` c ` di... |
| trisegint 36224 | A line segment between two... |
| funtransport 36227 | The ` TransportTo ` relati... |
| fvtransport 36228 | Calculate the value of the... |
| transportcl 36229 | Closure law for segment tr... |
| transportprops 36230 | Calculate the defining pro... |
| brifs 36239 | Binary relation form of th... |
| ifscgr 36240 | Inner five segment congrue... |
| cgrsub 36241 | Removing identical parts f... |
| brcgr3 36242 | Binary relation form of th... |
| cgr3permute3 36243 | Permutation law for three-... |
| cgr3permute1 36244 | Permutation law for three-... |
| cgr3permute2 36245 | Permutation law for three-... |
| cgr3permute4 36246 | Permutation law for three-... |
| cgr3permute5 36247 | Permutation law for three-... |
| cgr3tr4 36248 | Transitivity law for three... |
| cgr3com 36249 | Commutativity law for thre... |
| cgr3rflx 36250 | Identity law for three-pla... |
| cgrxfr 36251 | A line segment can be divi... |
| btwnxfr 36252 | A condition for extending ... |
| colinrel 36253 | Colinearity is a relations... |
| brcolinear2 36254 | Alternate colinearity bina... |
| brcolinear 36255 | The binary relation form o... |
| colinearex 36256 | The colinear predicate exi... |
| colineardim1 36257 | If ` A ` is colinear with ... |
| colinearperm1 36258 | Permutation law for coline... |
| colinearperm3 36259 | Permutation law for coline... |
| colinearperm2 36260 | Permutation law for coline... |
| colinearperm4 36261 | Permutation law for coline... |
| colinearperm5 36262 | Permutation law for coline... |
| colineartriv1 36263 | Trivial case of colinearit... |
| colineartriv2 36264 | Trivial case of colinearit... |
| btwncolinear1 36265 | Betweenness implies coline... |
| btwncolinear2 36266 | Betweenness implies coline... |
| btwncolinear3 36267 | Betweenness implies coline... |
| btwncolinear4 36268 | Betweenness implies coline... |
| btwncolinear5 36269 | Betweenness implies coline... |
| btwncolinear6 36270 | Betweenness implies coline... |
| colinearxfr 36271 | Transfer law for colineari... |
| lineext 36272 | Extend a line with a missi... |
| brofs2 36273 | Change some conditions for... |
| brifs2 36274 | Change some conditions for... |
| brfs 36275 | Binary relation form of th... |
| fscgr 36276 | Congruence law for the gen... |
| linecgr 36277 | Congruence rule for lines.... |
| linecgrand 36278 | Deduction form of ~ linecg... |
| lineid 36279 | Identity law for points on... |
| idinside 36280 | Law for finding a point in... |
| endofsegid 36281 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 36282 | Deduction form of ~ endofs... |
| btwnconn1lem1 36283 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 36284 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 36285 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 36286 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 36287 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 36288 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 36289 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 36290 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 36291 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 36292 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 36293 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 36294 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 36295 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 36296 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 36297 | Connectitivy law for betwe... |
| btwnconn2 36298 | Another connectivity law f... |
| btwnconn3 36299 | Inner connectivity law for... |
| midofsegid 36300 | If two points fall in the ... |
| segcon2 36301 | Generalization of ~ axsegc... |
| brsegle 36304 | Binary relation form of th... |
| brsegle2 36305 | Alternate characterization... |
| seglecgr12im 36306 | Substitution law for segme... |
| seglecgr12 36307 | Substitution law for segme... |
| seglerflx 36308 | Segment comparison is refl... |
| seglemin 36309 | Any segment is at least as... |
| segletr 36310 | Segment less than is trans... |
| segleantisym 36311 | Antisymmetry law for segme... |
| seglelin 36312 | Linearity law for segment ... |
| btwnsegle 36313 | If ` B ` falls between ` A... |
| colinbtwnle 36314 | Given three colinear point... |
| broutsideof 36317 | Binary relation form of ` ... |
| broutsideof2 36318 | Alternate form of ` Outsid... |
| outsidene1 36319 | Outsideness implies inequa... |
| outsidene2 36320 | Outsideness implies inequa... |
| btwnoutside 36321 | A principle linking outsid... |
| broutsideof3 36322 | Characterization of outsid... |
| outsideofrflx 36323 | Reflexivity of outsideness... |
| outsideofcom 36324 | Commutativity law for outs... |
| outsideoftr 36325 | Transitivity law for outsi... |
| outsideofeq 36326 | Uniqueness law for ` Outsi... |
| outsideofeu 36327 | Given a nondegenerate ray,... |
| outsidele 36328 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 36329 | Outside of implies colinea... |
| funray 36336 | Show that the ` Ray ` rela... |
| fvray 36337 | Calculate the value of the... |
| funline 36338 | Show that the ` Line ` rel... |
| linedegen 36339 | When ` Line ` is applied w... |
| fvline 36340 | Calculate the value of the... |
| liness 36341 | A line is a subset of the ... |
| fvline2 36342 | Alternate definition of a ... |
| lineunray 36343 | A line is composed of a po... |
| lineelsb2 36344 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 36345 | Reflexivity law for line m... |
| linecom 36346 | Commutativity law for line... |
| linerflx2 36347 | Reflexivity law for line m... |
| ellines 36348 | Membership in the set of a... |
| linethru 36349 | If ` A ` is a line contain... |
| hilbert1.1 36350 | There is a line through an... |
| hilbert1.2 36351 | There is at most one line ... |
| linethrueu 36352 | There is a unique line goi... |
| lineintmo 36353 | Two distinct lines interse... |
| fwddifval 36358 | Calculate the value of the... |
| fwddifnval 36359 | The value of the forward d... |
| fwddifn0 36360 | The value of the n-iterate... |
| fwddifnp1 36361 | The value of the n-iterate... |
| rankung 36362 | The rank of the union of t... |
| ranksng 36363 | The rank of a singleton. ... |
| rankelg 36364 | The membership relation is... |
| rankpwg 36365 | The rank of a power set. ... |
| rank0 36366 | The rank of the empty set ... |
| rankeq1o 36367 | The only set with rank ` 1... |
| elhf 36370 | Membership in the heredita... |
| elhf2 36371 | Alternate form of membersh... |
| elhf2g 36372 | Hereditarily finiteness vi... |
| 0hf 36373 | The empty set is a heredit... |
| hfun 36374 | The union of two HF sets i... |
| hfsn 36375 | The singleton of an HF set... |
| hfadj 36376 | Adjoining one HF element t... |
| hfelhf 36377 | Any member of an HF set is... |
| hftr 36378 | The class of all hereditar... |
| hfext 36379 | Extensionality for HF sets... |
| hfuni 36380 | The union of an HF set is ... |
| hfpw 36381 | The power class of an HF s... |
| hfninf 36382 | ` _om ` is not hereditaril... |
| rmoeqi 36383 | Equality inference for res... |
| rmoeqbii 36384 | Equality inference for res... |
| reueqi 36385 | Equality inference for res... |
| reueqbii 36386 | Equality inference for res... |
| sbceqbii 36387 | Formula-building inference... |
| disjeq1i 36388 | Equality theorem for disjo... |
| disjeq12i 36389 | Equality theorem for disjo... |
| rabeqbii 36390 | Equality theorem for restr... |
| iuneq12i 36391 | Equality theorem for index... |
| iineq1i 36392 | Equality theorem for index... |
| iineq12i 36393 | Equality theorem for index... |
| riotaeqbii 36394 | Equivalent wff's and equal... |
| riotaeqi 36395 | Equal domains yield equal ... |
| ixpeq1i 36396 | Equality inference for inf... |
| ixpeq12i 36397 | Equality inference for inf... |
| sumeq2si 36398 | Equality inference for sum... |
| sumeq12si 36399 | Equality inference for sum... |
| prodeq2si 36400 | Equality inference for pro... |
| prodeq12si 36401 | Equality inference for pro... |
| itgeq12i 36402 | Equality inference for an ... |
| itgeq1i 36403 | Equality inference for an ... |
| itgeq2i 36404 | Equality inference for an ... |
| ditgeq123i 36405 | Equality inference for the... |
| ditgeq12i 36406 | Equality inference for the... |
| ditgeq3i 36407 | Equality inference for the... |
| rmoeqdv 36408 | Formula-building rule for ... |
| rmoeqbidv 36409 | Formula-building rule for ... |
| sbequbidv 36410 | Deduction substituting bot... |
| disjeq12dv 36411 | Equality theorem for disjo... |
| ixpeq12dv 36412 | Equality theorem for infin... |
| sumeq12sdv 36413 | Equality deduction for sum... |
| prodeq12sdv 36414 | Equality deduction for pro... |
| itgeq12sdv 36415 | Equality theorem for an in... |
| itgeq2sdv 36416 | Equality theorem for an in... |
| ditgeq123dv 36417 | Equality theorem for the d... |
| ditgeq12d 36418 | Equality theorem for the d... |
| ditgeq3sdv 36419 | Equality theorem for the d... |
| in-ax8 36420 | A proof of ~ ax-8 that doe... |
| ss-ax8 36421 | A proof of ~ ax-8 that doe... |
| cbvralvw2 36422 | Change bound variable and ... |
| cbvrexvw2 36423 | Change bound variable and ... |
| cbvrmovw2 36424 | Change bound variable and ... |
| cbvreuvw2 36425 | Change bound variable and ... |
| cbvsbcvw2 36426 | Change bound variable of a... |
| cbvcsbvw2 36427 | Change bound variable of a... |
| cbviunvw2 36428 | Change bound variable and ... |
| cbviinvw2 36429 | Change bound variable and ... |
| cbvmptvw2 36430 | Change bound variable and ... |
| cbvdisjvw2 36431 | Change bound variable and ... |
| cbvriotavw2 36432 | Change bound variable and ... |
| cbvoprab1vw 36433 | Change the first bound var... |
| cbvoprab2vw 36434 | Change the second bound va... |
| cbvoprab123vw 36435 | Change all bound variables... |
| cbvoprab23vw 36436 | Change the second and thir... |
| cbvoprab13vw 36437 | Change the first and third... |
| cbvmpovw2 36438 | Change bound variables and... |
| cbvmpo1vw2 36439 | Change domains and the fir... |
| cbvmpo2vw2 36440 | Change domains and the sec... |
| cbvixpvw2 36441 | Change bound variable and ... |
| cbvsumvw2 36442 | Change bound variable and ... |
| cbvprodvw2 36443 | Change bound variable and ... |
| cbvitgvw2 36444 | Change bound variable and ... |
| cbvditgvw2 36445 | Change bound variable and ... |
| cbvmodavw 36446 | Change bound variable in t... |
| cbveudavw 36447 | Change bound variable in t... |
| cbvrmodavw 36448 | Change bound variable in t... |
| cbvreudavw 36449 | Change bound variable in t... |
| cbvsbdavw 36450 | Change bound variable in p... |
| cbvsbdavw2 36451 | Change bound variable in p... |
| cbvabdavw 36452 | Change bound variable in c... |
| cbvsbcdavw 36453 | Change bound variable of a... |
| cbvsbcdavw2 36454 | Change bound variable of a... |
| cbvcsbdavw 36455 | Change bound variable of a... |
| cbvcsbdavw2 36456 | Change bound variable of a... |
| cbvrabdavw 36457 | Change bound variable in r... |
| cbviundavw 36458 | Change bound variable in i... |
| cbviindavw 36459 | Change bound variable in i... |
| cbvopab1davw 36460 | Change the first bound var... |
| cbvopab2davw 36461 | Change the second bound va... |
| cbvopabdavw 36462 | Change bound variables in ... |
| cbvmptdavw 36463 | Change bound variable in a... |
| cbvdisjdavw 36464 | Change bound variable in a... |
| cbviotadavw 36465 | Change bound variable in a... |
| cbvriotadavw 36466 | Change bound variable in a... |
| cbvoprab1davw 36467 | Change the first bound var... |
| cbvoprab2davw 36468 | Change the second bound va... |
| cbvoprab3davw 36469 | Change the third bound var... |
| cbvoprab123davw 36470 | Change all bound variables... |
| cbvoprab12davw 36471 | Change the first and secon... |
| cbvoprab23davw 36472 | Change the second and thir... |
| cbvoprab13davw 36473 | Change the first and third... |
| cbvixpdavw 36474 | Change bound variable in a... |
| cbvsumdavw 36475 | Change bound variable in a... |
| cbvproddavw 36476 | Change bound variable in a... |
| cbvitgdavw 36477 | Change bound variable in a... |
| cbvditgdavw 36478 | Change bound variable in a... |
| cbvrmodavw2 36479 | Change bound variable and ... |
| cbvreudavw2 36480 | Change bound variable and ... |
| cbvrabdavw2 36481 | Change bound variable and ... |
| cbviundavw2 36482 | Change bound variable and ... |
| cbviindavw2 36483 | Change bound variable and ... |
| cbvmptdavw2 36484 | Change bound variable and ... |
| cbvdisjdavw2 36485 | Change bound variable and ... |
| cbvriotadavw2 36486 | Change bound variable and ... |
| cbvmpodavw2 36487 | Change bound variable and ... |
| cbvmpo1davw2 36488 | Change first bound variabl... |
| cbvmpo2davw2 36489 | Change second bound variab... |
| cbvixpdavw2 36490 | Change bound variable and ... |
| cbvsumdavw2 36491 | Change bound variable and ... |
| cbvproddavw2 36492 | Change bound variable and ... |
| cbvitgdavw2 36493 | Change bound variable and ... |
| cbvditgdavw2 36494 | Change bound variable and ... |
| mpomulnzcnf 36495 | Multiplication maps nonzer... |
| a1i14 36496 | Add two antecedents to a w... |
| a1i24 36497 | Add two antecedents to a w... |
| exp5d 36498 | An exportation inference. ... |
| exp5g 36499 | An exportation inference. ... |
| exp5k 36500 | An exportation inference. ... |
| exp56 36501 | An exportation inference. ... |
| exp58 36502 | An exportation inference. ... |
| exp510 36503 | An exportation inference. ... |
| exp511 36504 | An exportation inference. ... |
| exp512 36505 | An exportation inference. ... |
| 3com12d 36506 | Commutation in consequent.... |
| imp5p 36507 | A triple importation infer... |
| imp5q 36508 | A triple importation infer... |
| ecase13d 36509 | Deduction for elimination ... |
| subtr 36510 | Transitivity of implicit s... |
| subtr2 36511 | Transitivity of implicit s... |
| trer 36512 | A relation intersected wit... |
| elicc3 36513 | An equivalent membership c... |
| finminlem 36514 | A useful lemma about finit... |
| gtinf 36515 | Any number greater than an... |
| opnrebl 36516 | A set is open in the stand... |
| opnrebl2 36517 | A set is open in the stand... |
| nn0prpwlem 36518 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 36519 | Two nonnegative integers a... |
| topbnd 36520 | Two equivalent expressions... |
| opnbnd 36521 | A set is open iff it is di... |
| cldbnd 36522 | A set is closed iff it con... |
| ntruni 36523 | A union of interiors is a ... |
| clsun 36524 | A pairwise union of closur... |
| clsint2 36525 | The closure of an intersec... |
| opnregcld 36526 | A set is regularly closed ... |
| cldregopn 36527 | A set if regularly open if... |
| neiin 36528 | Two neighborhoods intersec... |
| hmeoclda 36529 | Homeomorphisms preserve cl... |
| hmeocldb 36530 | Homeomorphisms preserve cl... |
| ivthALT 36531 | An alternate proof of the ... |
| fnerel 36534 | Fineness is a relation. (... |
| isfne 36535 | The predicate " ` B ` is f... |
| isfne4 36536 | The predicate " ` B ` is f... |
| isfne4b 36537 | A condition for a topology... |
| isfne2 36538 | The predicate " ` B ` is f... |
| isfne3 36539 | The predicate " ` B ` is f... |
| fnebas 36540 | A finer cover covers the s... |
| fnetg 36541 | A finer cover generates a ... |
| fnessex 36542 | If ` B ` is finer than ` A... |
| fneuni 36543 | If ` B ` is finer than ` A... |
| fneint 36544 | If a cover is finer than a... |
| fness 36545 | A cover is finer than its ... |
| fneref 36546 | Reflexivity of the finenes... |
| fnetr 36547 | Transitivity of the finene... |
| fneval 36548 | Two covers are finer than ... |
| fneer 36549 | Fineness intersected with ... |
| topfne 36550 | Fineness for covers corres... |
| topfneec 36551 | A cover is equivalent to a... |
| topfneec2 36552 | A topology is precisely id... |
| fnessref 36553 | A cover is finer iff it ha... |
| refssfne 36554 | A cover is a refinement if... |
| neibastop1 36555 | A collection of neighborho... |
| neibastop2lem 36556 | Lemma for ~ neibastop2 . ... |
| neibastop2 36557 | In the topology generated ... |
| neibastop3 36558 | The topology generated by ... |
| topmtcl 36559 | The meet of a collection o... |
| topmeet 36560 | Two equivalent formulation... |
| topjoin 36561 | Two equivalent formulation... |
| fnemeet1 36562 | The meet of a collection o... |
| fnemeet2 36563 | The meet of equivalence cl... |
| fnejoin1 36564 | Join of equivalence classe... |
| fnejoin2 36565 | Join of equivalence classe... |
| fgmin 36566 | Minimality property of a g... |
| neifg 36567 | The neighborhood filter of... |
| tailfval 36568 | The tail function for a di... |
| tailval 36569 | The tail of an element in ... |
| eltail 36570 | An element of a tail. (Co... |
| tailf 36571 | The tail function of a dir... |
| tailini 36572 | A tail contains its initia... |
| tailfb 36573 | The collection of tails of... |
| filnetlem1 36574 | Lemma for ~ filnet . Chan... |
| filnetlem2 36575 | Lemma for ~ filnet . The ... |
| filnetlem3 36576 | Lemma for ~ filnet . (Con... |
| filnetlem4 36577 | Lemma for ~ filnet . (Con... |
| filnet 36578 | A filter has the same conv... |
| tb-ax1 36579 | The first of three axioms ... |
| tb-ax2 36580 | The second of three axioms... |
| tb-ax3 36581 | The third of three axioms ... |
| tbsyl 36582 | The weak syllogism from Ta... |
| re1ax2lem 36583 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 36584 | ~ ax-2 rederived from the ... |
| naim1 36585 | Constructor theorem for ` ... |
| naim2 36586 | Constructor theorem for ` ... |
| naim1i 36587 | Constructor rule for ` -/\... |
| naim2i 36588 | Constructor rule for ` -/\... |
| naim12i 36589 | Constructor rule for ` -/\... |
| nabi1i 36590 | Constructor rule for ` -/\... |
| nabi2i 36591 | Constructor rule for ` -/\... |
| nabi12i 36592 | Constructor rule for ` -/\... |
| df3nandALT1 36595 | The double nand expressed ... |
| df3nandALT2 36596 | The double nand expressed ... |
| andnand1 36597 | Double and in terms of dou... |
| imnand2 36598 | An ` -> ` nand relation. ... |
| nalfal 36599 | Not all sets hold ` F. ` a... |
| nexntru 36600 | There does not exist a set... |
| nexfal 36601 | There does not exist a set... |
| neufal 36602 | There does not exist exact... |
| neutru 36603 | There does not exist exact... |
| nmotru 36604 | There does not exist at mo... |
| mofal 36605 | There exist at most one se... |
| nrmo 36606 | "At most one" restricted e... |
| meran1 36607 | A single axiom for proposi... |
| meran2 36608 | A single axiom for proposi... |
| meran3 36609 | A single axiom for proposi... |
| waj-ax 36610 | A single axiom for proposi... |
| lukshef-ax2 36611 | A single axiom for proposi... |
| arg-ax 36612 | A single axiom for proposi... |
| negsym1 36613 | In the paper "On Variable ... |
| imsym1 36614 | A symmetry with ` -> ` . ... |
| bisym1 36615 | A symmetry with ` <-> ` . ... |
| consym1 36616 | A symmetry with ` /\ ` . ... |
| dissym1 36617 | A symmetry with ` \/ ` . ... |
| nandsym1 36618 | A symmetry with ` -/\ ` . ... |
| unisym1 36619 | A symmetry with ` A. ` . ... |
| exisym1 36620 | A symmetry with ` E. ` . ... |
| unqsym1 36621 | A symmetry with ` E! ` . ... |
| amosym1 36622 | A symmetry with ` E* ` . ... |
| subsym1 36623 | A symmetry with ` [ x / y ... |
| ontopbas 36624 | An ordinal number is a top... |
| onsstopbas 36625 | The class of ordinal numbe... |
| onpsstopbas 36626 | The class of ordinal numbe... |
| ontgval 36627 | The topology generated fro... |
| ontgsucval 36628 | The topology generated fro... |
| onsuctop 36629 | A successor ordinal number... |
| onsuctopon 36630 | One of the topologies on a... |
| ordtoplem 36631 | Membership of the class of... |
| ordtop 36632 | An ordinal is a topology i... |
| onsucconni 36633 | A successor ordinal number... |
| onsucconn 36634 | A successor ordinal number... |
| ordtopconn 36635 | An ordinal topology is con... |
| onintopssconn 36636 | An ordinal topology is con... |
| onsuct0 36637 | A successor ordinal number... |
| ordtopt0 36638 | An ordinal topology is T_0... |
| onsucsuccmpi 36639 | The successor of a success... |
| onsucsuccmp 36640 | The successor of a success... |
| limsucncmpi 36641 | The successor of a limit o... |
| limsucncmp 36642 | The successor of a limit o... |
| ordcmp 36643 | An ordinal topology is com... |
| ssoninhaus 36644 | The ordinal topologies ` 1... |
| onint1 36645 | The ordinal T_1 spaces are... |
| oninhaus 36646 | The ordinal Hausdorff spac... |
| fveleq 36647 | Please add description her... |
| findfvcl 36648 | Please add description her... |
| findreccl 36649 | Please add description her... |
| findabrcl 36650 | Please add description her... |
| nnssi2 36651 | Convert a theorem for real... |
| nnssi3 36652 | Convert a theorem for real... |
| nndivsub 36653 | Please add description her... |
| nndivlub 36654 | A factor of a positive int... |
| ee7.2aOLD 36657 | Lemma for Euclid's Element... |
| weiunval 36658 | Value of the relation cons... |
| weiunlem 36659 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 36660 | Lemma for ~ weiunfr . (Co... |
| weiunpo 36661 | A partial ordering on an i... |
| weiunso 36662 | A strict ordering on an in... |
| weiunfr 36663 | A well-founded relation on... |
| weiunse 36664 | The relation constructed i... |
| weiunwe 36665 | A well-ordering on an inde... |
| numiunnum 36666 | An indexed union of sets i... |
| exeltr 36667 | Every set is a member of a... |
| mh-setind 36668 | Principle of set induction... |
| mh-setindnd 36669 | A version of ~ mh-setind w... |
| regsfromregtr 36670 | Derivation of ~ ax-regs fr... |
| regsfromsetind 36671 | Derivation of ~ ax-regs fr... |
| regsfromunir1 36672 | Derivation of ~ ax-regs fr... |
| dnival 36673 | Value of the "distance to ... |
| dnicld1 36674 | Closure theorem for the "d... |
| dnicld2 36675 | Closure theorem for the "d... |
| dnif 36676 | The "distance to nearest i... |
| dnizeq0 36677 | The distance to nearest in... |
| dnizphlfeqhlf 36678 | The distance to nearest in... |
| rddif2 36679 | Variant of ~ rddif . (Con... |
| dnibndlem1 36680 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 36681 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 36682 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 36683 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 36684 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 36685 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 36686 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 36687 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 36688 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 36689 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 36690 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 36691 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 36692 | Lemma for ~ dnibnd . (Con... |
| dnibnd 36693 | The "distance to nearest i... |
| dnicn 36694 | The "distance to nearest i... |
| knoppcnlem1 36695 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 36696 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 36697 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 36698 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 36699 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 36700 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 36701 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 36702 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 36703 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 36704 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 36705 | Lemma for ~ knoppcn . (Co... |
| knoppcn 36706 | The continuous nowhere dif... |
| knoppcld 36707 | Closure theorem for Knopp'... |
| unblimceq0lem 36708 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 36709 | If ` F ` is unbounded near... |
| unbdqndv1 36710 | If the difference quotient... |
| unbdqndv2lem1 36711 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 36712 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 36713 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 36714 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 36715 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 36716 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 36717 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 36718 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 36719 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 36720 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 36721 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 36722 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 36723 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 36724 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 36725 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 36726 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 36727 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 36728 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 36729 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 36730 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 36731 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 36732 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 36733 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 36734 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 36735 | Lemma for ~ knoppndv . (C... |
| knoppndv 36736 | The continuous nowhere dif... |
| knoppf 36737 | Knopp's function is a func... |
| knoppcn2 36738 | Variant of ~ knoppcn with ... |
| cnndvlem1 36739 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 36740 | Lemma for ~ cnndv . (Cont... |
| cnndv 36741 | There exists a continuous ... |
| bj-mp2c 36742 | A double _modus ponens_ in... |
| bj-mp2d 36743 | A double _modus ponens_ in... |
| bj-0 36744 | A syntactic theorem. See ... |
| bj-1 36745 | In this proof, the use of ... |
| bj-a1k 36746 | Weakening of ~ ax-1 . As ... |
| bj-poni 36747 | Inference associated with ... |
| bj-nnclav 36748 | When ` F. ` is substituted... |
| bj-nnclavi 36749 | Inference associated with ... |
| bj-nnclavc 36750 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 36751 | Inference associated with ... |
| bj-jarrii 36752 | Inference associated with ... |
| bj-imim21 36753 | The propositional function... |
| bj-imim21i 36754 | Inference associated with ... |
| bj-peircestab 36755 | Over minimal implicational... |
| bj-stabpeirce 36756 | This minimal implicational... |
| bj-syl66ib 36757 | A mixed syllogism inferenc... |
| bj-orim2 36758 | Proof of ~ orim2 from the ... |
| bj-currypeirce 36759 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 36760 | Peirce's axiom ~ peirce im... |
| bj-animbi 36761 | Conjunction in terms of im... |
| bj-currypara 36762 | Curry's paradox. Note tha... |
| bj-con2com 36763 | A commuted form of the con... |
| bj-con2comi 36764 | Inference associated with ... |
| bj-nimn 36765 | If a formula is true, then... |
| bj-nimni 36766 | Inference associated with ... |
| bj-peircei 36767 | Inference associated with ... |
| bj-looinvi 36768 | Inference associated with ... |
| bj-looinvii 36769 | Inference associated with ... |
| bj-mt2bi 36770 | Version of ~ mt2 where the... |
| bj-ntrufal 36771 | The negation of a theorem ... |
| bj-fal 36772 | Shortening of ~ fal using ... |
| bj-jaoi1 36773 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 36774 | Shortens ~ consensus (110>... |
| bj-dfbi4 36775 | Alternate definition of th... |
| bj-dfbi5 36776 | Alternate definition of th... |
| bj-dfbi6 36777 | Alternate definition of th... |
| bj-bijust0ALT 36778 | Alternate proof of ~ bijus... |
| bj-bijust00 36779 | A self-implication does no... |
| bj-consensus 36780 | Version of ~ consensus exp... |
| bj-consensusALT 36781 | Alternate proof of ~ bj-co... |
| bj-df-ifc 36782 | Candidate definition for t... |
| bj-dfif 36783 | Alternate definition of th... |
| bj-ififc 36784 | A biconditional connecting... |
| bj-imbi12 36785 | Uncurried (imported) form ... |
| bj-falor 36786 | Dual of ~ truan (which has... |
| bj-falor2 36787 | Dual of ~ truan . (Contri... |
| bj-bibibi 36788 | A property of the bicondit... |
| bj-imn3ani 36789 | Duplication of ~ bnj1224 .... |
| bj-andnotim 36790 | Two ways of expressing a c... |
| bj-bi3ant 36791 | This used to be in the mai... |
| bj-bisym 36792 | This used to be in the mai... |
| bj-bixor 36793 | Equivalence of two ternary... |
| bj-axdd2 36794 | This implication, proved u... |
| bj-axd2d 36795 | This implication, proved u... |
| bj-axtd 36796 | This implication, proved f... |
| bj-gl4 36797 | In a normal modal logic, t... |
| bj-axc4 36798 | Over minimal calculus, the... |
| prvlem1 36803 | An elementary property of ... |
| prvlem2 36804 | An elementary property of ... |
| bj-babygodel 36805 | See the section header com... |
| bj-babylob 36806 | See the section header com... |
| bj-godellob 36807 | Proof of Gödel's theo... |
| bj-genr 36808 | Generalization rule on the... |
| bj-genl 36809 | Generalization rule on the... |
| bj-genan 36810 | Generalization rule on a c... |
| bj-mpgs 36811 | From a closed form theorem... |
| bj-2alim 36812 | Closed form of ~ 2alimi . ... |
| bj-2exim 36813 | Closed form of ~ 2eximi . ... |
| bj-alanim 36814 | Closed form of ~ alanimi .... |
| bj-2albi 36815 | Closed form of ~ 2albii . ... |
| bj-notalbii 36816 | Equivalence of universal q... |
| bj-2exbi 36817 | Closed form of ~ 2exbii . ... |
| bj-3exbi 36818 | Closed form of ~ 3exbii . ... |
| bj-sylggt 36819 | Stronger form of ~ sylgt ,... |
| bj-sylgt2 36820 | Uncurried (imported) form ... |
| bj-alrimg 36821 | The general form of the *a... |
| bj-alrimd 36822 | A slightly more general ~ ... |
| bj-sylget 36823 | Dual statement of ~ sylgt ... |
| bj-sylget2 36824 | Uncurried (imported) form ... |
| bj-exlimg 36825 | The general form of the *e... |
| bj-sylge 36826 | Dual statement of ~ sylg (... |
| bj-exlimd 36827 | A slightly more general ~ ... |
| bj-nfimexal 36828 | A weak from of nonfreeness... |
| bj-alexim 36829 | Closed form of ~ aleximi .... |
| bj-nexdh 36830 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 36831 | Uncurried (imported) form ... |
| bj-hbxfrbi 36832 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 36833 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 36834 | Distribute quantifiers ove... |
| bj-exalimi 36835 | An inference for distribut... |
| bj-exalims 36836 | Distributing quantifiers o... |
| bj-exalimsi 36837 | An inference for distribut... |
| bj-ax12ig 36838 | A lemma used to prove a we... |
| bj-ax12i 36839 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 36840 | Closed form of ~ nfim and ... |
| bj-cbvalimt 36841 | A lemma in closed form use... |
| bj-cbveximt 36842 | A lemma in closed form use... |
| bj-eximALT 36843 | Alternate proof of ~ exim ... |
| bj-aleximiALT 36844 | Alternate proof of ~ alexi... |
| bj-eximcom 36845 | A commuted form of ~ exim ... |
| bj-ax12wlem 36846 | A lemma used to prove a we... |
| bj-cbvalim 36847 | A lemma used to prove ~ bj... |
| bj-cbvexim 36848 | A lemma used to prove ~ bj... |
| bj-cbvalimi 36849 | An equality-free general i... |
| bj-cbveximi 36850 | An equality-free general i... |
| bj-cbval 36851 | Changing a bound variable ... |
| bj-cbvex 36852 | Changing a bound variable ... |
| bj-df-sb 36855 | Proposed definition to rep... |
| bj-ssbeq 36856 | Substitution in an equalit... |
| bj-ssblem1 36857 | A lemma for the definiens ... |
| bj-ssblem2 36858 | An instance of ~ ax-11 pro... |
| bj-ax12v 36859 | A weaker form of ~ ax-12 a... |
| bj-ax12 36860 | Remove a DV condition from... |
| bj-ax12ssb 36861 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 36862 | Special case of ~ 19.41 pr... |
| bj-equsexval 36863 | Special case of ~ equsexv ... |
| bj-subst 36864 | Proof of ~ sbalex from cor... |
| bj-ssbid2 36865 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 36866 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 36867 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 36868 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 36869 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 36870 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 36871 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 36872 | Closed form of ~ spimvw . ... |
| bj-spnfw 36873 | Theorem close to a closed ... |
| bj-cbvexiw 36874 | Change bound variable. Th... |
| bj-cbvexivw 36875 | Change bound variable. Th... |
| bj-modald 36876 | A short form of the axiom ... |
| bj-denot 36877 | A weakening of ~ ax-6 and ... |
| bj-eqs 36878 | A lemma for substitutions,... |
| bj-cbvexw 36879 | Change bound variable. Th... |
| bj-ax12w 36880 | The general statement that... |
| bj-ax89 36881 | A theorem which could be u... |
| bj-cleljusti 36882 | One direction of ~ cleljus... |
| bj-alcomexcom 36883 | Commutation of two existen... |
| bj-hbalt 36884 | Closed form of ~ hbal . W... |
| axc11n11 36885 | Proof of ~ axc11n from { ~... |
| axc11n11r 36886 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 36887 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 36888 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 36889 | Alternate proof of ~ bj-ax... |
| bj-sb 36890 | A weak variant of ~ sbid2 ... |
| bj-modalbe 36891 | The predicate-calculus ver... |
| bj-spst 36892 | Closed form of ~ sps . On... |
| bj-19.21bit 36893 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 36894 | Closed form of ~ 19.23bi .... |
| bj-nexrt 36895 | Closed form of ~ nexr . C... |
| bj-alrim 36896 | Closed form of ~ alrimi . ... |
| bj-alrim2 36897 | Uncurried (imported) form ... |
| bj-nfdt0 36898 | A theorem close to a close... |
| bj-nfdt 36899 | Closed form of ~ nf5d and ... |
| bj-nexdt 36900 | Closed form of ~ nexd . (... |
| bj-nexdvt 36901 | Closed form of ~ nexdv . ... |
| bj-alexbiex 36902 | Adding a second quantifier... |
| bj-exexbiex 36903 | Adding a second quantifier... |
| bj-alalbial 36904 | Adding a second quantifier... |
| bj-exalbial 36905 | Adding a second quantifier... |
| bj-19.9htbi 36906 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 36907 | Strengthening ~ hbnt by re... |
| bj-biexal1 36908 | A general FOL biconditiona... |
| bj-biexal2 36909 | When ` ph ` is substituted... |
| bj-biexal3 36910 | When ` ph ` is substituted... |
| bj-bialal 36911 | When ` ph ` is substituted... |
| bj-biexex 36912 | When ` ph ` is substituted... |
| bj-hbext 36913 | Closed form of ~ hbex . (... |
| bj-nfalt 36914 | Closed form of ~ nfal . (... |
| bj-nfext 36915 | Closed form of ~ nfex . (... |
| bj-eeanvw 36916 | Version of ~ exdistrv with... |
| bj-modal4 36917 | First-order logic form of ... |
| bj-modal4e 36918 | First-order logic form of ... |
| bj-modalb 36919 | A short form of the axiom ... |
| bj-wnf1 36920 | When ` ph ` is substituted... |
| bj-wnf2 36921 | When ` ph ` is substituted... |
| bj-wnfanf 36922 | When ` ph ` is substituted... |
| bj-wnfenf 36923 | When ` ph ` is substituted... |
| bj-substax12 36924 | Equivalent form of the axi... |
| bj-substw 36925 | Weak form of the LHS of ~ ... |
| bj-nnfbi 36928 | If two formulas are equiva... |
| bj-nnfbd 36929 | If two formulas are equiva... |
| bj-nnfbii 36930 | If two formulas are equiva... |
| bj-nnfa 36931 | Nonfreeness implies the eq... |
| bj-nnfad 36932 | Nonfreeness implies the eq... |
| bj-nnfai 36933 | Nonfreeness implies the eq... |
| bj-nnfe 36934 | Nonfreeness implies the eq... |
| bj-nnfed 36935 | Nonfreeness implies the eq... |
| bj-nnfei 36936 | Nonfreeness implies the eq... |
| bj-nnfea 36937 | Nonfreeness implies the eq... |
| bj-nnfead 36938 | Nonfreeness implies the eq... |
| bj-nnfeai 36939 | Nonfreeness implies the eq... |
| bj-dfnnf2 36940 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 36941 | New nonfreeness implies ol... |
| bj-wnfnf 36942 | When ` ph ` is substituted... |
| bj-nnfnt 36943 | A variable is nonfree in a... |
| bj-nnftht 36944 | A variable is nonfree in a... |
| bj-nnfth 36945 | A variable is nonfree in a... |
| bj-nnfnth 36946 | A variable is nonfree in t... |
| bj-nnfim1 36947 | A consequence of nonfreene... |
| bj-nnfim2 36948 | A consequence of nonfreene... |
| bj-nnfim 36949 | Nonfreeness in the anteced... |
| bj-nnfimd 36950 | Nonfreeness in the anteced... |
| bj-nnfan 36951 | Nonfreeness in both conjun... |
| bj-nnfand 36952 | Nonfreeness in both conjun... |
| bj-nnfor 36953 | Nonfreeness in both disjun... |
| bj-nnford 36954 | Nonfreeness in both disjun... |
| bj-nnfbit 36955 | Nonfreeness in both sides ... |
| bj-nnfbid 36956 | Nonfreeness in both sides ... |
| bj-nnfv 36957 | A non-occurring variable i... |
| bj-nnf-alrim 36958 | Proof of the closed form o... |
| bj-nnf-exlim 36959 | Proof of the closed form o... |
| bj-dfnnf3 36960 | Alternate definition of no... |
| bj-nfnnfTEMP 36961 | New nonfreeness is equival... |
| bj-nnfa1 36962 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 36963 | See ~ nfe1 . (Contributed... |
| bj-19.12 36964 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 36965 | One of four lemmas for non... |
| bj-nnflemee 36966 | One of four lemmas for non... |
| bj-nnflemae 36967 | One of four lemmas for non... |
| bj-nnflemea 36968 | One of four lemmas for non... |
| bj-nnfalt 36969 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 36970 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 36971 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 36972 | Statement ~ 19.21t proved ... |
| bj-19.23t 36973 | Statement ~ 19.23t proved ... |
| bj-19.36im 36974 | One direction of ~ 19.36 f... |
| bj-19.37im 36975 | One direction of ~ 19.37 f... |
| bj-19.42t 36976 | Closed form of ~ 19.42 fro... |
| bj-19.41t 36977 | Closed form of ~ 19.41 fro... |
| bj-sbft 36978 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 36979 | Version of ~ pm11.53v with... |
| bj-pm11.53v 36980 | Version of ~ pm11.53v with... |
| bj-pm11.53a 36981 | A variant of ~ pm11.53v . ... |
| bj-equsvt 36982 | A variant of ~ equsv . (C... |
| bj-equsalvwd 36983 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 36984 | Variant of ~ equsexvw . (... |
| bj-sbievwd 36985 | Variant of ~ sbievw . (Co... |
| bj-axc10 36986 | Alternate proof of ~ axc10... |
| bj-alequex 36987 | A fol lemma. See ~ aleque... |
| bj-spimt2 36988 | A step in the proof of ~ s... |
| bj-cbv3ta 36989 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 36990 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 36991 | A theorem close to a close... |
| bj-hbsb3 36992 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 36993 | A theorem close to a close... |
| bj-nfs1t2 36994 | A theorem close to a close... |
| bj-nfs1 36995 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 36996 | Version of ~ axc10 with a ... |
| bj-spimtv 36997 | Version of ~ spimt with a ... |
| bj-cbv3hv2 36998 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 36999 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 37000 | Version of ~ cbv2h with a ... |
| bj-cbv2v 37001 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 37002 | Version of ~ cbvald with a... |
| bj-cbvexdv 37003 | Version of ~ cbvexd with a... |
| bj-cbval2vv 37004 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 37005 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 37006 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 37007 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 37008 | Version of ~ cbvex4v with ... |
| bj-equsalhv 37009 | Version of ~ equsalh with ... |
| bj-axc11nv 37010 | Version of ~ axc11n with a... |
| bj-aecomsv 37011 | Version of ~ aecoms with a... |
| bj-axc11v 37012 | Version of ~ axc11 with a ... |
| bj-drnf2v 37013 | Version of ~ drnf2 with a ... |
| bj-equs45fv 37014 | Version of ~ equs45f with ... |
| bj-hbs1 37015 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 37016 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 37017 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 37018 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 37019 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 37020 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 37021 | Biconditional version of a... |
| bj-hbaeb 37022 | Biconditional version of ~... |
| bj-hbnaeb 37023 | Biconditional version of ~... |
| bj-dvv 37024 | A special instance of ~ bj... |
| bj-equsal1t 37025 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 37026 | Inference associated with ... |
| bj-equsal1 37027 | One direction of ~ equsal ... |
| bj-equsal2 37028 | One direction of ~ equsal ... |
| bj-equsal 37029 | Shorter proof of ~ equsal ... |
| stdpc5t 37030 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 37031 | More direct proof of ~ std... |
| 2stdpc5 37032 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 37033 | Proof of ~ 19.21t from ~ s... |
| exlimii 37034 | Inference associated with ... |
| ax11-pm 37035 | Proof of ~ ax-11 similar t... |
| ax6er 37036 | Commuted form of ~ ax6e . ... |
| exlimiieq1 37037 | Inferring a theorem when i... |
| exlimiieq2 37038 | Inferring a theorem when i... |
| ax11-pm2 37039 | Proof of ~ ax-11 from the ... |
| bj-sbsb 37040 | Biconditional showing two ... |
| bj-dfsb2 37041 | Alternate (dual) definitio... |
| bj-sbf3 37042 | Substitution has no effect... |
| bj-sbf4 37043 | Substitution has no effect... |
| bj-eu3f 37044 | Version of ~ eu3v where th... |
| bj-sblem1 37045 | Lemma for substitution. (... |
| bj-sblem2 37046 | Lemma for substitution. (... |
| bj-sblem 37047 | Lemma for substitution. (... |
| bj-sbievw1 37048 | Lemma for substitution. (... |
| bj-sbievw2 37049 | Lemma for substitution. (... |
| bj-sbievw 37050 | Lemma for substitution. C... |
| bj-sbievv 37051 | Version of ~ sbie with a s... |
| bj-moeub 37052 | Uniqueness is equivalent t... |
| bj-sbidmOLD 37053 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 37054 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 37055 | Curried (exported) form of... |
| bj-dvelimv 37056 | A version of ~ dvelim usin... |
| bj-nfeel2 37057 | Nonfreeness in a membershi... |
| bj-axc14nf 37058 | Proof of a version of ~ ax... |
| bj-axc14 37059 | Alternate proof of ~ axc14... |
| mobidvALT 37060 | Alternate proof of ~ mobid... |
| sbn1ALT 37061 | Alternate proof of ~ sbn1 ... |
| eliminable1 37062 | A theorem used to prove th... |
| eliminable2a 37063 | A theorem used to prove th... |
| eliminable2b 37064 | A theorem used to prove th... |
| eliminable2c 37065 | A theorem used to prove th... |
| eliminable3a 37066 | A theorem used to prove th... |
| eliminable3b 37067 | A theorem used to prove th... |
| eliminable-velab 37068 | A theorem used to prove th... |
| eliminable-veqab 37069 | A theorem used to prove th... |
| eliminable-abeqv 37070 | A theorem used to prove th... |
| eliminable-abeqab 37071 | A theorem used to prove th... |
| eliminable-abelv 37072 | A theorem used to prove th... |
| eliminable-abelab 37073 | A theorem used to prove th... |
| bj-denoteslem 37074 | Duplicate of ~ issettru an... |
| bj-denotesALTV 37075 | Moved to main as ~ iseqset... |
| bj-issettruALTV 37076 | Moved to main as ~ issettr... |
| bj-elabtru 37077 | This is as close as we can... |
| bj-issetwt 37078 | Closed form of ~ bj-issetw... |
| bj-issetw 37079 | The closest one can get to... |
| bj-issetiv 37080 | Version of ~ bj-isseti wit... |
| bj-isseti 37081 | Version of ~ isseti with a... |
| bj-ralvw 37082 | A weak version of ~ ralv n... |
| bj-rexvw 37083 | A weak version of ~ rexv n... |
| bj-rababw 37084 | A weak version of ~ rabab ... |
| bj-rexcom4bv 37085 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 37086 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 37087 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 37088 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 37089 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 37090 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 37091 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 37092 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 37093 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 37094 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 37095 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 37096 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 37097 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 37098 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 37099 | Remove from ~ spcimdv depe... |
| elelb 37100 | Equivalence between two co... |
| bj-pwvrelb 37101 | Characterization of the el... |
| bj-nfcsym 37102 | The nonfreeness quantifier... |
| bj-sbeqALT 37103 | Substitution in an equalit... |
| bj-sbeq 37104 | Distribute proper substitu... |
| bj-sbceqgALT 37105 | Distribute proper substitu... |
| bj-csbsnlem 37106 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 37107 | Substitution in a singleto... |
| bj-sbel1 37108 | Version of ~ sbcel1g when ... |
| bj-abv 37109 | The class of sets verifyin... |
| bj-abvALT 37110 | Alternate version of ~ bj-... |
| bj-ab0 37111 | The class of sets verifyin... |
| bj-abf 37112 | Shorter proof of ~ abf (wh... |
| bj-csbprc 37113 | More direct proof of ~ csb... |
| bj-exlimvmpi 37114 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 37115 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 37116 | Lemma for theorems of the ... |
| bj-exlimmpbir 37117 | Lemma for theorems of the ... |
| bj-vtoclf 37118 | Remove dependency on ~ ax-... |
| bj-vtocl 37119 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 37120 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 37121 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 37122 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 37123 | A version of ~ vtoclg with... |
| bj-rabeqbid 37124 | Version of ~ rabeqbidv wit... |
| bj-seex 37125 | Version of ~ seex with a d... |
| bj-nfcf 37126 | Version of ~ df-nfc with a... |
| bj-zfauscl 37127 | General version of ~ zfaus... |
| bj-elabd2ALT 37128 | Alternate proof of ~ elabd... |
| bj-unrab 37129 | Generalization of ~ unrab ... |
| bj-inrab 37130 | Generalization of ~ inrab ... |
| bj-inrab2 37131 | Shorter proof of ~ inrab .... |
| bj-inrab3 37132 | Generalization of ~ dfrab3... |
| bj-rabtr 37133 | Restricted class abstracti... |
| bj-rabtrALT 37134 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 37135 | Proof of ~ bj-rabtr found ... |
| bj-gabss 37138 | Inclusion of generalized c... |
| bj-gabssd 37139 | Inclusion of generalized c... |
| bj-gabeqd 37140 | Equality of generalized cl... |
| bj-gabeqis 37141 | Equality of generalized cl... |
| bj-elgab 37142 | Elements of a generalized ... |
| bj-gabima 37143 | Generalized class abstract... |
| bj-ru1 37146 | A version of Russell's par... |
| bj-ru 37147 | Remove dependency on ~ ax-... |
| currysetlem 37148 | Lemma for ~ currysetlem , ... |
| curryset 37149 | Curry's paradox in set the... |
| currysetlem1 37150 | Lemma for ~ currysetALT . ... |
| currysetlem2 37151 | Lemma for ~ currysetALT . ... |
| currysetlem3 37152 | Lemma for ~ currysetALT . ... |
| currysetALT 37153 | Alternate proof of ~ curry... |
| bj-n0i 37154 | Inference associated with ... |
| bj-disjsn01 37155 | Disjointness of the single... |
| bj-0nel1 37156 | The empty set does not bel... |
| bj-1nel0 37157 | ` 1o ` does not belong to ... |
| bj-xpimasn 37158 | The image of a singleton, ... |
| bj-xpima1sn 37159 | The image of a singleton b... |
| bj-xpima1snALT 37160 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 37161 | The image of a singleton b... |
| bj-xpnzex 37162 | If the first factor of a p... |
| bj-xpexg2 37163 | Curried (exported) form of... |
| bj-xpnzexb 37164 | If the first factor of a p... |
| bj-cleq 37165 | Substitution property for ... |
| bj-snsetex 37166 | The class of sets "whose s... |
| bj-clexab 37167 | Sethood of certain classes... |
| bj-sngleq 37170 | Substitution property for ... |
| bj-elsngl 37171 | Characterization of the el... |
| bj-snglc 37172 | Characterization of the el... |
| bj-snglss 37173 | The singletonization of a ... |
| bj-0nelsngl 37174 | The empty set is not a mem... |
| bj-snglinv 37175 | Inverse of singletonizatio... |
| bj-snglex 37176 | A class is a set if and on... |
| bj-tageq 37179 | Substitution property for ... |
| bj-eltag 37180 | Characterization of the el... |
| bj-0eltag 37181 | The empty set belongs to t... |
| bj-tagn0 37182 | The tagging of a class is ... |
| bj-tagss 37183 | The tagging of a class is ... |
| bj-snglsstag 37184 | The singletonization is in... |
| bj-sngltagi 37185 | The singletonization is in... |
| bj-sngltag 37186 | The singletonization and t... |
| bj-tagci 37187 | Characterization of the el... |
| bj-tagcg 37188 | Characterization of the el... |
| bj-taginv 37189 | Inverse of tagging. (Cont... |
| bj-tagex 37190 | A class is a set if and on... |
| bj-xtageq 37191 | The products of a given cl... |
| bj-xtagex 37192 | The product of a set and t... |
| bj-projeq 37195 | Substitution property for ... |
| bj-projeq2 37196 | Substitution property for ... |
| bj-projun 37197 | The class projection on a ... |
| bj-projex 37198 | Sethood of the class proje... |
| bj-projval 37199 | Value of the class project... |
| bj-1upleq 37202 | Substitution property for ... |
| bj-pr1eq 37205 | Substitution property for ... |
| bj-pr1un 37206 | The first projection prese... |
| bj-pr1val 37207 | Value of the first project... |
| bj-pr11val 37208 | Value of the first project... |
| bj-pr1ex 37209 | Sethood of the first proje... |
| bj-1uplth 37210 | The characteristic propert... |
| bj-1uplex 37211 | A monuple is a set if and ... |
| bj-1upln0 37212 | A monuple is nonempty. (C... |
| bj-2upleq 37215 | Substitution property for ... |
| bj-pr21val 37216 | Value of the first project... |
| bj-pr2eq 37219 | Substitution property for ... |
| bj-pr2un 37220 | The second projection pres... |
| bj-pr2val 37221 | Value of the second projec... |
| bj-pr22val 37222 | Value of the second projec... |
| bj-pr2ex 37223 | Sethood of the second proj... |
| bj-2uplth 37224 | The characteristic propert... |
| bj-2uplex 37225 | A couple is a set if and o... |
| bj-2upln0 37226 | A couple is nonempty. (Co... |
| bj-2upln1upl 37227 | A couple is never equal to... |
| bj-rcleqf 37228 | Relative version of ~ cleq... |
| bj-rcleq 37229 | Relative version of ~ dfcl... |
| bj-reabeq 37230 | Relative form of ~ eqabb .... |
| bj-disj2r 37231 | Relative version of ~ ssdi... |
| bj-sscon 37232 | Contraposition law for rel... |
| bj-abex 37233 | Two ways of stating that t... |
| bj-clex 37234 | Two ways of stating that a... |
| bj-axsn 37235 | Two ways of stating the ax... |
| bj-snexg 37237 | A singleton built on a set... |
| bj-snex 37238 | A singleton is a set. See... |
| bj-axbun 37239 | Two ways of stating the ax... |
| bj-unexg 37241 | Existence of binary unions... |
| bj-prexg 37242 | Existence of unordered pai... |
| bj-prex 37243 | Existence of unordered pai... |
| bj-axadj 37244 | Two ways of stating the ax... |
| bj-adjg1 37246 | Existence of the result of... |
| bj-snfromadj 37247 | Singleton from adjunction ... |
| bj-prfromadj 37248 | Unordered pair from adjunc... |
| bj-adjfrombun 37249 | Adjunction from singleton ... |
| eleq2w2ALT 37250 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 37251 | Alternate proof of ~ clel3... |
| bj-pw0ALT 37252 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 37253 | Quantitative version of ~ ... |
| bj-unirel 37254 | Quantitative version of ~ ... |
| bj-elpwg 37255 | If the intersection of two... |
| bj-velpwALT 37256 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 37257 | Alternate proof of ~ elpwg... |
| bj-vjust 37258 | Justification theorem for ... |
| bj-nul 37259 | Two formulations of the ax... |
| bj-nuliota 37260 | Definition of the empty se... |
| bj-nuliotaALT 37261 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 37262 | Alternate proof of ~ vtocl... |
| bj-elsn12g 37263 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 37264 | Biconditional version of ~... |
| bj-pwcfsdom 37265 | Remove hypothesis from ~ p... |
| bj-grur1 37266 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 37267 | The extension of a predica... |
| bj-dfid2ALT 37268 | Alternate version of ~ dfi... |
| bj-0nelopab 37269 | The empty set is never an ... |
| bj-brrelex12ALT 37270 | Two classes related by a b... |
| bj-epelg 37271 | The membership relation an... |
| bj-epelb 37272 | Two classes are related by... |
| bj-nsnid 37273 | A set does not contain the... |
| bj-rdg0gALT 37274 | Alternate proof of ~ rdg0g... |
| bj-axdd2ALT 37275 | Alternate proof of ~ bj-ax... |
| bj-spvw 37276 | Version of ~ spvw proved f... |
| bj-cbvexvv 37277 | Existentially quantifying ... |
| bj-axnul 37278 | Over the base theory ~ ax-... |
| bj-evaleq 37279 | Equality theorem for the `... |
| bj-evalfun 37280 | The evaluation at a class ... |
| bj-evalfn 37281 | The evaluation at a class ... |
| bj-evalval 37282 | Value of the evaluation at... |
| bj-evalid 37283 | The evaluation at a set of... |
| bj-ndxarg 37284 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 37285 | Closed general form of ~ s... |
| bj-rest00 37288 | An elementwise intersectio... |
| bj-restsn 37289 | An elementwise intersectio... |
| bj-restsnss 37290 | Special case of ~ bj-rests... |
| bj-restsnss2 37291 | Special case of ~ bj-rests... |
| bj-restsn0 37292 | An elementwise intersectio... |
| bj-restsn10 37293 | Special case of ~ bj-rests... |
| bj-restsnid 37294 | The elementwise intersecti... |
| bj-rest10 37295 | An elementwise intersectio... |
| bj-rest10b 37296 | Alternate version of ~ bj-... |
| bj-restn0 37297 | An elementwise intersectio... |
| bj-restn0b 37298 | Alternate version of ~ bj-... |
| bj-restpw 37299 | The elementwise intersecti... |
| bj-rest0 37300 | An elementwise intersectio... |
| bj-restb 37301 | An elementwise intersectio... |
| bj-restv 37302 | An elementwise intersectio... |
| bj-resta 37303 | An elementwise intersectio... |
| bj-restuni 37304 | The union of an elementwis... |
| bj-restuni2 37305 | The union of an elementwis... |
| bj-restreg 37306 | A reformulation of the axi... |
| bj-raldifsn 37307 | All elements in a set sati... |
| bj-0int 37308 | If ` A ` is a collection o... |
| bj-mooreset 37309 | A Moore collection is a se... |
| bj-ismoore 37312 | Characterization of Moore ... |
| bj-ismoored0 37313 | Necessary condition to be ... |
| bj-ismoored 37314 | Necessary condition to be ... |
| bj-ismoored2 37315 | Necessary condition to be ... |
| bj-ismooredr 37316 | Sufficient condition to be... |
| bj-ismooredr2 37317 | Sufficient condition to be... |
| bj-discrmoore 37318 | The powerclass ` ~P A ` is... |
| bj-0nmoore 37319 | The empty set is not a Moo... |
| bj-snmoore 37320 | A singleton is a Moore col... |
| bj-snmooreb 37321 | A singleton is a Moore col... |
| bj-prmoore 37322 | A pair formed of two neste... |
| bj-0nelmpt 37323 | The empty set is not an el... |
| bj-mptval 37324 | Value of a function given ... |
| bj-dfmpoa 37325 | An equivalent definition o... |
| bj-mpomptALT 37326 | Alternate proof of ~ mpomp... |
| setsstrset 37343 | Relation between ~ df-sets... |
| bj-nfald 37344 | Variant of ~ nfald . (Con... |
| bj-nfexd 37345 | Variant of ~ nfexd . (Con... |
| copsex2d 37346 | Implicit substitution dedu... |
| copsex2b 37347 | Biconditional form of ~ co... |
| opelopabd 37348 | Membership of an ordere pa... |
| opelopabb 37349 | Membership of an ordered p... |
| opelopabbv 37350 | Membership of an ordered p... |
| bj-opelrelex 37351 | The coordinates of an orde... |
| bj-opelresdm 37352 | If an ordered pair is in a... |
| bj-brresdm 37353 | If two classes are related... |
| brabd0 37354 | Expressing that two sets a... |
| brabd 37355 | Expressing that two sets a... |
| bj-brab2a1 37356 | "Unbounded" version of ~ b... |
| bj-opabssvv 37357 | A variant of ~ relopabiv (... |
| bj-funidres 37358 | The restricted identity re... |
| bj-opelidb 37359 | Characterization of the or... |
| bj-opelidb1 37360 | Characterization of the or... |
| bj-inexeqex 37361 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 37362 | If the intersection of two... |
| bj-opelid 37363 | Characterization of the or... |
| bj-ideqg 37364 | Characterization of the cl... |
| bj-ideqgALT 37365 | Alternate proof of ~ bj-id... |
| bj-ideqb 37366 | Characterization of classe... |
| bj-idres 37367 | Alternate expression for t... |
| bj-opelidres 37368 | Characterization of the or... |
| bj-idreseq 37369 | Sufficient condition for t... |
| bj-idreseqb 37370 | Characterization for two c... |
| bj-ideqg1 37371 | For sets, the identity rel... |
| bj-ideqg1ALT 37372 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 37373 | Characterization of the co... |
| bj-elid3 37374 | Characterization of the co... |
| bj-elid4 37375 | Characterization of the el... |
| bj-elid5 37376 | Characterization of the el... |
| bj-elid6 37377 | Characterization of the el... |
| bj-elid7 37378 | Characterization of the el... |
| bj-diagval 37381 | Value of the functionalize... |
| bj-diagval2 37382 | Value of the functionalize... |
| bj-eldiag 37383 | Characterization of the el... |
| bj-eldiag2 37384 | Characterization of the el... |
| bj-imdirvallem 37387 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 37388 | Value of the functionalize... |
| bj-imdirval2lem 37389 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 37390 | Value of the functionalize... |
| bj-imdirval3 37391 | Value of the functionalize... |
| bj-imdiridlem 37392 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 37393 | Functorial property of the... |
| bj-opelopabid 37394 | Membership in an ordered-p... |
| bj-opabco 37395 | Composition of ordered-pai... |
| bj-xpcossxp 37396 | The composition of two Car... |
| bj-imdirco 37397 | Functorial property of the... |
| bj-iminvval 37400 | Value of the functionalize... |
| bj-iminvval2 37401 | Value of the functionalize... |
| bj-iminvid 37402 | Functorial property of the... |
| bj-inftyexpitaufo 37409 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 37412 | An element of the circle a... |
| bj-inftyexpiinv 37415 | Utility theorem for the in... |
| bj-inftyexpiinj 37416 | Injectivity of the paramet... |
| bj-inftyexpidisj 37417 | An element of the circle a... |
| bj-ccinftydisj 37420 | The circle at infinity is ... |
| bj-elccinfty 37421 | A lemma for infinite exten... |
| bj-ccssccbar 37424 | Complex numbers are extend... |
| bj-ccinftyssccbar 37425 | Infinite extended complex ... |
| bj-pinftyccb 37428 | The class ` pinfty ` is an... |
| bj-pinftynrr 37429 | The extended complex numbe... |
| bj-minftyccb 37432 | The class ` minfty ` is an... |
| bj-minftynrr 37433 | The extended complex numbe... |
| bj-pinftynminfty 37434 | The extended complex numbe... |
| bj-rrhatsscchat 37443 | The real projective line i... |
| bj-imafv 37458 | If the direct image of a s... |
| bj-funun 37459 | Value of a function expres... |
| bj-fununsn1 37460 | Value of a function expres... |
| bj-fununsn2 37461 | Value of a function expres... |
| bj-fvsnun1 37462 | The value of a function wi... |
| bj-fvsnun2 37463 | The value of a function wi... |
| bj-fvmptunsn1 37464 | Value of a function expres... |
| bj-fvmptunsn2 37465 | Value of a function expres... |
| bj-iomnnom 37466 | The canonical bijection fr... |
| bj-smgrpssmgm 37475 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 37476 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 37477 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 37478 | Monoids are semigroups (el... |
| bj-cmnssmnd 37479 | Commutative monoids are mo... |
| bj-cmnssmndel 37480 | Commutative monoids are mo... |
| bj-grpssmnd 37481 | Groups are monoids. (Cont... |
| bj-grpssmndel 37482 | Groups are monoids (elemen... |
| bj-ablssgrp 37483 | Abelian groups are groups.... |
| bj-ablssgrpel 37484 | Abelian groups are groups ... |
| bj-ablsscmn 37485 | Abelian groups are commuta... |
| bj-ablsscmnel 37486 | Abelian groups are commuta... |
| bj-modssabl 37487 | (The additive groups of) m... |
| bj-vecssmod 37488 | Vector spaces are modules.... |
| bj-vecssmodel 37489 | Vector spaces are modules ... |
| bj-finsumval0 37492 | Value of a finite sum. (C... |
| bj-fvimacnv0 37493 | Variant of ~ fvimacnv wher... |
| bj-isvec 37494 | The predicate "is a vector... |
| bj-fldssdrng 37495 | Fields are division rings.... |
| bj-flddrng 37496 | Fields are division rings ... |
| bj-rrdrg 37497 | The field of real numbers ... |
| bj-isclm 37498 | The predicate "is a subcom... |
| bj-isrvec 37501 | The predicate "is a real v... |
| bj-rvecmod 37502 | Real vector spaces are mod... |
| bj-rvecssmod 37503 | Real vector spaces are mod... |
| bj-rvecrr 37504 | The field of scalars of a ... |
| bj-isrvecd 37505 | The predicate "is a real v... |
| bj-rvecvec 37506 | Real vector spaces are vec... |
| bj-isrvec2 37507 | The predicate "is a real v... |
| bj-rvecssvec 37508 | Real vector spaces are vec... |
| bj-rveccmod 37509 | Real vector spaces are sub... |
| bj-rvecsscmod 37510 | Real vector spaces are sub... |
| bj-rvecsscvec 37511 | Real vector spaces are sub... |
| bj-rveccvec 37512 | Real vector spaces are sub... |
| bj-rvecssabl 37513 | (The additive groups of) r... |
| bj-rvecabl 37514 | (The additive groups of) r... |
| bj-subcom 37515 | A consequence of commutati... |
| bj-lineqi 37516 | Solution of a (scalar) lin... |
| bj-bary1lem 37517 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 37518 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 37519 | Barycentric coordinates in... |
| bj-endval 37522 | Value of the monoid of end... |
| bj-endbase 37523 | Base set of the monoid of ... |
| bj-endcomp 37524 | Composition law of the mon... |
| bj-endmnd 37525 | The monoid of endomorphism... |
| taupilem3 37526 | Lemma for tau-related theo... |
| taupilemrplb 37527 | A set of positive reals ha... |
| taupilem1 37528 | Lemma for ~ taupi . A pos... |
| taupilem2 37529 | Lemma for ~ taupi . The s... |
| taupi 37530 | Relationship between ` _ta... |
| dfgcd3 37531 | Alternate definition of th... |
| irrdifflemf 37532 | Lemma for ~ irrdiff . The... |
| irrdiff 37533 | The irrationals are exactl... |
| iccioo01 37534 | The closed unit interval i... |
| csbrecsg 37535 | Move class substitution in... |
| csbrdgg 37536 | Move class substitution in... |
| csboprabg 37537 | Move class substitution in... |
| csbmpo123 37538 | Move class substitution in... |
| con1bii2 37539 | A contraposition inference... |
| con2bii2 37540 | A contraposition inference... |
| vtoclefex 37541 | Implicit substitution of a... |
| rnmptsn 37542 | The range of a function ma... |
| f1omptsnlem 37543 | This is the core of the pr... |
| f1omptsn 37544 | A function mapping to sing... |
| mptsnunlem 37545 | This is the core of the pr... |
| mptsnun 37546 | A class ` B ` is equal to ... |
| dissneqlem 37547 | This is the core of the pr... |
| dissneq 37548 | Any topology that contains... |
| exlimim 37549 | Closed form of ~ exlimimd ... |
| exlimimd 37550 | Existential elimination ru... |
| exellim 37551 | Closed form of ~ exellimdd... |
| exellimddv 37552 | Eliminate an antecedent wh... |
| topdifinfindis 37553 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 37554 | This is the core of the pr... |
| topdifinffin 37555 | Part of Exercise 3 of [Mun... |
| topdifinf 37556 | Part of Exercise 3 of [Mun... |
| topdifinfeq 37557 | Two different ways of defi... |
| icorempo 37558 | Closed-below, open-above i... |
| icoreresf 37559 | Closed-below, open-above i... |
| icoreval 37560 | Value of the closed-below,... |
| icoreelrnab 37561 | Elementhood in the set of ... |
| isbasisrelowllem1 37562 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 37563 | Lemma for ~ isbasisrelowl ... |
| icoreclin 37564 | The set of closed-below, o... |
| isbasisrelowl 37565 | The set of all closed-belo... |
| icoreunrn 37566 | The union of all closed-be... |
| istoprelowl 37567 | The set of all closed-belo... |
| icoreelrn 37568 | A class abstraction which ... |
| iooelexlt 37569 | An element of an open inte... |
| relowlssretop 37570 | The lower limit topology o... |
| relowlpssretop 37571 | The lower limit topology o... |
| sucneqond 37572 | Inequality of an ordinal s... |
| sucneqoni 37573 | Inequality of an ordinal s... |
| onsucuni3 37574 | If an ordinal number has a... |
| 1oequni2o 37575 | The ordinal number ` 1o ` ... |
| rdgsucuni 37576 | If an ordinal number has a... |
| rdgeqoa 37577 | If a recursive function wi... |
| elxp8 37578 | Membership in a Cartesian ... |
| cbveud 37579 | Deduction used to change b... |
| cbvreud 37580 | Deduction used to change b... |
| difunieq 37581 | The difference of unions i... |
| inunissunidif 37582 | Theorem about subsets of t... |
| rdgellim 37583 | Elementhood in a recursive... |
| rdglimss 37584 | A recursive definition at ... |
| rdgssun 37585 | In a recursive definition ... |
| exrecfnlem 37586 | Lemma for ~ exrecfn . (Co... |
| exrecfn 37587 | Theorem about the existenc... |
| exrecfnpw 37588 | For any base set, a set wh... |
| finorwe 37589 | If the Axiom of Infinity i... |
| dffinxpf 37592 | This theorem is the same a... |
| finxpeq1 37593 | Equality theorem for Carte... |
| finxpeq2 37594 | Equality theorem for Carte... |
| csbfinxpg 37595 | Distribute proper substitu... |
| finxpreclem1 37596 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 37597 | Lemma for ` ^^ ` recursion... |
| finxp0 37598 | The value of Cartesian exp... |
| finxp1o 37599 | The value of Cartesian exp... |
| finxpreclem3 37600 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 37601 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 37602 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 37603 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 37604 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 37605 | The value of Cartesian exp... |
| finxp2o 37606 | The value of Cartesian exp... |
| finxp3o 37607 | The value of Cartesian exp... |
| finxpnom 37608 | Cartesian exponentiation w... |
| finxp00 37609 | Cartesian exponentiation o... |
| iunctb2 37610 | Using the axiom of countab... |
| domalom 37611 | A class which dominates ev... |
| isinf2 37612 | The converse of ~ isinf . ... |
| ctbssinf 37613 | Using the axiom of choice,... |
| ralssiun 37614 | The index set of an indexe... |
| nlpineqsn 37615 | For every point ` p ` of a... |
| nlpfvineqsn 37616 | Given a subset ` A ` of ` ... |
| fvineqsnf1 37617 | A theorem about functions ... |
| fvineqsneu 37618 | A theorem about functions ... |
| fvineqsneq 37619 | A theorem about functions ... |
| pibp16 37620 | Property P000016 of pi-bas... |
| pibp19 37621 | Property P000019 of pi-bas... |
| pibp21 37622 | Property P000021 of pi-bas... |
| pibt1 37623 | Theorem T000001 of pi-base... |
| pibt2 37624 | Theorem T000002 of pi-base... |
| wl-section-prop 37625 | Intuitionistic logic is no... |
| wl-section-boot 37629 | In this section, I provide... |
| wl-luk-imim1i 37630 | Inference adding common co... |
| wl-luk-syl 37631 | An inference version of th... |
| wl-luk-imtrid 37632 | A syllogism rule of infere... |
| wl-luk-pm2.18d 37633 | Deduction based on reducti... |
| wl-luk-con4i 37634 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 37635 | Inference rule. Copy of ~... |
| wl-luk-a1i 37636 | Inference rule. Copy of ~... |
| wl-luk-mpi 37637 | A nested _modus ponens_ in... |
| wl-luk-imim2i 37638 | Inference adding common an... |
| wl-luk-imtrdi 37639 | A syllogism rule of infere... |
| wl-luk-ax3 37640 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 37641 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 37642 | This theorem, called "Asse... |
| wl-luk-com12 37643 | Inference that swaps (comm... |
| wl-luk-pm2.21 37644 | From a wff and its negatio... |
| wl-luk-con1i 37645 | A contraposition inference... |
| wl-luk-ja 37646 | Inference joining the ante... |
| wl-luk-imim2 37647 | A closed form of syllogism... |
| wl-luk-a1d 37648 | Deduction introducing an e... |
| wl-luk-ax2 37649 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 37650 | Principle of identity. Th... |
| wl-luk-notnotr 37651 | Converse of double negatio... |
| wl-luk-pm2.04 37652 | Swap antecedents. Theorem... |
| wl-section-impchain 37653 | An implication like ` ( ps... |
| wl-impchain-mp-x 37654 | This series of theorems pr... |
| wl-impchain-mp-0 37655 | This theorem is the start ... |
| wl-impchain-mp-1 37656 | This theorem is in fact a ... |
| wl-impchain-mp-2 37657 | This theorem is in fact a ... |
| wl-impchain-com-1.x 37658 | It is often convenient to ... |
| wl-impchain-com-1.1 37659 | A degenerate form of antec... |
| wl-impchain-com-1.2 37660 | This theorem is in fact a ... |
| wl-impchain-com-1.3 37661 | This theorem is in fact a ... |
| wl-impchain-com-1.4 37662 | This theorem is in fact a ... |
| wl-impchain-com-n.m 37663 | This series of theorems al... |
| wl-impchain-com-2.3 37664 | This theorem is in fact a ... |
| wl-impchain-com-2.4 37665 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 37666 | This theorem is in fact a ... |
| wl-impchain-a1-x 37667 | If an implication chain is... |
| wl-impchain-a1-1 37668 | Inference rule, a copy of ... |
| wl-impchain-a1-2 37669 | Inference rule, a copy of ... |
| wl-impchain-a1-3 37670 | Inference rule, a copy of ... |
| wl-ifp-ncond1 37671 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 37672 | If one case of an ` if- ` ... |
| wl-ifpimpr 37673 | If one case of an ` if- ` ... |
| wl-ifp4impr 37674 | If one case of an ` if- ` ... |
| wl-df-3xor 37675 | Alternative definition of ... |
| wl-df3xor2 37676 | Alternative definition of ... |
| wl-df3xor3 37677 | Alternative form of ~ wl-d... |
| wl-3xortru 37678 | If the first input is true... |
| wl-3xorfal 37679 | If the first input is fals... |
| wl-3xorbi 37680 | Triple xor can be replaced... |
| wl-3xorbi2 37681 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 37682 | Equivalence theorem for tr... |
| wl-3xorbi123i 37683 | Equivalence theorem for tr... |
| wl-3xorrot 37684 | Rotation law for triple xo... |
| wl-3xorcoma 37685 | Commutative law for triple... |
| wl-3xorcomb 37686 | Commutative law for triple... |
| wl-3xornot1 37687 | Flipping the first input f... |
| wl-3xornot 37688 | Triple xor distributes ove... |
| wl-1xor 37689 | In the recursive scheme ... |
| wl-2xor 37690 | In the recursive scheme ... |
| wl-df-3mintru2 37691 | Alternative definition of ... |
| wl-df2-3mintru2 37692 | The adder carry in disjunc... |
| wl-df3-3mintru2 37693 | The adder carry in conjunc... |
| wl-df4-3mintru2 37694 | An alternative definition ... |
| wl-1mintru1 37695 | Using the recursion formul... |
| wl-1mintru2 37696 | Using the recursion formul... |
| wl-2mintru1 37697 | Using the recursion formul... |
| wl-2mintru2 37698 | Using the recursion formul... |
| wl-df3maxtru1 37699 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 37701 | A version of ~ ax-wl-13v w... |
| wl-cleq-0 37702 |
Disclaimer: |
| wl-cleq-1 37703 |
Disclaimer: |
| wl-cleq-2 37704 |
Disclaimer: |
| wl-cleq-3 37705 |
Disclaimer: |
| wl-cleq-4 37706 |
Disclaimer: |
| wl-cleq-5 37707 |
Disclaimer: |
| wl-cleq-6 37708 |
Disclaimer: |
| wl-df-clab 37711 | Disclaimer: The material ... |
| wl-isseteq 37712 | A class equal to a set var... |
| wl-ax12v2cl 37713 | The class version of ~ ax1... |
| wl-mps 37714 | Replacing a nested consequ... |
| wl-syls1 37715 | Replacing a nested consequ... |
| wl-syls2 37716 | Replacing a nested anteced... |
| wl-embant 37717 | A true wff can always be a... |
| wl-orel12 37718 | In a conjunctive normal fo... |
| wl-cases2-dnf 37719 | A particular instance of ~... |
| wl-cbvmotv 37720 | Change bound variable. Us... |
| wl-moteq 37721 | Change bound variable. Us... |
| wl-motae 37722 | Change bound variable. Us... |
| wl-moae 37723 | Two ways to express "at mo... |
| wl-euae 37724 | Two ways to express "exact... |
| wl-nax6im 37725 | The following series of th... |
| wl-hbae1 37726 | This specialization of ~ h... |
| wl-naevhba1v 37727 | An instance of ~ hbn1w app... |
| wl-spae 37728 | Prove an instance of ~ sp ... |
| wl-speqv 37729 | Under the assumption ` -. ... |
| wl-19.8eqv 37730 | Under the assumption ` -. ... |
| wl-19.2reqv 37731 | Under the assumption ` -. ... |
| wl-nfalv 37732 | If ` x ` is not present in... |
| wl-nfimf1 37733 | An antecedent is irrelevan... |
| wl-nfae1 37734 | Unlike ~ nfae , this speci... |
| wl-nfnae1 37735 | Unlike ~ nfnae , this spec... |
| wl-aetr 37736 | A transitive law for varia... |
| wl-axc11r 37737 | Same as ~ axc11r , but usi... |
| wl-dral1d 37738 | A version of ~ dral1 with ... |
| wl-cbvalnaed 37739 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 37740 | A more general version of ... |
| wl-exeq 37741 | The semantics of ` E. x y ... |
| wl-aleq 37742 | The semantics of ` A. x y ... |
| wl-nfeqfb 37743 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 37744 | If ` y ` is not free in ` ... |
| wl-equsalvw 37745 | Version of ~ equsalv with ... |
| wl-equsald 37746 | Deduction version of ~ equ... |
| wl-equsaldv 37747 | Deduction version of ~ equ... |
| wl-equsal 37748 | A useful equivalence relat... |
| wl-equsal1t 37749 | The expression ` x = y ` i... |
| wl-equsalcom 37750 | This simple equivalence ea... |
| wl-equsal1i 37751 | The antecedent ` x = y ` i... |
| wl-sbid2ft 37752 | A more general version of ... |
| wl-cbvalsbi 37753 | Change bounded variables i... |
| wl-sbrimt 37754 | Substitution with a variab... |
| wl-sblimt 37755 | Substitution with a variab... |
| wl-sb9v 37756 | Commutation of quantificat... |
| wl-sb8ft 37757 | Substitution of variable i... |
| wl-sb8eft 37758 | Substitution of variable i... |
| wl-sb8t 37759 | Substitution of variable i... |
| wl-sb8et 37760 | Substitution of variable i... |
| wl-sbhbt 37761 | Closed form of ~ sbhb . C... |
| wl-sbnf1 37762 | Two ways expressing that `... |
| wl-equsb3 37763 | ~ equsb3 with a distinctor... |
| wl-equsb4 37764 | Substitution applied to an... |
| wl-2sb6d 37765 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 37766 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 37767 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 37768 | Version of ~ sbcom2 with a... |
| wl-sbalnae 37769 | A theorem used in eliminat... |
| wl-sbal1 37770 | A theorem used in eliminat... |
| wl-sbal2 37771 | Move quantifier in and out... |
| wl-2spsbbi 37772 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 37773 | This theorem provides a ba... |
| wl-lem-nexmo 37774 | This theorem provides a ba... |
| wl-lem-moexsb 37775 | The antecedent ` A. x ( ph... |
| wl-alanbii 37776 | This theorem extends ~ ala... |
| wl-mo2df 37777 | Version of ~ mof with a co... |
| wl-mo2tf 37778 | Closed form of ~ mof with ... |
| wl-eudf 37779 | Version of ~ eu6 with a co... |
| wl-eutf 37780 | Closed form of ~ eu6 with ... |
| wl-euequf 37781 | ~ euequ proved with a dist... |
| wl-mo2t 37782 | Closed form of ~ mof . (C... |
| wl-mo3t 37783 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 37784 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 37785 | Substitution of variable i... |
| wl-sb8eutv 37786 | Substitution of variable i... |
| wl-sb8mot 37787 | Substitution of variable i... |
| wl-sb8motv 37788 | Substitution of variable i... |
| wl-issetft 37789 | A closed form of ~ issetf ... |
| wl-axc11rc11 37790 | Proving ~ axc11r from ~ ax... |
| wl-clabv 37791 | Variant of ~ df-clab , whe... |
| wl-dfclab 37792 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 37793 | Using class abstraction in... |
| wl-clabt 37794 | Using class abstraction in... |
| wl-eujustlem1 37795 | Version of ~ cbvexvw with ... |
| rabiun 37796 | Abstraction restricted to ... |
| iundif1 37797 | Indexed union of class dif... |
| imadifss 37798 | The difference of images i... |
| cureq 37799 | Equality theorem for curry... |
| unceq 37800 | Equality theorem for uncur... |
| curf 37801 | Functional property of cur... |
| uncf 37802 | Functional property of unc... |
| curfv 37803 | Value of currying. (Contr... |
| uncov 37804 | Value of uncurrying. (Con... |
| curunc 37805 | Currying of uncurrying. (... |
| unccur 37806 | Uncurrying of currying. (... |
| phpreu 37807 | Theorem related to pigeonh... |
| finixpnum 37808 | A finite Cartesian product... |
| fin2solem 37809 | Lemma for ~ fin2so . (Con... |
| fin2so 37810 | Any totally ordered Tarski... |
| ltflcei 37811 | Theorem to move the floor ... |
| leceifl 37812 | Theorem to move the floor ... |
| sin2h 37813 | Half-angle rule for sine. ... |
| cos2h 37814 | Half-angle rule for cosine... |
| tan2h 37815 | Half-angle rule for tangen... |
| lindsadd 37816 | In a vector space, the uni... |
| lindsdom 37817 | A linearly independent set... |
| lindsenlbs 37818 | A maximal linearly indepen... |
| matunitlindflem1 37819 | One direction of ~ matunit... |
| matunitlindflem2 37820 | One direction of ~ matunit... |
| matunitlindf 37821 | A matrix over a field is i... |
| ptrest 37822 | Expressing a restriction o... |
| ptrecube 37823 | Any point in an open set o... |
| poimirlem1 37824 | Lemma for ~ poimir - the v... |
| poimirlem2 37825 | Lemma for ~ poimir - conse... |
| poimirlem3 37826 | Lemma for ~ poimir to add ... |
| poimirlem4 37827 | Lemma for ~ poimir connect... |
| poimirlem5 37828 | Lemma for ~ poimir to esta... |
| poimirlem6 37829 | Lemma for ~ poimir establi... |
| poimirlem7 37830 | Lemma for ~ poimir , simil... |
| poimirlem8 37831 | Lemma for ~ poimir , estab... |
| poimirlem9 37832 | Lemma for ~ poimir , estab... |
| poimirlem10 37833 | Lemma for ~ poimir establi... |
| poimirlem11 37834 | Lemma for ~ poimir connect... |
| poimirlem12 37835 | Lemma for ~ poimir connect... |
| poimirlem13 37836 | Lemma for ~ poimir - for a... |
| poimirlem14 37837 | Lemma for ~ poimir - for a... |
| poimirlem15 37838 | Lemma for ~ poimir , that ... |
| poimirlem16 37839 | Lemma for ~ poimir establi... |
| poimirlem17 37840 | Lemma for ~ poimir establi... |
| poimirlem18 37841 | Lemma for ~ poimir stating... |
| poimirlem19 37842 | Lemma for ~ poimir establi... |
| poimirlem20 37843 | Lemma for ~ poimir establi... |
| poimirlem21 37844 | Lemma for ~ poimir stating... |
| poimirlem22 37845 | Lemma for ~ poimir , that ... |
| poimirlem23 37846 | Lemma for ~ poimir , two w... |
| poimirlem24 37847 | Lemma for ~ poimir , two w... |
| poimirlem25 37848 | Lemma for ~ poimir stating... |
| poimirlem26 37849 | Lemma for ~ poimir showing... |
| poimirlem27 37850 | Lemma for ~ poimir showing... |
| poimirlem28 37851 | Lemma for ~ poimir , a var... |
| poimirlem29 37852 | Lemma for ~ poimir connect... |
| poimirlem30 37853 | Lemma for ~ poimir combini... |
| poimirlem31 37854 | Lemma for ~ poimir , assig... |
| poimirlem32 37855 | Lemma for ~ poimir , combi... |
| poimir 37856 | Poincare-Miranda theorem. ... |
| broucube 37857 | Brouwer - or as Kulpa call... |
| heicant 37858 | Heine-Cantor theorem: a co... |
| opnmbllem0 37859 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 37860 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 37861 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 37862 | The difference between two... |
| mblfinlem4 37863 | Backward direction of ~ is... |
| ismblfin 37864 | Measurability in terms of ... |
| ovoliunnfl 37865 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 37866 | Demonstration of ~ ovoliun... |
| voliunnfl 37867 | ~ voliun is incompatible w... |
| volsupnfl 37868 | ~ volsup is incompatible w... |
| mbfresfi 37869 | Measurability of a piecewi... |
| mbfposadd 37870 | If the sum of two measurab... |
| cnambfre 37871 | A real-valued, a.e. contin... |
| dvtanlem 37872 | Lemma for ~ dvtan - the do... |
| dvtan 37873 | Derivative of tangent. (C... |
| itg2addnclem 37874 | An alternate expression fo... |
| itg2addnclem2 37875 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 37876 | Lemma incomprehensible in ... |
| itg2addnc 37877 | Alternate proof of ~ itg2a... |
| itg2gt0cn 37878 | ~ itg2gt0 holds on functio... |
| ibladdnclem 37879 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 37880 | Choice-free analogue of ~ ... |
| itgaddnclem1 37881 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 37882 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 37883 | Choice-free analogue of ~ ... |
| iblsubnc 37884 | Choice-free analogue of ~ ... |
| itgsubnc 37885 | Choice-free analogue of ~ ... |
| iblabsnclem 37886 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 37887 | Choice-free analogue of ~ ... |
| iblmulc2nc 37888 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 37889 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 37890 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 37891 | Choice-free analogue of ~ ... |
| itgabsnc 37892 | Choice-free analogue of ~ ... |
| itggt0cn 37893 | ~ itggt0 holds for continu... |
| ftc1cnnclem 37894 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 37895 | Choice-free proof of ~ ftc... |
| ftc1anclem1 37896 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 37897 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 37898 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 37899 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 37900 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 37901 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 37902 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 37903 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 37904 | ~ ftc1a holds for function... |
| ftc2nc 37905 | Choice-free proof of ~ ftc... |
| asindmre 37906 | Real part of domain of dif... |
| dvasin 37907 | Derivative of arcsine. (C... |
| dvacos 37908 | Derivative of arccosine. ... |
| dvreasin 37909 | Real derivative of arcsine... |
| dvreacos 37910 | Real derivative of arccosi... |
| areacirclem1 37911 | Antiderivative of cross-se... |
| areacirclem2 37912 | Endpoint-inclusive continu... |
| areacirclem3 37913 | Integrability of cross-sec... |
| areacirclem4 37914 | Endpoint-inclusive continu... |
| areacirclem5 37915 | Finding the cross-section ... |
| areacirc 37916 | The area of a circle of ra... |
| unirep 37917 | Define a quantity whose de... |
| cover2 37918 | Two ways of expressing the... |
| cover2g 37919 | Two ways of expressing the... |
| brabg2 37920 | Relation by a binary relat... |
| opelopab3 37921 | Ordered pair membership in... |
| cocanfo 37922 | Cancellation of a surjecti... |
| brresi2 37923 | Restriction of a binary re... |
| fnopabeqd 37924 | Equality deduction for fun... |
| fvopabf4g 37925 | Function value of an opera... |
| fnopabco 37926 | Composition of a function ... |
| opropabco 37927 | Composition of an operator... |
| cocnv 37928 | Composition with a functio... |
| f1ocan1fv 37929 | Cancel a composition by a ... |
| f1ocan2fv 37930 | Cancel a composition by th... |
| inixp 37931 | Intersection of Cartesian ... |
| upixp 37932 | Universal property of the ... |
| abrexdom 37933 | An indexed set is dominate... |
| abrexdom2 37934 | An indexed set is dominate... |
| ac6gf 37935 | Axiom of Choice. (Contrib... |
| indexa 37936 | If for every element of an... |
| indexdom 37937 | If for every element of an... |
| frinfm 37938 | A subset of a well-founded... |
| welb 37939 | A nonempty subset of a wel... |
| supex2g 37940 | Existence of supremum. (C... |
| supclt 37941 | Closure of supremum. (Con... |
| supubt 37942 | Upper bound property of su... |
| filbcmb 37943 | Combine a finite set of lo... |
| fzmul 37944 | Membership of a product in... |
| sdclem2 37945 | Lemma for ~ sdc . (Contri... |
| sdclem1 37946 | Lemma for ~ sdc . (Contri... |
| sdc 37947 | Strong dependent choice. ... |
| fdc 37948 | Finite version of dependen... |
| fdc1 37949 | Variant of ~ fdc with no s... |
| seqpo 37950 | Two ways to say that a seq... |
| incsequz 37951 | An increasing sequence of ... |
| incsequz2 37952 | An increasing sequence of ... |
| nnubfi 37953 | A bounded above set of pos... |
| nninfnub 37954 | An infinite set of positiv... |
| subspopn 37955 | An open set is open in the... |
| neificl 37956 | Neighborhoods are closed u... |
| lpss2 37957 | Limit points of a subset a... |
| metf1o 37958 | Use a bijection with a met... |
| blssp 37959 | A ball in the subspace met... |
| mettrifi 37960 | Generalized triangle inequ... |
| lmclim2 37961 | A sequence in a metric spa... |
| geomcau 37962 | If the distance between co... |
| caures 37963 | The restriction of a Cauch... |
| caushft 37964 | A shifted Cauchy sequence ... |
| constcncf 37965 | A constant function is a c... |
| cnres2 37966 | The restriction of a conti... |
| cnresima 37967 | A continuous function is c... |
| cncfres 37968 | A continuous function on c... |
| istotbnd 37972 | The predicate "is a totall... |
| istotbnd2 37973 | The predicate "is a totall... |
| istotbnd3 37974 | A metric space is totally ... |
| totbndmet 37975 | The predicate "totally bou... |
| 0totbnd 37976 | The metric (there is only ... |
| sstotbnd2 37977 | Condition for a subset of ... |
| sstotbnd 37978 | Condition for a subset of ... |
| sstotbnd3 37979 | Use a net that is not nece... |
| totbndss 37980 | A subset of a totally boun... |
| equivtotbnd 37981 | If the metric ` M ` is "st... |
| isbnd 37983 | The predicate "is a bounde... |
| bndmet 37984 | A bounded metric space is ... |
| isbndx 37985 | A "bounded extended metric... |
| isbnd2 37986 | The predicate "is a bounde... |
| isbnd3 37987 | A metric space is bounded ... |
| isbnd3b 37988 | A metric space is bounded ... |
| bndss 37989 | A subset of a bounded metr... |
| blbnd 37990 | A ball is bounded. (Contr... |
| ssbnd 37991 | A subset of a metric space... |
| totbndbnd 37992 | A totally bounded metric s... |
| equivbnd 37993 | If the metric ` M ` is "st... |
| bnd2lem 37994 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 37995 | If balls are totally bound... |
| prdsbnd 37996 | The product metric over fi... |
| prdstotbnd 37997 | The product metric over fi... |
| prdsbnd2 37998 | If balls are totally bound... |
| cntotbnd 37999 | A subset of the complex nu... |
| cnpwstotbnd 38000 | A subset of ` A ^ I ` , wh... |
| ismtyval 38003 | The set of isometries betw... |
| isismty 38004 | The condition "is an isome... |
| ismtycnv 38005 | The inverse of an isometry... |
| ismtyima 38006 | The image of a ball under ... |
| ismtyhmeolem 38007 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 38008 | An isometry is a homeomorp... |
| ismtybndlem 38009 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 38010 | Isometries preserve bounde... |
| ismtyres 38011 | A restriction of an isomet... |
| heibor1lem 38012 | Lemma for ~ heibor1 . A c... |
| heibor1 38013 | One half of ~ heibor , tha... |
| heiborlem1 38014 | Lemma for ~ heibor . We w... |
| heiborlem2 38015 | Lemma for ~ heibor . Subs... |
| heiborlem3 38016 | Lemma for ~ heibor . Usin... |
| heiborlem4 38017 | Lemma for ~ heibor . Usin... |
| heiborlem5 38018 | Lemma for ~ heibor . The ... |
| heiborlem6 38019 | Lemma for ~ heibor . Sinc... |
| heiborlem7 38020 | Lemma for ~ heibor . Sinc... |
| heiborlem8 38021 | Lemma for ~ heibor . The ... |
| heiborlem9 38022 | Lemma for ~ heibor . Disc... |
| heiborlem10 38023 | Lemma for ~ heibor . The ... |
| heibor 38024 | Generalized Heine-Borel Th... |
| bfplem1 38025 | Lemma for ~ bfp . The seq... |
| bfplem2 38026 | Lemma for ~ bfp . Using t... |
| bfp 38027 | Banach fixed point theorem... |
| rrnval 38030 | The n-dimensional Euclidea... |
| rrnmval 38031 | The value of the Euclidean... |
| rrnmet 38032 | Euclidean space is a metri... |
| rrndstprj1 38033 | The distance between two p... |
| rrndstprj2 38034 | Bound on the distance betw... |
| rrncmslem 38035 | Lemma for ~ rrncms . (Con... |
| rrncms 38036 | Euclidean space is complet... |
| repwsmet 38037 | The supremum metric on ` R... |
| rrnequiv 38038 | The supremum metric on ` R... |
| rrntotbnd 38039 | A set in Euclidean space i... |
| rrnheibor 38040 | Heine-Borel theorem for Eu... |
| ismrer1 38041 | An isometry between ` RR `... |
| reheibor 38042 | Heine-Borel theorem for re... |
| iccbnd 38043 | A closed interval in ` RR ... |
| icccmpALT 38044 | A closed interval in ` RR ... |
| isass 38049 | The predicate "is an assoc... |
| isexid 38050 | The predicate ` G ` has a ... |
| ismgmOLD 38053 | Obsolete version of ~ ismg... |
| clmgmOLD 38054 | Obsolete version of ~ mgmc... |
| opidonOLD 38055 | Obsolete version of ~ mndp... |
| rngopidOLD 38056 | Obsolete version of ~ mndp... |
| opidon2OLD 38057 | Obsolete version of ~ mndp... |
| isexid2 38058 | If ` G e. ( Magma i^i ExId... |
| exidu1 38059 | Uniqueness of the left and... |
| idrval 38060 | The value of the identity ... |
| iorlid 38061 | A magma right and left ide... |
| cmpidelt 38062 | A magma right and left ide... |
| smgrpismgmOLD 38065 | Obsolete version of ~ sgrp... |
| issmgrpOLD 38066 | Obsolete version of ~ issg... |
| smgrpmgm 38067 | A semigroup is a magma. (... |
| smgrpassOLD 38068 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 38071 | Obsolete version of ~ mnds... |
| mndoisexid 38072 | A monoid has an identity e... |
| mndoismgmOLD 38073 | Obsolete version of ~ mndm... |
| mndomgmid 38074 | A monoid is a magma with a... |
| ismndo 38075 | The predicate "is a monoid... |
| ismndo1 38076 | The predicate "is a monoid... |
| ismndo2 38077 | The predicate "is a monoid... |
| grpomndo 38078 | A group is a monoid. (Con... |
| exidcl 38079 | Closure of the binary oper... |
| exidreslem 38080 | Lemma for ~ exidres and ~ ... |
| exidres 38081 | The restriction of a binar... |
| exidresid 38082 | The restriction of a binar... |
| ablo4pnp 38083 | A commutative/associative ... |
| grpoeqdivid 38084 | Two group elements are equ... |
| grposnOLD 38085 | The group operation for th... |
| elghomlem1OLD 38088 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 38089 | Obsolete as of 15-Mar-2020... |
| elghomOLD 38090 | Obsolete version of ~ isgh... |
| ghomlinOLD 38091 | Obsolete version of ~ ghml... |
| ghomidOLD 38092 | Obsolete version of ~ ghmi... |
| ghomf 38093 | Mapping property of a grou... |
| ghomco 38094 | The composition of two gro... |
| ghomdiv 38095 | Group homomorphisms preser... |
| grpokerinj 38096 | A group homomorphism is in... |
| relrngo 38099 | The class of all unital ri... |
| isrngo 38100 | The predicate "is a (unita... |
| isrngod 38101 | Conditions that determine ... |
| rngoi 38102 | The properties of a unital... |
| rngosm 38103 | Functionality of the multi... |
| rngocl 38104 | Closure of the multiplicat... |
| rngoid 38105 | The multiplication operati... |
| rngoideu 38106 | The unity element of a rin... |
| rngodi 38107 | Distributive law for the m... |
| rngodir 38108 | Distributive law for the m... |
| rngoass 38109 | Associative law for the mu... |
| rngo2 38110 | A ring element plus itself... |
| rngoablo 38111 | A ring's addition operatio... |
| rngoablo2 38112 | In a unital ring the addit... |
| rngogrpo 38113 | A ring's addition operatio... |
| rngone0 38114 | The base set of a ring is ... |
| rngogcl 38115 | Closure law for the additi... |
| rngocom 38116 | The addition operation of ... |
| rngoaass 38117 | The addition operation of ... |
| rngoa32 38118 | The addition operation of ... |
| rngoa4 38119 | Rearrangement of 4 terms i... |
| rngorcan 38120 | Right cancellation law for... |
| rngolcan 38121 | Left cancellation law for ... |
| rngo0cl 38122 | A ring has an additive ide... |
| rngo0rid 38123 | The additive identity of a... |
| rngo0lid 38124 | The additive identity of a... |
| rngolz 38125 | The zero of a unital ring ... |
| rngorz 38126 | The zero of a unital ring ... |
| rngosn3 38127 | Obsolete as of 25-Jan-2020... |
| rngosn4 38128 | Obsolete as of 25-Jan-2020... |
| rngosn6 38129 | Obsolete as of 25-Jan-2020... |
| rngonegcl 38130 | A ring is closed under neg... |
| rngoaddneg1 38131 | Adding the negative in a r... |
| rngoaddneg2 38132 | Adding the negative in a r... |
| rngosub 38133 | Subtraction in a ring, in ... |
| rngmgmbs4 38134 | The range of an internal o... |
| rngodm1dm2 38135 | In a unital ring the domai... |
| rngorn1 38136 | In a unital ring the range... |
| rngorn1eq 38137 | In a unital ring the range... |
| rngomndo 38138 | In a unital ring the multi... |
| rngoidmlem 38139 | The unity element of a rin... |
| rngolidm 38140 | The unity element of a rin... |
| rngoridm 38141 | The unity element of a rin... |
| rngo1cl 38142 | The unity element of a rin... |
| rngoueqz 38143 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 38144 | Negation in a ring is the ... |
| rngonegmn1r 38145 | Negation in a ring is the ... |
| rngoneglmul 38146 | Negation of a product in a... |
| rngonegrmul 38147 | Negation of a product in a... |
| rngosubdi 38148 | Ring multiplication distri... |
| rngosubdir 38149 | Ring multiplication distri... |
| zerdivemp1x 38150 | In a unital ring a left in... |
| isdivrngo 38153 | The predicate "is a divisi... |
| drngoi 38154 | The properties of a divisi... |
| gidsn 38155 | Obsolete as of 23-Jan-2020... |
| zrdivrng 38156 | The zero ring is not a div... |
| dvrunz 38157 | In a division ring the rin... |
| isgrpda 38158 | Properties that determine ... |
| isdrngo1 38159 | The predicate "is a divisi... |
| divrngcl 38160 | The product of two nonzero... |
| isdrngo2 38161 | A division ring is a ring ... |
| isdrngo3 38162 | A division ring is a ring ... |
| rngohomval 38167 | The set of ring homomorphi... |
| isrngohom 38168 | The predicate "is a ring h... |
| rngohomf 38169 | A ring homomorphism is a f... |
| rngohomcl 38170 | Closure law for a ring hom... |
| rngohom1 38171 | A ring homomorphism preser... |
| rngohomadd 38172 | Ring homomorphisms preserv... |
| rngohommul 38173 | Ring homomorphisms preserv... |
| rngogrphom 38174 | A ring homomorphism is a g... |
| rngohom0 38175 | A ring homomorphism preser... |
| rngohomsub 38176 | Ring homomorphisms preserv... |
| rngohomco 38177 | The composition of two rin... |
| rngokerinj 38178 | A ring homomorphism is inj... |
| rngoisoval 38180 | The set of ring isomorphis... |
| isrngoiso 38181 | The predicate "is a ring i... |
| rngoiso1o 38182 | A ring isomorphism is a bi... |
| rngoisohom 38183 | A ring isomorphism is a ri... |
| rngoisocnv 38184 | The inverse of a ring isom... |
| rngoisoco 38185 | The composition of two rin... |
| isriscg 38187 | The ring isomorphism relat... |
| isrisc 38188 | The ring isomorphism relat... |
| risc 38189 | The ring isomorphism relat... |
| risci 38190 | Determine that two rings a... |
| riscer 38191 | Ring isomorphism is an equ... |
| iscom2 38198 | A device to add commutativ... |
| iscrngo 38199 | The predicate "is a commut... |
| iscrngo2 38200 | The predicate "is a commut... |
| iscringd 38201 | Conditions that determine ... |
| flddivrng 38202 | A field is a division ring... |
| crngorngo 38203 | A commutative ring is a ri... |
| crngocom 38204 | The multiplication operati... |
| crngm23 38205 | Commutative/associative la... |
| crngm4 38206 | Commutative/associative la... |
| fldcrngo 38207 | A field is a commutative r... |
| isfld2 38208 | The predicate "is a field"... |
| crngohomfo 38209 | The image of a homomorphis... |
| idlval 38216 | The class of ideals of a r... |
| isidl 38217 | The predicate "is an ideal... |
| isidlc 38218 | The predicate "is an ideal... |
| idlss 38219 | An ideal of ` R ` is a sub... |
| idlcl 38220 | An element of an ideal is ... |
| idl0cl 38221 | An ideal contains ` 0 ` . ... |
| idladdcl 38222 | An ideal is closed under a... |
| idllmulcl 38223 | An ideal is closed under m... |
| idlrmulcl 38224 | An ideal is closed under m... |
| idlnegcl 38225 | An ideal is closed under n... |
| idlsubcl 38226 | An ideal is closed under s... |
| rngoidl 38227 | A ring ` R ` is an ` R ` i... |
| 0idl 38228 | The set containing only ` ... |
| 1idl 38229 | Two ways of expressing the... |
| 0rngo 38230 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 38231 | The only ideals in a divis... |
| intidl 38232 | The intersection of a none... |
| inidl 38233 | The intersection of two id... |
| unichnidl 38234 | The union of a nonempty ch... |
| keridl 38235 | The kernel of a ring homom... |
| pridlval 38236 | The class of prime ideals ... |
| ispridl 38237 | The predicate "is a prime ... |
| pridlidl 38238 | A prime ideal is an ideal.... |
| pridlnr 38239 | A prime ideal is a proper ... |
| pridl 38240 | The main property of a pri... |
| ispridl2 38241 | A condition that shows an ... |
| maxidlval 38242 | The set of maximal ideals ... |
| ismaxidl 38243 | The predicate "is a maxima... |
| maxidlidl 38244 | A maximal ideal is an idea... |
| maxidlnr 38245 | A maximal ideal is proper.... |
| maxidlmax 38246 | A maximal ideal is a maxim... |
| maxidln1 38247 | One is not contained in an... |
| maxidln0 38248 | A ring with a maximal idea... |
| isprrngo 38253 | The predicate "is a prime ... |
| prrngorngo 38254 | A prime ring is a ring. (... |
| smprngopr 38255 | A simple ring (one whose o... |
| divrngpr 38256 | A division ring is a prime... |
| isdmn 38257 | The predicate "is a domain... |
| isdmn2 38258 | The predicate "is a domain... |
| dmncrng 38259 | A domain is a commutative ... |
| dmnrngo 38260 | A domain is a ring. (Cont... |
| flddmn 38261 | A field is a domain. (Con... |
| igenval 38264 | The ideal generated by a s... |
| igenss 38265 | A set is a subset of the i... |
| igenidl 38266 | The ideal generated by a s... |
| igenmin 38267 | The ideal generated by a s... |
| igenidl2 38268 | The ideal generated by an ... |
| igenval2 38269 | The ideal generated by a s... |
| prnc 38270 | A principal ideal (an idea... |
| isfldidl 38271 | Determine if a ring is a f... |
| isfldidl2 38272 | Determine if a ring is a f... |
| ispridlc 38273 | The predicate "is a prime ... |
| pridlc 38274 | Property of a prime ideal ... |
| pridlc2 38275 | Property of a prime ideal ... |
| pridlc3 38276 | Property of a prime ideal ... |
| isdmn3 38277 | The predicate "is a domain... |
| dmnnzd 38278 | A domain has no zero-divis... |
| dmncan1 38279 | Cancellation law for domai... |
| dmncan2 38280 | Cancellation law for domai... |
| efald2 38281 | A proof by contradiction. ... |
| notbinot1 38282 | Simplification rule of neg... |
| bicontr 38283 | Biconditional of its own n... |
| impor 38284 | An equivalent formula for ... |
| orfa 38285 | The falsum ` F. ` can be r... |
| notbinot2 38286 | Commutation rule between n... |
| biimpor 38287 | A rewriting rule for bicon... |
| orfa1 38288 | Add a contradicting disjun... |
| orfa2 38289 | Remove a contradicting dis... |
| bifald 38290 | Infer the equivalence to a... |
| orsild 38291 | A lemma for not-or-not eli... |
| orsird 38292 | A lemma for not-or-not eli... |
| cnf1dd 38293 | A lemma for Conjunctive No... |
| cnf2dd 38294 | A lemma for Conjunctive No... |
| cnfn1dd 38295 | A lemma for Conjunctive No... |
| cnfn2dd 38296 | A lemma for Conjunctive No... |
| or32dd 38297 | A rearrangement of disjunc... |
| notornotel1 38298 | A lemma for not-or-not eli... |
| notornotel2 38299 | A lemma for not-or-not eli... |
| contrd 38300 | A proof by contradiction, ... |
| an12i 38301 | An inference from commutin... |
| exmid2 38302 | An excluded middle law. (... |
| selconj 38303 | An inference for selecting... |
| truconj 38304 | Add true as a conjunct. (... |
| orel 38305 | An inference for disjuncti... |
| negel 38306 | An inference for negation ... |
| botel 38307 | An inference for bottom el... |
| tradd 38308 | Add top ad a conjunct. (C... |
| gm-sbtru 38309 | Substitution does not chan... |
| sbfal 38310 | Substitution does not chan... |
| sbcani 38311 | Distribution of class subs... |
| sbcori 38312 | Distribution of class subs... |
| sbcimi 38313 | Distribution of class subs... |
| sbcni 38314 | Move class substitution in... |
| sbali 38315 | Discard class substitution... |
| sbexi 38316 | Discard class substitution... |
| sbcalf 38317 | Move universal quantifier ... |
| sbcexf 38318 | Move existential quantifie... |
| sbcalfi 38319 | Move universal quantifier ... |
| sbcexfi 38320 | Move existential quantifie... |
| spsbcdi 38321 | A lemma for eliminating a ... |
| alrimii 38322 | A lemma for introducing a ... |
| spesbcdi 38323 | A lemma for introducing an... |
| exlimddvf 38324 | A lemma for eliminating an... |
| exlimddvfi 38325 | A lemma for eliminating an... |
| sbceq1ddi 38326 | A lemma for eliminating in... |
| sbccom2lem 38327 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 38328 | Commutative law for double... |
| sbccom2f 38329 | Commutative law for double... |
| sbccom2fi 38330 | Commutative law for double... |
| csbcom2fi 38331 | Commutative law for double... |
| fald 38332 | Refutation of falsity, in ... |
| tsim1 38333 | A Tseitin axiom for logica... |
| tsim2 38334 | A Tseitin axiom for logica... |
| tsim3 38335 | A Tseitin axiom for logica... |
| tsbi1 38336 | A Tseitin axiom for logica... |
| tsbi2 38337 | A Tseitin axiom for logica... |
| tsbi3 38338 | A Tseitin axiom for logica... |
| tsbi4 38339 | A Tseitin axiom for logica... |
| tsxo1 38340 | A Tseitin axiom for logica... |
| tsxo2 38341 | A Tseitin axiom for logica... |
| tsxo3 38342 | A Tseitin axiom for logica... |
| tsxo4 38343 | A Tseitin axiom for logica... |
| tsan1 38344 | A Tseitin axiom for logica... |
| tsan2 38345 | A Tseitin axiom for logica... |
| tsan3 38346 | A Tseitin axiom for logica... |
| tsna1 38347 | A Tseitin axiom for logica... |
| tsna2 38348 | A Tseitin axiom for logica... |
| tsna3 38349 | A Tseitin axiom for logica... |
| tsor1 38350 | A Tseitin axiom for logica... |
| tsor2 38351 | A Tseitin axiom for logica... |
| tsor3 38352 | A Tseitin axiom for logica... |
| ts3an1 38353 | A Tseitin axiom for triple... |
| ts3an2 38354 | A Tseitin axiom for triple... |
| ts3an3 38355 | A Tseitin axiom for triple... |
| ts3or1 38356 | A Tseitin axiom for triple... |
| ts3or2 38357 | A Tseitin axiom for triple... |
| ts3or3 38358 | A Tseitin axiom for triple... |
| iuneq2f 38359 | Equality deduction for ind... |
| rabeq12f 38360 | Equality deduction for res... |
| csbeq12 38361 | Equality deduction for sub... |
| sbeqi 38362 | Equality deduction for sub... |
| ralbi12f 38363 | Equality deduction for res... |
| oprabbi 38364 | Equality deduction for cla... |
| mpobi123f 38365 | Equality deduction for map... |
| iuneq12f 38366 | Equality deduction for ind... |
| iineq12f 38367 | Equality deduction for ind... |
| opabbi 38368 | Equality deduction for cla... |
| mptbi12f 38369 | Equality deduction for map... |
| orcomdd 38370 | Commutativity of logic dis... |
| scottexf 38371 | A version of ~ scottex wit... |
| scott0f 38372 | A version of ~ scott0 with... |
| scottn0f 38373 | A version of ~ scott0f wit... |
| ac6s3f 38374 | Generalization of the Axio... |
| ac6s6 38375 | Generalization of the Axio... |
| ac6s6f 38376 | Generalization of the Axio... |
| el2v1 38428 | New way ( ~ elv , and the ... |
| el3v1 38429 | New way ( ~ elv , and the ... |
| el3v2 38430 | New way ( ~ elv , and the ... |
| el3v12 38431 | New way ( ~ elv , and the ... |
| el3v13 38432 | New way ( ~ elv , and the ... |
| el3v23 38433 | New way ( ~ elv , and the ... |
| anan 38434 | Multiple commutations in c... |
| triantru3 38435 | A wff is equivalent to its... |
| biorfd 38436 | A wff is equivalent to its... |
| eqbrtr 38437 | Substitution of equal clas... |
| eqbrb 38438 | Substitution of equal clas... |
| eqeltr 38439 | Substitution of equal clas... |
| eqelb 38440 | Substitution of equal clas... |
| eqeqan2d 38441 | Implication of introducing... |
| disjresin 38442 | The restriction to a disjo... |
| disjresdisj 38443 | The intersection of restri... |
| disjresdif 38444 | The difference between res... |
| disjresundif 38445 | Lemma for ~ ressucdifsn2 .... |
| inres2 38446 | Two ways of expressing the... |
| coideq 38447 | Equality theorem for compo... |
| nexmo1 38448 | If there is no case where ... |
| eqab2 38449 | Implication of a class abs... |
| r2alan 38450 | Double restricted universa... |
| ssrabi 38451 | Inference of restricted ab... |
| rabimbieq 38452 | Restricted equivalent wff'... |
| abeqin 38453 | Intersection with class ab... |
| abeqinbi 38454 | Intersection with class ab... |
| rabeqel 38455 | Class element of a restric... |
| eqrelf 38456 | The equality connective be... |
| br1cnvinxp 38457 | Binary relation on the con... |
| releleccnv 38458 | Elementhood in a converse ... |
| releccnveq 38459 | Equality of converse ` R `... |
| xpv 38460 | Cartesian product of a cla... |
| vxp 38461 | Cartesian product of the u... |
| opelvvdif 38462 | Negated elementhood of ord... |
| vvdifopab 38463 | Ordered-pair class abstrac... |
| brvdif 38464 | Binary relation with unive... |
| brvdif2 38465 | Binary relation with unive... |
| brvvdif 38466 | Binary relation with the c... |
| brvbrvvdif 38467 | Binary relation with the c... |
| brcnvep 38468 | The converse of the binary... |
| elecALTV 38469 | Elementhood in the ` R ` -... |
| brcnvepres 38470 | Restricted converse epsilo... |
| brres2 38471 | Binary relation on a restr... |
| br1cnvres 38472 | Binary relation on the con... |
| elec1cnvres 38473 | Elementhood in the convers... |
| ec1cnvres 38474 | Converse restricted coset ... |
| eldmres 38475 | Elementhood in the domain ... |
| elrnres 38476 | Element of the range of a ... |
| eldmressnALTV 38477 | Element of the domain of a... |
| elrnressn 38478 | Element of the range of a ... |
| eldm4 38479 | Elementhood in a domain. ... |
| eldmres2 38480 | Elementhood in the domain ... |
| eldmres3 38481 | Elementhood in the domain ... |
| eceq1i 38482 | Equality theorem for ` C `... |
| ecres 38483 | Restricted coset of ` B ` ... |
| eccnvepres 38484 | Restricted converse epsilo... |
| eleccnvep 38485 | Elementhood in the convers... |
| eccnvep 38486 | The converse epsilon coset... |
| extep 38487 | Property of epsilon relati... |
| disjeccnvep 38488 | Property of the epsilon re... |
| eccnvepres2 38489 | The restricted converse ep... |
| eccnvepres3 38490 | Condition for a restricted... |
| eldmqsres 38491 | Elementhood in a restricte... |
| eldmqsres2 38492 | Elementhood in a restricte... |
| qsss1 38493 | Subclass theorem for quoti... |
| qseq1i 38494 | Equality theorem for quoti... |
| brinxprnres 38495 | Binary relation on a restr... |
| inxprnres 38496 | Restriction of a class as ... |
| dfres4 38497 | Alternate definition of th... |
| exan3 38498 | Equivalent expressions wit... |
| exanres 38499 | Equivalent expressions wit... |
| exanres3 38500 | Equivalent expressions wit... |
| exanres2 38501 | Equivalent expressions wit... |
| cnvepres 38502 | Restricted converse epsilo... |
| eqrel2 38503 | Equality of relations. (C... |
| rncnv 38504 | Range of converse is the d... |
| dfdm6 38505 | Alternate definition of do... |
| dfrn6 38506 | Alternate definition of ra... |
| rncnvepres 38507 | The range of the restricte... |
| dmecd 38508 | Equality of the coset of `... |
| dmec2d 38509 | Equality of the coset of `... |
| brid 38510 | Property of the identity b... |
| ideq2 38511 | For sets, the identity bin... |
| idresssidinxp 38512 | Condition for the identity... |
| idreseqidinxp 38513 | Condition for the identity... |
| extid 38514 | Property of identity relat... |
| inxpss 38515 | Two ways to say that an in... |
| idinxpss 38516 | Two ways to say that an in... |
| ref5 38517 | Two ways to say that an in... |
| inxpss3 38518 | Two ways to say that an in... |
| inxpss2 38519 | Two ways to say that inter... |
| inxpssidinxp 38520 | Two ways to say that inter... |
| idinxpssinxp 38521 | Two ways to say that inter... |
| idinxpssinxp2 38522 | Identity intersection with... |
| idinxpssinxp3 38523 | Identity intersection with... |
| idinxpssinxp4 38524 | Identity intersection with... |
| relcnveq3 38525 | Two ways of saying a relat... |
| relcnveq 38526 | Two ways of saying a relat... |
| relcnveq2 38527 | Two ways of saying a relat... |
| relcnveq4 38528 | Two ways of saying a relat... |
| qsresid 38529 | Simplification of a specia... |
| n0elqs 38530 | Two ways of expressing tha... |
| n0elqs2 38531 | Two ways of expressing tha... |
| rnresequniqs 38532 | The range of a restriction... |
| n0el2 38533 | Two ways of expressing tha... |
| cnvepresex 38534 | Sethood condition for the ... |
| cnvepima 38535 | The image of converse epsi... |
| inex3 38536 | Sufficient condition for t... |
| inxpex 38537 | Sufficient condition for a... |
| eqres 38538 | Converting a class constan... |
| brrabga 38539 | The law of concretion for ... |
| brcnvrabga 38540 | The law of concretion for ... |
| opideq 38541 | Equality conditions for or... |
| iss2 38542 | A subclass of the identity... |
| eldmcnv 38543 | Elementhood in a domain of... |
| dfrel5 38544 | Alternate definition of th... |
| dfrel6 38545 | Alternate definition of th... |
| cnvresrn 38546 | Converse restricted to ran... |
| relssinxpdmrn 38547 | Subset of restriction, spe... |
| cnvref4 38548 | Two ways to say that a rel... |
| cnvref5 38549 | Two ways to say that a rel... |
| ecin0 38550 | Two ways of saying that th... |
| ecinn0 38551 | Two ways of saying that th... |
| ineleq 38552 | Equivalence of restricted ... |
| inecmo 38553 | Equivalence of a double re... |
| inecmo2 38554 | Equivalence of a double re... |
| ineccnvmo 38555 | Equivalence of a double re... |
| alrmomorn 38556 | Equivalence of an "at most... |
| alrmomodm 38557 | Equivalence of an "at most... |
| ralmo 38558 | "At most one" can be restr... |
| ralrnmo 38559 | On the range, "at most one... |
| dmqsex 38560 | Sethood of the domain quot... |
| raldmqsmo 38561 | On the quotient carrier, "... |
| ralrmo3 38562 | Pull a restricted universa... |
| raldmqseu 38563 | Equivalence between "exact... |
| rsp3 38564 | From a restricted universa... |
| rsp3eq 38565 | From a restricted universa... |
| ineccnvmo2 38566 | Equivalence of a double un... |
| inecmo3 38567 | Equivalence of a double un... |
| moeu2 38568 | Uniqueness is equivalent t... |
| mopickr 38569 | "At most one" picks a vari... |
| moantr 38570 | Sufficient condition for t... |
| brabidgaw 38571 | The law of concretion for ... |
| brabidga 38572 | The law of concretion for ... |
| inxp2 38573 | Intersection with a Cartes... |
| opabf 38574 | A class abstraction of a c... |
| ec0 38575 | The empty-coset of a class... |
| brcnvin 38576 | Intersection with a conver... |
| ssdmral 38577 | Subclass of a domain. (Co... |
| xrnss3v 38579 | A range Cartesian product ... |
| xrnrel 38580 | A range Cartesian product ... |
| brxrn 38581 | Characterize a ternary rel... |
| brxrn2 38582 | A characterization of the ... |
| dfxrn2 38583 | Alternate definition of th... |
| brxrncnvep 38584 | The range product with con... |
| dmxrn 38585 | Domain of the range produc... |
| dmcnvep 38586 | Domain of converse epsilon... |
| dmxrncnvep 38587 | Domain of the range produc... |
| dmcnvepres 38588 | Domain of the restricted c... |
| dmuncnvepres 38589 | Domain of the union with t... |
| dmxrnuncnvepres 38590 | Domain of the range Cartes... |
| ecun 38591 | The union coset of ` A ` .... |
| ecunres 38592 | The restricted union coset... |
| ecuncnvepres 38593 | The restricted union with ... |
| xrneq1 38594 | Equality theorem for the r... |
| xrneq1i 38595 | Equality theorem for the r... |
| xrneq1d 38596 | Equality theorem for the r... |
| xrneq2 38597 | Equality theorem for the r... |
| xrneq2i 38598 | Equality theorem for the r... |
| xrneq2d 38599 | Equality theorem for the r... |
| xrneq12 38600 | Equality theorem for the r... |
| xrneq12i 38601 | Equality theorem for the r... |
| xrneq12d 38602 | Equality theorem for the r... |
| elecxrn 38603 | Elementhood in the ` ( R |... |
| ecxrn 38604 | The ` ( R |X. S ) ` -coset... |
| relecxrn 38605 | The ` ( R |X. S ) ` -coset... |
| ecxrn2 38606 | The ` ( R |X. S ) ` -coset... |
| ecxrncnvep 38607 | The ` ( R |X. ``' _E ) ` -... |
| ecxrncnvep2 38608 | The ` ( R |X. ``' _E ) ` -... |
| disjressuc2 38609 | Double restricted quantifi... |
| disjecxrn 38610 | Two ways of saying that ` ... |
| disjecxrncnvep 38611 | Two ways of saying that co... |
| disjsuc2 38612 | Double restricted quantifi... |
| xrninxp 38613 | Intersection of a range Ca... |
| xrninxp2 38614 | Intersection of a range Ca... |
| xrninxpex 38615 | Sufficient condition for t... |
| inxpxrn 38616 | Two ways to express the in... |
| br1cnvxrn2 38617 | The converse of a binary r... |
| elec1cnvxrn2 38618 | Elementhood in the convers... |
| rnxrn 38619 | Range of the range Cartesi... |
| rnxrnres 38620 | Range of a range Cartesian... |
| rnxrncnvepres 38621 | Range of a range Cartesian... |
| rnxrnidres 38622 | Range of a range Cartesian... |
| xrnres 38623 | Two ways to express restri... |
| xrnres2 38624 | Two ways to express restri... |
| xrnres3 38625 | Two ways to express restri... |
| xrnres4 38626 | Two ways to express restri... |
| xrnresex 38627 | Sufficient condition for a... |
| xrnidresex 38628 | Sufficient condition for a... |
| xrncnvepresex 38629 | Sufficient condition for a... |
| dmxrncnvepres 38630 | Domain of the range produc... |
| dmxrncnvepres2 38631 | Domain of the range produc... |
| eldmxrncnvepres 38632 | Element of the domain of t... |
| eldmxrncnvepres2 38633 | Element of the domain of t... |
| eceldmqsxrncnvepres 38634 | An ` ( R |X. ( ``' _E |`` ... |
| eceldmqsxrncnvepres2 38635 | An ` ( R |X. ( ``' _E |`` ... |
| brin2 38636 | Binary relation on an inte... |
| brin3 38637 | Binary relation on an inte... |
| elrels2 38639 | The element of the relatio... |
| elrelsrel 38640 | The element of the relatio... |
| elrelsrelim 38641 | The element of the relatio... |
| elrels5 38642 | Equivalent expressions for... |
| elrels6 38643 | Equivalent expressions for... |
| dfqmap2 38645 | Alternate definition of th... |
| dfqmap3 38646 | Alternate definition of th... |
| ecqmap 38647 | ` QMap ` fibers are single... |
| ecqmap2 38648 | Fiber of ` QMap ` equals s... |
| qmapex 38649 | Quotient map exists if ` R... |
| relqmap 38650 | Quotient map is a relation... |
| dmqmap 38651 | ` QMap ` preserves the dom... |
| rnqmap 38652 | The range of the quotient ... |
| dfadjliftmap2 38654 | Alternate definition of th... |
| blockadjliftmap 38655 | A "two-stage" construction... |
| dfblockliftmap2 38657 | Alternate definition of th... |
| dfsucmap3 38659 | Alternate definition of th... |
| dfsucmap2 38660 | Alternate definition of th... |
| dfsucmap4 38661 | Alternate definition of th... |
| brsucmap 38662 | Binary relation form of th... |
| relsucmap 38663 | The successor map is a rel... |
| dmsucmap 38664 | The domain of the successo... |
| dfsuccl2 38666 | Alternate definition of th... |
| mopre 38667 | There is at most one prede... |
| exeupre2 38668 | Whenever a predecessor exi... |
| dfsuccl3 38669 | Alternate definition of th... |
| dfsuccl4 38670 | Alternate definition that ... |
| dfpre 38672 | Alternate definition of th... |
| dfpre2 38673 | Alternate definition of th... |
| dfpre3 38674 | Alternate definition of th... |
| dfpred4 38675 | Alternate definition of th... |
| dfpre4 38676 | Alternate definition of th... |
| suceqsneq 38679 | One-to-one relationship be... |
| sucdifsn2 38680 | Absorption of union with a... |
| sucdifsn 38681 | The difference between the... |
| ressucdifsn2 38682 | The difference between res... |
| ressucdifsn 38683 | The difference between res... |
| sucmapsuc 38684 | A set is succeeded by its ... |
| sucmapleftuniq 38685 | Left uniqueness of the suc... |
| exeupre 38686 | Whenever a predecessor exi... |
| preex 38687 | The successor-predecessor ... |
| eupre2 38688 | Unique predecessor exists ... |
| eupre 38689 | Unique predecessor exists ... |
| presucmap 38690 | ` pre ` is really a predec... |
| preuniqval 38691 | Uniqueness/canonicity of `... |
| sucpre 38692 | ` suc ` is a right-inverse... |
| presuc 38693 | ` pre ` is a left-inverse ... |
| press 38694 | Predecessor is a subset of... |
| preel 38695 | Predecessor is a subset of... |
| dfcoss2 38698 | Alternate definition of th... |
| dfcoss3 38699 | Alternate definition of th... |
| dfcoss4 38700 | Alternate definition of th... |
| cosscnv 38701 | Class of cosets by the con... |
| coss1cnvres 38702 | Class of cosets by the con... |
| coss2cnvepres 38703 | Special case of ~ coss1cnv... |
| cossex 38704 | If ` A ` is a set then the... |
| cosscnvex 38705 | If ` A ` is a set then the... |
| 1cosscnvepresex 38706 | Sufficient condition for a... |
| 1cossxrncnvepresex 38707 | Sufficient condition for a... |
| relcoss 38708 | Cosets by ` R ` is a relat... |
| relcoels 38709 | Coelements on ` A ` is a r... |
| cossss 38710 | Subclass theorem for the c... |
| cosseq 38711 | Equality theorem for the c... |
| cosseqi 38712 | Equality theorem for the c... |
| cosseqd 38713 | Equality theorem for the c... |
| 1cossres 38714 | The class of cosets by a r... |
| dfcoels 38715 | Alternate definition of th... |
| brcoss 38716 | ` A ` and ` B ` are cosets... |
| brcoss2 38717 | Alternate form of the ` A ... |
| brcoss3 38718 | Alternate form of the ` A ... |
| brcosscnvcoss 38719 | For sets, the ` A ` and ` ... |
| brcoels 38720 | ` B ` and ` C ` are coelem... |
| cocossss 38721 | Two ways of saying that co... |
| cnvcosseq 38722 | The converse of cosets by ... |
| br2coss 38723 | Cosets by ` ,~ R ` binary ... |
| br1cossres 38724 | ` B ` and ` C ` are cosets... |
| br1cossres2 38725 | ` B ` and ` C ` are cosets... |
| brressn 38726 | Binary relation on a restr... |
| ressn2 38727 | A class ' R ' restricted t... |
| refressn 38728 | Any class ' R ' restricted... |
| antisymressn 38729 | Every class ' R ' restrict... |
| trressn 38730 | Any class ' R ' restricted... |
| relbrcoss 38731 | ` A ` and ` B ` are cosets... |
| br1cossinres 38732 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 38733 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 38734 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 38735 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 38736 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 38737 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 38738 | The domain of cosets is th... |
| dmcoss2 38739 | The domain of cosets is th... |
| rncossdmcoss 38740 | The range of cosets is the... |
| dm1cosscnvepres 38741 | The domain of cosets of th... |
| dmcoels 38742 | The domain of coelements i... |
| eldmcoss 38743 | Elementhood in the domain ... |
| eldmcoss2 38744 | Elementhood in the domain ... |
| eldm1cossres 38745 | Elementhood in the domain ... |
| eldm1cossres2 38746 | Elementhood in the domain ... |
| refrelcosslem 38747 | Lemma for the left side of... |
| refrelcoss3 38748 | The class of cosets by ` R... |
| refrelcoss2 38749 | The class of cosets by ` R... |
| symrelcoss3 38750 | The class of cosets by ` R... |
| symrelcoss2 38751 | The class of cosets by ` R... |
| cossssid 38752 | Equivalent expressions for... |
| cossssid2 38753 | Equivalent expressions for... |
| cossssid3 38754 | Equivalent expressions for... |
| cossssid4 38755 | Equivalent expressions for... |
| cossssid5 38756 | Equivalent expressions for... |
| brcosscnv 38757 | ` A ` and ` B ` are cosets... |
| brcosscnv2 38758 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 38759 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 38760 | Cosets by the converse ran... |
| cosscnvssid3 38761 | Equivalent expressions for... |
| cosscnvssid4 38762 | Equivalent expressions for... |
| cosscnvssid5 38763 | Equivalent expressions for... |
| coss0 38764 | Cosets by the empty set ar... |
| cossid 38765 | Cosets by the identity rel... |
| cosscnvid 38766 | Cosets by the converse ide... |
| trcoss 38767 | Sufficient condition for t... |
| eleccossin 38768 | Two ways of saying that th... |
| trcoss2 38769 | Equivalent expressions for... |
| cosselrels 38770 | Cosets of sets are element... |
| cnvelrels 38771 | The converse of a set is a... |
| cosscnvelrels 38772 | Cosets of converse sets ar... |
| dfssr2 38774 | Alternate definition of th... |
| relssr 38775 | The subset relation is a r... |
| brssr 38776 | The subset relation and su... |
| brssrid 38777 | Any set is a subset of its... |
| issetssr 38778 | Two ways of expressing set... |
| brssrres 38779 | Restricted subset binary r... |
| br1cnvssrres 38780 | Restricted converse subset... |
| brcnvssr 38781 | The converse of a subset r... |
| brcnvssrid 38782 | Any set is a converse subs... |
| br1cossxrncnvssrres 38783 | ` <. B , C >. ` and ` <. D... |
| extssr 38784 | Property of subset relatio... |
| dfrefrels2 38788 | Alternate definition of th... |
| dfrefrels3 38789 | Alternate definition of th... |
| dfrefrel2 38790 | Alternate definition of th... |
| dfrefrel3 38791 | Alternate definition of th... |
| dfrefrel5 38792 | Alternate definition of th... |
| elrefrels2 38793 | Element of the class of re... |
| elrefrels3 38794 | Element of the class of re... |
| elrefrelsrel 38795 | For sets, being an element... |
| refreleq 38796 | Equality theorem for refle... |
| refrelid 38797 | Identity relation is refle... |
| refrelcoss 38798 | The class of cosets by ` R... |
| refrelressn 38799 | Any class ' R ' restricted... |
| dfcnvrefrels2 38803 | Alternate definition of th... |
| dfcnvrefrels3 38804 | Alternate definition of th... |
| dfcnvrefrel2 38805 | Alternate definition of th... |
| dfcnvrefrel3 38806 | Alternate definition of th... |
| dfcnvrefrel4 38807 | Alternate definition of th... |
| dfcnvrefrel5 38808 | Alternate definition of th... |
| elcnvrefrels2 38809 | Element of the class of co... |
| elcnvrefrels3 38810 | Element of the class of co... |
| elcnvrefrelsrel 38811 | For sets, being an element... |
| cnvrefrelcoss2 38812 | Necessary and sufficient c... |
| cosselcnvrefrels2 38813 | Necessary and sufficient c... |
| cosselcnvrefrels3 38814 | Necessary and sufficient c... |
| cosselcnvrefrels4 38815 | Necessary and sufficient c... |
| cosselcnvrefrels5 38816 | Necessary and sufficient c... |
| dfsymrels2 38820 | Alternate definition of th... |
| dfsymrels3 38821 | Alternate definition of th... |
| elrelscnveq3 38822 | Two ways of saying a relat... |
| elrelscnveq 38823 | Two ways of saying a relat... |
| elrelscnveq2 38824 | Two ways of saying a relat... |
| elrelscnveq4 38825 | Two ways of saying a relat... |
| dfsymrels4 38826 | Alternate definition of th... |
| dfsymrels5 38827 | Alternate definition of th... |
| dfsymrel2 38828 | Alternate definition of th... |
| dfsymrel3 38829 | Alternate definition of th... |
| dfsymrel4 38830 | Alternate definition of th... |
| dfsymrel5 38831 | Alternate definition of th... |
| elsymrels2 38832 | Element of the class of sy... |
| elsymrels3 38833 | Element of the class of sy... |
| elsymrels4 38834 | Element of the class of sy... |
| elsymrels5 38835 | Element of the class of sy... |
| elsymrelsrel 38836 | For sets, being an element... |
| symreleq 38837 | Equality theorem for symme... |
| symrelim 38838 | Symmetric relation implies... |
| symrelcoss 38839 | The class of cosets by ` R... |
| idsymrel 38840 | The identity relation is s... |
| epnsymrel 38841 | The membership (epsilon) r... |
| symrefref2 38842 | Symmetry is a sufficient c... |
| symrefref3 38843 | Symmetry is a sufficient c... |
| refsymrels2 38844 | Elements of the class of r... |
| refsymrels3 38845 | Elements of the class of r... |
| refsymrel2 38846 | A relation which is reflex... |
| refsymrel3 38847 | A relation which is reflex... |
| elrefsymrels2 38848 | Elements of the class of r... |
| elrefsymrels3 38849 | Elements of the class of r... |
| elrefsymrelsrel 38850 | For sets, being an element... |
| dftrrels2 38854 | Alternate definition of th... |
| dftrrels3 38855 | Alternate definition of th... |
| dftrrel2 38856 | Alternate definition of th... |
| dftrrel3 38857 | Alternate definition of th... |
| eltrrels2 38858 | Element of the class of tr... |
| eltrrels3 38859 | Element of the class of tr... |
| eltrrelsrel 38860 | For sets, being an element... |
| trreleq 38861 | Equality theorem for the t... |
| trrelressn 38862 | Any class ' R ' restricted... |
| dfeqvrels2 38867 | Alternate definition of th... |
| dfeqvrels3 38868 | Alternate definition of th... |
| dfeqvrel2 38869 | Alternate definition of th... |
| dfeqvrel3 38870 | Alternate definition of th... |
| eleqvrels2 38871 | Element of the class of eq... |
| eleqvrels3 38872 | Element of the class of eq... |
| eleqvrelsrel 38873 | For sets, being an element... |
| elcoeleqvrels 38874 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 38875 | For sets, being an element... |
| eqvrelrel 38876 | An equivalence relation is... |
| eqvrelrefrel 38877 | An equivalence relation is... |
| eqvrelsymrel 38878 | An equivalence relation is... |
| eqvreltrrel 38879 | An equivalence relation is... |
| eqvrelim 38880 | Equivalence relation impli... |
| eqvreleq 38881 | Equality theorem for equiv... |
| eqvreleqi 38882 | Equality theorem for equiv... |
| eqvreleqd 38883 | Equality theorem for equiv... |
| eqvrelsym 38884 | An equivalence relation is... |
| eqvrelsymb 38885 | An equivalence relation is... |
| eqvreltr 38886 | An equivalence relation is... |
| eqvreltrd 38887 | A transitivity relation fo... |
| eqvreltr4d 38888 | A transitivity relation fo... |
| eqvrelref 38889 | An equivalence relation is... |
| eqvrelth 38890 | Basic property of equivale... |
| eqvrelcl 38891 | Elementhood in the field o... |
| eqvrelthi 38892 | Basic property of equivale... |
| eqvreldisj 38893 | Equivalence classes do not... |
| qsdisjALTV 38894 | Elements of a quotient set... |
| eqvrelqsel 38895 | If an element of a quotien... |
| eqvrelcoss 38896 | Two ways to express equiva... |
| eqvrelcoss3 38897 | Two ways to express equiva... |
| eqvrelcoss2 38898 | Two ways to express equiva... |
| eqvrelcoss4 38899 | Two ways to express equiva... |
| dfcoeleqvrels 38900 | Alternate definition of th... |
| dfcoeleqvrel 38901 | Alternate definition of th... |
| brredunds 38905 | Binary relation on the cla... |
| brredundsredund 38906 | For sets, binary relation ... |
| redundss3 38907 | Implication of redundancy ... |
| redundeq1 38908 | Equivalence of redundancy ... |
| redundpim3 38909 | Implication of redundancy ... |
| redundpbi1 38910 | Equivalence of redundancy ... |
| refrelsredund4 38911 | The naive version of the c... |
| refrelsredund2 38912 | The naive version of the c... |
| refrelsredund3 38913 | The naive version of the c... |
| refrelredund4 38914 | The naive version of the d... |
| refrelredund2 38915 | The naive version of the d... |
| refrelredund3 38916 | The naive version of the d... |
| dfblockliftfix2 38919 | Alternate definition of th... |
| dmqseq 38920 | Equality theorem for domai... |
| dmqseqi 38921 | Equality theorem for domai... |
| dmqseqd 38922 | Equality theorem for domai... |
| dmqseqeq1 38923 | Equality theorem for domai... |
| dmqseqeq1i 38924 | Equality theorem for domai... |
| dmqseqeq1d 38925 | Equality theorem for domai... |
| brdmqss 38926 | The domain quotient binary... |
| brdmqssqs 38927 | If ` A ` and ` R ` are set... |
| n0eldmqs 38928 | The empty set is not an el... |
| qseq 38929 | The quotient set equal to ... |
| n0eldmqseq 38930 | The empty set is not an el... |
| n0elim 38931 | Implication of that the em... |
| n0el3 38932 | Two ways of expressing tha... |
| cnvepresdmqss 38933 | The domain quotient binary... |
| cnvepresdmqs 38934 | The domain quotient predic... |
| unidmqs 38935 | The range of a relation is... |
| unidmqseq 38936 | The union of the domain qu... |
| dmqseqim 38937 | If the domain quotient of ... |
| dmqseqim2 38938 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 38939 | Elementhood in the domain ... |
| eldmqs1cossres 38940 | Elementhood in the domain ... |
| releldmqscoss 38941 | Elementhood in the domain ... |
| dmqscoelseq 38942 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 38943 | Two ways to express the eq... |
| brers 38948 | Binary equivalence relatio... |
| dferALTV2 38949 | Equivalence relation with ... |
| erALTVeq1 38950 | Equality theorem for equiv... |
| erALTVeq1i 38951 | Equality theorem for equiv... |
| erALTVeq1d 38952 | Equality theorem for equiv... |
| dfcomember 38953 | Alternate definition of th... |
| dfcomember2 38954 | Alternate definition of th... |
| dfcomember3 38955 | Alternate definition of th... |
| eqvreldmqs 38956 | Two ways to express comemb... |
| eqvreldmqs2 38957 | Two ways to express comemb... |
| brerser 38958 | Binary equivalence relatio... |
| erimeq2 38959 | Equivalence relation on it... |
| erimeq 38960 | Equivalence relation on it... |
| dffunsALTV 38964 | Alternate definition of th... |
| dffunsALTV2 38965 | Alternate definition of th... |
| dffunsALTV3 38966 | Alternate definition of th... |
| dffunsALTV4 38967 | Alternate definition of th... |
| dffunsALTV5 38968 | Alternate definition of th... |
| dffunALTV2 38969 | Alternate definition of th... |
| dffunALTV3 38970 | Alternate definition of th... |
| dffunALTV4 38971 | Alternate definition of th... |
| dffunALTV5 38972 | Alternate definition of th... |
| elfunsALTV 38973 | Elementhood in the class o... |
| elfunsALTV2 38974 | Elementhood in the class o... |
| elfunsALTV3 38975 | Elementhood in the class o... |
| elfunsALTV4 38976 | Elementhood in the class o... |
| elfunsALTV5 38977 | Elementhood in the class o... |
| elfunsALTVfunALTV 38978 | The element of the class o... |
| funALTVfun 38979 | Our definition of the func... |
| funALTVss 38980 | Subclass theorem for funct... |
| funALTVeq 38981 | Equality theorem for funct... |
| funALTVeqi 38982 | Equality inference for the... |
| funALTVeqd 38983 | Equality deduction for the... |
| dfdisjs 38989 | Alternate definition of th... |
| dfdisjs2 38990 | Alternate definition of th... |
| dfdisjs3 38991 | Alternate definition of th... |
| dfdisjs4 38992 | Alternate definition of th... |
| dfdisjs5 38993 | Alternate definition of th... |
| dfdisjALTV 38994 | Alternate definition of th... |
| dfdisjALTV2 38995 | Alternate definition of th... |
| dfdisjALTV3 38996 | Alternate definition of th... |
| dfdisjALTV4 38997 | Alternate definition of th... |
| dfdisjALTV5 38998 | Alternate definition of th... |
| dfeldisj2 38999 | Alternate definition of th... |
| dfeldisj3 39000 | Alternate definition of th... |
| dfeldisj4 39001 | Alternate definition of th... |
| dfeldisj5 39002 | Alternate definition of th... |
| eldisjs 39003 | Elementhood in the class o... |
| eldisjs2 39004 | Elementhood in the class o... |
| eldisjs3 39005 | Elementhood in the class o... |
| eldisjs4 39006 | Elementhood in the class o... |
| eldisjs5 39007 | Elementhood in the class o... |
| eldisjsdisj 39008 | The element of the class o... |
| eleldisjs 39009 | Elementhood in the disjoin... |
| eleldisjseldisj 39010 | The element of the disjoin... |
| disjrel 39011 | Disjoint relation is a rel... |
| disjss 39012 | Subclass theorem for disjo... |
| disjssi 39013 | Subclass theorem for disjo... |
| disjssd 39014 | Subclass theorem for disjo... |
| disjeq 39015 | Equality theorem for disjo... |
| disjeqi 39016 | Equality theorem for disjo... |
| disjeqd 39017 | Equality theorem for disjo... |
| disjdmqseqeq1 39018 | Lemma for the equality the... |
| eldisjss 39019 | Subclass theorem for disjo... |
| eldisjssi 39020 | Subclass theorem for disjo... |
| eldisjssd 39021 | Subclass theorem for disjo... |
| eldisjeq 39022 | Equality theorem for disjo... |
| eldisjeqi 39023 | Equality theorem for disjo... |
| eldisjeqd 39024 | Equality theorem for disjo... |
| disjres 39025 | Disjoint restriction. (Co... |
| eldisjn0elb 39026 | Two forms of disjoint elem... |
| disjxrn 39027 | Two ways of saying that a ... |
| disjxrnres5 39028 | Disjoint range Cartesian p... |
| disjorimxrn 39029 | Disjointness condition for... |
| disjimxrn 39030 | Disjointness condition for... |
| disjimres 39031 | Disjointness condition for... |
| disjimin 39032 | Disjointness condition for... |
| disjiminres 39033 | Disjointness condition for... |
| disjimxrnres 39034 | Disjointness condition for... |
| disjALTV0 39035 | The null class is disjoint... |
| disjALTVid 39036 | The class of identity rela... |
| disjALTVidres 39037 | The class of identity rela... |
| disjALTVinidres 39038 | The intersection with rest... |
| disjALTVxrnidres 39039 | The class of range Cartesi... |
| disjsuc 39040 | Disjoint range Cartesian p... |
| dfantisymrel4 39042 | Alternate definition of th... |
| dfantisymrel5 39043 | Alternate definition of th... |
| antisymrelres 39044 | (Contributed by Peter Mazs... |
| antisymrelressn 39045 | (Contributed by Peter Mazs... |
| dfpart2 39050 | Alternate definition of th... |
| dfmembpart2 39051 | Alternate definition of th... |
| brparts 39052 | Binary partitions relation... |
| brparts2 39053 | Binary partitions relation... |
| brpartspart 39054 | Binary partition and the p... |
| parteq1 39055 | Equality theorem for parti... |
| parteq2 39056 | Equality theorem for parti... |
| parteq12 39057 | Equality theorem for parti... |
| parteq1i 39058 | Equality theorem for parti... |
| parteq1d 39059 | Equality theorem for parti... |
| partsuc2 39060 | Property of the partition.... |
| partsuc 39061 | Property of the partition.... |
| disjim 39062 | The "Divide et Aequivalere... |
| disjimi 39063 | Every disjoint relation ge... |
| detlem 39064 | If a relation is disjoint,... |
| eldisjim 39065 | If the elements of ` A ` a... |
| eldisjim2 39066 | Alternate form of ~ eldisj... |
| eqvrel0 39067 | The null class is an equiv... |
| det0 39068 | The cosets by the null cla... |
| eqvrelcoss0 39069 | The cosets by the null cla... |
| eqvrelid 39070 | The identity relation is a... |
| eqvrel1cossidres 39071 | The cosets by a restricted... |
| eqvrel1cossinidres 39072 | The cosets by an intersect... |
| eqvrel1cossxrnidres 39073 | The cosets by a range Cart... |
| detid 39074 | The cosets by the identity... |
| eqvrelcossid 39075 | The cosets by the identity... |
| detidres 39076 | The cosets by the restrict... |
| detinidres 39077 | The cosets by the intersec... |
| detxrnidres 39078 | The cosets by the range Ca... |
| disjlem14 39079 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 39080 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 39081 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 39082 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 39083 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 39084 | Lemma for ~ disjdmqseq via... |
| disjdmqs 39085 | If a relation is disjoint,... |
| disjdmqseq 39086 | If a relation is disjoint,... |
| eldisjn0el 39087 | Special case of ~ disjdmqs... |
| partim2 39088 | Disjoint relation on its n... |
| partim 39089 | Partition implies equivale... |
| partimeq 39090 | Partition implies that the... |
| eldisjlem19 39091 | Special case of ~ disjlem1... |
| membpartlem19 39092 | Together with ~ disjlem19 ... |
| petlem 39093 | If you can prove that the ... |
| petlemi 39094 | If you can prove disjointn... |
| pet02 39095 | Class ` A ` is a partition... |
| pet0 39096 | Class ` A ` is a partition... |
| petid2 39097 | Class ` A ` is a partition... |
| petid 39098 | A class is a partition by ... |
| petidres2 39099 | Class ` A ` is a partition... |
| petidres 39100 | A class is a partition by ... |
| petinidres2 39101 | Class ` A ` is a partition... |
| petinidres 39102 | A class is a partition by ... |
| petxrnidres2 39103 | Class ` A ` is a partition... |
| petxrnidres 39104 | A class is a partition by ... |
| eqvreldisj1 39105 | The elements of the quotie... |
| eqvreldisj2 39106 | The elements of the quotie... |
| eqvreldisj3 39107 | The elements of the quotie... |
| eqvreldisj4 39108 | Intersection with the conv... |
| eqvreldisj5 39109 | Range Cartesian product wi... |
| eqvrelqseqdisj2 39110 | Implication of ~ eqvreldis... |
| fences3 39111 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 39112 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 39113 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 39114 | Lemma for the Partition-Eq... |
| mainer 39115 | The Main Theorem of Equiva... |
| partimcomember 39116 | Partition with general ` R... |
| mpet3 39117 | Member Partition-Equivalen... |
| cpet2 39118 | The conventional form of t... |
| cpet 39119 | The conventional form of M... |
| mpet 39120 | Member Partition-Equivalen... |
| mpet2 39121 | Member Partition-Equivalen... |
| mpets2 39122 | Member Partition-Equivalen... |
| mpets 39123 | Member Partition-Equivalen... |
| mainpart 39124 | Partition with general ` R... |
| fences 39125 | The Theorem of Fences by E... |
| fences2 39126 | The Theorem of Fences by E... |
| mainer2 39127 | The Main Theorem of Equiva... |
| mainerim 39128 | Every equivalence relation... |
| petincnvepres2 39129 | A partition-equivalence th... |
| petincnvepres 39130 | The shortest form of a par... |
| pet2 39131 | Partition-Equivalence Theo... |
| pet 39132 | Partition-Equivalence Theo... |
| pets 39133 | Partition-Equivalence Theo... |
| dmqsblocks 39134 | If the ~ pet span ` ( R |X... |
| prtlem60 39135 | Lemma for ~ prter3 . (Con... |
| bicomdd 39136 | Commute two sides of a bic... |
| jca2r 39137 | Inference conjoining the c... |
| jca3 39138 | Inference conjoining the c... |
| prtlem70 39139 | Lemma for ~ prter3 : a rea... |
| ibdr 39140 | Reverse of ~ ibd . (Contr... |
| prtlem100 39141 | Lemma for ~ prter3 . (Con... |
| prtlem5 39142 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 39143 | Lemma for ~ prter2 . (Con... |
| brabsb2 39144 | A closed form of ~ brabsb ... |
| eqbrrdv2 39145 | Other version of ~ eqbrrdi... |
| prtlem9 39146 | Lemma for ~ prter3 . (Con... |
| prtlem10 39147 | Lemma for ~ prter3 . (Con... |
| prtlem11 39148 | Lemma for ~ prter2 . (Con... |
| prtlem12 39149 | Lemma for ~ prtex and ~ pr... |
| prtlem13 39150 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 39151 | Lemma for ~ prtex , ~ prte... |
| prtlem400 39152 | Lemma for ~ prter2 and als... |
| erprt 39155 | The quotient set of an equ... |
| prtlem14 39156 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 39157 | Lemma for ~ prter1 and ~ p... |
| prtlem17 39158 | Lemma for ~ prter2 . (Con... |
| prtlem18 39159 | Lemma for ~ prter2 . (Con... |
| prtlem19 39160 | Lemma for ~ prter2 . (Con... |
| prter1 39161 | Every partition generates ... |
| prtex 39162 | The equivalence relation g... |
| prter2 39163 | The quotient set of the eq... |
| prter3 39164 | For every partition there ... |
| axc5 39175 | This theorem repeats ~ sp ... |
| ax4fromc4 39176 | Rederivation of Axiom ~ ax... |
| ax10fromc7 39177 | Rederivation of Axiom ~ ax... |
| ax6fromc10 39178 | Rederivation of Axiom ~ ax... |
| hba1-o 39179 | The setvar ` x ` is not fr... |
| axc4i-o 39180 | Inference version of ~ ax-... |
| equid1 39181 | Proof of ~ equid from our ... |
| equcomi1 39182 | Proof of ~ equcomi from ~ ... |
| aecom-o 39183 | Commutation law for identi... |
| aecoms-o 39184 | A commutation rule for ide... |
| hbae-o 39185 | All variables are effectiv... |
| dral1-o 39186 | Formula-building lemma for... |
| ax12fromc15 39187 | Rederivation of Axiom ~ ax... |
| ax13fromc9 39188 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 39189 | Axiom to quantify a variab... |
| sps-o 39190 | Generalization of antecede... |
| hbequid 39191 | Bound-variable hypothesis ... |
| nfequid-o 39192 | Bound-variable hypothesis ... |
| axc5c7 39193 | Proof of a single axiom th... |
| axc5c7toc5 39194 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 39195 | Rederivation of ~ ax-c7 fr... |
| axc711 39196 | Proof of a single axiom th... |
| nfa1-o 39197 | ` x ` is not free in ` A. ... |
| axc711toc7 39198 | Rederivation of ~ ax-c7 fr... |
| axc711to11 39199 | Rederivation of ~ ax-11 fr... |
| axc5c711 39200 | Proof of a single axiom th... |
| axc5c711toc5 39201 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 39202 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 39203 | Rederivation of ~ ax-11 fr... |
| equidqe 39204 | ~ equid with existential q... |
| axc5sp1 39205 | A special case of ~ ax-c5 ... |
| equidq 39206 | ~ equid with universal qua... |
| equid1ALT 39207 | Alternate proof of ~ equid... |
| axc11nfromc11 39208 | Rederivation of ~ ax-c11n ... |
| naecoms-o 39209 | A commutation rule for dis... |
| hbnae-o 39210 | All variables are effectiv... |
| dvelimf-o 39211 | Proof of ~ dvelimh that us... |
| dral2-o 39212 | Formula-building lemma for... |
| aev-o 39213 | A "distinctor elimination"... |
| ax5eq 39214 | Theorem to add distinct qu... |
| dveeq2-o 39215 | Quantifier introduction wh... |
| axc16g-o 39216 | A generalization of Axiom ... |
| dveeq1-o 39217 | Quantifier introduction wh... |
| dveeq1-o16 39218 | Version of ~ dveeq1 using ... |
| ax5el 39219 | Theorem to add distinct qu... |
| axc11n-16 39220 | This theorem shows that, g... |
| dveel2ALT 39221 | Alternate proof of ~ dveel... |
| ax12f 39222 | Basis step for constructin... |
| ax12eq 39223 | Basis step for constructin... |
| ax12el 39224 | Basis step for constructin... |
| ax12indn 39225 | Induction step for constru... |
| ax12indi 39226 | Induction step for constru... |
| ax12indalem 39227 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 39228 | Alternate proof of ~ ax12i... |
| ax12inda2 39229 | Induction step for constru... |
| ax12inda 39230 | Induction step for constru... |
| ax12v2-o 39231 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 39232 | Derive ~ ax-c15 from a hyp... |
| axc11-o 39233 | Show that ~ ax-c11 can be ... |
| fsumshftd 39234 | Index shift of a finite su... |
| riotaclbgBAD 39236 | Closure of restricted iota... |
| riotaclbBAD 39237 | Closure of restricted iota... |
| riotasvd 39238 | Deduction version of ~ rio... |
| riotasv2d 39239 | Value of description binde... |
| riotasv2s 39240 | The value of description b... |
| riotasv 39241 | Value of description binde... |
| riotasv3d 39242 | A property ` ch ` holding ... |
| elimhyps 39243 | A version of ~ elimhyp usi... |
| dedths 39244 | A version of weak deductio... |
| renegclALT 39245 | Closure law for negative o... |
| elimhyps2 39246 | Generalization of ~ elimhy... |
| dedths2 39247 | Generalization of ~ dedths... |
| nfcxfrdf 39248 | A utility lemma to transfe... |
| nfded 39249 | A deduction theorem that c... |
| nfded2 39250 | A deduction theorem that c... |
| nfunidALT2 39251 | Deduction version of ~ nfu... |
| nfunidALT 39252 | Deduction version of ~ nfu... |
| nfopdALT 39253 | Deduction version of bound... |
| cnaddcom 39254 | Recover the commutative la... |
| toycom 39255 | Show the commutative law f... |
| lshpset 39260 | The set of all hyperplanes... |
| islshp 39261 | The predicate "is a hyperp... |
| islshpsm 39262 | Hyperplane properties expr... |
| lshplss 39263 | A hyperplane is a subspace... |
| lshpne 39264 | A hyperplane is not equal ... |
| lshpnel 39265 | A hyperplane's generating ... |
| lshpnelb 39266 | The subspace sum of a hype... |
| lshpnel2N 39267 | Condition that determines ... |
| lshpne0 39268 | The member of the span in ... |
| lshpdisj 39269 | A hyperplane and the span ... |
| lshpcmp 39270 | If two hyperplanes are com... |
| lshpinN 39271 | The intersection of two di... |
| lsatset 39272 | The set of all 1-dim subsp... |
| islsat 39273 | The predicate "is a 1-dim ... |
| lsatlspsn2 39274 | The span of a nonzero sing... |
| lsatlspsn 39275 | The span of a nonzero sing... |
| islsati 39276 | A 1-dim subspace (atom) (o... |
| lsateln0 39277 | A 1-dim subspace (atom) (o... |
| lsatlss 39278 | The set of 1-dim subspaces... |
| lsatlssel 39279 | An atom is a subspace. (C... |
| lsatssv 39280 | An atom is a set of vector... |
| lsatn0 39281 | A 1-dim subspace (atom) of... |
| lsatspn0 39282 | The span of a vector is an... |
| lsator0sp 39283 | The span of a vector is ei... |
| lsatssn0 39284 | A subspace (or any class) ... |
| lsatcmp 39285 | If two atoms are comparabl... |
| lsatcmp2 39286 | If an atom is included in ... |
| lsatel 39287 | A nonzero vector in an ato... |
| lsatelbN 39288 | A nonzero vector in an ato... |
| lsat2el 39289 | Two atoms sharing a nonzer... |
| lsmsat 39290 | Convert comparison of atom... |
| lsatfixedN 39291 | Show equality with the spa... |
| lsmsatcv 39292 | Subspace sum has the cover... |
| lssatomic 39293 | The lattice of subspaces i... |
| lssats 39294 | The lattice of subspaces i... |
| lpssat 39295 | Two subspaces in a proper ... |
| lrelat 39296 | Subspaces are relatively a... |
| lssatle 39297 | The ordering of two subspa... |
| lssat 39298 | Two subspaces in a proper ... |
| islshpat 39299 | Hyperplane properties expr... |
| lcvfbr 39302 | The covers relation for a ... |
| lcvbr 39303 | The covers relation for a ... |
| lcvbr2 39304 | The covers relation for a ... |
| lcvbr3 39305 | The covers relation for a ... |
| lcvpss 39306 | The covers relation implie... |
| lcvnbtwn 39307 | The covers relation implie... |
| lcvntr 39308 | The covers relation is not... |
| lcvnbtwn2 39309 | The covers relation implie... |
| lcvnbtwn3 39310 | The covers relation implie... |
| lsmcv2 39311 | Subspace sum has the cover... |
| lcvat 39312 | If a subspace covers anoth... |
| lsatcv0 39313 | An atom covers the zero su... |
| lsatcveq0 39314 | A subspace covered by an a... |
| lsat0cv 39315 | A subspace is an atom iff ... |
| lcvexchlem1 39316 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 39317 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 39318 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 39319 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 39320 | Lemma for ~ lcvexch . (Co... |
| lcvexch 39321 | Subspaces satisfy the exch... |
| lcvp 39322 | Covering property of Defin... |
| lcv1 39323 | Covering property of a sub... |
| lcv2 39324 | Covering property of a sub... |
| lsatexch 39325 | The atom exchange property... |
| lsatnle 39326 | The meet of a subspace and... |
| lsatnem0 39327 | The meet of distinct atoms... |
| lsatexch1 39328 | The atom exch1ange propert... |
| lsatcv0eq 39329 | If the sum of two atoms co... |
| lsatcv1 39330 | Two atoms covering the zer... |
| lsatcvatlem 39331 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 39332 | A nonzero subspace less th... |
| lsatcvat2 39333 | A subspace covered by the ... |
| lsatcvat3 39334 | A condition implying that ... |
| islshpcv 39335 | Hyperplane properties expr... |
| l1cvpat 39336 | A subspace covered by the ... |
| l1cvat 39337 | Create an atom under an el... |
| lshpat 39338 | Create an atom under a hyp... |
| lflset 39341 | The set of linear function... |
| islfl 39342 | The predicate "is a linear... |
| lfli 39343 | Property of a linear funct... |
| islfld 39344 | Properties that determine ... |
| lflf 39345 | A linear functional is a f... |
| lflcl 39346 | A linear functional value ... |
| lfl0 39347 | A linear functional is zer... |
| lfladd 39348 | Property of a linear funct... |
| lflsub 39349 | Property of a linear funct... |
| lflmul 39350 | Property of a linear funct... |
| lfl0f 39351 | The zero function is a fun... |
| lfl1 39352 | A nonzero functional has a... |
| lfladdcl 39353 | Closure of addition of two... |
| lfladdcom 39354 | Commutativity of functiona... |
| lfladdass 39355 | Associativity of functiona... |
| lfladd0l 39356 | Functional addition with t... |
| lflnegcl 39357 | Closure of the negative of... |
| lflnegl 39358 | A functional plus its nega... |
| lflvscl 39359 | Closure of a scalar produc... |
| lflvsdi1 39360 | Distributive law for (righ... |
| lflvsdi2 39361 | Reverse distributive law f... |
| lflvsdi2a 39362 | Reverse distributive law f... |
| lflvsass 39363 | Associative law for (right... |
| lfl0sc 39364 | The (right vector space) s... |
| lflsc0N 39365 | The scalar product with th... |
| lfl1sc 39366 | The (right vector space) s... |
| lkrfval 39369 | The kernel of a functional... |
| lkrval 39370 | Value of the kernel of a f... |
| ellkr 39371 | Membership in the kernel o... |
| lkrval2 39372 | Value of the kernel of a f... |
| ellkr2 39373 | Membership in the kernel o... |
| lkrcl 39374 | A member of the kernel of ... |
| lkrf0 39375 | The value of a functional ... |
| lkr0f 39376 | The kernel of the zero fun... |
| lkrlss 39377 | The kernel of a linear fun... |
| lkrssv 39378 | The kernel of a linear fun... |
| lkrsc 39379 | The kernel of a nonzero sc... |
| lkrscss 39380 | The kernel of a scalar pro... |
| eqlkr 39381 | Two functionals with the s... |
| eqlkr2 39382 | Two functionals with the s... |
| eqlkr3 39383 | Two functionals with the s... |
| lkrlsp 39384 | The subspace sum of a kern... |
| lkrlsp2 39385 | The subspace sum of a kern... |
| lkrlsp3 39386 | The subspace sum of a kern... |
| lkrshp 39387 | The kernel of a nonzero fu... |
| lkrshp3 39388 | The kernels of nonzero fun... |
| lkrshpor 39389 | The kernel of a functional... |
| lkrshp4 39390 | A kernel is a hyperplane i... |
| lshpsmreu 39391 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 39392 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 39393 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 39394 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 39395 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 39396 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 39397 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 39398 | The set ` G ` defined by h... |
| lshpkr 39399 | The kernel of functional `... |
| lshpkrex 39400 | There exists a functional ... |
| lshpset2N 39401 | The set of all hyperplanes... |
| islshpkrN 39402 | The predicate "is a hyperp... |
| lfl1dim 39403 | Equivalent expressions for... |
| lfl1dim2N 39404 | Equivalent expressions for... |
| ldualset 39407 | Define the (left) dual of ... |
| ldualvbase 39408 | The vectors of a dual spac... |
| ldualelvbase 39409 | Utility theorem for conver... |
| ldualfvadd 39410 | Vector addition in the dua... |
| ldualvadd 39411 | Vector addition in the dua... |
| ldualvaddcl 39412 | The value of vector additi... |
| ldualvaddval 39413 | The value of the value of ... |
| ldualsca 39414 | The ring of scalars of the... |
| ldualsbase 39415 | Base set of scalar ring fo... |
| ldualsaddN 39416 | Scalar addition for the du... |
| ldualsmul 39417 | Scalar multiplication for ... |
| ldualfvs 39418 | Scalar product operation f... |
| ldualvs 39419 | Scalar product operation v... |
| ldualvsval 39420 | Value of scalar product op... |
| ldualvscl 39421 | The scalar product operati... |
| ldualvaddcom 39422 | Commutative law for vector... |
| ldualvsass 39423 | Associative law for scalar... |
| ldualvsass2 39424 | Associative law for scalar... |
| ldualvsdi1 39425 | Distributive law for scala... |
| ldualvsdi2 39426 | Reverse distributive law f... |
| ldualgrplem 39427 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 39428 | The dual of a vector space... |
| ldual0 39429 | The zero scalar of the dua... |
| ldual1 39430 | The unit scalar of the dua... |
| ldualneg 39431 | The negative of a scalar o... |
| ldual0v 39432 | The zero vector of the dua... |
| ldual0vcl 39433 | The dual zero vector is a ... |
| lduallmodlem 39434 | Lemma for ~ lduallmod . (... |
| lduallmod 39435 | The dual of a left module ... |
| lduallvec 39436 | The dual of a left vector ... |
| ldualvsub 39437 | The value of vector subtra... |
| ldualvsubcl 39438 | Closure of vector subtract... |
| ldualvsubval 39439 | The value of the value of ... |
| ldualssvscl 39440 | Closure of scalar product ... |
| ldualssvsubcl 39441 | Closure of vector subtract... |
| ldual0vs 39442 | Scalar zero times a functi... |
| lkr0f2 39443 | The kernel of the zero fun... |
| lduallkr3 39444 | The kernels of nonzero fun... |
| lkrpssN 39445 | Proper subset relation bet... |
| lkrin 39446 | Intersection of the kernel... |
| eqlkr4 39447 | Two functionals with the s... |
| ldual1dim 39448 | Equivalent expressions for... |
| ldualkrsc 39449 | The kernel of a nonzero sc... |
| lkrss 39450 | The kernel of a scalar pro... |
| lkrss2N 39451 | Two functionals with kerne... |
| lkreqN 39452 | Proportional functionals h... |
| lkrlspeqN 39453 | Condition for colinear fun... |
| isopos 39462 | The predicate "is an ortho... |
| opposet 39463 | Every orthoposet is a pose... |
| oposlem 39464 | Lemma for orthoposet prope... |
| op01dm 39465 | Conditions necessary for z... |
| op0cl 39466 | An orthoposet has a zero e... |
| op1cl 39467 | An orthoposet has a unity ... |
| op0le 39468 | Orthoposet zero is less th... |
| ople0 39469 | An element less than or eq... |
| opnlen0 39470 | An element not less than a... |
| lub0N 39471 | The least upper bound of t... |
| opltn0 39472 | A lattice element greater ... |
| ople1 39473 | Any element is less than t... |
| op1le 39474 | If the orthoposet unity is... |
| glb0N 39475 | The greatest lower bound o... |
| opoccl 39476 | Closure of orthocomplement... |
| opococ 39477 | Double negative law for or... |
| opcon3b 39478 | Contraposition law for ort... |
| opcon2b 39479 | Orthocomplement contraposi... |
| opcon1b 39480 | Orthocomplement contraposi... |
| oplecon3 39481 | Contraposition law for ort... |
| oplecon3b 39482 | Contraposition law for ort... |
| oplecon1b 39483 | Contraposition law for str... |
| opoc1 39484 | Orthocomplement of orthopo... |
| opoc0 39485 | Orthocomplement of orthopo... |
| opltcon3b 39486 | Contraposition law for str... |
| opltcon1b 39487 | Contraposition law for str... |
| opltcon2b 39488 | Contraposition law for str... |
| opexmid 39489 | Law of excluded middle for... |
| opnoncon 39490 | Law of contradiction for o... |
| riotaocN 39491 | The orthocomplement of the... |
| cmtfvalN 39492 | Value of commutes relation... |
| cmtvalN 39493 | Equivalence for commutes r... |
| isolat 39494 | The predicate "is an ortho... |
| ollat 39495 | An ortholattice is a latti... |
| olop 39496 | An ortholattice is an orth... |
| olposN 39497 | An ortholattice is a poset... |
| isolatiN 39498 | Properties that determine ... |
| oldmm1 39499 | De Morgan's law for meet i... |
| oldmm2 39500 | De Morgan's law for meet i... |
| oldmm3N 39501 | De Morgan's law for meet i... |
| oldmm4 39502 | De Morgan's law for meet i... |
| oldmj1 39503 | De Morgan's law for join i... |
| oldmj2 39504 | De Morgan's law for join i... |
| oldmj3 39505 | De Morgan's law for join i... |
| oldmj4 39506 | De Morgan's law for join i... |
| olj01 39507 | An ortholattice element jo... |
| olj02 39508 | An ortholattice element jo... |
| olm11 39509 | The meet of an ortholattic... |
| olm12 39510 | The meet of an ortholattic... |
| latmassOLD 39511 | Ortholattice meet is assoc... |
| latm12 39512 | A rearrangement of lattice... |
| latm32 39513 | A rearrangement of lattice... |
| latmrot 39514 | Rotate lattice meet of 3 c... |
| latm4 39515 | Rearrangement of lattice m... |
| latmmdiN 39516 | Lattice meet distributes o... |
| latmmdir 39517 | Lattice meet distributes o... |
| olm01 39518 | Meet with lattice zero is ... |
| olm02 39519 | Meet with lattice zero is ... |
| isoml 39520 | The predicate "is an ortho... |
| isomliN 39521 | Properties that determine ... |
| omlol 39522 | An orthomodular lattice is... |
| omlop 39523 | An orthomodular lattice is... |
| omllat 39524 | An orthomodular lattice is... |
| omllaw 39525 | The orthomodular law. (Co... |
| omllaw2N 39526 | Variation of orthomodular ... |
| omllaw3 39527 | Orthomodular law equivalen... |
| omllaw4 39528 | Orthomodular law equivalen... |
| omllaw5N 39529 | The orthomodular law. Rem... |
| cmtcomlemN 39530 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 39531 | Commutation is symmetric. ... |
| cmt2N 39532 | Commutation with orthocomp... |
| cmt3N 39533 | Commutation with orthocomp... |
| cmt4N 39534 | Commutation with orthocomp... |
| cmtbr2N 39535 | Alternate definition of th... |
| cmtbr3N 39536 | Alternate definition for t... |
| cmtbr4N 39537 | Alternate definition for t... |
| lecmtN 39538 | Ordered elements commute. ... |
| cmtidN 39539 | Any element commutes with ... |
| omlfh1N 39540 | Foulis-Holland Theorem, pa... |
| omlfh3N 39541 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 39542 | Analogue of modular law ~ ... |
| omlspjN 39543 | Contraction of a Sasaki pr... |
| cvrfval 39550 | Value of covers relation "... |
| cvrval 39551 | Binary relation expressing... |
| cvrlt 39552 | The covers relation implie... |
| cvrnbtwn 39553 | There is no element betwee... |
| ncvr1 39554 | No element covers the latt... |
| cvrletrN 39555 | Property of an element abo... |
| cvrval2 39556 | Binary relation expressing... |
| cvrnbtwn2 39557 | The covers relation implie... |
| cvrnbtwn3 39558 | The covers relation implie... |
| cvrcon3b 39559 | Contraposition law for the... |
| cvrle 39560 | The covers relation implie... |
| cvrnbtwn4 39561 | The covers relation implie... |
| cvrnle 39562 | The covers relation implie... |
| cvrne 39563 | The covers relation implie... |
| cvrnrefN 39564 | The covers relation is not... |
| cvrcmp 39565 | If two lattice elements th... |
| cvrcmp2 39566 | If two lattice elements co... |
| pats 39567 | The set of atoms in a pose... |
| isat 39568 | The predicate "is an atom"... |
| isat2 39569 | The predicate "is an atom"... |
| atcvr0 39570 | An atom covers zero. ( ~ ... |
| atbase 39571 | An atom is a member of the... |
| atssbase 39572 | The set of atoms is a subs... |
| 0ltat 39573 | An atom is greater than ze... |
| leatb 39574 | A poset element less than ... |
| leat 39575 | A poset element less than ... |
| leat2 39576 | A nonzero poset element le... |
| leat3 39577 | A poset element less than ... |
| meetat 39578 | The meet of any element wi... |
| meetat2 39579 | The meet of any element wi... |
| isatl 39581 | The predicate "is an atomi... |
| atllat 39582 | An atomic lattice is a lat... |
| atlpos 39583 | An atomic lattice is a pos... |
| atl0dm 39584 | Condition necessary for ze... |
| atl0cl 39585 | An atomic lattice has a ze... |
| atl0le 39586 | Orthoposet zero is less th... |
| atlle0 39587 | An element less than or eq... |
| atlltn0 39588 | A lattice element greater ... |
| isat3 39589 | The predicate "is an atom"... |
| atn0 39590 | An atom is not zero. ( ~ ... |
| atnle0 39591 | An atom is not less than o... |
| atlen0 39592 | A lattice element is nonze... |
| atcmp 39593 | If two atoms are comparabl... |
| atncmp 39594 | Frequently-used variation ... |
| atnlt 39595 | Two atoms cannot satisfy t... |
| atcvreq0 39596 | An element covered by an a... |
| atncvrN 39597 | Two atoms cannot satisfy t... |
| atlex 39598 | Every nonzero element of a... |
| atnle 39599 | Two ways of expressing "an... |
| atnem0 39600 | The meet of distinct atoms... |
| atlatmstc 39601 | An atomic, complete, ortho... |
| atlatle 39602 | The ordering of two Hilber... |
| atlrelat1 39603 | An atomistic lattice with ... |
| iscvlat 39605 | The predicate "is an atomi... |
| iscvlat2N 39606 | The predicate "is an atomi... |
| cvlatl 39607 | An atomic lattice with the... |
| cvllat 39608 | An atomic lattice with the... |
| cvlposN 39609 | An atomic lattice with the... |
| cvlexch1 39610 | An atomic covering lattice... |
| cvlexch2 39611 | An atomic covering lattice... |
| cvlexchb1 39612 | An atomic covering lattice... |
| cvlexchb2 39613 | An atomic covering lattice... |
| cvlexch3 39614 | An atomic covering lattice... |
| cvlexch4N 39615 | An atomic covering lattice... |
| cvlatexchb1 39616 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 39617 | A version of ~ cvlexchb2 f... |
| cvlatexch1 39618 | Atom exchange property. (... |
| cvlatexch2 39619 | Atom exchange property. (... |
| cvlatexch3 39620 | Atom exchange property. (... |
| cvlcvr1 39621 | The covering property. Pr... |
| cvlcvrp 39622 | A Hilbert lattice satisfie... |
| cvlatcvr1 39623 | An atom is covered by its ... |
| cvlatcvr2 39624 | An atom is covered by its ... |
| cvlsupr2 39625 | Two equivalent ways of exp... |
| cvlsupr3 39626 | Two equivalent ways of exp... |
| cvlsupr4 39627 | Consequence of superpositi... |
| cvlsupr5 39628 | Consequence of superpositi... |
| cvlsupr6 39629 | Consequence of superpositi... |
| cvlsupr7 39630 | Consequence of superpositi... |
| cvlsupr8 39631 | Consequence of superpositi... |
| ishlat1 39634 | The predicate "is a Hilber... |
| ishlat2 39635 | The predicate "is a Hilber... |
| ishlat3N 39636 | The predicate "is a Hilber... |
| ishlatiN 39637 | Properties that determine ... |
| hlomcmcv 39638 | A Hilbert lattice is ortho... |
| hloml 39639 | A Hilbert lattice is ortho... |
| hlclat 39640 | A Hilbert lattice is compl... |
| hlcvl 39641 | A Hilbert lattice is an at... |
| hlatl 39642 | A Hilbert lattice is atomi... |
| hlol 39643 | A Hilbert lattice is an or... |
| hlop 39644 | A Hilbert lattice is an or... |
| hllat 39645 | A Hilbert lattice is a lat... |
| hllatd 39646 | Deduction form of ~ hllat ... |
| hlomcmat 39647 | A Hilbert lattice is ortho... |
| hlpos 39648 | A Hilbert lattice is a pos... |
| hlatjcl 39649 | Closure of join operation.... |
| hlatjcom 39650 | Commutatitivity of join op... |
| hlatjidm 39651 | Idempotence of join operat... |
| hlatjass 39652 | Lattice join is associativ... |
| hlatj12 39653 | Swap 1st and 2nd members o... |
| hlatj32 39654 | Swap 2nd and 3rd members o... |
| hlatjrot 39655 | Rotate lattice join of 3 c... |
| hlatj4 39656 | Rearrangement of lattice j... |
| hlatlej1 39657 | A join's first argument is... |
| hlatlej2 39658 | A join's second argument i... |
| glbconN 39659 | De Morgan's law for GLB an... |
| glbconxN 39660 | De Morgan's law for GLB an... |
| atnlej1 39661 | If an atom is not less tha... |
| atnlej2 39662 | If an atom is not less tha... |
| hlsuprexch 39663 | A Hilbert lattice has the ... |
| hlexch1 39664 | A Hilbert lattice has the ... |
| hlexch2 39665 | A Hilbert lattice has the ... |
| hlexchb1 39666 | A Hilbert lattice has the ... |
| hlexchb2 39667 | A Hilbert lattice has the ... |
| hlsupr 39668 | A Hilbert lattice has the ... |
| hlsupr2 39669 | A Hilbert lattice has the ... |
| hlhgt4 39670 | A Hilbert lattice has a he... |
| hlhgt2 39671 | A Hilbert lattice has a he... |
| hl0lt1N 39672 | Lattice 0 is less than lat... |
| hlexch3 39673 | A Hilbert lattice has the ... |
| hlexch4N 39674 | A Hilbert lattice has the ... |
| hlatexchb1 39675 | A version of ~ hlexchb1 fo... |
| hlatexchb2 39676 | A version of ~ hlexchb2 fo... |
| hlatexch1 39677 | Atom exchange property. (... |
| hlatexch2 39678 | Atom exchange property. (... |
| hlatmstcOLDN 39679 | An atomic, complete, ortho... |
| hlatle 39680 | The ordering of two Hilber... |
| hlateq 39681 | The equality of two Hilber... |
| hlrelat1 39682 | An atomistic lattice with ... |
| hlrelat5N 39683 | An atomistic lattice with ... |
| hlrelat 39684 | A Hilbert lattice is relat... |
| hlrelat2 39685 | A consequence of relative ... |
| exatleN 39686 | A condition for an atom to... |
| hl2at 39687 | A Hilbert lattice has at l... |
| atex 39688 | At least one atom exists. ... |
| intnatN 39689 | If the intersection with a... |
| 2llnne2N 39690 | Condition implying that tw... |
| 2llnneN 39691 | Condition implying that tw... |
| cvr1 39692 | A Hilbert lattice has the ... |
| cvr2N 39693 | Less-than and covers equiv... |
| hlrelat3 39694 | The Hilbert lattice is rel... |
| cvrval3 39695 | Binary relation expressing... |
| cvrval4N 39696 | Binary relation expressing... |
| cvrval5 39697 | Binary relation expressing... |
| cvrp 39698 | A Hilbert lattice satisfie... |
| atcvr1 39699 | An atom is covered by its ... |
| atcvr2 39700 | An atom is covered by its ... |
| cvrexchlem 39701 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 39702 | A Hilbert lattice satisfie... |
| cvratlem 39703 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 39704 | A nonzero Hilbert lattice ... |
| ltltncvr 39705 | A chained strong ordering ... |
| ltcvrntr 39706 | Non-transitive condition f... |
| cvrntr 39707 | The covers relation is not... |
| atcvr0eq 39708 | The covers relation is not... |
| lnnat 39709 | A line (the join of two di... |
| atcvrj0 39710 | Two atoms covering the zer... |
| cvrat2 39711 | A Hilbert lattice element ... |
| atcvrneN 39712 | Inequality derived from at... |
| atcvrj1 39713 | Condition for an atom to b... |
| atcvrj2b 39714 | Condition for an atom to b... |
| atcvrj2 39715 | Condition for an atom to b... |
| atleneN 39716 | Inequality derived from at... |
| atltcvr 39717 | An equivalence of less-tha... |
| atle 39718 | Any nonzero element has an... |
| atlt 39719 | Two atoms are unequal iff ... |
| atlelt 39720 | Transfer less-than relatio... |
| 2atlt 39721 | Given an atom less than an... |
| atexchcvrN 39722 | Atom exchange property. V... |
| atexchltN 39723 | Atom exchange property. V... |
| cvrat3 39724 | A condition implying that ... |
| cvrat4 39725 | A condition implying exist... |
| cvrat42 39726 | Commuted version of ~ cvra... |
| 2atjm 39727 | The meet of a line (expres... |
| atbtwn 39728 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 39729 | There exists a 3rd atom ` ... |
| atbtwnex 39730 | Given atoms ` P ` in ` X `... |
| 3noncolr2 39731 | Two ways to express 3 non-... |
| 3noncolr1N 39732 | Two ways to express 3 non-... |
| hlatcon3 39733 | Atom exchange combined wit... |
| hlatcon2 39734 | Atom exchange combined wit... |
| 4noncolr3 39735 | A way to express 4 non-col... |
| 4noncolr2 39736 | A way to express 4 non-col... |
| 4noncolr1 39737 | A way to express 4 non-col... |
| athgt 39738 | A Hilbert lattice, whose h... |
| 3dim0 39739 | There exists a 3-dimension... |
| 3dimlem1 39740 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 39741 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 39742 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 39743 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 39744 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 39745 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 39746 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 39747 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 39748 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 39749 | Construct a 3-dimensional ... |
| 3dim2 39750 | Construct 2 new layers on ... |
| 3dim3 39751 | Construct a new layer on t... |
| 2dim 39752 | Generate a height-3 elemen... |
| 1dimN 39753 | An atom is covered by a he... |
| 1cvrco 39754 | The orthocomplement of an ... |
| 1cvratex 39755 | There exists an atom less ... |
| 1cvratlt 39756 | An atom less than or equal... |
| 1cvrjat 39757 | An element covered by the ... |
| 1cvrat 39758 | Create an atom under an el... |
| ps-1 39759 | The join of two atoms ` R ... |
| ps-2 39760 | Lattice analogue for the p... |
| 2atjlej 39761 | Two atoms are different if... |
| hlatexch3N 39762 | Rearrange join of atoms in... |
| hlatexch4 39763 | Exchange 2 atoms. (Contri... |
| ps-2b 39764 | Variation of projective ge... |
| 3atlem1 39765 | Lemma for ~ 3at . (Contri... |
| 3atlem2 39766 | Lemma for ~ 3at . (Contri... |
| 3atlem3 39767 | Lemma for ~ 3at . (Contri... |
| 3atlem4 39768 | Lemma for ~ 3at . (Contri... |
| 3atlem5 39769 | Lemma for ~ 3at . (Contri... |
| 3atlem6 39770 | Lemma for ~ 3at . (Contri... |
| 3atlem7 39771 | Lemma for ~ 3at . (Contri... |
| 3at 39772 | Any three non-colinear ato... |
| llnset 39787 | The set of lattice lines i... |
| islln 39788 | The predicate "is a lattic... |
| islln4 39789 | The predicate "is a lattic... |
| llni 39790 | Condition implying a latti... |
| llnbase 39791 | A lattice line is a lattic... |
| islln3 39792 | The predicate "is a lattic... |
| islln2 39793 | The predicate "is a lattic... |
| llni2 39794 | The join of two different ... |
| llnnleat 39795 | An atom cannot majorize a ... |
| llnneat 39796 | A lattice line is not an a... |
| 2atneat 39797 | The join of two distinct a... |
| llnn0 39798 | A lattice line is nonzero.... |
| islln2a 39799 | The predicate "is a lattic... |
| llnle 39800 | Any element greater than 0... |
| atcvrlln2 39801 | An atom under a line is co... |
| atcvrlln 39802 | An element covering an ato... |
| llnexatN 39803 | Given an atom on a line, t... |
| llncmp 39804 | If two lattice lines are c... |
| llnnlt 39805 | Two lattice lines cannot s... |
| 2llnmat 39806 | Two intersecting lines int... |
| 2at0mat0 39807 | Special case of ~ 2atmat0 ... |
| 2atmat0 39808 | The meet of two unequal li... |
| 2atm 39809 | An atom majorized by two d... |
| ps-2c 39810 | Variation of projective ge... |
| lplnset 39811 | The set of lattice planes ... |
| islpln 39812 | The predicate "is a lattic... |
| islpln4 39813 | The predicate "is a lattic... |
| lplni 39814 | Condition implying a latti... |
| islpln3 39815 | The predicate "is a lattic... |
| lplnbase 39816 | A lattice plane is a latti... |
| islpln5 39817 | The predicate "is a lattic... |
| islpln2 39818 | The predicate "is a lattic... |
| lplni2 39819 | The join of 3 different at... |
| lvolex3N 39820 | There is an atom outside o... |
| llnmlplnN 39821 | The intersection of a line... |
| lplnle 39822 | Any element greater than 0... |
| lplnnle2at 39823 | A lattice line (or atom) c... |
| lplnnleat 39824 | A lattice plane cannot maj... |
| lplnnlelln 39825 | A lattice plane is not les... |
| 2atnelpln 39826 | The join of two atoms is n... |
| lplnneat 39827 | No lattice plane is an ato... |
| lplnnelln 39828 | No lattice plane is a latt... |
| lplnn0N 39829 | A lattice plane is nonzero... |
| islpln2a 39830 | The predicate "is a lattic... |
| islpln2ah 39831 | The predicate "is a lattic... |
| lplnriaN 39832 | Property of a lattice plan... |
| lplnribN 39833 | Property of a lattice plan... |
| lplnric 39834 | Property of a lattice plan... |
| lplnri1 39835 | Property of a lattice plan... |
| lplnri2N 39836 | Property of a lattice plan... |
| lplnri3N 39837 | Property of a lattice plan... |
| lplnllnneN 39838 | Two lattice lines defined ... |
| llncvrlpln2 39839 | A lattice line under a lat... |
| llncvrlpln 39840 | An element covering a latt... |
| 2lplnmN 39841 | If the join of two lattice... |
| 2llnmj 39842 | The meet of two lattice li... |
| 2atmat 39843 | The meet of two intersecti... |
| lplncmp 39844 | If two lattice planes are ... |
| lplnexatN 39845 | Given a lattice line on a ... |
| lplnexllnN 39846 | Given an atom on a lattice... |
| lplnnlt 39847 | Two lattice planes cannot ... |
| 2llnjaN 39848 | The join of two different ... |
| 2llnjN 39849 | The join of two different ... |
| 2llnm2N 39850 | The meet of two different ... |
| 2llnm3N 39851 | Two lattice lines in a lat... |
| 2llnm4 39852 | Two lattice lines that maj... |
| 2llnmeqat 39853 | An atom equals the interse... |
| lvolset 39854 | The set of 3-dim lattice v... |
| islvol 39855 | The predicate "is a 3-dim ... |
| islvol4 39856 | The predicate "is a 3-dim ... |
| lvoli 39857 | Condition implying a 3-dim... |
| islvol3 39858 | The predicate "is a 3-dim ... |
| lvoli3 39859 | Condition implying a 3-dim... |
| lvolbase 39860 | A 3-dim lattice volume is ... |
| islvol5 39861 | The predicate "is a 3-dim ... |
| islvol2 39862 | The predicate "is a 3-dim ... |
| lvoli2 39863 | The join of 4 different at... |
| lvolnle3at 39864 | A lattice plane (or lattic... |
| lvolnleat 39865 | An atom cannot majorize a ... |
| lvolnlelln 39866 | A lattice line cannot majo... |
| lvolnlelpln 39867 | A lattice plane cannot maj... |
| 3atnelvolN 39868 | The join of 3 atoms is not... |
| 2atnelvolN 39869 | The join of two atoms is n... |
| lvolneatN 39870 | No lattice volume is an at... |
| lvolnelln 39871 | No lattice volume is a lat... |
| lvolnelpln 39872 | No lattice volume is a lat... |
| lvoln0N 39873 | A lattice volume is nonzer... |
| islvol2aN 39874 | The predicate "is a lattic... |
| 4atlem0a 39875 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 39876 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 39877 | Lemma for ~ 4at . (Contri... |
| 4atlem3 39878 | Lemma for ~ 4at . Break i... |
| 4atlem3a 39879 | Lemma for ~ 4at . Break i... |
| 4atlem3b 39880 | Lemma for ~ 4at . Break i... |
| 4atlem4a 39881 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 39882 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 39883 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 39884 | Lemma for ~ 4at . Frequen... |
| 4atlem9 39885 | Lemma for ~ 4at . Substit... |
| 4atlem10a 39886 | Lemma for ~ 4at . Substit... |
| 4atlem10b 39887 | Lemma for ~ 4at . Substit... |
| 4atlem10 39888 | Lemma for ~ 4at . Combine... |
| 4atlem11a 39889 | Lemma for ~ 4at . Substit... |
| 4atlem11b 39890 | Lemma for ~ 4at . Substit... |
| 4atlem11 39891 | Lemma for ~ 4at . Combine... |
| 4atlem12a 39892 | Lemma for ~ 4at . Substit... |
| 4atlem12b 39893 | Lemma for ~ 4at . Substit... |
| 4atlem12 39894 | Lemma for ~ 4at . Combine... |
| 4at 39895 | Four atoms determine a lat... |
| 4at2 39896 | Four atoms determine a lat... |
| lplncvrlvol2 39897 | A lattice line under a lat... |
| lplncvrlvol 39898 | An element covering a latt... |
| lvolcmp 39899 | If two lattice planes are ... |
| lvolnltN 39900 | Two lattice volumes cannot... |
| 2lplnja 39901 | The join of two different ... |
| 2lplnj 39902 | The join of two different ... |
| 2lplnm2N 39903 | The meet of two different ... |
| 2lplnmj 39904 | The meet of two lattice pl... |
| dalemkehl 39905 | Lemma for ~ dath . Freque... |
| dalemkelat 39906 | Lemma for ~ dath . Freque... |
| dalemkeop 39907 | Lemma for ~ dath . Freque... |
| dalempea 39908 | Lemma for ~ dath . Freque... |
| dalemqea 39909 | Lemma for ~ dath . Freque... |
| dalemrea 39910 | Lemma for ~ dath . Freque... |
| dalemsea 39911 | Lemma for ~ dath . Freque... |
| dalemtea 39912 | Lemma for ~ dath . Freque... |
| dalemuea 39913 | Lemma for ~ dath . Freque... |
| dalemyeo 39914 | Lemma for ~ dath . Freque... |
| dalemzeo 39915 | Lemma for ~ dath . Freque... |
| dalemclpjs 39916 | Lemma for ~ dath . Freque... |
| dalemclqjt 39917 | Lemma for ~ dath . Freque... |
| dalemclrju 39918 | Lemma for ~ dath . Freque... |
| dalem-clpjq 39919 | Lemma for ~ dath . Freque... |
| dalemceb 39920 | Lemma for ~ dath . Freque... |
| dalempeb 39921 | Lemma for ~ dath . Freque... |
| dalemqeb 39922 | Lemma for ~ dath . Freque... |
| dalemreb 39923 | Lemma for ~ dath . Freque... |
| dalemseb 39924 | Lemma for ~ dath . Freque... |
| dalemteb 39925 | Lemma for ~ dath . Freque... |
| dalemueb 39926 | Lemma for ~ dath . Freque... |
| dalempjqeb 39927 | Lemma for ~ dath . Freque... |
| dalemsjteb 39928 | Lemma for ~ dath . Freque... |
| dalemtjueb 39929 | Lemma for ~ dath . Freque... |
| dalemqrprot 39930 | Lemma for ~ dath . Freque... |
| dalemyeb 39931 | Lemma for ~ dath . Freque... |
| dalemcnes 39932 | Lemma for ~ dath . Freque... |
| dalempnes 39933 | Lemma for ~ dath . Freque... |
| dalemqnet 39934 | Lemma for ~ dath . Freque... |
| dalempjsen 39935 | Lemma for ~ dath . Freque... |
| dalemply 39936 | Lemma for ~ dath . Freque... |
| dalemsly 39937 | Lemma for ~ dath . Freque... |
| dalemswapyz 39938 | Lemma for ~ dath . Swap t... |
| dalemrot 39939 | Lemma for ~ dath . Rotate... |
| dalemrotyz 39940 | Lemma for ~ dath . Rotate... |
| dalem1 39941 | Lemma for ~ dath . Show t... |
| dalemcea 39942 | Lemma for ~ dath . Freque... |
| dalem2 39943 | Lemma for ~ dath . Show t... |
| dalemdea 39944 | Lemma for ~ dath . Freque... |
| dalemeea 39945 | Lemma for ~ dath . Freque... |
| dalem3 39946 | Lemma for ~ dalemdnee . (... |
| dalem4 39947 | Lemma for ~ dalemdnee . (... |
| dalemdnee 39948 | Lemma for ~ dath . Axis o... |
| dalem5 39949 | Lemma for ~ dath . Atom `... |
| dalem6 39950 | Lemma for ~ dath . Analog... |
| dalem7 39951 | Lemma for ~ dath . Analog... |
| dalem8 39952 | Lemma for ~ dath . Plane ... |
| dalem-cly 39953 | Lemma for ~ dalem9 . Cent... |
| dalem9 39954 | Lemma for ~ dath . Since ... |
| dalem10 39955 | Lemma for ~ dath . Atom `... |
| dalem11 39956 | Lemma for ~ dath . Analog... |
| dalem12 39957 | Lemma for ~ dath . Analog... |
| dalem13 39958 | Lemma for ~ dalem14 . (Co... |
| dalem14 39959 | Lemma for ~ dath . Planes... |
| dalem15 39960 | Lemma for ~ dath . The ax... |
| dalem16 39961 | Lemma for ~ dath . The at... |
| dalem17 39962 | Lemma for ~ dath . When p... |
| dalem18 39963 | Lemma for ~ dath . Show t... |
| dalem19 39964 | Lemma for ~ dath . Show t... |
| dalemccea 39965 | Lemma for ~ dath . Freque... |
| dalemddea 39966 | Lemma for ~ dath . Freque... |
| dalem-ccly 39967 | Lemma for ~ dath . Freque... |
| dalem-ddly 39968 | Lemma for ~ dath . Freque... |
| dalemccnedd 39969 | Lemma for ~ dath . Freque... |
| dalemclccjdd 39970 | Lemma for ~ dath . Freque... |
| dalemcceb 39971 | Lemma for ~ dath . Freque... |
| dalemswapyzps 39972 | Lemma for ~ dath . Swap t... |
| dalemrotps 39973 | Lemma for ~ dath . Rotate... |
| dalemcjden 39974 | Lemma for ~ dath . Show t... |
| dalem20 39975 | Lemma for ~ dath . Show t... |
| dalem21 39976 | Lemma for ~ dath . Show t... |
| dalem22 39977 | Lemma for ~ dath . Show t... |
| dalem23 39978 | Lemma for ~ dath . Show t... |
| dalem24 39979 | Lemma for ~ dath . Show t... |
| dalem25 39980 | Lemma for ~ dath . Show t... |
| dalem27 39981 | Lemma for ~ dath . Show t... |
| dalem28 39982 | Lemma for ~ dath . Lemma ... |
| dalem29 39983 | Lemma for ~ dath . Analog... |
| dalem30 39984 | Lemma for ~ dath . Analog... |
| dalem31N 39985 | Lemma for ~ dath . Analog... |
| dalem32 39986 | Lemma for ~ dath . Analog... |
| dalem33 39987 | Lemma for ~ dath . Analog... |
| dalem34 39988 | Lemma for ~ dath . Analog... |
| dalem35 39989 | Lemma for ~ dath . Analog... |
| dalem36 39990 | Lemma for ~ dath . Analog... |
| dalem37 39991 | Lemma for ~ dath . Analog... |
| dalem38 39992 | Lemma for ~ dath . Plane ... |
| dalem39 39993 | Lemma for ~ dath . Auxili... |
| dalem40 39994 | Lemma for ~ dath . Analog... |
| dalem41 39995 | Lemma for ~ dath . (Contr... |
| dalem42 39996 | Lemma for ~ dath . Auxili... |
| dalem43 39997 | Lemma for ~ dath . Planes... |
| dalem44 39998 | Lemma for ~ dath . Dummy ... |
| dalem45 39999 | Lemma for ~ dath . Dummy ... |
| dalem46 40000 | Lemma for ~ dath . Analog... |
| dalem47 40001 | Lemma for ~ dath . Analog... |
| dalem48 40002 | Lemma for ~ dath . Analog... |
| dalem49 40003 | Lemma for ~ dath . Analog... |
| dalem50 40004 | Lemma for ~ dath . Analog... |
| dalem51 40005 | Lemma for ~ dath . Constr... |
| dalem52 40006 | Lemma for ~ dath . Lines ... |
| dalem53 40007 | Lemma for ~ dath . The au... |
| dalem54 40008 | Lemma for ~ dath . Line `... |
| dalem55 40009 | Lemma for ~ dath . Lines ... |
| dalem56 40010 | Lemma for ~ dath . Analog... |
| dalem57 40011 | Lemma for ~ dath . Axis o... |
| dalem58 40012 | Lemma for ~ dath . Analog... |
| dalem59 40013 | Lemma for ~ dath . Analog... |
| dalem60 40014 | Lemma for ~ dath . ` B ` i... |
| dalem61 40015 | Lemma for ~ dath . Show t... |
| dalem62 40016 | Lemma for ~ dath . Elimin... |
| dalem63 40017 | Lemma for ~ dath . Combin... |
| dath 40018 | Desargues's theorem of pro... |
| dath2 40019 | Version of Desargues's the... |
| lineset 40020 | The set of lines in a Hilb... |
| isline 40021 | The predicate "is a line".... |
| islinei 40022 | Condition implying "is a l... |
| pointsetN 40023 | The set of points in a Hil... |
| ispointN 40024 | The predicate "is a point"... |
| atpointN 40025 | The singleton of an atom i... |
| psubspset 40026 | The set of projective subs... |
| ispsubsp 40027 | The predicate "is a projec... |
| ispsubsp2 40028 | The predicate "is a projec... |
| psubspi 40029 | Property of a projective s... |
| psubspi2N 40030 | Property of a projective s... |
| 0psubN 40031 | The empty set is a project... |
| snatpsubN 40032 | The singleton of an atom i... |
| pointpsubN 40033 | A point (singleton of an a... |
| linepsubN 40034 | A line is a projective sub... |
| atpsubN 40035 | The set of all atoms is a ... |
| psubssat 40036 | A projective subspace cons... |
| psubatN 40037 | A member of a projective s... |
| pmapfval 40038 | The projective map of a Hi... |
| pmapval 40039 | Value of the projective ma... |
| elpmap 40040 | Member of a projective map... |
| pmapssat 40041 | The projective map of a Hi... |
| pmapssbaN 40042 | A weakening of ~ pmapssat ... |
| pmaple 40043 | The projective map of a Hi... |
| pmap11 40044 | The projective map of a Hi... |
| pmapat 40045 | The projective map of an a... |
| elpmapat 40046 | Member of the projective m... |
| pmap0 40047 | Value of the projective ma... |
| pmapeq0 40048 | A projective map value is ... |
| pmap1N 40049 | Value of the projective ma... |
| pmapsub 40050 | The projective map of a Hi... |
| pmapglbx 40051 | The projective map of the ... |
| pmapglb 40052 | The projective map of the ... |
| pmapglb2N 40053 | The projective map of the ... |
| pmapglb2xN 40054 | The projective map of the ... |
| pmapmeet 40055 | The projective map of a me... |
| isline2 40056 | Definition of line in term... |
| linepmap 40057 | A line described with a pr... |
| isline3 40058 | Definition of line in term... |
| isline4N 40059 | Definition of line in term... |
| lneq2at 40060 | A line equals the join of ... |
| lnatexN 40061 | There is an atom in a line... |
| lnjatN 40062 | Given an atom in a line, t... |
| lncvrelatN 40063 | A lattice element covered ... |
| lncvrat 40064 | A line covers the atoms it... |
| lncmp 40065 | If two lines are comparabl... |
| 2lnat 40066 | Two intersecting lines int... |
| 2atm2atN 40067 | Two joins with a common at... |
| 2llnma1b 40068 | Generalization of ~ 2llnma... |
| 2llnma1 40069 | Two different intersecting... |
| 2llnma3r 40070 | Two different intersecting... |
| 2llnma2 40071 | Two different intersecting... |
| 2llnma2rN 40072 | Two different intersecting... |
| cdlema1N 40073 | A condition for required f... |
| cdlema2N 40074 | A condition for required f... |
| cdlemblem 40075 | Lemma for ~ cdlemb . (Con... |
| cdlemb 40076 | Given two atoms not less t... |
| paddfval 40079 | Projective subspace sum op... |
| paddval 40080 | Projective subspace sum op... |
| elpadd 40081 | Member of a projective sub... |
| elpaddn0 40082 | Member of projective subsp... |
| paddvaln0N 40083 | Projective subspace sum op... |
| elpaddri 40084 | Condition implying members... |
| elpaddatriN 40085 | Condition implying members... |
| elpaddat 40086 | Membership in a projective... |
| elpaddatiN 40087 | Consequence of membership ... |
| elpadd2at 40088 | Membership in a projective... |
| elpadd2at2 40089 | Membership in a projective... |
| paddunssN 40090 | Projective subspace sum in... |
| elpadd0 40091 | Member of projective subsp... |
| paddval0 40092 | Projective subspace sum wi... |
| padd01 40093 | Projective subspace sum wi... |
| padd02 40094 | Projective subspace sum wi... |
| paddcom 40095 | Projective subspace sum co... |
| paddssat 40096 | A projective subspace sum ... |
| sspadd1 40097 | A projective subspace sum ... |
| sspadd2 40098 | A projective subspace sum ... |
| paddss1 40099 | Subset law for projective ... |
| paddss2 40100 | Subset law for projective ... |
| paddss12 40101 | Subset law for projective ... |
| paddasslem1 40102 | Lemma for ~ paddass . (Co... |
| paddasslem2 40103 | Lemma for ~ paddass . (Co... |
| paddasslem3 40104 | Lemma for ~ paddass . Res... |
| paddasslem4 40105 | Lemma for ~ paddass . Com... |
| paddasslem5 40106 | Lemma for ~ paddass . Sho... |
| paddasslem6 40107 | Lemma for ~ paddass . (Co... |
| paddasslem7 40108 | Lemma for ~ paddass . Com... |
| paddasslem8 40109 | Lemma for ~ paddass . (Co... |
| paddasslem9 40110 | Lemma for ~ paddass . Com... |
| paddasslem10 40111 | Lemma for ~ paddass . Use... |
| paddasslem11 40112 | Lemma for ~ paddass . The... |
| paddasslem12 40113 | Lemma for ~ paddass . The... |
| paddasslem13 40114 | Lemma for ~ paddass . The... |
| paddasslem14 40115 | Lemma for ~ paddass . Rem... |
| paddasslem15 40116 | Lemma for ~ paddass . Use... |
| paddasslem16 40117 | Lemma for ~ paddass . Use... |
| paddasslem17 40118 | Lemma for ~ paddass . The... |
| paddasslem18 40119 | Lemma for ~ paddass . Com... |
| paddass 40120 | Projective subspace sum is... |
| padd12N 40121 | Commutative/associative la... |
| padd4N 40122 | Rearrangement of 4 terms i... |
| paddidm 40123 | Projective subspace sum is... |
| paddclN 40124 | The projective sum of two ... |
| paddssw1 40125 | Subset law for projective ... |
| paddssw2 40126 | Subset law for projective ... |
| paddss 40127 | Subset law for projective ... |
| pmodlem1 40128 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 40129 | Lemma for ~ pmod1i . (Con... |
| pmod1i 40130 | The modular law holds in a... |
| pmod2iN 40131 | Dual of the modular law. ... |
| pmodN 40132 | The modular law for projec... |
| pmodl42N 40133 | Lemma derived from modular... |
| pmapjoin 40134 | The projective map of the ... |
| pmapjat1 40135 | The projective map of the ... |
| pmapjat2 40136 | The projective map of the ... |
| pmapjlln1 40137 | The projective map of the ... |
| hlmod1i 40138 | A version of the modular l... |
| atmod1i1 40139 | Version of modular law ~ p... |
| atmod1i1m 40140 | Version of modular law ~ p... |
| atmod1i2 40141 | Version of modular law ~ p... |
| llnmod1i2 40142 | Version of modular law ~ p... |
| atmod2i1 40143 | Version of modular law ~ p... |
| atmod2i2 40144 | Version of modular law ~ p... |
| llnmod2i2 40145 | Version of modular law ~ p... |
| atmod3i1 40146 | Version of modular law tha... |
| atmod3i2 40147 | Version of modular law tha... |
| atmod4i1 40148 | Version of modular law tha... |
| atmod4i2 40149 | Version of modular law tha... |
| llnexchb2lem 40150 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 40151 | Line exchange property (co... |
| llnexch2N 40152 | Line exchange property (co... |
| dalawlem1 40153 | Lemma for ~ dalaw . Speci... |
| dalawlem2 40154 | Lemma for ~ dalaw . Utili... |
| dalawlem3 40155 | Lemma for ~ dalaw . First... |
| dalawlem4 40156 | Lemma for ~ dalaw . Secon... |
| dalawlem5 40157 | Lemma for ~ dalaw . Speci... |
| dalawlem6 40158 | Lemma for ~ dalaw . First... |
| dalawlem7 40159 | Lemma for ~ dalaw . Secon... |
| dalawlem8 40160 | Lemma for ~ dalaw . Speci... |
| dalawlem9 40161 | Lemma for ~ dalaw . Speci... |
| dalawlem10 40162 | Lemma for ~ dalaw . Combi... |
| dalawlem11 40163 | Lemma for ~ dalaw . First... |
| dalawlem12 40164 | Lemma for ~ dalaw . Secon... |
| dalawlem13 40165 | Lemma for ~ dalaw . Speci... |
| dalawlem14 40166 | Lemma for ~ dalaw . Combi... |
| dalawlem15 40167 | Lemma for ~ dalaw . Swap ... |
| dalaw 40168 | Desargues's law, derived f... |
| pclfvalN 40171 | The projective subspace cl... |
| pclvalN 40172 | Value of the projective su... |
| pclclN 40173 | Closure of the projective ... |
| elpclN 40174 | Membership in the projecti... |
| elpcliN 40175 | Implication of membership ... |
| pclssN 40176 | Ordering is preserved by s... |
| pclssidN 40177 | A set of atoms is included... |
| pclidN 40178 | The projective subspace cl... |
| pclbtwnN 40179 | A projective subspace sand... |
| pclunN 40180 | The projective subspace cl... |
| pclun2N 40181 | The projective subspace cl... |
| pclfinN 40182 | The projective subspace cl... |
| pclcmpatN 40183 | The set of projective subs... |
| polfvalN 40186 | The projective subspace po... |
| polvalN 40187 | Value of the projective su... |
| polval2N 40188 | Alternate expression for v... |
| polsubN 40189 | The polarity of a set of a... |
| polssatN 40190 | The polarity of a set of a... |
| pol0N 40191 | The polarity of the empty ... |
| pol1N 40192 | The polarity of the whole ... |
| 2pol0N 40193 | The closed subspace closur... |
| polpmapN 40194 | The polarity of a projecti... |
| 2polpmapN 40195 | Double polarity of a proje... |
| 2polvalN 40196 | Value of double polarity. ... |
| 2polssN 40197 | A set of atoms is a subset... |
| 3polN 40198 | Triple polarity cancels to... |
| polcon3N 40199 | Contraposition law for pol... |
| 2polcon4bN 40200 | Contraposition law for pol... |
| polcon2N 40201 | Contraposition law for pol... |
| polcon2bN 40202 | Contraposition law for pol... |
| pclss2polN 40203 | The projective subspace cl... |
| pcl0N 40204 | The projective subspace cl... |
| pcl0bN 40205 | The projective subspace cl... |
| pmaplubN 40206 | The LUB of a projective ma... |
| sspmaplubN 40207 | A set of atoms is a subset... |
| 2pmaplubN 40208 | Double projective map of a... |
| paddunN 40209 | The closure of the project... |
| poldmj1N 40210 | De Morgan's law for polari... |
| pmapj2N 40211 | The projective map of the ... |
| pmapocjN 40212 | The projective map of the ... |
| polatN 40213 | The polarity of the single... |
| 2polatN 40214 | Double polarity of the sin... |
| pnonsingN 40215 | The intersection of a set ... |
| psubclsetN 40218 | The set of closed projecti... |
| ispsubclN 40219 | The predicate "is a closed... |
| psubcliN 40220 | Property of a closed proje... |
| psubcli2N 40221 | Property of a closed proje... |
| psubclsubN 40222 | A closed projective subspa... |
| psubclssatN 40223 | A closed projective subspa... |
| pmapidclN 40224 | Projective map of the LUB ... |
| 0psubclN 40225 | The empty set is a closed ... |
| 1psubclN 40226 | The set of all atoms is a ... |
| atpsubclN 40227 | A point (singleton of an a... |
| pmapsubclN 40228 | A projective map value is ... |
| ispsubcl2N 40229 | Alternate predicate for "i... |
| psubclinN 40230 | The intersection of two cl... |
| paddatclN 40231 | The projective sum of a cl... |
| pclfinclN 40232 | The projective subspace cl... |
| linepsubclN 40233 | A line is a closed project... |
| polsubclN 40234 | A polarity is a closed pro... |
| poml4N 40235 | Orthomodular law for proje... |
| poml5N 40236 | Orthomodular law for proje... |
| poml6N 40237 | Orthomodular law for proje... |
| osumcllem1N 40238 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 40239 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 40240 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 40241 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 40242 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 40243 | Lemma for ~ osumclN . Use... |
| osumcllem7N 40244 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 40245 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 40246 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 40247 | Lemma for ~ osumclN . Con... |
| osumcllem11N 40248 | Lemma for ~ osumclN . (Co... |
| osumclN 40249 | Closure of orthogonal sum.... |
| pmapojoinN 40250 | For orthogonal elements, p... |
| pexmidN 40251 | Excluded middle law for cl... |
| pexmidlem1N 40252 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 40253 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 40254 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 40255 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 40256 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 40257 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 40258 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 40259 | Lemma for ~ pexmidN . The... |
| pexmidALTN 40260 | Excluded middle law for cl... |
| pl42lem1N 40261 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 40262 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 40263 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 40264 | Lemma for ~ pl42N . (Cont... |
| pl42N 40265 | Law holding in a Hilbert l... |
| watfvalN 40274 | The W atoms function. (Co... |
| watvalN 40275 | Value of the W atoms funct... |
| iswatN 40276 | The predicate "is a W atom... |
| lhpset 40277 | The set of co-atoms (latti... |
| islhp 40278 | The predicate "is a co-ato... |
| islhp2 40279 | The predicate "is a co-ato... |
| lhpbase 40280 | A co-atom is a member of t... |
| lhp1cvr 40281 | The lattice unity covers a... |
| lhplt 40282 | An atom under a co-atom is... |
| lhp2lt 40283 | The join of two atoms unde... |
| lhpexlt 40284 | There exists an atom less ... |
| lhp0lt 40285 | A co-atom is greater than ... |
| lhpn0 40286 | A co-atom is nonzero. TOD... |
| lhpexle 40287 | There exists an atom under... |
| lhpexnle 40288 | There exists an atom not u... |
| lhpexle1lem 40289 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 40290 | There exists an atom under... |
| lhpexle2lem 40291 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 40292 | There exists atom under a ... |
| lhpexle3lem 40293 | There exists atom under a ... |
| lhpexle3 40294 | There exists atom under a ... |
| lhpex2leN 40295 | There exist at least two d... |
| lhpoc 40296 | The orthocomplement of a c... |
| lhpoc2N 40297 | The orthocomplement of an ... |
| lhpocnle 40298 | The orthocomplement of a c... |
| lhpocat 40299 | The orthocomplement of a c... |
| lhpocnel 40300 | The orthocomplement of a c... |
| lhpocnel2 40301 | The orthocomplement of a c... |
| lhpjat1 40302 | The join of a co-atom (hyp... |
| lhpjat2 40303 | The join of a co-atom (hyp... |
| lhpj1 40304 | The join of a co-atom (hyp... |
| lhpmcvr 40305 | The meet of a lattice hype... |
| lhpmcvr2 40306 | Alternate way to express t... |
| lhpmcvr3 40307 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 40308 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 40309 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 40310 | Specialization of ~ lhpmcv... |
| lhpm0atN 40311 | If the meet of a lattice h... |
| lhpmat 40312 | An element covered by the ... |
| lhpmatb 40313 | An element covered by the ... |
| lhp2at0 40314 | Join and meet with differe... |
| lhp2atnle 40315 | Inequality for 2 different... |
| lhp2atne 40316 | Inequality for joins with ... |
| lhp2at0nle 40317 | Inequality for 2 different... |
| lhp2at0ne 40318 | Inequality for joins with ... |
| lhpelim 40319 | Eliminate an atom not unde... |
| lhpmod2i2 40320 | Modular law for hyperplane... |
| lhpmod6i1 40321 | Modular law for hyperplane... |
| lhprelat3N 40322 | The Hilbert lattice is rel... |
| cdlemb2 40323 | Given two atoms not under ... |
| lhple 40324 | Property of a lattice elem... |
| lhpat 40325 | Create an atom under a co-... |
| lhpat4N 40326 | Property of an atom under ... |
| lhpat2 40327 | Create an atom under a co-... |
| lhpat3 40328 | There is only one atom und... |
| 4atexlemk 40329 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 40330 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 40331 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 40332 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 40333 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 40334 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 40335 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 40336 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 40337 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 40338 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 40339 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 40340 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 40341 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 40342 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 40343 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 40344 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 40345 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 40346 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 40347 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 40348 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 40349 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 40350 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 40351 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 40352 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 40353 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 40354 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 40355 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 40356 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 40357 | Whenever there are at leas... |
| 4atex 40358 | Whenever there are at leas... |
| 4atex2 40359 | More general version of ~ ... |
| 4atex2-0aOLDN 40360 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 40361 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 40362 | Same as ~ 4atex2 except th... |
| 4atex3 40363 | More general version of ~ ... |
| lautset 40364 | The set of lattice automor... |
| islaut 40365 | The predicate "is a lattic... |
| lautle 40366 | Less-than or equal propert... |
| laut1o 40367 | A lattice automorphism is ... |
| laut11 40368 | One-to-one property of a l... |
| lautcl 40369 | A lattice automorphism val... |
| lautcnvclN 40370 | Reverse closure of a latti... |
| lautcnvle 40371 | Less-than or equal propert... |
| lautcnv 40372 | The converse of a lattice ... |
| lautlt 40373 | Less-than property of a la... |
| lautcvr 40374 | Covering property of a lat... |
| lautj 40375 | Meet property of a lattice... |
| lautm 40376 | Meet property of a lattice... |
| lauteq 40377 | A lattice automorphism arg... |
| idlaut 40378 | The identity function is a... |
| lautco 40379 | The composition of two lat... |
| pautsetN 40380 | The set of projective auto... |
| ispautN 40381 | The predicate "is a projec... |
| ldilfset 40390 | The mapping from fiducial ... |
| ldilset 40391 | The set of lattice dilatio... |
| isldil 40392 | The predicate "is a lattic... |
| ldillaut 40393 | A lattice dilation is an a... |
| ldil1o 40394 | A lattice dilation is a on... |
| ldilval 40395 | Value of a lattice dilatio... |
| idldil 40396 | The identity function is a... |
| ldilcnv 40397 | The converse of a lattice ... |
| ldilco 40398 | The composition of two lat... |
| ltrnfset 40399 | The set of all lattice tra... |
| ltrnset 40400 | The set of lattice transla... |
| isltrn 40401 | The predicate "is a lattic... |
| isltrn2N 40402 | The predicate "is a lattic... |
| ltrnu 40403 | Uniqueness property of a l... |
| ltrnldil 40404 | A lattice translation is a... |
| ltrnlaut 40405 | A lattice translation is a... |
| ltrn1o 40406 | A lattice translation is a... |
| ltrncl 40407 | Closure of a lattice trans... |
| ltrn11 40408 | One-to-one property of a l... |
| ltrncnvnid 40409 | If a translation is differ... |
| ltrncoidN 40410 | Two translations are equal... |
| ltrnle 40411 | Less-than or equal propert... |
| ltrncnvleN 40412 | Less-than or equal propert... |
| ltrnm 40413 | Lattice translation of a m... |
| ltrnj 40414 | Lattice translation of a m... |
| ltrncvr 40415 | Covering property of a lat... |
| ltrnval1 40416 | Value of a lattice transla... |
| ltrnid 40417 | A lattice translation is t... |
| ltrnnid 40418 | If a lattice translation i... |
| ltrnatb 40419 | The lattice translation of... |
| ltrncnvatb 40420 | The converse of the lattic... |
| ltrnel 40421 | The lattice translation of... |
| ltrnat 40422 | The lattice translation of... |
| ltrncnvat 40423 | The converse of the lattic... |
| ltrncnvel 40424 | The converse of the lattic... |
| ltrncoelN 40425 | Composition of lattice tra... |
| ltrncoat 40426 | Composition of lattice tra... |
| ltrncoval 40427 | Two ways to express value ... |
| ltrncnv 40428 | The converse of a lattice ... |
| ltrn11at 40429 | Frequently used one-to-one... |
| ltrneq2 40430 | The equality of two transl... |
| ltrneq 40431 | The equality of two transl... |
| idltrn 40432 | The identity function is a... |
| ltrnmw 40433 | Property of lattice transl... |
| dilfsetN 40434 | The mapping from fiducial ... |
| dilsetN 40435 | The set of dilations for a... |
| isdilN 40436 | The predicate "is a dilati... |
| trnfsetN 40437 | The mapping from fiducial ... |
| trnsetN 40438 | The set of translations fo... |
| istrnN 40439 | The predicate "is a transl... |
| trlfset 40442 | The set of all traces of l... |
| trlset 40443 | The set of traces of latti... |
| trlval 40444 | The value of the trace of ... |
| trlval2 40445 | The value of the trace of ... |
| trlcl 40446 | Closure of the trace of a ... |
| trlcnv 40447 | The trace of the converse ... |
| trljat1 40448 | The value of a translation... |
| trljat2 40449 | The value of a translation... |
| trljat3 40450 | The value of a translation... |
| trlat 40451 | If an atom differs from it... |
| trl0 40452 | If an atom not under the f... |
| trlator0 40453 | The trace of a lattice tra... |
| trlatn0 40454 | The trace of a lattice tra... |
| trlnidat 40455 | The trace of a lattice tra... |
| ltrnnidn 40456 | If a lattice translation i... |
| ltrnideq 40457 | Property of the identity l... |
| trlid0 40458 | The trace of the identity ... |
| trlnidatb 40459 | A lattice translation is n... |
| trlid0b 40460 | A lattice translation is t... |
| trlnid 40461 | Different translations wit... |
| ltrn2ateq 40462 | Property of the equality o... |
| ltrnateq 40463 | If any atom (under ` W ` )... |
| ltrnatneq 40464 | If any atom (under ` W ` )... |
| ltrnatlw 40465 | If the value of an atom eq... |
| trlle 40466 | The trace of a lattice tra... |
| trlne 40467 | The trace of a lattice tra... |
| trlnle 40468 | The atom not under the fid... |
| trlval3 40469 | The value of the trace of ... |
| trlval4 40470 | The value of the trace of ... |
| trlval5 40471 | The value of the trace of ... |
| arglem1N 40472 | Lemma for Desargues's law.... |
| cdlemc1 40473 | Part of proof of Lemma C i... |
| cdlemc2 40474 | Part of proof of Lemma C i... |
| cdlemc3 40475 | Part of proof of Lemma C i... |
| cdlemc4 40476 | Part of proof of Lemma C i... |
| cdlemc5 40477 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 40478 | Lemma for ~ cdlemc . (Con... |
| cdlemc 40479 | Lemma C in [Crawley] p. 11... |
| cdlemd1 40480 | Part of proof of Lemma D i... |
| cdlemd2 40481 | Part of proof of Lemma D i... |
| cdlemd3 40482 | Part of proof of Lemma D i... |
| cdlemd4 40483 | Part of proof of Lemma D i... |
| cdlemd5 40484 | Part of proof of Lemma D i... |
| cdlemd6 40485 | Part of proof of Lemma D i... |
| cdlemd7 40486 | Part of proof of Lemma D i... |
| cdlemd8 40487 | Part of proof of Lemma D i... |
| cdlemd9 40488 | Part of proof of Lemma D i... |
| cdlemd 40489 | If two translations agree ... |
| ltrneq3 40490 | Two translations agree at ... |
| cdleme00a 40491 | Part of proof of Lemma E i... |
| cdleme0aa 40492 | Part of proof of Lemma E i... |
| cdleme0a 40493 | Part of proof of Lemma E i... |
| cdleme0b 40494 | Part of proof of Lemma E i... |
| cdleme0c 40495 | Part of proof of Lemma E i... |
| cdleme0cp 40496 | Part of proof of Lemma E i... |
| cdleme0cq 40497 | Part of proof of Lemma E i... |
| cdleme0dN 40498 | Part of proof of Lemma E i... |
| cdleme0e 40499 | Part of proof of Lemma E i... |
| cdleme0fN 40500 | Part of proof of Lemma E i... |
| cdleme0gN 40501 | Part of proof of Lemma E i... |
| cdlemeulpq 40502 | Part of proof of Lemma E i... |
| cdleme01N 40503 | Part of proof of Lemma E i... |
| cdleme02N 40504 | Part of proof of Lemma E i... |
| cdleme0ex1N 40505 | Part of proof of Lemma E i... |
| cdleme0ex2N 40506 | Part of proof of Lemma E i... |
| cdleme0moN 40507 | Part of proof of Lemma E i... |
| cdleme1b 40508 | Part of proof of Lemma E i... |
| cdleme1 40509 | Part of proof of Lemma E i... |
| cdleme2 40510 | Part of proof of Lemma E i... |
| cdleme3b 40511 | Part of proof of Lemma E i... |
| cdleme3c 40512 | Part of proof of Lemma E i... |
| cdleme3d 40513 | Part of proof of Lemma E i... |
| cdleme3e 40514 | Part of proof of Lemma E i... |
| cdleme3fN 40515 | Part of proof of Lemma E i... |
| cdleme3g 40516 | Part of proof of Lemma E i... |
| cdleme3h 40517 | Part of proof of Lemma E i... |
| cdleme3fa 40518 | Part of proof of Lemma E i... |
| cdleme3 40519 | Part of proof of Lemma E i... |
| cdleme4 40520 | Part of proof of Lemma E i... |
| cdleme4a 40521 | Part of proof of Lemma E i... |
| cdleme5 40522 | Part of proof of Lemma E i... |
| cdleme6 40523 | Part of proof of Lemma E i... |
| cdleme7aa 40524 | Part of proof of Lemma E i... |
| cdleme7a 40525 | Part of proof of Lemma E i... |
| cdleme7b 40526 | Part of proof of Lemma E i... |
| cdleme7c 40527 | Part of proof of Lemma E i... |
| cdleme7d 40528 | Part of proof of Lemma E i... |
| cdleme7e 40529 | Part of proof of Lemma E i... |
| cdleme7ga 40530 | Part of proof of Lemma E i... |
| cdleme7 40531 | Part of proof of Lemma E i... |
| cdleme8 40532 | Part of proof of Lemma E i... |
| cdleme9a 40533 | Part of proof of Lemma E i... |
| cdleme9b 40534 | Utility lemma for Lemma E ... |
| cdleme9 40535 | Part of proof of Lemma E i... |
| cdleme10 40536 | Part of proof of Lemma E i... |
| cdleme8tN 40537 | Part of proof of Lemma E i... |
| cdleme9taN 40538 | Part of proof of Lemma E i... |
| cdleme9tN 40539 | Part of proof of Lemma E i... |
| cdleme10tN 40540 | Part of proof of Lemma E i... |
| cdleme16aN 40541 | Part of proof of Lemma E i... |
| cdleme11a 40542 | Part of proof of Lemma E i... |
| cdleme11c 40543 | Part of proof of Lemma E i... |
| cdleme11dN 40544 | Part of proof of Lemma E i... |
| cdleme11e 40545 | Part of proof of Lemma E i... |
| cdleme11fN 40546 | Part of proof of Lemma E i... |
| cdleme11g 40547 | Part of proof of Lemma E i... |
| cdleme11h 40548 | Part of proof of Lemma E i... |
| cdleme11j 40549 | Part of proof of Lemma E i... |
| cdleme11k 40550 | Part of proof of Lemma E i... |
| cdleme11l 40551 | Part of proof of Lemma E i... |
| cdleme11 40552 | Part of proof of Lemma E i... |
| cdleme12 40553 | Part of proof of Lemma E i... |
| cdleme13 40554 | Part of proof of Lemma E i... |
| cdleme14 40555 | Part of proof of Lemma E i... |
| cdleme15a 40556 | Part of proof of Lemma E i... |
| cdleme15b 40557 | Part of proof of Lemma E i... |
| cdleme15c 40558 | Part of proof of Lemma E i... |
| cdleme15d 40559 | Part of proof of Lemma E i... |
| cdleme15 40560 | Part of proof of Lemma E i... |
| cdleme16b 40561 | Part of proof of Lemma E i... |
| cdleme16c 40562 | Part of proof of Lemma E i... |
| cdleme16d 40563 | Part of proof of Lemma E i... |
| cdleme16e 40564 | Part of proof of Lemma E i... |
| cdleme16f 40565 | Part of proof of Lemma E i... |
| cdleme16g 40566 | Part of proof of Lemma E i... |
| cdleme16 40567 | Part of proof of Lemma E i... |
| cdleme17a 40568 | Part of proof of Lemma E i... |
| cdleme17b 40569 | Lemma leading to ~ cdleme1... |
| cdleme17c 40570 | Part of proof of Lemma E i... |
| cdleme17d1 40571 | Part of proof of Lemma E i... |
| cdleme0nex 40572 | Part of proof of Lemma E i... |
| cdleme18a 40573 | Part of proof of Lemma E i... |
| cdleme18b 40574 | Part of proof of Lemma E i... |
| cdleme18c 40575 | Part of proof of Lemma E i... |
| cdleme22gb 40576 | Utility lemma for Lemma E ... |
| cdleme18d 40577 | Part of proof of Lemma E i... |
| cdlemesner 40578 | Part of proof of Lemma E i... |
| cdlemedb 40579 | Part of proof of Lemma E i... |
| cdlemeda 40580 | Part of proof of Lemma E i... |
| cdlemednpq 40581 | Part of proof of Lemma E i... |
| cdlemednuN 40582 | Part of proof of Lemma E i... |
| cdleme20zN 40583 | Part of proof of Lemma E i... |
| cdleme20y 40584 | Part of proof of Lemma E i... |
| cdleme19a 40585 | Part of proof of Lemma E i... |
| cdleme19b 40586 | Part of proof of Lemma E i... |
| cdleme19c 40587 | Part of proof of Lemma E i... |
| cdleme19d 40588 | Part of proof of Lemma E i... |
| cdleme19e 40589 | Part of proof of Lemma E i... |
| cdleme19f 40590 | Part of proof of Lemma E i... |
| cdleme20aN 40591 | Part of proof of Lemma E i... |
| cdleme20bN 40592 | Part of proof of Lemma E i... |
| cdleme20c 40593 | Part of proof of Lemma E i... |
| cdleme20d 40594 | Part of proof of Lemma E i... |
| cdleme20e 40595 | Part of proof of Lemma E i... |
| cdleme20f 40596 | Part of proof of Lemma E i... |
| cdleme20g 40597 | Part of proof of Lemma E i... |
| cdleme20h 40598 | Part of proof of Lemma E i... |
| cdleme20i 40599 | Part of proof of Lemma E i... |
| cdleme20j 40600 | Part of proof of Lemma E i... |
| cdleme20k 40601 | Part of proof of Lemma E i... |
| cdleme20l1 40602 | Part of proof of Lemma E i... |
| cdleme20l2 40603 | Part of proof of Lemma E i... |
| cdleme20l 40604 | Part of proof of Lemma E i... |
| cdleme20m 40605 | Part of proof of Lemma E i... |
| cdleme20 40606 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 40607 | Part of proof of Lemma E i... |
| cdleme21b 40608 | Part of proof of Lemma E i... |
| cdleme21c 40609 | Part of proof of Lemma E i... |
| cdleme21at 40610 | Part of proof of Lemma E i... |
| cdleme21ct 40611 | Part of proof of Lemma E i... |
| cdleme21d 40612 | Part of proof of Lemma E i... |
| cdleme21e 40613 | Part of proof of Lemma E i... |
| cdleme21f 40614 | Part of proof of Lemma E i... |
| cdleme21g 40615 | Part of proof of Lemma E i... |
| cdleme21h 40616 | Part of proof of Lemma E i... |
| cdleme21i 40617 | Part of proof of Lemma E i... |
| cdleme21j 40618 | Combine ~ cdleme20 and ~ c... |
| cdleme21 40619 | Part of proof of Lemma E i... |
| cdleme21k 40620 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 40621 | Part of proof of Lemma E i... |
| cdleme22a 40622 | Part of proof of Lemma E i... |
| cdleme22b 40623 | Part of proof of Lemma E i... |
| cdleme22cN 40624 | Part of proof of Lemma E i... |
| cdleme22d 40625 | Part of proof of Lemma E i... |
| cdleme22e 40626 | Part of proof of Lemma E i... |
| cdleme22eALTN 40627 | Part of proof of Lemma E i... |
| cdleme22f 40628 | Part of proof of Lemma E i... |
| cdleme22f2 40629 | Part of proof of Lemma E i... |
| cdleme22g 40630 | Part of proof of Lemma E i... |
| cdleme23a 40631 | Part of proof of Lemma E i... |
| cdleme23b 40632 | Part of proof of Lemma E i... |
| cdleme23c 40633 | Part of proof of Lemma E i... |
| cdleme24 40634 | Quantified version of ~ cd... |
| cdleme25a 40635 | Lemma for ~ cdleme25b . (... |
| cdleme25b 40636 | Transform ~ cdleme24 . TO... |
| cdleme25c 40637 | Transform ~ cdleme25b . (... |
| cdleme25dN 40638 | Transform ~ cdleme25c . (... |
| cdleme25cl 40639 | Show closure of the unique... |
| cdleme25cv 40640 | Change bound variables in ... |
| cdleme26e 40641 | Part of proof of Lemma E i... |
| cdleme26ee 40642 | Part of proof of Lemma E i... |
| cdleme26eALTN 40643 | Part of proof of Lemma E i... |
| cdleme26fALTN 40644 | Part of proof of Lemma E i... |
| cdleme26f 40645 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 40646 | Part of proof of Lemma E i... |
| cdleme26f2 40647 | Part of proof of Lemma E i... |
| cdleme27cl 40648 | Part of proof of Lemma E i... |
| cdleme27a 40649 | Part of proof of Lemma E i... |
| cdleme27b 40650 | Lemma for ~ cdleme27N . (... |
| cdleme27N 40651 | Part of proof of Lemma E i... |
| cdleme28a 40652 | Lemma for ~ cdleme25b . T... |
| cdleme28b 40653 | Lemma for ~ cdleme25b . T... |
| cdleme28c 40654 | Part of proof of Lemma E i... |
| cdleme28 40655 | Quantified version of ~ cd... |
| cdleme29ex 40656 | Lemma for ~ cdleme29b . (... |
| cdleme29b 40657 | Transform ~ cdleme28 . (C... |
| cdleme29c 40658 | Transform ~ cdleme28b . (... |
| cdleme29cl 40659 | Show closure of the unique... |
| cdleme30a 40660 | Part of proof of Lemma E i... |
| cdleme31so 40661 | Part of proof of Lemma E i... |
| cdleme31sn 40662 | Part of proof of Lemma E i... |
| cdleme31sn1 40663 | Part of proof of Lemma E i... |
| cdleme31se 40664 | Part of proof of Lemma D i... |
| cdleme31se2 40665 | Part of proof of Lemma D i... |
| cdleme31sc 40666 | Part of proof of Lemma E i... |
| cdleme31sde 40667 | Part of proof of Lemma D i... |
| cdleme31snd 40668 | Part of proof of Lemma D i... |
| cdleme31sdnN 40669 | Part of proof of Lemma E i... |
| cdleme31sn1c 40670 | Part of proof of Lemma E i... |
| cdleme31sn2 40671 | Part of proof of Lemma E i... |
| cdleme31fv 40672 | Part of proof of Lemma E i... |
| cdleme31fv1 40673 | Part of proof of Lemma E i... |
| cdleme31fv1s 40674 | Part of proof of Lemma E i... |
| cdleme31fv2 40675 | Part of proof of Lemma E i... |
| cdleme31id 40676 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 40677 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 40678 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 40679 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 40680 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 40681 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 40682 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 40683 | Part of proof of Lemma E i... |
| cdlemefr29exN 40684 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 40685 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 40686 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 40687 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 40688 | TODO fix comment. (Contri... |
| cdlemefr29clN 40689 | Show closure of the unique... |
| cdleme43frv1snN 40690 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 40691 | Part of proof of Lemma E i... |
| cdlemefr32fva1 40692 | Part of proof of Lemma E i... |
| cdlemefr31fv1 40693 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 40694 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 40695 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 40696 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 40697 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 40698 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 40699 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 40700 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 40701 | Show closure of the unique... |
| cdleme43fsv1snlem 40702 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 40703 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 40704 | Part of proof of Lemma E i... |
| cdlemefs32fva1 40705 | Part of proof of Lemma E i... |
| cdlemefs31fv1 40706 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 40707 | Value of f(r) when r is an... |
| cdlemefs44 40708 | Value of f_s(r) when r is ... |
| cdlemefr45 40709 | Value of f(r) when r is an... |
| cdlemefr45e 40710 | Explicit expansion of ~ cd... |
| cdlemefs45 40711 | Value of f_s(r) when r is ... |
| cdlemefs45ee 40712 | Explicit expansion of ~ cd... |
| cdlemefs45eN 40713 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 40714 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 40715 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 40716 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 40717 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 40718 | Show closure of ` [_ R / s... |
| cdleme32fva 40719 | Part of proof of Lemma D i... |
| cdleme32fva1 40720 | Part of proof of Lemma D i... |
| cdleme32fvaw 40721 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 40722 | Part of proof of Lemma D i... |
| cdleme32a 40723 | Part of proof of Lemma D i... |
| cdleme32b 40724 | Part of proof of Lemma D i... |
| cdleme32c 40725 | Part of proof of Lemma D i... |
| cdleme32d 40726 | Part of proof of Lemma D i... |
| cdleme32e 40727 | Part of proof of Lemma D i... |
| cdleme32f 40728 | Part of proof of Lemma D i... |
| cdleme32le 40729 | Part of proof of Lemma D i... |
| cdleme35a 40730 | Part of proof of Lemma E i... |
| cdleme35fnpq 40731 | Part of proof of Lemma E i... |
| cdleme35b 40732 | Part of proof of Lemma E i... |
| cdleme35c 40733 | Part of proof of Lemma E i... |
| cdleme35d 40734 | Part of proof of Lemma E i... |
| cdleme35e 40735 | Part of proof of Lemma E i... |
| cdleme35f 40736 | Part of proof of Lemma E i... |
| cdleme35g 40737 | Part of proof of Lemma E i... |
| cdleme35h 40738 | Part of proof of Lemma E i... |
| cdleme35h2 40739 | Part of proof of Lemma E i... |
| cdleme35sn2aw 40740 | Part of proof of Lemma E i... |
| cdleme35sn3a 40741 | Part of proof of Lemma E i... |
| cdleme36a 40742 | Part of proof of Lemma E i... |
| cdleme36m 40743 | Part of proof of Lemma E i... |
| cdleme37m 40744 | Part of proof of Lemma E i... |
| cdleme38m 40745 | Part of proof of Lemma E i... |
| cdleme38n 40746 | Part of proof of Lemma E i... |
| cdleme39a 40747 | Part of proof of Lemma E i... |
| cdleme39n 40748 | Part of proof of Lemma E i... |
| cdleme40m 40749 | Part of proof of Lemma E i... |
| cdleme40n 40750 | Part of proof of Lemma E i... |
| cdleme40v 40751 | Part of proof of Lemma E i... |
| cdleme40w 40752 | Part of proof of Lemma E i... |
| cdleme42a 40753 | Part of proof of Lemma E i... |
| cdleme42c 40754 | Part of proof of Lemma E i... |
| cdleme42d 40755 | Part of proof of Lemma E i... |
| cdleme41sn3aw 40756 | Part of proof of Lemma E i... |
| cdleme41sn4aw 40757 | Part of proof of Lemma E i... |
| cdleme41snaw 40758 | Part of proof of Lemma E i... |
| cdleme41fva11 40759 | Part of proof of Lemma E i... |
| cdleme42b 40760 | Part of proof of Lemma E i... |
| cdleme42e 40761 | Part of proof of Lemma E i... |
| cdleme42f 40762 | Part of proof of Lemma E i... |
| cdleme42g 40763 | Part of proof of Lemma E i... |
| cdleme42h 40764 | Part of proof of Lemma E i... |
| cdleme42i 40765 | Part of proof of Lemma E i... |
| cdleme42k 40766 | Part of proof of Lemma E i... |
| cdleme42ke 40767 | Part of proof of Lemma E i... |
| cdleme42keg 40768 | Part of proof of Lemma E i... |
| cdleme42mN 40769 | Part of proof of Lemma E i... |
| cdleme42mgN 40770 | Part of proof of Lemma E i... |
| cdleme43aN 40771 | Part of proof of Lemma E i... |
| cdleme43bN 40772 | Lemma for Lemma E in [Craw... |
| cdleme43cN 40773 | Part of proof of Lemma E i... |
| cdleme43dN 40774 | Part of proof of Lemma E i... |
| cdleme46f2g2 40775 | Conversion for ` G ` to re... |
| cdleme46f2g1 40776 | Conversion for ` G ` to re... |
| cdleme17d2 40777 | Part of proof of Lemma E i... |
| cdleme17d3 40778 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 40779 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 40780 | Part of proof of Lemma E i... |
| cdleme48fv 40781 | Part of proof of Lemma D i... |
| cdleme48fvg 40782 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 40783 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 40784 | TODO: fix comment. TODO: ... |
| cdleme48b 40785 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 40786 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 40787 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 40788 | TODO: fix comment. (Contr... |
| cdleme4gfv 40789 | Part of proof of Lemma D i... |
| cdlemeg47b 40790 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 40791 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 40792 | Value of g_s(r) when r is ... |
| cdlemeg49le 40793 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 40794 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 40795 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 40796 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 40797 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 40798 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 40799 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 40800 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 40801 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 40802 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 40803 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 40804 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 40805 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 40806 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 40807 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 40808 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 40809 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 40810 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 40811 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 40812 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 40813 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 40814 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 40815 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 40816 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 40817 | TODO: fix comment. (Contr... |
| cdleme48gfv1 40818 | TODO: fix comment. (Contr... |
| cdleme48gfv 40819 | TODO: fix comment. (Contr... |
| cdleme48fgv 40820 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 40821 | Part of proof of Lemma D i... |
| cdleme50lebi 40822 | Part of proof of Lemma D i... |
| cdleme50eq 40823 | Part of proof of Lemma D i... |
| cdleme50f 40824 | Part of proof of Lemma D i... |
| cdleme50f1 40825 | Part of proof of Lemma D i... |
| cdleme50rnlem 40826 | Part of proof of Lemma D i... |
| cdleme50rn 40827 | Part of proof of Lemma D i... |
| cdleme50f1o 40828 | Part of proof of Lemma D i... |
| cdleme50laut 40829 | Part of proof of Lemma D i... |
| cdleme50ldil 40830 | Part of proof of Lemma D i... |
| cdleme50trn1 40831 | Part of proof that ` F ` i... |
| cdleme50trn2a 40832 | Part of proof that ` F ` i... |
| cdleme50trn2 40833 | Part of proof that ` F ` i... |
| cdleme50trn12 40834 | Part of proof that ` F ` i... |
| cdleme50trn3 40835 | Part of proof that ` F ` i... |
| cdleme50trn123 40836 | Part of proof that ` F ` i... |
| cdleme51finvfvN 40837 | Part of proof of Lemma E i... |
| cdleme51finvN 40838 | Part of proof of Lemma E i... |
| cdleme50ltrn 40839 | Part of proof of Lemma E i... |
| cdleme51finvtrN 40840 | Part of proof of Lemma E i... |
| cdleme50ex 40841 | Part of Lemma E in [Crawle... |
| cdleme 40842 | Lemma E in [Crawley] p. 11... |
| cdlemf1 40843 | Part of Lemma F in [Crawle... |
| cdlemf2 40844 | Part of Lemma F in [Crawle... |
| cdlemf 40845 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 40846 | ~ cdlemf with additional c... |
| cdlemftr3 40847 | Special case of ~ cdlemf s... |
| cdlemftr2 40848 | Special case of ~ cdlemf s... |
| cdlemftr1 40849 | Part of proof of Lemma G o... |
| cdlemftr0 40850 | Special case of ~ cdlemf s... |
| trlord 40851 | The ordering of two Hilber... |
| cdlemg1a 40852 | Shorter expression for ` G... |
| cdlemg1b2 40853 | This theorem can be used t... |
| cdlemg1idlemN 40854 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 40855 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 40856 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 40857 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 40858 | This theorem can be used t... |
| cdlemg1idN 40859 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 40860 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 40861 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 40862 | Version of ~ cdleme51finvt... |
| ltrniotaval 40863 | Value of the unique transl... |
| ltrniotacnvval 40864 | Converse value of the uniq... |
| ltrniotaidvalN 40865 | Value of the unique transl... |
| ltrniotavalbN 40866 | Value of the unique transl... |
| cdlemeiota 40867 | A translation is uniquely ... |
| cdlemg1ci2 40868 | Any function of the form o... |
| cdlemg1cN 40869 | Any translation belongs to... |
| cdlemg1cex 40870 | Any translation is one of ... |
| cdlemg2cN 40871 | Any translation belongs to... |
| cdlemg2dN 40872 | This theorem can be used t... |
| cdlemg2cex 40873 | Any translation is one of ... |
| cdlemg2ce 40874 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 40875 | Part of proof of Lemma E i... |
| cdlemg2fvlem 40876 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 40877 | ~ cdleme42keg with simpler... |
| cdlemg2idN 40878 | Version of ~ cdleme31id wi... |
| cdlemg3a 40879 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 40880 | TODO: Replace this with ~... |
| cdlemg2fv 40881 | Value of a translation in ... |
| cdlemg2fv2 40882 | Value of a translation in ... |
| cdlemg2k 40883 | ~ cdleme42keg with simpler... |
| cdlemg2kq 40884 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 40885 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 40886 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 40887 | TODO: Is there a simpler ... |
| cdlemb3 40888 | Given two atoms not under ... |
| cdlemg7fvbwN 40889 | Properties of a translatio... |
| cdlemg4a 40890 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 40891 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 40892 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 40893 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 40894 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 40895 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 40896 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 40897 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 40898 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 40899 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 40900 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 40901 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 40902 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 40903 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 40904 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 40905 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 40906 | Value of a translation com... |
| cdlemg7aN 40907 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 40908 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 40909 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 40910 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 40911 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 40912 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 40913 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 40914 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 40915 | The triples ` <. P , ( F `... |
| cdlemg9 40916 | The triples ` <. P , ( F `... |
| cdlemg10b 40917 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 40918 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 40919 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 40920 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 40921 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 40922 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 40923 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 40924 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 40925 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 40926 | The triples ` <. P , ( F `... |
| cdlemg12c 40927 | The triples ` <. P , ( F `... |
| cdlemg12d 40928 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 40929 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 40930 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 40931 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 40932 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 40933 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 40934 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 40935 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 40936 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 40937 | Eliminate the ` ( F `` P )... |
| cdlemg15 40938 | Eliminate the ` ( (... |
| cdlemg16 40939 | Part of proof of Lemma G o... |
| cdlemg16ALTN 40940 | This version of ~ cdlemg16... |
| cdlemg16z 40941 | Eliminate ` ( ( F `... |
| cdlemg16zz 40942 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 40943 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 40944 | Part of proof of Lemma G i... |
| cdlemg17dN 40945 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 40946 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 40947 | TODO: fix comment. (Contr... |
| cdlemg17f 40948 | TODO: fix comment. (Contr... |
| cdlemg17g 40949 | TODO: fix comment. (Contr... |
| cdlemg17h 40950 | TODO: fix comment. (Contr... |
| cdlemg17i 40951 | TODO: fix comment. (Contr... |
| cdlemg17ir 40952 | TODO: fix comment. (Contr... |
| cdlemg17j 40953 | TODO: fix comment. (Contr... |
| cdlemg17pq 40954 | Utility theorem for swappi... |
| cdlemg17bq 40955 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 40956 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 40957 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 40958 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 40959 | Part of Lemma G of [Crawle... |
| cdlemg18a 40960 | Show two lines are differe... |
| cdlemg18b 40961 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 40962 | Show two lines intersect a... |
| cdlemg18d 40963 | Show two lines intersect a... |
| cdlemg18 40964 | Show two lines intersect a... |
| cdlemg19a 40965 | Show two lines intersect a... |
| cdlemg19 40966 | Show two lines intersect a... |
| cdlemg20 40967 | Show two lines intersect a... |
| cdlemg21 40968 | Version of cdlemg19 with `... |
| cdlemg22 40969 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 40970 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 40971 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 40972 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 40973 | ~ cdlemg16zz restated for ... |
| cdlemg27a 40974 | For use with case when ` (... |
| cdlemg28a 40975 | Part of proof of Lemma G o... |
| cdlemg31b0N 40976 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 40977 | TODO: Fix comment. (Cont... |
| cdlemg27b 40978 | TODO: Fix comment. (Cont... |
| cdlemg31a 40979 | TODO: fix comment. (Contr... |
| cdlemg31b 40980 | TODO: fix comment. (Contr... |
| cdlemg31c 40981 | Show that when ` N ` is an... |
| cdlemg31d 40982 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 40983 | TODO: Fix comment. (Cont... |
| cdlemg33c0 40984 | TODO: Fix comment. (Cont... |
| cdlemg28b 40985 | Part of proof of Lemma G o... |
| cdlemg28 40986 | Part of proof of Lemma G o... |
| cdlemg29 40987 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 40988 | TODO: Fix comment. (Cont... |
| cdlemg33b 40989 | TODO: Fix comment. (Cont... |
| cdlemg33c 40990 | TODO: Fix comment. (Cont... |
| cdlemg33d 40991 | TODO: Fix comment. (Cont... |
| cdlemg33e 40992 | TODO: Fix comment. (Cont... |
| cdlemg33 40993 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 40994 | Use cdlemg33 to eliminate ... |
| cdlemg35 40995 | TODO: Fix comment. TODO:... |
| cdlemg36 40996 | Use cdlemg35 to eliminate ... |
| cdlemg38 40997 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 40998 | Eliminate ` =/= ` conditio... |
| cdlemg40 40999 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 41000 | Convert ~ cdlemg40 to func... |
| ltrnco 41001 | The composition of two tra... |
| trlcocnv 41002 | Swap the arguments of the ... |
| trlcoabs 41003 | Absorption into a composit... |
| trlcoabs2N 41004 | Absorption of the trace of... |
| trlcoat 41005 | The trace of a composition... |
| trlcocnvat 41006 | Commonly used special case... |
| trlconid 41007 | The composition of two dif... |
| trlcolem 41008 | Lemma for ~ trlco . (Cont... |
| trlco 41009 | The trace of a composition... |
| trlcone 41010 | If two translations have d... |
| cdlemg42 41011 | Part of proof of Lemma G o... |
| cdlemg43 41012 | Part of proof of Lemma G o... |
| cdlemg44a 41013 | Part of proof of Lemma G o... |
| cdlemg44b 41014 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 41015 | Part of proof of Lemma G o... |
| cdlemg47a 41016 | TODO: fix comment. TODO: ... |
| cdlemg46 41017 | Part of proof of Lemma G o... |
| cdlemg47 41018 | Part of proof of Lemma G o... |
| cdlemg48 41019 | Eliminate ` h ` from ~ cdl... |
| ltrncom 41020 | Composition is commutative... |
| ltrnco4 41021 | Rearrange a composition of... |
| trljco 41022 | Trace joined with trace of... |
| trljco2 41023 | Trace joined with trace of... |
| tgrpfset 41026 | The translation group maps... |
| tgrpset 41027 | The translation group for ... |
| tgrpbase 41028 | The base set of the transl... |
| tgrpopr 41029 | The group operation of the... |
| tgrpov 41030 | The group operation value ... |
| tgrpgrplem 41031 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 41032 | The translation group is a... |
| tgrpabl 41033 | The translation group is a... |
| tendofset 41040 | The set of all trace-prese... |
| tendoset 41041 | The set of trace-preservin... |
| istendo 41042 | The predicate "is a trace-... |
| tendotp 41043 | Trace-preserving property ... |
| istendod 41044 | Deduce the predicate "is a... |
| tendof 41045 | Functionality of a trace-p... |
| tendoeq1 41046 | Condition determining equa... |
| tendovalco 41047 | Value of composition of tr... |
| tendocoval 41048 | Value of composition of en... |
| tendocl 41049 | Closure of a trace-preserv... |
| tendoco2 41050 | Distribution of compositio... |
| tendoidcl 41051 | The identity is a trace-pr... |
| tendo1mul 41052 | Multiplicative identity mu... |
| tendo1mulr 41053 | Multiplicative identity mu... |
| tendococl 41054 | The composition of two tra... |
| tendoid 41055 | The identity value of a tr... |
| tendoeq2 41056 | Condition determining equa... |
| tendoplcbv 41057 | Define sum operation for t... |
| tendopl 41058 | Value of endomorphism sum ... |
| tendopl2 41059 | Value of result of endomor... |
| tendoplcl2 41060 | Value of result of endomor... |
| tendoplco2 41061 | Value of result of endomor... |
| tendopltp 41062 | Trace-preserving property ... |
| tendoplcl 41063 | Endomorphism sum is a trac... |
| tendoplcom 41064 | The endomorphism sum opera... |
| tendoplass 41065 | The endomorphism sum opera... |
| tendodi1 41066 | Endomorphism composition d... |
| tendodi2 41067 | Endomorphism composition d... |
| tendo0cbv 41068 | Define additive identity f... |
| tendo02 41069 | Value of additive identity... |
| tendo0co2 41070 | The additive identity trac... |
| tendo0tp 41071 | Trace-preserving property ... |
| tendo0cl 41072 | The additive identity is a... |
| tendo0pl 41073 | Property of the additive i... |
| tendo0plr 41074 | Property of the additive i... |
| tendoicbv 41075 | Define inverse function fo... |
| tendoi 41076 | Value of inverse endomorph... |
| tendoi2 41077 | Value of additive inverse ... |
| tendoicl 41078 | Closure of the additive in... |
| tendoipl 41079 | Property of the additive i... |
| tendoipl2 41080 | Property of the additive i... |
| erngfset 41081 | The division rings on trac... |
| erngset 41082 | The division ring on trace... |
| erngbase 41083 | The base set of the divisi... |
| erngfplus 41084 | Ring addition operation. ... |
| erngplus 41085 | Ring addition operation. ... |
| erngplus2 41086 | Ring addition operation. ... |
| erngfmul 41087 | Ring multiplication operat... |
| erngmul 41088 | Ring addition operation. ... |
| erngfset-rN 41089 | The division rings on trac... |
| erngset-rN 41090 | The division ring on trace... |
| erngbase-rN 41091 | The base set of the divisi... |
| erngfplus-rN 41092 | Ring addition operation. ... |
| erngplus-rN 41093 | Ring addition operation. ... |
| erngplus2-rN 41094 | Ring addition operation. ... |
| erngfmul-rN 41095 | Ring multiplication operat... |
| erngmul-rN 41096 | Ring addition operation. ... |
| cdlemh1 41097 | Part of proof of Lemma H o... |
| cdlemh2 41098 | Part of proof of Lemma H o... |
| cdlemh 41099 | Lemma H of [Crawley] p. 11... |
| cdlemi1 41100 | Part of proof of Lemma I o... |
| cdlemi2 41101 | Part of proof of Lemma I o... |
| cdlemi 41102 | Lemma I of [Crawley] p. 11... |
| cdlemj1 41103 | Part of proof of Lemma J o... |
| cdlemj2 41104 | Part of proof of Lemma J o... |
| cdlemj3 41105 | Part of proof of Lemma J o... |
| tendocan 41106 | Cancellation law: if the v... |
| tendoid0 41107 | A trace-preserving endomor... |
| tendo0mul 41108 | Additive identity multipli... |
| tendo0mulr 41109 | Additive identity multipli... |
| tendo1ne0 41110 | The identity (unity) is no... |
| tendoconid 41111 | The composition (product) ... |
| tendotr 41112 | The trace of the value of ... |
| cdlemk1 41113 | Part of proof of Lemma K o... |
| cdlemk2 41114 | Part of proof of Lemma K o... |
| cdlemk3 41115 | Part of proof of Lemma K o... |
| cdlemk4 41116 | Part of proof of Lemma K o... |
| cdlemk5a 41117 | Part of proof of Lemma K o... |
| cdlemk5 41118 | Part of proof of Lemma K o... |
| cdlemk6 41119 | Part of proof of Lemma K o... |
| cdlemk8 41120 | Part of proof of Lemma K o... |
| cdlemk9 41121 | Part of proof of Lemma K o... |
| cdlemk9bN 41122 | Part of proof of Lemma K o... |
| cdlemki 41123 | Part of proof of Lemma K o... |
| cdlemkvcl 41124 | Part of proof of Lemma K o... |
| cdlemk10 41125 | Part of proof of Lemma K o... |
| cdlemksv 41126 | Part of proof of Lemma K o... |
| cdlemksel 41127 | Part of proof of Lemma K o... |
| cdlemksat 41128 | Part of proof of Lemma K o... |
| cdlemksv2 41129 | Part of proof of Lemma K o... |
| cdlemk7 41130 | Part of proof of Lemma K o... |
| cdlemk11 41131 | Part of proof of Lemma K o... |
| cdlemk12 41132 | Part of proof of Lemma K o... |
| cdlemkoatnle 41133 | Utility lemma. (Contribut... |
| cdlemk13 41134 | Part of proof of Lemma K o... |
| cdlemkole 41135 | Utility lemma. (Contribut... |
| cdlemk14 41136 | Part of proof of Lemma K o... |
| cdlemk15 41137 | Part of proof of Lemma K o... |
| cdlemk16a 41138 | Part of proof of Lemma K o... |
| cdlemk16 41139 | Part of proof of Lemma K o... |
| cdlemk17 41140 | Part of proof of Lemma K o... |
| cdlemk1u 41141 | Part of proof of Lemma K o... |
| cdlemk5auN 41142 | Part of proof of Lemma K o... |
| cdlemk5u 41143 | Part of proof of Lemma K o... |
| cdlemk6u 41144 | Part of proof of Lemma K o... |
| cdlemkj 41145 | Part of proof of Lemma K o... |
| cdlemkuvN 41146 | Part of proof of Lemma K o... |
| cdlemkuel 41147 | Part of proof of Lemma K o... |
| cdlemkuat 41148 | Part of proof of Lemma K o... |
| cdlemkuv2 41149 | Part of proof of Lemma K o... |
| cdlemk18 41150 | Part of proof of Lemma K o... |
| cdlemk19 41151 | Part of proof of Lemma K o... |
| cdlemk7u 41152 | Part of proof of Lemma K o... |
| cdlemk11u 41153 | Part of proof of Lemma K o... |
| cdlemk12u 41154 | Part of proof of Lemma K o... |
| cdlemk21N 41155 | Part of proof of Lemma K o... |
| cdlemk20 41156 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 41157 | Utility lemma. (Contribut... |
| cdlemk13-2N 41158 | Part of proof of Lemma K o... |
| cdlemkole-2N 41159 | Utility lemma. (Contribut... |
| cdlemk14-2N 41160 | Part of proof of Lemma K o... |
| cdlemk15-2N 41161 | Part of proof of Lemma K o... |
| cdlemk16-2N 41162 | Part of proof of Lemma K o... |
| cdlemk17-2N 41163 | Part of proof of Lemma K o... |
| cdlemkj-2N 41164 | Part of proof of Lemma K o... |
| cdlemkuv-2N 41165 | Part of proof of Lemma K o... |
| cdlemkuel-2N 41166 | Part of proof of Lemma K o... |
| cdlemkuv2-2 41167 | Part of proof of Lemma K o... |
| cdlemk18-2N 41168 | Part of proof of Lemma K o... |
| cdlemk19-2N 41169 | Part of proof of Lemma K o... |
| cdlemk7u-2N 41170 | Part of proof of Lemma K o... |
| cdlemk11u-2N 41171 | Part of proof of Lemma K o... |
| cdlemk12u-2N 41172 | Part of proof of Lemma K o... |
| cdlemk21-2N 41173 | Part of proof of Lemma K o... |
| cdlemk20-2N 41174 | Part of proof of Lemma K o... |
| cdlemk22 41175 | Part of proof of Lemma K o... |
| cdlemk30 41176 | Part of proof of Lemma K o... |
| cdlemkuu 41177 | Convert between function a... |
| cdlemk31 41178 | Part of proof of Lemma K o... |
| cdlemk32 41179 | Part of proof of Lemma K o... |
| cdlemkuel-3 41180 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 41181 | Part of proof of Lemma K o... |
| cdlemk18-3N 41182 | Part of proof of Lemma K o... |
| cdlemk22-3 41183 | Part of proof of Lemma K o... |
| cdlemk23-3 41184 | Part of proof of Lemma K o... |
| cdlemk24-3 41185 | Part of proof of Lemma K o... |
| cdlemk25-3 41186 | Part of proof of Lemma K o... |
| cdlemk26b-3 41187 | Part of proof of Lemma K o... |
| cdlemk26-3 41188 | Part of proof of Lemma K o... |
| cdlemk27-3 41189 | Part of proof of Lemma K o... |
| cdlemk28-3 41190 | Part of proof of Lemma K o... |
| cdlemk33N 41191 | Part of proof of Lemma K o... |
| cdlemk34 41192 | Part of proof of Lemma K o... |
| cdlemk29-3 41193 | Part of proof of Lemma K o... |
| cdlemk35 41194 | Part of proof of Lemma K o... |
| cdlemk36 41195 | Part of proof of Lemma K o... |
| cdlemk37 41196 | Part of proof of Lemma K o... |
| cdlemk38 41197 | Part of proof of Lemma K o... |
| cdlemk39 41198 | Part of proof of Lemma K o... |
| cdlemk40 41199 | TODO: fix comment. (Contr... |
| cdlemk40t 41200 | TODO: fix comment. (Contr... |
| cdlemk40f 41201 | TODO: fix comment. (Contr... |
| cdlemk41 41202 | Part of proof of Lemma K o... |
| cdlemkfid1N 41203 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 41204 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 41205 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 41206 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 41207 | TODO: is this useful or sh... |
| cdlemky 41208 | Part of proof of Lemma K o... |
| cdlemkyu 41209 | Convert between function a... |
| cdlemkyuu 41210 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 41211 | Part of proof of Lemma K o... |
| cdlemk19ylem 41212 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 41213 | Part of proof of Lemma K o... |
| cdlemk19y 41214 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 41215 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 41216 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 41217 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 41218 | The value of the tau funct... |
| cdlemk35s 41219 | Substitution version of ~ ... |
| cdlemk35s-id 41220 | Substitution version of ~ ... |
| cdlemk39s 41221 | Substitution version of ~ ... |
| cdlemk39s-id 41222 | Substitution version of ~ ... |
| cdlemk42 41223 | Part of proof of Lemma K o... |
| cdlemk19xlem 41224 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 41225 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 41226 | Part of proof of Lemma K o... |
| cdlemk11tc 41227 | Part of proof of Lemma K o... |
| cdlemk11t 41228 | Part of proof of Lemma K o... |
| cdlemk45 41229 | Part of proof of Lemma K o... |
| cdlemk46 41230 | Part of proof of Lemma K o... |
| cdlemk47 41231 | Part of proof of Lemma K o... |
| cdlemk48 41232 | Part of proof of Lemma K o... |
| cdlemk49 41233 | Part of proof of Lemma K o... |
| cdlemk50 41234 | Part of proof of Lemma K o... |
| cdlemk51 41235 | Part of proof of Lemma K o... |
| cdlemk52 41236 | Part of proof of Lemma K o... |
| cdlemk53a 41237 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 41238 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 41239 | Part of proof of Lemma K o... |
| cdlemk54 41240 | Part of proof of Lemma K o... |
| cdlemk55a 41241 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 41242 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 41243 | Part of proof of Lemma K o... |
| cdlemkyyN 41244 | Part of proof of Lemma K o... |
| cdlemk43N 41245 | Part of proof of Lemma K o... |
| cdlemk35u 41246 | Substitution version of ~ ... |
| cdlemk55u1 41247 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 41248 | Part of proof of Lemma K o... |
| cdlemk39u1 41249 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 41250 | Part of proof of Lemma K o... |
| cdlemk19u1 41251 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 41252 | Part of Lemma K of [Crawle... |
| cdlemk56 41253 | Part of Lemma K of [Crawle... |
| cdlemk19w 41254 | Use a fixed element to eli... |
| cdlemk56w 41255 | Use a fixed element to eli... |
| cdlemk 41256 | Lemma K of [Crawley] p. 11... |
| tendoex 41257 | Generalization of Lemma K ... |
| cdleml1N 41258 | Part of proof of Lemma L o... |
| cdleml2N 41259 | Part of proof of Lemma L o... |
| cdleml3N 41260 | Part of proof of Lemma L o... |
| cdleml4N 41261 | Part of proof of Lemma L o... |
| cdleml5N 41262 | Part of proof of Lemma L o... |
| cdleml6 41263 | Part of proof of Lemma L o... |
| cdleml7 41264 | Part of proof of Lemma L o... |
| cdleml8 41265 | Part of proof of Lemma L o... |
| cdleml9 41266 | Part of proof of Lemma L o... |
| dva1dim 41267 | Two expressions for the 1-... |
| dvhb1dimN 41268 | Two expressions for the 1-... |
| erng1lem 41269 | Value of the endomorphism ... |
| erngdvlem1 41270 | Lemma for ~ eringring . (... |
| erngdvlem2N 41271 | Lemma for ~ eringring . (... |
| erngdvlem3 41272 | Lemma for ~ eringring . (... |
| erngdvlem4 41273 | Lemma for ~ erngdv . (Con... |
| eringring 41274 | An endomorphism ring is a ... |
| erngdv 41275 | An endomorphism ring is a ... |
| erng0g 41276 | The division ring zero of ... |
| erng1r 41277 | The division ring unity of... |
| erngdvlem1-rN 41278 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 41279 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 41280 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 41281 | Lemma for ~ erngdv . (Con... |
| erngring-rN 41282 | An endomorphism ring is a ... |
| erngdv-rN 41283 | An endomorphism ring is a ... |
| dvafset 41286 | The constructed partial ve... |
| dvaset 41287 | The constructed partial ve... |
| dvasca 41288 | The ring base set of the c... |
| dvabase 41289 | The ring base set of the c... |
| dvafplusg 41290 | Ring addition operation fo... |
| dvaplusg 41291 | Ring addition operation fo... |
| dvaplusgv 41292 | Ring addition operation fo... |
| dvafmulr 41293 | Ring multiplication operat... |
| dvamulr 41294 | Ring multiplication operat... |
| dvavbase 41295 | The vectors (vector base s... |
| dvafvadd 41296 | The vector sum operation f... |
| dvavadd 41297 | Ring addition operation fo... |
| dvafvsca 41298 | Ring addition operation fo... |
| dvavsca 41299 | Ring addition operation fo... |
| tendospcl 41300 | Closure of endomorphism sc... |
| tendospass 41301 | Associative law for endomo... |
| tendospdi1 41302 | Forward distributive law f... |
| tendocnv 41303 | Converse of a trace-preser... |
| tendospdi2 41304 | Reverse distributive law f... |
| tendospcanN 41305 | Cancellation law for trace... |
| dvaabl 41306 | The constructed partial ve... |
| dvalveclem 41307 | Lemma for ~ dvalvec . (Co... |
| dvalvec 41308 | The constructed partial ve... |
| dva0g 41309 | The zero vector of partial... |
| diaffval 41312 | The partial isomorphism A ... |
| diafval 41313 | The partial isomorphism A ... |
| diaval 41314 | The partial isomorphism A ... |
| diaelval 41315 | Member of the partial isom... |
| diafn 41316 | Functionality and domain o... |
| diadm 41317 | Domain of the partial isom... |
| diaeldm 41318 | Member of domain of the pa... |
| diadmclN 41319 | A member of domain of the ... |
| diadmleN 41320 | A member of domain of the ... |
| dian0 41321 | The value of the partial i... |
| dia0eldmN 41322 | The lattice zero belongs t... |
| dia1eldmN 41323 | The fiducial hyperplane (t... |
| diass 41324 | The value of the partial i... |
| diael 41325 | A member of the value of t... |
| diatrl 41326 | Trace of a member of the p... |
| diaelrnN 41327 | Any value of the partial i... |
| dialss 41328 | The value of partial isomo... |
| diaord 41329 | The partial isomorphism A ... |
| dia11N 41330 | The partial isomorphism A ... |
| diaf11N 41331 | The partial isomorphism A ... |
| diaclN 41332 | Closure of partial isomorp... |
| diacnvclN 41333 | Closure of partial isomorp... |
| dia0 41334 | The value of the partial i... |
| dia1N 41335 | The value of the partial i... |
| dia1elN 41336 | The largest subspace in th... |
| diaglbN 41337 | Partial isomorphism A of a... |
| diameetN 41338 | Partial isomorphism A of a... |
| diainN 41339 | Inverse partial isomorphis... |
| diaintclN 41340 | The intersection of partia... |
| diasslssN 41341 | The partial isomorphism A ... |
| diassdvaN 41342 | The partial isomorphism A ... |
| dia1dim 41343 | Two expressions for the 1-... |
| dia1dim2 41344 | Two expressions for a 1-di... |
| dia1dimid 41345 | A vector (translation) bel... |
| dia2dimlem1 41346 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 41347 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 41348 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 41349 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 41350 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 41351 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 41352 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 41353 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 41354 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 41355 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 41356 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 41357 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 41358 | Lemma for ~ dia2dim . Eli... |
| dia2dim 41359 | A two-dimensional subspace... |
| dvhfset 41362 | The constructed full vecto... |
| dvhset 41363 | The constructed full vecto... |
| dvhsca 41364 | The ring of scalars of the... |
| dvhbase 41365 | The ring base set of the c... |
| dvhfplusr 41366 | Ring addition operation fo... |
| dvhfmulr 41367 | Ring multiplication operat... |
| dvhmulr 41368 | Ring multiplication operat... |
| dvhvbase 41369 | The vectors (vector base s... |
| dvhelvbasei 41370 | Vector membership in the c... |
| dvhvaddcbv 41371 | Change bound variables to ... |
| dvhvaddval 41372 | The vector sum operation f... |
| dvhfvadd 41373 | The vector sum operation f... |
| dvhvadd 41374 | The vector sum operation f... |
| dvhopvadd 41375 | The vector sum operation f... |
| dvhopvadd2 41376 | The vector sum operation f... |
| dvhvaddcl 41377 | Closure of the vector sum ... |
| dvhvaddcomN 41378 | Commutativity of vector su... |
| dvhvaddass 41379 | Associativity of vector su... |
| dvhvscacbv 41380 | Change bound variables to ... |
| dvhvscaval 41381 | The scalar product operati... |
| dvhfvsca 41382 | Scalar product operation f... |
| dvhvsca 41383 | Scalar product operation f... |
| dvhopvsca 41384 | Scalar product operation f... |
| dvhvscacl 41385 | Closure of the scalar prod... |
| tendoinvcl 41386 | Closure of multiplicative ... |
| tendolinv 41387 | Left multiplicative invers... |
| tendorinv 41388 | Right multiplicative inver... |
| dvhgrp 41389 | The full vector space ` U ... |
| dvhlveclem 41390 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 41391 | The full vector space ` U ... |
| dvhlmod 41392 | The full vector space ` U ... |
| dvh0g 41393 | The zero vector of vector ... |
| dvheveccl 41394 | Properties of a unit vecto... |
| dvhopclN 41395 | Closure of a ` DVecH ` vec... |
| dvhopaddN 41396 | Sum of ` DVecH ` vectors e... |
| dvhopspN 41397 | Scalar product of ` DVecH ... |
| dvhopN 41398 | Decompose a ` DVecH ` vect... |
| dvhopellsm 41399 | Ordered pair membership in... |
| cdlemm10N 41400 | The image of the map ` G `... |
| docaffvalN 41403 | Subspace orthocomplement f... |
| docafvalN 41404 | Subspace orthocomplement f... |
| docavalN 41405 | Subspace orthocomplement f... |
| docaclN 41406 | Closure of subspace orthoc... |
| diaocN 41407 | Value of partial isomorphi... |
| doca2N 41408 | Double orthocomplement of ... |
| doca3N 41409 | Double orthocomplement of ... |
| dvadiaN 41410 | Any closed subspace is a m... |
| diarnN 41411 | Partial isomorphism A maps... |
| diaf1oN 41412 | The partial isomorphism A ... |
| djaffvalN 41415 | Subspace join for ` DVecA ... |
| djafvalN 41416 | Subspace join for ` DVecA ... |
| djavalN 41417 | Subspace join for ` DVecA ... |
| djaclN 41418 | Closure of subspace join f... |
| djajN 41419 | Transfer lattice join to `... |
| dibffval 41422 | The partial isomorphism B ... |
| dibfval 41423 | The partial isomorphism B ... |
| dibval 41424 | The partial isomorphism B ... |
| dibopelvalN 41425 | Member of the partial isom... |
| dibval2 41426 | Value of the partial isomo... |
| dibopelval2 41427 | Member of the partial isom... |
| dibval3N 41428 | Value of the partial isomo... |
| dibelval3 41429 | Member of the partial isom... |
| dibopelval3 41430 | Member of the partial isom... |
| dibelval1st 41431 | Membership in value of the... |
| dibelval1st1 41432 | Membership in value of the... |
| dibelval1st2N 41433 | Membership in value of the... |
| dibelval2nd 41434 | Membership in value of the... |
| dibn0 41435 | The value of the partial i... |
| dibfna 41436 | Functionality and domain o... |
| dibdiadm 41437 | Domain of the partial isom... |
| dibfnN 41438 | Functionality and domain o... |
| dibdmN 41439 | Domain of the partial isom... |
| dibeldmN 41440 | Member of domain of the pa... |
| dibord 41441 | The isomorphism B for a la... |
| dib11N 41442 | The isomorphism B for a la... |
| dibf11N 41443 | The partial isomorphism A ... |
| dibclN 41444 | Closure of partial isomorp... |
| dibvalrel 41445 | The value of partial isomo... |
| dib0 41446 | The value of partial isomo... |
| dib1dim 41447 | Two expressions for the 1-... |
| dibglbN 41448 | Partial isomorphism B of a... |
| dibintclN 41449 | The intersection of partia... |
| dib1dim2 41450 | Two expressions for a 1-di... |
| dibss 41451 | The partial isomorphism B ... |
| diblss 41452 | The value of partial isomo... |
| diblsmopel 41453 | Membership in subspace sum... |
| dicffval 41456 | The partial isomorphism C ... |
| dicfval 41457 | The partial isomorphism C ... |
| dicval 41458 | The partial isomorphism C ... |
| dicopelval 41459 | Membership in value of the... |
| dicelvalN 41460 | Membership in value of the... |
| dicval2 41461 | The partial isomorphism C ... |
| dicelval3 41462 | Member of the partial isom... |
| dicopelval2 41463 | Membership in value of the... |
| dicelval2N 41464 | Membership in value of the... |
| dicfnN 41465 | Functionality and domain o... |
| dicdmN 41466 | Domain of the partial isom... |
| dicvalrelN 41467 | The value of partial isomo... |
| dicssdvh 41468 | The partial isomorphism C ... |
| dicelval1sta 41469 | Membership in value of the... |
| dicelval1stN 41470 | Membership in value of the... |
| dicelval2nd 41471 | Membership in value of the... |
| dicvaddcl 41472 | Membership in value of the... |
| dicvscacl 41473 | Membership in value of the... |
| dicn0 41474 | The value of the partial i... |
| diclss 41475 | The value of partial isomo... |
| diclspsn 41476 | The value of isomorphism C... |
| cdlemn2 41477 | Part of proof of Lemma N o... |
| cdlemn2a 41478 | Part of proof of Lemma N o... |
| cdlemn3 41479 | Part of proof of Lemma N o... |
| cdlemn4 41480 | Part of proof of Lemma N o... |
| cdlemn4a 41481 | Part of proof of Lemma N o... |
| cdlemn5pre 41482 | Part of proof of Lemma N o... |
| cdlemn5 41483 | Part of proof of Lemma N o... |
| cdlemn6 41484 | Part of proof of Lemma N o... |
| cdlemn7 41485 | Part of proof of Lemma N o... |
| cdlemn8 41486 | Part of proof of Lemma N o... |
| cdlemn9 41487 | Part of proof of Lemma N o... |
| cdlemn10 41488 | Part of proof of Lemma N o... |
| cdlemn11a 41489 | Part of proof of Lemma N o... |
| cdlemn11b 41490 | Part of proof of Lemma N o... |
| cdlemn11c 41491 | Part of proof of Lemma N o... |
| cdlemn11pre 41492 | Part of proof of Lemma N o... |
| cdlemn11 41493 | Part of proof of Lemma N o... |
| cdlemn 41494 | Lemma N of [Crawley] p. 12... |
| dihordlem6 41495 | Part of proof of Lemma N o... |
| dihordlem7 41496 | Part of proof of Lemma N o... |
| dihordlem7b 41497 | Part of proof of Lemma N o... |
| dihjustlem 41498 | Part of proof after Lemma ... |
| dihjust 41499 | Part of proof after Lemma ... |
| dihord1 41500 | Part of proof after Lemma ... |
| dihord2a 41501 | Part of proof after Lemma ... |
| dihord2b 41502 | Part of proof after Lemma ... |
| dihord2cN 41503 | Part of proof after Lemma ... |
| dihord11b 41504 | Part of proof after Lemma ... |
| dihord10 41505 | Part of proof after Lemma ... |
| dihord11c 41506 | Part of proof after Lemma ... |
| dihord2pre 41507 | Part of proof after Lemma ... |
| dihord2pre2 41508 | Part of proof after Lemma ... |
| dihord2 41509 | Part of proof after Lemma ... |
| dihffval 41512 | The isomorphism H for a la... |
| dihfval 41513 | Isomorphism H for a lattic... |
| dihval 41514 | Value of isomorphism H for... |
| dihvalc 41515 | Value of isomorphism H for... |
| dihlsscpre 41516 | Closure of isomorphism H f... |
| dihvalcqpre 41517 | Value of isomorphism H for... |
| dihvalcq 41518 | Value of isomorphism H for... |
| dihvalb 41519 | Value of isomorphism H for... |
| dihopelvalbN 41520 | Ordered pair member of the... |
| dihvalcqat 41521 | Value of isomorphism H for... |
| dih1dimb 41522 | Two expressions for a 1-di... |
| dih1dimb2 41523 | Isomorphism H at an atom u... |
| dih1dimc 41524 | Isomorphism H at an atom n... |
| dib2dim 41525 | Extend ~ dia2dim to partia... |
| dih2dimb 41526 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 41527 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 41528 | Ordered pair member of the... |
| dihvalcq2 41529 | Value of isomorphism H for... |
| dihopelvalcpre 41530 | Member of value of isomorp... |
| dihopelvalc 41531 | Member of value of isomorp... |
| dihlss 41532 | The value of isomorphism H... |
| dihss 41533 | The value of isomorphism H... |
| dihssxp 41534 | An isomorphism H value is ... |
| dihopcl 41535 | Closure of an ordered pair... |
| xihopellsmN 41536 | Ordered pair membership in... |
| dihopellsm 41537 | Ordered pair membership in... |
| dihord6apre 41538 | Part of proof that isomorp... |
| dihord3 41539 | The isomorphism H for a la... |
| dihord4 41540 | The isomorphism H for a la... |
| dihord5b 41541 | Part of proof that isomorp... |
| dihord6b 41542 | Part of proof that isomorp... |
| dihord6a 41543 | Part of proof that isomorp... |
| dihord5apre 41544 | Part of proof that isomorp... |
| dihord5a 41545 | Part of proof that isomorp... |
| dihord 41546 | The isomorphism H is order... |
| dih11 41547 | The isomorphism H is one-t... |
| dihf11lem 41548 | Functionality of the isomo... |
| dihf11 41549 | The isomorphism H for a la... |
| dihfn 41550 | Functionality and domain o... |
| dihdm 41551 | Domain of isomorphism H. (... |
| dihcl 41552 | Closure of isomorphism H. ... |
| dihcnvcl 41553 | Closure of isomorphism H c... |
| dihcnvid1 41554 | The converse isomorphism o... |
| dihcnvid2 41555 | The isomorphism of a conve... |
| dihcnvord 41556 | Ordering property for conv... |
| dihcnv11 41557 | The converse of isomorphis... |
| dihsslss 41558 | The isomorphism H maps to ... |
| dihrnlss 41559 | The isomorphism H maps to ... |
| dihrnss 41560 | The isomorphism H maps to ... |
| dihvalrel 41561 | The value of isomorphism H... |
| dih0 41562 | The value of isomorphism H... |
| dih0bN 41563 | A lattice element is zero ... |
| dih0vbN 41564 | A vector is zero iff its s... |
| dih0cnv 41565 | The isomorphism H converse... |
| dih0rn 41566 | The zero subspace belongs ... |
| dih0sb 41567 | A subspace is zero iff the... |
| dih1 41568 | The value of isomorphism H... |
| dih1rn 41569 | The full vector space belo... |
| dih1cnv 41570 | The isomorphism H converse... |
| dihwN 41571 | Value of isomorphism H at ... |
| dihmeetlem1N 41572 | Isomorphism H of a conjunc... |
| dihglblem5apreN 41573 | A conjunction property of ... |
| dihglblem5aN 41574 | A conjunction property of ... |
| dihglblem2aN 41575 | Lemma for isomorphism H of... |
| dihglblem2N 41576 | The GLB of a set of lattic... |
| dihglblem3N 41577 | Isomorphism H of a lattice... |
| dihglblem3aN 41578 | Isomorphism H of a lattice... |
| dihglblem4 41579 | Isomorphism H of a lattice... |
| dihglblem5 41580 | Isomorphism H of a lattice... |
| dihmeetlem2N 41581 | Isomorphism H of a conjunc... |
| dihglbcpreN 41582 | Isomorphism H of a lattice... |
| dihglbcN 41583 | Isomorphism H of a lattice... |
| dihmeetcN 41584 | Isomorphism H of a lattice... |
| dihmeetbN 41585 | Isomorphism H of a lattice... |
| dihmeetbclemN 41586 | Lemma for isomorphism H of... |
| dihmeetlem3N 41587 | Lemma for isomorphism H of... |
| dihmeetlem4preN 41588 | Lemma for isomorphism H of... |
| dihmeetlem4N 41589 | Lemma for isomorphism H of... |
| dihmeetlem5 41590 | Part of proof that isomorp... |
| dihmeetlem6 41591 | Lemma for isomorphism H of... |
| dihmeetlem7N 41592 | Lemma for isomorphism H of... |
| dihjatc1 41593 | Lemma for isomorphism H of... |
| dihjatc2N 41594 | Isomorphism H of join with... |
| dihjatc3 41595 | Isomorphism H of join with... |
| dihmeetlem8N 41596 | Lemma for isomorphism H of... |
| dihmeetlem9N 41597 | Lemma for isomorphism H of... |
| dihmeetlem10N 41598 | Lemma for isomorphism H of... |
| dihmeetlem11N 41599 | Lemma for isomorphism H of... |
| dihmeetlem12N 41600 | Lemma for isomorphism H of... |
| dihmeetlem13N 41601 | Lemma for isomorphism H of... |
| dihmeetlem14N 41602 | Lemma for isomorphism H of... |
| dihmeetlem15N 41603 | Lemma for isomorphism H of... |
| dihmeetlem16N 41604 | Lemma for isomorphism H of... |
| dihmeetlem17N 41605 | Lemma for isomorphism H of... |
| dihmeetlem18N 41606 | Lemma for isomorphism H of... |
| dihmeetlem19N 41607 | Lemma for isomorphism H of... |
| dihmeetlem20N 41608 | Lemma for isomorphism H of... |
| dihmeetALTN 41609 | Isomorphism H of a lattice... |
| dih1dimatlem0 41610 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 41611 | Lemma for ~ dih1dimat . (... |
| dih1dimat 41612 | Any 1-dimensional subspace... |
| dihlsprn 41613 | The span of a vector belon... |
| dihlspsnssN 41614 | A subspace included in a 1... |
| dihlspsnat 41615 | The inverse isomorphism H ... |
| dihatlat 41616 | The isomorphism H of an at... |
| dihat 41617 | There exists at least one ... |
| dihpN 41618 | The value of isomorphism H... |
| dihlatat 41619 | The reverse isomorphism H ... |
| dihatexv 41620 | There is a nonzero vector ... |
| dihatexv2 41621 | There is a nonzero vector ... |
| dihglblem6 41622 | Isomorphism H of a lattice... |
| dihglb 41623 | Isomorphism H of a lattice... |
| dihglb2 41624 | Isomorphism H of a lattice... |
| dihmeet 41625 | Isomorphism H of a lattice... |
| dihintcl 41626 | The intersection of closed... |
| dihmeetcl 41627 | Closure of closed subspace... |
| dihmeet2 41628 | Reverse isomorphism H of a... |
| dochffval 41631 | Subspace orthocomplement f... |
| dochfval 41632 | Subspace orthocomplement f... |
| dochval 41633 | Subspace orthocomplement f... |
| dochval2 41634 | Subspace orthocomplement f... |
| dochcl 41635 | Closure of subspace orthoc... |
| dochlss 41636 | A subspace orthocomplement... |
| dochssv 41637 | A subspace orthocomplement... |
| dochfN 41638 | Domain and codomain of the... |
| dochvalr 41639 | Orthocomplement of a close... |
| doch0 41640 | Orthocomplement of the zer... |
| doch1 41641 | Orthocomplement of the uni... |
| dochoc0 41642 | The zero subspace is close... |
| dochoc1 41643 | The unit subspace (all vec... |
| dochvalr2 41644 | Orthocomplement of a close... |
| dochvalr3 41645 | Orthocomplement of a close... |
| doch2val2 41646 | Double orthocomplement for... |
| dochss 41647 | Subset law for orthocomple... |
| dochocss 41648 | Double negative law for or... |
| dochoc 41649 | Double negative law for or... |
| dochsscl 41650 | If a set of vectors is inc... |
| dochoccl 41651 | A set of vectors is closed... |
| dochord 41652 | Ordering law for orthocomp... |
| dochord2N 41653 | Ordering law for orthocomp... |
| dochord3 41654 | Ordering law for orthocomp... |
| doch11 41655 | Orthocomplement is one-to-... |
| dochsordN 41656 | Strict ordering law for or... |
| dochn0nv 41657 | An orthocomplement is nonz... |
| dihoml4c 41658 | Version of ~ dihoml4 with ... |
| dihoml4 41659 | Orthomodular law for const... |
| dochspss 41660 | The span of a set of vecto... |
| dochocsp 41661 | The span of an orthocomple... |
| dochspocN 41662 | The span of an orthocomple... |
| dochocsn 41663 | The double orthocomplement... |
| dochsncom 41664 | Swap vectors in an orthoco... |
| dochsat 41665 | The double orthocomplement... |
| dochshpncl 41666 | If a hyperplane is not clo... |
| dochlkr 41667 | Equivalent conditions for ... |
| dochkrshp 41668 | The closure of a kernel is... |
| dochkrshp2 41669 | Properties of the closure ... |
| dochkrshp3 41670 | Properties of the closure ... |
| dochkrshp4 41671 | Properties of the closure ... |
| dochdmj1 41672 | De Morgan-like law for sub... |
| dochnoncon 41673 | Law of noncontradiction. ... |
| dochnel2 41674 | A nonzero member of a subs... |
| dochnel 41675 | A nonzero vector doesn't b... |
| djhffval 41678 | Subspace join for ` DVecH ... |
| djhfval 41679 | Subspace join for ` DVecH ... |
| djhval 41680 | Subspace join for ` DVecH ... |
| djhval2 41681 | Value of subspace join for... |
| djhcl 41682 | Closure of subspace join f... |
| djhlj 41683 | Transfer lattice join to `... |
| djhljjN 41684 | Lattice join in terms of `... |
| djhjlj 41685 | ` DVecH ` vector space clo... |
| djhj 41686 | ` DVecH ` vector space clo... |
| djhcom 41687 | Subspace join commutes. (... |
| djhspss 41688 | Subspace span of union is ... |
| djhsumss 41689 | Subspace sum is a subset o... |
| dihsumssj 41690 | The subspace sum of two is... |
| djhunssN 41691 | Subspace union is a subset... |
| dochdmm1 41692 | De Morgan-like law for clo... |
| djhexmid 41693 | Excluded middle property o... |
| djh01 41694 | Closed subspace join with ... |
| djh02 41695 | Closed subspace join with ... |
| djhlsmcl 41696 | A closed subspace sum equa... |
| djhcvat42 41697 | A covering property. ( ~ ... |
| dihjatb 41698 | Isomorphism H of lattice j... |
| dihjatc 41699 | Isomorphism H of lattice j... |
| dihjatcclem1 41700 | Lemma for isomorphism H of... |
| dihjatcclem2 41701 | Lemma for isomorphism H of... |
| dihjatcclem3 41702 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 41703 | Lemma for isomorphism H of... |
| dihjatcc 41704 | Isomorphism H of lattice j... |
| dihjat 41705 | Isomorphism H of lattice j... |
| dihprrnlem1N 41706 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 41707 | Lemma for ~ dihprrn . (Co... |
| dihprrn 41708 | The span of a vector pair ... |
| djhlsmat 41709 | The sum of two subspace at... |
| dihjat1lem 41710 | Subspace sum of a closed s... |
| dihjat1 41711 | Subspace sum of a closed s... |
| dihsmsprn 41712 | Subspace sum of a closed s... |
| dihjat2 41713 | The subspace sum of a clos... |
| dihjat3 41714 | Isomorphism H of lattice j... |
| dihjat4 41715 | Transfer the subspace sum ... |
| dihjat6 41716 | Transfer the subspace sum ... |
| dihsmsnrn 41717 | The subspace sum of two si... |
| dihsmatrn 41718 | The subspace sum of a clos... |
| dihjat5N 41719 | Transfer lattice join with... |
| dvh4dimat 41720 | There is an atom that is o... |
| dvh3dimatN 41721 | There is an atom that is o... |
| dvh2dimatN 41722 | Given an atom, there exist... |
| dvh1dimat 41723 | There exists an atom. (Co... |
| dvh1dim 41724 | There exists a nonzero vec... |
| dvh4dimlem 41725 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 41726 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 41727 | There is a vector that is ... |
| dvh3dim 41728 | There is a vector that is ... |
| dvh4dimN 41729 | There is a vector that is ... |
| dvh3dim2 41730 | There is a vector that is ... |
| dvh3dim3N 41731 | There is a vector that is ... |
| dochsnnz 41732 | The orthocomplement of a s... |
| dochsatshp 41733 | The orthocomplement of a s... |
| dochsatshpb 41734 | The orthocomplement of a s... |
| dochsnshp 41735 | The orthocomplement of a n... |
| dochshpsat 41736 | A hyperplane is closed iff... |
| dochkrsat 41737 | The orthocomplement of a k... |
| dochkrsat2 41738 | The orthocomplement of a k... |
| dochsat0 41739 | The orthocomplement of a k... |
| dochkrsm 41740 | The subspace sum of a clos... |
| dochexmidat 41741 | Special case of excluded m... |
| dochexmidlem1 41742 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 41743 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 41744 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 41745 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 41746 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 41747 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 41748 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 41749 | Lemma for ~ dochexmid . T... |
| dochexmid 41750 | Excluded middle law for cl... |
| dochsnkrlem1 41751 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 41752 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 41753 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 41754 | A (closed) kernel expresse... |
| dochsnkr2 41755 | Kernel of the explicit fun... |
| dochsnkr2cl 41756 | The ` X ` determining func... |
| dochflcl 41757 | Closure of the explicit fu... |
| dochfl1 41758 | The value of the explicit ... |
| dochfln0 41759 | The value of a functional ... |
| dochkr1 41760 | A nonzero functional has a... |
| dochkr1OLDN 41761 | A nonzero functional has a... |
| lpolsetN 41764 | The set of polarities of a... |
| islpolN 41765 | The predicate "is a polari... |
| islpoldN 41766 | Properties that determine ... |
| lpolfN 41767 | Functionality of a polarit... |
| lpolvN 41768 | The polarity of the whole ... |
| lpolconN 41769 | Contraposition property of... |
| lpolsatN 41770 | The polarity of an atomic ... |
| lpolpolsatN 41771 | Property of a polarity. (... |
| dochpolN 41772 | The subspace orthocompleme... |
| lcfl1lem 41773 | Property of a functional w... |
| lcfl1 41774 | Property of a functional w... |
| lcfl2 41775 | Property of a functional w... |
| lcfl3 41776 | Property of a functional w... |
| lcfl4N 41777 | Property of a functional w... |
| lcfl5 41778 | Property of a functional w... |
| lcfl5a 41779 | Property of a functional w... |
| lcfl6lem 41780 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 41781 | Lemma for ~ lcfl7N . If t... |
| lcfl6 41782 | Property of a functional w... |
| lcfl7N 41783 | Property of a functional w... |
| lcfl8 41784 | Property of a functional w... |
| lcfl8a 41785 | Property of a functional w... |
| lcfl8b 41786 | Property of a nonzero func... |
| lcfl9a 41787 | Property implying that a f... |
| lclkrlem1 41788 | The set of functionals hav... |
| lclkrlem2a 41789 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 41790 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 41791 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 41792 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 41793 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 41794 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 41795 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 41796 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 41797 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 41798 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 41799 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 41800 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 41801 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 41802 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 41803 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 41804 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 41805 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 41806 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 41807 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 41808 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 41809 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 41810 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 41811 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 41812 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 41813 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 41814 | The set of functionals hav... |
| lclkr 41815 | The set of functionals wit... |
| lcfls1lem 41816 | Property of a functional w... |
| lcfls1N 41817 | Property of a functional w... |
| lcfls1c 41818 | Property of a functional w... |
| lclkrslem1 41819 | The set of functionals hav... |
| lclkrslem2 41820 | The set of functionals hav... |
| lclkrs 41821 | The set of functionals hav... |
| lclkrs2 41822 | The set of functionals wit... |
| lcfrvalsnN 41823 | Reconstruction from the du... |
| lcfrlem1 41824 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 41825 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 41826 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 41827 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 41828 | Lemma for ~ lcfr . The se... |
| lcfrlem6 41829 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 41830 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 41831 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 41832 | Lemma for ~ lcf1o . (This... |
| lcf1o 41833 | Define a function ` J ` th... |
| lcfrlem10 41834 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 41835 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 41836 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 41837 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 41838 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 41839 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 41840 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 41841 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 41842 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 41843 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 41844 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 41845 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 41846 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 41847 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 41848 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 41849 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 41850 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 41851 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 41852 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 41853 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 41854 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 41855 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 41856 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 41857 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 41858 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 41859 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 41860 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 41861 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 41862 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 41863 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 41864 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 41865 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 41866 | Lemma for ~ lcfr . Elimin... |
| lcfr 41867 | Reconstruction of a subspa... |
| lcdfval 41870 | Dual vector space of funct... |
| lcdval 41871 | Dual vector space of funct... |
| lcdval2 41872 | Dual vector space of funct... |
| lcdlvec 41873 | The dual vector space of f... |
| lcdlmod 41874 | The dual vector space of f... |
| lcdvbase 41875 | Vector base set of a dual ... |
| lcdvbasess 41876 | The vector base set of the... |
| lcdvbaselfl 41877 | A vector in the base set o... |
| lcdvbasecl 41878 | Closure of the value of a ... |
| lcdvadd 41879 | Vector addition for the cl... |
| lcdvaddval 41880 | The value of the value of ... |
| lcdsca 41881 | The ring of scalars of the... |
| lcdsbase 41882 | Base set of scalar ring fo... |
| lcdsadd 41883 | Scalar addition for the cl... |
| lcdsmul 41884 | Scalar multiplication for ... |
| lcdvs 41885 | Scalar product for the clo... |
| lcdvsval 41886 | Value of scalar product op... |
| lcdvscl 41887 | The scalar product operati... |
| lcdlssvscl 41888 | Closure of scalar product ... |
| lcdvsass 41889 | Associative law for scalar... |
| lcd0 41890 | The zero scalar of the clo... |
| lcd1 41891 | The unit scalar of the clo... |
| lcdneg 41892 | The unit scalar of the clo... |
| lcd0v 41893 | The zero functional in the... |
| lcd0v2 41894 | The zero functional in the... |
| lcd0vvalN 41895 | Value of the zero function... |
| lcd0vcl 41896 | Closure of the zero functi... |
| lcd0vs 41897 | A scalar zero times a func... |
| lcdvs0N 41898 | A scalar times the zero fu... |
| lcdvsub 41899 | The value of vector subtra... |
| lcdvsubval 41900 | The value of the value of ... |
| lcdlss 41901 | Subspaces of a dual vector... |
| lcdlss2N 41902 | Subspaces of a dual vector... |
| lcdlsp 41903 | Span in the set of functio... |
| lcdlkreqN 41904 | Colinear functionals have ... |
| lcdlkreq2N 41905 | Colinear functionals have ... |
| mapdffval 41908 | Projectivity from vector s... |
| mapdfval 41909 | Projectivity from vector s... |
| mapdval 41910 | Value of projectivity from... |
| mapdvalc 41911 | Value of projectivity from... |
| mapdval2N 41912 | Value of projectivity from... |
| mapdval3N 41913 | Value of projectivity from... |
| mapdval4N 41914 | Value of projectivity from... |
| mapdval5N 41915 | Value of projectivity from... |
| mapdordlem1a 41916 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 41917 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 41918 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 41919 | Lemma for ~ mapdord . Ord... |
| mapdord 41920 | Ordering property of the m... |
| mapd11 41921 | The map defined by ~ df-ma... |
| mapddlssN 41922 | The mapping of a subspace ... |
| mapdsn 41923 | Value of the map defined b... |
| mapdsn2 41924 | Value of the map defined b... |
| mapdsn3 41925 | Value of the map defined b... |
| mapd1dim2lem1N 41926 | Value of the map defined b... |
| mapdrvallem2 41927 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 41928 | Lemma for ~ mapdrval . (C... |
| mapdrval 41929 | Given a dual subspace ` R ... |
| mapd1o 41930 | The map defined by ~ df-ma... |
| mapdrn 41931 | Range of the map defined b... |
| mapdunirnN 41932 | Union of the range of the ... |
| mapdrn2 41933 | Range of the map defined b... |
| mapdcnvcl 41934 | Closure of the converse of... |
| mapdcl 41935 | Closure the value of the m... |
| mapdcnvid1N 41936 | Converse of the value of t... |
| mapdsord 41937 | Strong ordering property o... |
| mapdcl2 41938 | The mapping of a subspace ... |
| mapdcnvid2 41939 | Value of the converse of t... |
| mapdcnvordN 41940 | Ordering property of the c... |
| mapdcnv11N 41941 | The converse of the map de... |
| mapdcv 41942 | Covering property of the c... |
| mapdincl 41943 | Closure of dual subspace i... |
| mapdin 41944 | Subspace intersection is p... |
| mapdlsmcl 41945 | Closure of dual subspace s... |
| mapdlsm 41946 | Subspace sum is preserved ... |
| mapd0 41947 | Projectivity map of the ze... |
| mapdcnvatN 41948 | Atoms are preserved by the... |
| mapdat 41949 | Atoms are preserved by the... |
| mapdspex 41950 | The map of a span equals t... |
| mapdn0 41951 | Transfer nonzero property ... |
| mapdncol 41952 | Transfer non-colinearity f... |
| mapdindp 41953 | Transfer (part of) vector ... |
| mapdpglem1 41954 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 41955 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 41956 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 41957 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 41958 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 41959 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 41960 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 41961 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 41962 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 41963 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 41964 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 41965 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 41966 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 41967 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 41968 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 41969 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 41970 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 41971 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 41972 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 41973 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 41974 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 41975 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 41976 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 41977 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 41978 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 41979 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 41980 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 41981 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 41982 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 41983 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 41984 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 41985 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 41986 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 41987 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 41988 | Part 1 of proof of the fir... |
| baerlem3lem1 41989 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 41990 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 41991 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 41992 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 41993 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 41994 | Lemma for ~ baerlem5b . (... |
| baerlem3 41995 | An equality that holds whe... |
| baerlem5a 41996 | An equality that holds whe... |
| baerlem5b 41997 | An equality that holds whe... |
| baerlem5amN 41998 | An equality that holds whe... |
| baerlem5bmN 41999 | An equality that holds whe... |
| baerlem5abmN 42000 | An equality that holds whe... |
| mapdindp0 42001 | Vector independence lemma.... |
| mapdindp1 42002 | Vector independence lemma.... |
| mapdindp2 42003 | Vector independence lemma.... |
| mapdindp3 42004 | Vector independence lemma.... |
| mapdindp4 42005 | Vector independence lemma.... |
| mapdhval 42006 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 42007 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 42008 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 42009 | Lemmma for ~~? mapdh . (C... |
| mapdheq 42010 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 42011 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 42012 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 42013 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 42014 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 42015 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 42016 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 42017 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 42018 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 42019 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 42020 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 42021 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 42022 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 42023 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 42024 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 42025 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 42026 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 42027 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 42028 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 42029 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 42030 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 42031 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 42032 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 42033 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 42034 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 42035 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 42036 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 42037 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 42040 | Map from nonzero vectors t... |
| hvmapfval 42041 | Map from nonzero vectors t... |
| hvmapval 42042 | Value of map from nonzero ... |
| hvmapvalvalN 42043 | Value of value of map (i.e... |
| hvmapidN 42044 | The value of the vector to... |
| hvmap1o 42045 | The vector to functional m... |
| hvmapclN 42046 | Closure of the vector to f... |
| hvmap1o2 42047 | The vector to functional m... |
| hvmapcl2 42048 | Closure of the vector to f... |
| hvmaplfl 42049 | The vector to functional m... |
| hvmaplkr 42050 | Kernel of the vector to fu... |
| mapdhvmap 42051 | Relationship between ` map... |
| lspindp5 42052 | Obtain an independent vect... |
| hdmaplem1 42053 | Lemma to convert a frequen... |
| hdmaplem2N 42054 | Lemma to convert a frequen... |
| hdmaplem3 42055 | Lemma to convert a frequen... |
| hdmaplem4 42056 | Lemma to convert a frequen... |
| mapdh8a 42057 | Part of Part (8) in [Baer]... |
| mapdh8aa 42058 | Part of Part (8) in [Baer]... |
| mapdh8ab 42059 | Part of Part (8) in [Baer]... |
| mapdh8ac 42060 | Part of Part (8) in [Baer]... |
| mapdh8ad 42061 | Part of Part (8) in [Baer]... |
| mapdh8b 42062 | Part of Part (8) in [Baer]... |
| mapdh8c 42063 | Part of Part (8) in [Baer]... |
| mapdh8d0N 42064 | Part of Part (8) in [Baer]... |
| mapdh8d 42065 | Part of Part (8) in [Baer]... |
| mapdh8e 42066 | Part of Part (8) in [Baer]... |
| mapdh8g 42067 | Part of Part (8) in [Baer]... |
| mapdh8i 42068 | Part of Part (8) in [Baer]... |
| mapdh8j 42069 | Part of Part (8) in [Baer]... |
| mapdh8 42070 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 42071 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 42072 | Lemma for part (9) in [Bae... |
| hdmap1ffval 42077 | Preliminary map from vecto... |
| hdmap1fval 42078 | Preliminary map from vecto... |
| hdmap1vallem 42079 | Value of preliminary map f... |
| hdmap1val 42080 | Value of preliminary map f... |
| hdmap1val0 42081 | Value of preliminary map f... |
| hdmap1val2 42082 | Value of preliminary map f... |
| hdmap1eq 42083 | The defining equation for ... |
| hdmap1cbv 42084 | Frequently used lemma to c... |
| hdmap1valc 42085 | Connect the value of the p... |
| hdmap1cl 42086 | Convert closure theorem ~ ... |
| hdmap1eq2 42087 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 42088 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 42089 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 42090 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 42091 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 42092 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 42093 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 42094 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 42095 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 42096 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 42097 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 42098 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 42099 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 42100 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 42101 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 42102 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 42103 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 42104 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 42105 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 42106 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 42107 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 42108 | Map from vectors to functi... |
| hdmapfval 42109 | Map from vectors to functi... |
| hdmapval 42110 | Value of map from vectors ... |
| hdmapfnN 42111 | Functionality of map from ... |
| hdmapcl 42112 | Closure of map from vector... |
| hdmapval2lem 42113 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 42114 | Value of map from vectors ... |
| hdmapval0 42115 | Value of map from vectors ... |
| hdmapeveclem 42116 | Lemma for ~ hdmapevec . T... |
| hdmapevec 42117 | Value of map from vectors ... |
| hdmapevec2 42118 | The inner product of the r... |
| hdmapval3lemN 42119 | Value of map from vectors ... |
| hdmapval3N 42120 | Value of map from vectors ... |
| hdmap10lem 42121 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 42122 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 42123 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 42124 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 42125 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 42126 | Part of proof of part 12 i... |
| hdmapnzcl 42127 | Nonzero vector closure of ... |
| hdmapneg 42128 | Part of proof of part 12 i... |
| hdmapsub 42129 | Part of proof of part 12 i... |
| hdmap11 42130 | Part of proof of part 12 i... |
| hdmaprnlem1N 42131 | Part of proof of part 12 i... |
| hdmaprnlem3N 42132 | Part of proof of part 12 i... |
| hdmaprnlem3uN 42133 | Part of proof of part 12 i... |
| hdmaprnlem4tN 42134 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 42135 | Part of proof of part 12 i... |
| hdmaprnlem6N 42136 | Part of proof of part 12 i... |
| hdmaprnlem7N 42137 | Part of proof of part 12 i... |
| hdmaprnlem8N 42138 | Part of proof of part 12 i... |
| hdmaprnlem9N 42139 | Part of proof of part 12 i... |
| hdmaprnlem3eN 42140 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 42141 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 42142 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 42143 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 42144 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 42145 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 42146 | Part of proof of part 12 i... |
| hdmapf1oN 42147 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 42148 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 42149 | Prior to part 14 in [Baer]... |
| hdmap14lem1 42150 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 42151 | Prior to part 14 in [Baer]... |
| hdmap14lem3 42152 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 42153 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 42154 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 42155 | Case where ` F ` is zero. ... |
| hdmap14lem7 42156 | Combine cases of ` F ` . ... |
| hdmap14lem8 42157 | Part of proof of part 14 i... |
| hdmap14lem9 42158 | Part of proof of part 14 i... |
| hdmap14lem10 42159 | Part of proof of part 14 i... |
| hdmap14lem11 42160 | Part of proof of part 14 i... |
| hdmap14lem12 42161 | Lemma for proof of part 14... |
| hdmap14lem13 42162 | Lemma for proof of part 14... |
| hdmap14lem14 42163 | Part of proof of part 14 i... |
| hdmap14lem15 42164 | Part of proof of part 14 i... |
| hgmapffval 42167 | Map from the scalar divisi... |
| hgmapfval 42168 | Map from the scalar divisi... |
| hgmapval 42169 | Value of map from the scal... |
| hgmapfnN 42170 | Functionality of scalar si... |
| hgmapcl 42171 | Closure of scalar sigma ma... |
| hgmapdcl 42172 | Closure of the vector spac... |
| hgmapvs 42173 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 42174 | Value of the scalar sigma ... |
| hgmapval1 42175 | Value of the scalar sigma ... |
| hgmapadd 42176 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 42177 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 42178 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 42179 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 42180 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 42181 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 42182 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 42183 | Part of proof of part 16 i... |
| hgmap11 42184 | The scalar sigma map is on... |
| hgmapf1oN 42185 | The scalar sigma map is a ... |
| hgmapeq0 42186 | The scalar sigma map is ze... |
| hdmapipcl 42187 | The inner product (Hermiti... |
| hdmapln1 42188 | Linearity property that wi... |
| hdmaplna1 42189 | Additive property of first... |
| hdmaplns1 42190 | Subtraction property of fi... |
| hdmaplnm1 42191 | Multiplicative property of... |
| hdmaplna2 42192 | Additive property of secon... |
| hdmapglnm2 42193 | g-linear property of secon... |
| hdmapgln2 42194 | g-linear property that wil... |
| hdmaplkr 42195 | Kernel of the vector to du... |
| hdmapellkr 42196 | Membership in the kernel (... |
| hdmapip0 42197 | Zero property that will be... |
| hdmapip1 42198 | Construct a proportional v... |
| hdmapip0com 42199 | Commutation property of Ba... |
| hdmapinvlem1 42200 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 42201 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 42202 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 42203 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 42204 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 42205 | Involution property of sca... |
| hgmapvvlem2 42206 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 42207 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 42208 | Value of a double involuti... |
| hdmapglem7a 42209 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 42210 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 42211 | Lemma for ~ hdmapg . Line... |
| hdmapg 42212 | Apply the scalar sigma fun... |
| hdmapoc 42213 | Express our constructed or... |
| hlhilset 42216 | The final Hilbert space co... |
| hlhilsca 42217 | The scalar of the final co... |
| hlhilbase 42218 | The base set of the final ... |
| hlhilplus 42219 | The vector addition for th... |
| hlhilslem 42220 | Lemma for ~ hlhilsbase etc... |
| hlhilsbase 42221 | The scalar base set of the... |
| hlhilsplus 42222 | Scalar addition for the fi... |
| hlhilsmul 42223 | Scalar multiplication for ... |
| hlhilsbase2 42224 | The scalar base set of the... |
| hlhilsplus2 42225 | Scalar addition for the fi... |
| hlhilsmul2 42226 | Scalar multiplication for ... |
| hlhils0 42227 | The scalar ring zero for t... |
| hlhils1N 42228 | The scalar ring unity for ... |
| hlhilvsca 42229 | The scalar product for the... |
| hlhilip 42230 | Inner product operation fo... |
| hlhilipval 42231 | Value of inner product ope... |
| hlhilnvl 42232 | The involution operation o... |
| hlhillvec 42233 | The final constructed Hilb... |
| hlhildrng 42234 | The star division ring for... |
| hlhilsrnglem 42235 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 42236 | The star division ring for... |
| hlhil0 42237 | The zero vector for the fi... |
| hlhillsm 42238 | The vector sum operation f... |
| hlhilocv 42239 | The orthocomplement for th... |
| hlhillcs 42240 | The closed subspaces of th... |
| hlhilphllem 42241 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 42242 | Lemma for ~ hlhil . (Cont... |
| hlathil 42243 | Construction of a Hilbert ... |
| iscsrg 42246 | A commutative semiring is ... |
| rhmzrhval 42247 | Evaluation of integers acr... |
| zndvdchrrhm 42248 | Construction of a ring hom... |
| relogbcld 42249 | Closure of the general log... |
| relogbexpd 42250 | Identity law for general l... |
| relogbzexpd 42251 | Power law for the general ... |
| logblebd 42252 | The general logarithm is m... |
| uzindd 42253 | Induction on the upper int... |
| fzadd2d 42254 | Membership of a sum in a f... |
| zltp1led 42255 | Integer ordering relation,... |
| fzne2d 42256 | Elementhood in a finite se... |
| eqfnfv2d2 42257 | Equality of functions is d... |
| fzsplitnd 42258 | Split a finite interval of... |
| fzsplitnr 42259 | Split a finite interval of... |
| addassnni 42260 | Associative law for additi... |
| addcomnni 42261 | Commutative law for additi... |
| mulassnni 42262 | Associative law for multip... |
| mulcomnni 42263 | Commutative law for multip... |
| gcdcomnni 42264 | Commutative law for gcd. ... |
| gcdnegnni 42265 | Negation invariance for gc... |
| neggcdnni 42266 | Negation invariance for gc... |
| bccl2d 42267 | Closure of the binomial co... |
| recbothd 42268 | Take reciprocal on both si... |
| gcdmultiplei 42269 | The GCD of a multiple of a... |
| gcdaddmzz2nni 42270 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 42271 | Adding a multiple of one o... |
| gcdnncli 42272 | Closure of the gcd operato... |
| muldvds1d 42273 | If a product divides an in... |
| muldvds2d 42274 | If a product divides an in... |
| nndivdvdsd 42275 | A positive integer divides... |
| nnproddivdvdsd 42276 | A product of natural numbe... |
| coprmdvds2d 42277 | If an integer is divisible... |
| imadomfi 42278 | An image of a function und... |
| 12gcd5e1 42279 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 42280 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 42281 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 42282 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 42283 | Calculate the least common... |
| 12lcm5e60 42284 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 42285 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 42286 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 42287 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 42288 | Useful equation to calcula... |
| lcm1un 42289 | Least common multiple of n... |
| lcm2un 42290 | Least common multiple of n... |
| lcm3un 42291 | Least common multiple of n... |
| lcm4un 42292 | Least common multiple of n... |
| lcm5un 42293 | Least common multiple of n... |
| lcm6un 42294 | Least common multiple of n... |
| lcm7un 42295 | Least common multiple of n... |
| lcm8un 42296 | Least common multiple of n... |
| 3factsumint1 42297 | Move constants out of inte... |
| 3factsumint2 42298 | Move constants out of inte... |
| 3factsumint3 42299 | Move constants out of inte... |
| 3factsumint4 42300 | Move constants out of inte... |
| 3factsumint 42301 | Helpful equation for lcm i... |
| resopunitintvd 42302 | Restrict continuous functi... |
| resclunitintvd 42303 | Restrict continuous functi... |
| resdvopclptsd 42304 | Restrict derivative on uni... |
| lcmineqlem1 42305 | Part of lcm inequality lem... |
| lcmineqlem2 42306 | Part of lcm inequality lem... |
| lcmineqlem3 42307 | Part of lcm inequality lem... |
| lcmineqlem4 42308 | Part of lcm inequality lem... |
| lcmineqlem5 42309 | Technical lemma for recipr... |
| lcmineqlem6 42310 | Part of lcm inequality lem... |
| lcmineqlem7 42311 | Derivative of 1-x for chai... |
| lcmineqlem8 42312 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 42313 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 42314 | Induction step of ~ lcmine... |
| lcmineqlem11 42315 | Induction step, continuati... |
| lcmineqlem12 42316 | Base case for induction. ... |
| lcmineqlem13 42317 | Induction proof for lcm in... |
| lcmineqlem14 42318 | Technical lemma for inequa... |
| lcmineqlem15 42319 | F times the least common m... |
| lcmineqlem16 42320 | Technical divisibility lem... |
| lcmineqlem17 42321 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 42322 | Technical lemma to shift f... |
| lcmineqlem19 42323 | Dividing implies inequalit... |
| lcmineqlem20 42324 | Inequality for lcm lemma. ... |
| lcmineqlem21 42325 | The lcm inequality lemma w... |
| lcmineqlem22 42326 | The lcm inequality lemma w... |
| lcmineqlem23 42327 | Penultimate step to the lc... |
| lcmineqlem 42328 | The least common multiple ... |
| 3exp7 42329 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 42330 | First inequality in inequa... |
| 3lexlogpow5ineq2 42331 | Second inequality in inequ... |
| 3lexlogpow5ineq4 42332 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 42333 | Combined inequality chain ... |
| 3lexlogpow2ineq1 42334 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 42335 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 42336 | Result for bound in AKS in... |
| intlewftc 42337 | Inequality inference by in... |
| aks4d1lem1 42338 | Technical lemma to reduce ... |
| aks4d1p1p1 42339 | Exponential law for finite... |
| dvrelog2 42340 | The derivative of the loga... |
| dvrelog3 42341 | The derivative of the loga... |
| dvrelog2b 42342 | Derivative of the binary l... |
| 0nonelalab 42343 | Technical lemma for open i... |
| dvrelogpow2b 42344 | Derivative of the power of... |
| aks4d1p1p3 42345 | Bound of a ceiling of the ... |
| aks4d1p1p2 42346 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 42347 | Technical step for inequal... |
| dvle2 42348 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 42349 | Inequality lift to differe... |
| aks4d1p1p7 42350 | Bound of intermediary of i... |
| aks4d1p1p5 42351 | Show inequality for existe... |
| aks4d1p1 42352 | Show inequality for existe... |
| aks4d1p2 42353 | Technical lemma for existe... |
| aks4d1p3 42354 | There exists a small enoug... |
| aks4d1p4 42355 | There exists a small enoug... |
| aks4d1p5 42356 | Show that ` N ` and ` R ` ... |
| aks4d1p6 42357 | The maximal prime power ex... |
| aks4d1p7d1 42358 | Technical step in AKS lemm... |
| aks4d1p7 42359 | Technical step in AKS lemm... |
| aks4d1p8d1 42360 | If a prime divides one num... |
| aks4d1p8d2 42361 | Any prime power dividing a... |
| aks4d1p8d3 42362 | The remainder of a divisio... |
| aks4d1p8 42363 | Show that ` N ` and ` R ` ... |
| aks4d1p9 42364 | Show that the order is bou... |
| aks4d1 42365 | Lemma 4.1 from ~ https://w... |
| fldhmf1 42366 | A field homomorphism is in... |
| isprimroot 42369 | The value of a primitive r... |
| isprimroot2 42370 | Alternative way of creatin... |
| mndmolinv 42371 | An element of a monoid tha... |
| linvh 42372 | If an element has a unique... |
| primrootsunit1 42373 | Primitive roots have left ... |
| primrootsunit 42374 | Primitive roots have left ... |
| primrootscoprmpow 42375 | Coprime powers of primitiv... |
| posbezout 42376 | Bezout's identity restrict... |
| primrootscoprf 42377 | Coprime powers of primitiv... |
| primrootscoprbij 42378 | A bijection between coprim... |
| primrootscoprbij2 42379 | A bijection between coprim... |
| remexz 42380 | Division with rest. (Cont... |
| primrootlekpowne0 42381 | There is no smaller power ... |
| primrootspoweq0 42382 | The power of a ` R ` -th p... |
| aks6d1c1p1 42383 | Definition of the introspe... |
| aks6d1c1p1rcl 42384 | Reverse closure of the int... |
| aks6d1c1p2 42385 | ` P ` and linear factors a... |
| aks6d1c1p3 42386 | In a field with a Frobeniu... |
| aks6d1c1p4 42387 | The product of polynomials... |
| aks6d1c1p5 42388 | The product of exponents i... |
| aks6d1c1p7 42389 | ` X ` is introspective to ... |
| aks6d1c1p6 42390 | If a polynomials ` F ` is ... |
| aks6d1c1p8 42391 | If a number ` E ` is intro... |
| aks6d1c1 42392 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 42393 | Polynomial evaluation buil... |
| aks6d1c2p1 42394 | In the AKS-theorem the sub... |
| aks6d1c2p2 42395 | Injective condition for co... |
| hashscontpowcl 42396 | Closure of E for ~ https:/... |
| hashscontpow1 42397 | Helper lemma for to prove ... |
| hashscontpow 42398 | If a set contains all ` N ... |
| aks6d1c3 42399 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 42400 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 42401 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 42402 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 42403 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 42404 | If the number of elements ... |
| hashnexinjle 42405 | If the number of elements ... |
| aks6d1c2 42406 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 42407 | Special case related to ~ ... |
| idomnnzpownz 42408 | A non-zero power in an int... |
| idomnnzgmulnz 42409 | A finite product of non-ze... |
| ringexp0nn 42410 | Zero to the power of a pos... |
| aks6d1c5lem0 42411 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 42412 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 42413 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 42414 | Lemma for Claim 5, contrad... |
| aks6d1c5 42415 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 42416 | Degree multiplication is a... |
| deg1pow 42417 | Exact degree of a power of... |
| 5bc2eq10 42418 | The value of 5 choose 2. ... |
| facp2 42419 | The factorial of a success... |
| 2np3bcnp1 42420 | Part of induction step for... |
| 2ap1caineq 42421 | Inequality for Theorem 6.6... |
| sticksstones1 42422 | Different strictly monoton... |
| sticksstones2 42423 | The range function on stri... |
| sticksstones3 42424 | The range function on stri... |
| sticksstones4 42425 | Equinumerosity lemma for s... |
| sticksstones5 42426 | Count the number of strict... |
| sticksstones6 42427 | Function induces an order ... |
| sticksstones7 42428 | Closure property of sticks... |
| sticksstones8 42429 | Establish mapping between ... |
| sticksstones9 42430 | Establish mapping between ... |
| sticksstones10 42431 | Establish mapping between ... |
| sticksstones11 42432 | Establish bijective mappin... |
| sticksstones12a 42433 | Establish bijective mappin... |
| sticksstones12 42434 | Establish bijective mappin... |
| sticksstones13 42435 | Establish bijective mappin... |
| sticksstones14 42436 | Sticks and stones with def... |
| sticksstones15 42437 | Sticks and stones with alm... |
| sticksstones16 42438 | Sticks and stones with col... |
| sticksstones17 42439 | Extend sticks and stones t... |
| sticksstones18 42440 | Extend sticks and stones t... |
| sticksstones19 42441 | Extend sticks and stones t... |
| sticksstones20 42442 | Lift sticks and stones to ... |
| sticksstones21 42443 | Lift sticks and stones to ... |
| sticksstones22 42444 | Non-exhaustive sticks and ... |
| sticksstones23 42445 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 42446 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 42447 | Every primitive root is ro... |
| aks6d1c6lem3 42448 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 42449 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 42450 | Lemma to construct the map... |
| aks6d1c6isolem2 42451 | Lemma to construct the gro... |
| aks6d1c6isolem3 42452 | The preimage of a map send... |
| aks6d1c6lem5 42453 | Eliminate the size hypothe... |
| bcled 42454 | Inequality for binomial co... |
| bcle2d 42455 | Inequality for binomial co... |
| aks6d1c7lem1 42456 | The last set of inequaliti... |
| aks6d1c7lem2 42457 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 42458 | Remove lots of hypotheses ... |
| aks6d1c7lem4 42459 | In the AKS algorithm there... |
| aks6d1c7 42460 | ` N ` is a prime power if ... |
| rhmqusspan 42461 | Ring homomorphism out of a... |
| aks5lem1 42462 | Section 5 of ~ https://www... |
| aks5lem2 42463 | Lemma for section 5 ~ http... |
| ply1asclzrhval 42464 | Transfer results from alge... |
| aks5lem3a 42465 | Lemma for AKS section 5. ... |
| aks5lem4a 42466 | Lemma for AKS section 5, r... |
| aks5lem5a 42467 | Lemma for AKS, section 5, ... |
| aks5lem6 42468 | Connect results of section... |
| indstrd 42469 | Strong induction, deductio... |
| grpods 42470 | Relate sums of elements of... |
| unitscyglem1 42471 | Lemma for unitscyg. (Cont... |
| unitscyglem2 42472 | Lemma for unitscyg. (Cont... |
| unitscyglem3 42473 | Lemma for unitscyg. (Cont... |
| unitscyglem4 42474 | Lemma for unitscyg (Contri... |
| unitscyglem5 42475 | Lemma for unitscyg (Contri... |
| aks5lem7 42476 | Lemma for aks5. We clean ... |
| aks5lem8 42477 | Lemma for aks5. Clean up ... |
| exfinfldd 42479 | For any prime ` P ` and an... |
| aks5 42480 | The AKS Primality test, gi... |
| jarrii 42481 | Inference associated with ... |
| intnanrt 42482 | Introduction of conjunct i... |
| ioin9i8 42483 | Miscellaneous inference cr... |
| jaodd 42484 | Double deduction form of ~... |
| syl3an12 42485 | A double syllogism inferen... |
| exbiii 42486 | Inference associated with ... |
| sbtd 42487 | A true statement is true u... |
| sbor2 42488 | One direction of ~ sbor , ... |
| sbalexi 42489 | Inference form of ~ sbalex... |
| nfalh 42490 | Version of ~ nfal with an ... |
| nfe2 42491 | An inner existential quant... |
| nfale2 42492 | An inner existential quant... |
| 19.9dev 42493 | ~ 19.9d in the case of an ... |
| 3rspcedvd 42494 | Triple application of ~ rs... |
| sn-axrep5v 42495 | A condensed form of ~ axre... |
| sn-axprlem3 42496 | ~ axprlem3 using only Tars... |
| sn-exelALT 42497 | Alternate proof of ~ exel ... |
| ssabdv 42498 | Deduction of abstraction s... |
| sn-iotalem 42499 | An unused lemma showing th... |
| sn-iotalemcor 42500 | Corollary of ~ sn-iotalem ... |
| abbi1sn 42501 | Originally part of ~ uniab... |
| brif2 42502 | Move a relation inside and... |
| brif12 42503 | Move a relation inside and... |
| pssexg 42504 | The proper subset of a set... |
| pssn0 42505 | A proper superset is nonem... |
| psspwb 42506 | Classes are proper subclas... |
| xppss12 42507 | Proper subset theorem for ... |
| elpwbi 42508 | Membership in a power set,... |
| imaopab 42509 | The image of a class of or... |
| eqresfnbd 42510 | Property of being the rest... |
| f1o2d2 42511 | Sufficient condition for a... |
| fmpocos 42512 | Composition of two functio... |
| ovmpogad 42513 | Value of an operation give... |
| ofun 42514 | A function operation of un... |
| dfqs3 42515 | Alternate definition of qu... |
| qseq12d 42516 | Equality theorem for quoti... |
| qsalrel 42517 | The quotient set is equal ... |
| supinf 42518 | The supremum is the infimu... |
| mapcod 42519 | Compose two mappings. (Co... |
| fisdomnn 42520 | A finite set is dominated ... |
| ltex 42521 | The less-than relation is ... |
| leex 42522 | The less-than-or-equal-to ... |
| subex 42523 | The subtraction operation ... |
| absex 42524 | The absolute value functio... |
| cjex 42525 | The conjugate function is ... |
| fzosumm1 42526 | Separate out the last term... |
| ccatcan2d 42527 | Cancellation law for conca... |
| c0exALT 42528 | Alternate proof of ~ c0ex ... |
| 0cnALT3 42529 | Alternate proof of ~ 0cn u... |
| elre0re 42530 | Specialized version of ~ 0... |
| 1t1e1ALT 42531 | Alternate proof of ~ 1t1e1... |
| lttrii 42532 | 'Less than' is transitive.... |
| remulcan2d 42533 | ~ mulcan2d for real number... |
| readdridaddlidd 42534 | Given some real number ` B... |
| 1p3e4 42535 | 1 + 3 = 4. (Contributed b... |
| 5ne0 42536 | The number 5 is nonzero. ... |
| 6ne0 42537 | The number 6 is nonzero. ... |
| 7ne0 42538 | The number 7 is nonzero. ... |
| 8ne0 42539 | The number 8 is nonzero. ... |
| 9ne0 42540 | The number 9 is nonzero. ... |
| sn-1ne2 42541 | A proof of ~ 1ne2 without ... |
| nnn1suc 42542 | A positive integer that is... |
| nnadd1com 42543 | Addition with 1 is commuta... |
| nnaddcom 42544 | Addition is commutative fo... |
| nnaddcomli 42545 | Version of ~ addcomli for ... |
| nnadddir 42546 | Right-distributivity for n... |
| nnmul1com 42547 | Multiplication with 1 is c... |
| nnmulcom 42548 | Multiplication is commutat... |
| readdrcl2d 42549 | Reverse closure for additi... |
| mvrrsubd 42550 | Move a subtraction in the ... |
| laddrotrd 42551 | Rotate the variables right... |
| raddswap12d 42552 | Swap the first two variabl... |
| lsubrotld 42553 | Rotate the variables left ... |
| rsubrotld 42554 | Rotate the variables left ... |
| lsubswap23d 42555 | Swap the second and third ... |
| addsubeq4com 42556 | Relation between sums and ... |
| sqsumi 42557 | A sum squared. (Contribut... |
| negn0nposznnd 42558 | Lemma for ~ dffltz . (Con... |
| sqmid3api 42559 | Value of the square of the... |
| decaddcom 42560 | Commute ones place in addi... |
| sqn5i 42561 | The square of a number end... |
| sqn5ii 42562 | The square of a number end... |
| decpmulnc 42563 | Partial products algorithm... |
| decpmul 42564 | Partial products algorithm... |
| sqdeccom12 42565 | The square of a number in ... |
| sq3deccom12 42566 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 42567 | 4 times 5 equals 20. (Con... |
| 3rdpwhole 42568 | A third of a number plus t... |
| sq4 42569 | The square of 4 is 16. (C... |
| sq5 42570 | The square of 5 is 25. (C... |
| sq6 42571 | The square of 6 is 36. (C... |
| sq7 42572 | The square of 7 is 49. (C... |
| sq8 42573 | The square of 8 is 64. (C... |
| sq9 42574 | The square of 9 is 81. (C... |
| rpsscn 42575 | The positive reals are a s... |
| 4rp 42576 | 4 is a positive real. (Co... |
| 6rp 42577 | 6 is a positive real. (Co... |
| 7rp 42578 | 7 is a positive real. (Co... |
| 8rp 42579 | 8 is a positive real. (Co... |
| 9rp 42580 | 9 is a positive real. (Co... |
| 235t711 42581 | Calculate a product by lon... |
| ex-decpmul 42582 | Example usage of ~ decpmul... |
| eluzp1 42583 | Membership in a successor ... |
| sn-eluzp1l 42584 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 42585 | The sum of ` N ` constant ... |
| fz1sump1 42586 | Add one more term to a sum... |
| oddnumth 42587 | The Odd Number Theorem. T... |
| nicomachus 42588 | Nicomachus's Theorem. The... |
| sumcubes 42589 | The sum of the first ` N `... |
| ine1 42590 | ` _i ` is not 1. (Contrib... |
| 0tie0 42591 | 0 times ` _i ` equals 0. ... |
| it1ei 42592 | ` _i ` times 1 equals ` _i... |
| 1tiei 42593 | 1 times ` _i ` equals ` _i... |
| itrere 42594 | ` _i ` times a real is rea... |
| retire 42595 | A real times ` _i ` is rea... |
| iocioodisjd 42596 | Adjacent intervals where t... |
| rpabsid 42597 | A positive real is its own... |
| oexpreposd 42598 | Lemma for ~ dffltz . For ... |
| explt1d 42599 | A nonnegative real number ... |
| expeq1d 42600 | A nonnegative real number ... |
| expeqidd 42601 | A nonnegative real number ... |
| exp11d 42602 | ~ exp11nnd for nonzero int... |
| 0dvds0 42603 | 0 divides 0. (Contributed... |
| absdvdsabsb 42604 | Divisibility is invariant ... |
| gcdnn0id 42605 | The ` gcd ` of a nonnegati... |
| gcdle1d 42606 | The greatest common diviso... |
| gcdle2d 42607 | The greatest common diviso... |
| dvdsexpad 42608 | Deduction associated with ... |
| dvdsexpnn 42609 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 42610 | ~ dvdsexpnn generalized to... |
| dvdsexpb 42611 | ~ dvdssq generalized to po... |
| posqsqznn 42612 | When a positive rational s... |
| zdivgd 42613 | Two ways to express " ` N ... |
| efsubd 42614 | Difference of exponents la... |
| ef11d 42615 | General condition for the ... |
| logccne0d 42616 | The logarithm isn't 0 if i... |
| cxp112d 42617 | General condition for comp... |
| cxp111d 42618 | General condition for comp... |
| cxpi11d 42619 | ` _i ` to the powers of ` ... |
| logne0d 42620 | Deduction form of ~ logne0... |
| rxp112d 42621 | Real exponentiation is one... |
| log11d 42622 | The natural logarithm is o... |
| rplog11d 42623 | The natural logarithm is o... |
| rxp11d 42624 | Real exponentiation is one... |
| tanhalfpim 42625 | The tangent of ` _pi / 2 `... |
| sinpim 42626 | Sine of a number subtracte... |
| cospim 42627 | Cosine of a number subtrac... |
| tan3rdpi 42628 | The tangent of ` _pi / 3 `... |
| sin2t3rdpi 42629 | The sine of ` 2 x. ( _pi /... |
| cos2t3rdpi 42630 | The cosine of ` 2 x. ( _pi... |
| sin4t3rdpi 42631 | The sine of ` 4 x. ( _pi /... |
| cos4t3rdpi 42632 | The cosine of ` 4 x. ( _pi... |
| asin1half 42633 | The arcsine of ` 1 / 2 ` i... |
| acos1half 42634 | The arccosine of ` 1 / 2 `... |
| dvun 42635 | Condition for the union of... |
| redvmptabs 42636 | The derivative of the abso... |
| readvrec2 42637 | The antiderivative of 1/x ... |
| readvrec 42638 | For real numbers, the anti... |
| resuppsinopn 42639 | The support of sin ( ~ df-... |
| readvcot 42640 | Real antiderivative of cot... |
| resubval 42643 | Value of real subtraction,... |
| renegeulemv 42644 | Lemma for ~ renegeu and si... |
| renegeulem 42645 | Lemma for ~ renegeu and si... |
| renegeu 42646 | Existential uniqueness of ... |
| rernegcl 42647 | Closure law for negative r... |
| renegadd 42648 | Relationship between real ... |
| renegid 42649 | Addition of a real number ... |
| reneg0addlid 42650 | Negative zero is a left ad... |
| resubeulem1 42651 | Lemma for ~ resubeu . A v... |
| resubeulem2 42652 | Lemma for ~ resubeu . A v... |
| resubeu 42653 | Existential uniqueness of ... |
| rersubcl 42654 | Closure for real subtracti... |
| resubadd 42655 | Relation between real subt... |
| resubaddd 42656 | Relationship between subtr... |
| resubf 42657 | Real subtraction is an ope... |
| repncan2 42658 | Addition and subtraction o... |
| repncan3 42659 | Addition and subtraction o... |
| readdsub 42660 | Law for addition and subtr... |
| reladdrsub 42661 | Move LHS of a sum into RHS... |
| reltsub1 42662 | Subtraction from both side... |
| reltsubadd2 42663 | 'Less than' relationship b... |
| resubcan2 42664 | Cancellation law for real ... |
| resubsub4 42665 | Law for double subtraction... |
| rennncan2 42666 | Cancellation law for real ... |
| renpncan3 42667 | Cancellation law for real ... |
| repnpcan 42668 | Cancellation law for addit... |
| reppncan 42669 | Cancellation law for mixed... |
| resubidaddlidlem 42670 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 42671 | Any real number subtracted... |
| resubdi 42672 | Distribution of multiplica... |
| re1m1e0m0 42673 | Equality of two left-addit... |
| sn-00idlem1 42674 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 42675 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 42676 | Lemma for ~ sn-00id . (Co... |
| sn-00id 42677 | ~ 00id proven without ~ ax... |
| re0m0e0 42678 | Real number version of ~ 0... |
| readdlid 42679 | Real number version of ~ a... |
| sn-addlid 42680 | ~ addlid without ~ ax-mulc... |
| remul02 42681 | Real number version of ~ m... |
| sn-0ne2 42682 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 42683 | Real number version of ~ m... |
| sn-remul0ord 42684 | A product is zero iff one ... |
| resubid 42685 | Subtraction of a real numb... |
| readdrid 42686 | Real number version of ~ a... |
| resubid1 42687 | Real number version of ~ s... |
| renegneg 42688 | A real number is equal to ... |
| readdcan2 42689 | Commuted version of ~ read... |
| renegid2 42690 | Commuted version of ~ rene... |
| remulneg2d 42691 | Product with negative is n... |
| sn-it0e0 42692 | Proof of ~ it0e0 without ~... |
| sn-negex12 42693 | A combination of ~ cnegex ... |
| sn-negex 42694 | Proof of ~ cnegex without ... |
| sn-negex2 42695 | Proof of ~ cnegex2 without... |
| sn-addcand 42696 | ~ addcand without ~ ax-mul... |
| sn-addrid 42697 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 42698 | ~ addcan2d without ~ ax-mu... |
| reixi 42699 | ~ ixi without ~ ax-mulcom ... |
| rei4 42700 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 42701 | A number that sums to itse... |
| sn-mul01 42702 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 42703 | ~ negeu without ~ ax-mulco... |
| sn-subcl 42704 | ~ subcl without ~ ax-mulco... |
| sn-subf 42705 | ~ subf without ~ ax-mulcom... |
| resubeqsub 42706 | Equivalence between real s... |
| subresre 42707 | Subtraction restricted to ... |
| addinvcom 42708 | A number commutes with its... |
| remulinvcom 42709 | A left multiplicative inve... |
| remullid 42710 | Commuted version of ~ ax-1... |
| sn-1ticom 42711 | Lemma for ~ sn-mullid and ... |
| sn-mullid 42712 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 42713 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 42714 | The multiplicative inverse... |
| remulcand 42715 | Commuted version of ~ remu... |
| redivvald 42718 | Value of real division, wh... |
| rediveud 42719 | Existential uniqueness of ... |
| sn-redivcld 42720 | Closure law for real divis... |
| redivmuld 42721 | Relationship between divis... |
| redivcan2d 42722 | A cancellation law for div... |
| redivcan3d 42723 | A cancellation law for div... |
| sn-rereccld 42724 | Closure law for reciprocal... |
| rerecid 42725 | Multiplication of a number... |
| rerecid2 42726 | Multiplication of a number... |
| sn-0tie0 42727 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 42728 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 42729 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 42730 | ~ ltaddneg without ~ ax-mu... |
| reposdif 42731 | Comparison of two numbers ... |
| relt0neg1 42732 | Comparison of a real and i... |
| relt0neg2 42733 | Comparison of a real and i... |
| sn-addlt0d 42734 | The sum of negative number... |
| sn-addgt0d 42735 | The sum of positive number... |
| sn-nnne0 42736 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 42737 | ~ elznn0nn restated using ... |
| nn0addcom 42738 | Addition is commutative fo... |
| zaddcomlem 42739 | Lemma for ~ zaddcom . (Co... |
| zaddcom 42740 | Addition is commutative fo... |
| renegmulnnass 42741 | Move multiplication by a n... |
| nn0mulcom 42742 | Multiplication is commutat... |
| zmulcomlem 42743 | Lemma for ~ zmulcom . (Co... |
| zmulcom 42744 | Multiplication is commutat... |
| mulgt0con1dlem 42745 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 42746 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 42747 | Lemma for ~ mulgt0b1d and ... |
| mulgt0b1d 42748 | Biconditional, deductive f... |
| sn-ltmul2d 42749 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 42750 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 42751 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 42752 | ~ ltp1 without ~ ax-mulcom... |
| sn-recgt0d 42753 | The reciprocal of a positi... |
| mulgt0b2d 42754 | Biconditional, deductive f... |
| sn-mulgt1d 42755 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 42756 | Negative one is a negative... |
| sn-reclt0d 42757 | The reciprocal of a negati... |
| mulltgt0d 42758 | Negative times positive is... |
| mullt0b1d 42759 | When the first term is neg... |
| mullt0b2d 42760 | When the second term is ne... |
| sn-mullt0d 42761 | The product of two negativ... |
| sn-msqgt0d 42762 | A nonzero square is positi... |
| sn-inelr 42763 | ~ inelr without ~ ax-mulco... |
| sn-itrere 42764 | ` _i ` times a real is rea... |
| sn-retire 42765 | Commuted version of ~ sn-i... |
| cnreeu 42766 | The reals in the expressio... |
| sn-sup2 42767 | ~ sup2 with exactly the sa... |
| sn-sup3d 42768 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 42769 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 42770 | ~ suprubd without ~ ax-mul... |
| sn-base0 42771 | Avoid axioms in ~ base0 by... |
| nelsubginvcld 42772 | The inverse of a non-subgr... |
| nelsubgcld 42773 | A non-subgroup-member plus... |
| nelsubgsubcld 42774 | A non-subgroup-member minu... |
| rnasclg 42775 | The set of injected scalar... |
| frlmfielbas 42776 | The vectors of a finite fr... |
| frlmfzwrd 42777 | A vector of a module with ... |
| frlmfzowrd 42778 | A vector of a module with ... |
| frlmfzolen 42779 | The dimension of a vector ... |
| frlmfzowrdb 42780 | The vectors of a module wi... |
| frlmfzoccat 42781 | The concatenation of two v... |
| frlmvscadiccat 42782 | Scalar multiplication dist... |
| grpasscan2d 42783 | An associative cancellatio... |
| grpcominv1 42784 | If two elements commute, t... |
| grpcominv2 42785 | If two elements commute, t... |
| finsubmsubg 42786 | A submonoid of a finite gr... |
| opprmndb 42787 | A class is a monoid if and... |
| opprgrpb 42788 | A class is a group if and ... |
| opprablb 42789 | A class is an Abelian grou... |
| imacrhmcl 42790 | The image of a commutative... |
| rimrcl1 42791 | Reverse closure of a ring ... |
| rimrcl2 42792 | Reverse closure of a ring ... |
| rimcnv 42793 | The converse of a ring iso... |
| rimco 42794 | The composition of ring is... |
| ricsym 42795 | Ring isomorphism is symmet... |
| rictr 42796 | Ring isomorphism is transi... |
| riccrng1 42797 | Ring isomorphism preserves... |
| riccrng 42798 | A ring is commutative if a... |
| domnexpgn0cl 42799 | In a domain, a (nonnegativ... |
| drnginvrn0d 42800 | A multiplicative inverse i... |
| drngmullcan 42801 | Cancellation of a nonzero ... |
| drngmulrcan 42802 | Cancellation of a nonzero ... |
| drnginvmuld 42803 | Inverse of a nonzero produ... |
| ricdrng1 42804 | A ring isomorphism maps a ... |
| ricdrng 42805 | A ring is a division ring ... |
| ricfld 42806 | A ring is a field if and o... |
| asclf1 42807 | Two ways of saying the sca... |
| abvexp 42808 | Move exponentiation in and... |
| fimgmcyclem 42809 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 42810 | Version of ~ odcl2 for fin... |
| fidomncyc 42811 | Version of ~ odcl2 for mul... |
| fiabv 42812 | In a finite domain (a fini... |
| lvecgrp 42813 | A vector space is a group.... |
| lvecring 42814 | The scalar component of a ... |
| frlm0vald 42815 | All coordinates of the zer... |
| frlmsnic 42816 | Given a free module with a... |
| uvccl 42817 | A unit vector is a vector.... |
| uvcn0 42818 | A unit vector is nonzero. ... |
| psrmnd 42819 | The ring of power series i... |
| psrbagres 42820 | Restrict a bag of variable... |
| mplcrngd 42821 | The polynomial ring is a c... |
| mplsubrgcl 42822 | An element of a polynomial... |
| mhmcopsr 42823 | The composition of a monoi... |
| mhmcoaddpsr 42824 | Show that the ring homomor... |
| rhmcomulpsr 42825 | Show that the ring homomor... |
| rhmpsr 42826 | Provide a ring homomorphis... |
| rhmpsr1 42827 | Provide a ring homomorphis... |
| mplmapghm 42828 | The function ` H ` mapping... |
| evl0 42829 | The zero polynomial evalua... |
| evlscl 42830 | A polynomial over the ring... |
| evlsscaval 42831 | Polynomial evaluation buil... |
| evlsvarval 42832 | Polynomial evaluation buil... |
| evlsbagval 42833 | Polynomial evaluation buil... |
| evlsexpval 42834 | Polynomial evaluation buil... |
| evlsaddval 42835 | Polynomial evaluation buil... |
| evlsmulval 42836 | Polynomial evaluation buil... |
| evlsmaprhm 42837 | The function ` F ` mapping... |
| evlsevl 42838 | Evaluation in a subring is... |
| evlvvval 42839 | Give a formula for the eva... |
| evlvvvallem 42840 | Lemma for theorems using ~... |
| selvcllem1 42841 | ` T ` is an associative al... |
| selvcllem2 42842 | ` D ` is a ring homomorphi... |
| selvcllem3 42843 | The third argument passed ... |
| selvcllemh 42844 | Apply the third argument (... |
| selvcllem4 42845 | The fourth argument passed... |
| selvcllem5 42846 | The fifth argument passed ... |
| selvcl 42847 | Closure of the "variable s... |
| selvval2 42848 | Value of the "variable sel... |
| selvvvval 42849 | Recover the original polyn... |
| evlselvlem 42850 | Lemma for ~ evlselv . Use... |
| evlselv 42851 | Evaluating a selection of ... |
| selvadd 42852 | The "variable selection" f... |
| selvmul 42853 | The "variable selection" f... |
| fsuppind 42854 | Induction on functions ` F... |
| fsuppssindlem1 42855 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 42856 | Lemma for ~ fsuppssind . ... |
| fsuppssind 42857 | Induction on functions ` F... |
| mhpind 42858 | The homogeneous polynomial... |
| evlsmhpvvval 42859 | Give a formula for the eva... |
| mhphflem 42860 | Lemma for ~ mhphf . Add s... |
| mhphf 42861 | A homogeneous polynomial d... |
| mhphf2 42862 | A homogeneous polynomial d... |
| mhphf3 42863 | A homogeneous polynomial d... |
| mhphf4 42864 | A homogeneous polynomial d... |
| prjspval 42867 | Value of the projective sp... |
| prjsprel 42868 | Utility theorem regarding ... |
| prjspertr 42869 | The relation in ` PrjSp ` ... |
| prjsperref 42870 | The relation in ` PrjSp ` ... |
| prjspersym 42871 | The relation in ` PrjSp ` ... |
| prjsper 42872 | The relation used to defin... |
| prjspreln0 42873 | Two nonzero vectors are eq... |
| prjspvs 42874 | A nonzero multiple of a ve... |
| prjsprellsp 42875 | Two vectors are equivalent... |
| prjspeclsp 42876 | The vectors equivalent to ... |
| prjspval2 42877 | Alternate definition of pr... |
| prjspnval 42880 | Value of the n-dimensional... |
| prjspnerlem 42881 | A lemma showing that the e... |
| prjspnval2 42882 | Value of the n-dimensional... |
| prjspner 42883 | The relation used to defin... |
| prjspnvs 42884 | A nonzero multiple of a ve... |
| prjspnssbas 42885 | A projective point spans a... |
| prjspnn0 42886 | A projective point is none... |
| 0prjspnlem 42887 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 42888 | Any vector is equivalent t... |
| prjspner01 42889 | Any vector is equivalent t... |
| prjspner1 42890 | Two vectors whose zeroth c... |
| 0prjspnrel 42891 | In the zero-dimensional pr... |
| 0prjspn 42892 | A zero-dimensional project... |
| prjcrvfval 42895 | Value of the projective cu... |
| prjcrvval 42896 | Value of the projective cu... |
| prjcrv0 42897 | The "curve" (zero set) cor... |
| dffltz 42898 | Fermat's Last Theorem (FLT... |
| fltmul 42899 | A counterexample to FLT st... |
| fltdiv 42900 | A counterexample to FLT st... |
| flt0 42901 | A counterexample for FLT d... |
| fltdvdsabdvdsc 42902 | Any factor of both ` A ` a... |
| fltabcoprmex 42903 | A counterexample to FLT im... |
| fltaccoprm 42904 | A counterexample to FLT wi... |
| fltbccoprm 42905 | A counterexample to FLT wi... |
| fltabcoprm 42906 | A counterexample to FLT wi... |
| infdesc 42907 | Infinite descent. The hyp... |
| fltne 42908 | If a counterexample to FLT... |
| flt4lem 42909 | Raising a number to the fo... |
| flt4lem1 42910 | Satisfy the antecedent use... |
| flt4lem2 42911 | If ` A ` is even, ` B ` is... |
| flt4lem3 42912 | Equivalent to ~ pythagtrip... |
| flt4lem4 42913 | If the product of two copr... |
| flt4lem5 42914 | In the context of the lemm... |
| flt4lem5elem 42915 | Version of ~ fltaccoprm an... |
| flt4lem5a 42916 | Part 1 of Equation 1 of ... |
| flt4lem5b 42917 | Part 2 of Equation 1 of ... |
| flt4lem5c 42918 | Part 2 of Equation 2 of ... |
| flt4lem5d 42919 | Part 3 of Equation 2 of ... |
| flt4lem5e 42920 | Satisfy the hypotheses of ... |
| flt4lem5f 42921 | Final equation of ~... |
| flt4lem6 42922 | Remove shared factors in a... |
| flt4lem7 42923 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 42924 | Strengthening of Fermat's ... |
| fltltc 42925 | ` ( C ^ N ) ` is the large... |
| fltnltalem 42926 | Lemma for ~ fltnlta . A l... |
| fltnlta 42927 | In a Fermat counterexample... |
| iddii 42928 | Version of ~ a1ii with the... |
| bicomdALT 42929 | Alternate proof of ~ bicom... |
| alan 42930 | Alias for ~ 19.26 for easi... |
| exor 42931 | Alias for ~ 19.43 for easi... |
| rexor 42932 | Alias for ~ r19.43 for eas... |
| ruvALT 42933 | Alternate proof of ~ ruv w... |
| sn-wcdeq 42934 | Alternative to ~ wcdeq and... |
| sq45 42935 | 45 squared is 2025. (Cont... |
| sum9cubes 42936 | The sum of the first nine ... |
| sn-isghm 42937 | Longer proof of ~ isghm , ... |
| aprilfools2025 42938 | An abuse of notation. (Co... |
| nfa1w 42939 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 42940 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 42941 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 42942 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 42943 | Replace ~ ax-10 , ~ ax-11 ... |
| cu3addd 42944 | Cube of sum of three numbe... |
| negexpidd 42945 | The sum of a real number t... |
| rexlimdv3d 42946 | An extended version of ~ r... |
| 3cubeslem1 42947 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 42948 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 42949 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 42950 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 42951 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 42952 | Lemma for ~ 3cubes . This... |
| 3cubes 42953 | Every rational number is a... |
| rntrclfvOAI 42954 | The range of the transitiv... |
| moxfr 42955 | Transfer at-most-one betwe... |
| imaiinfv 42956 | Indexed intersection of an... |
| elrfi 42957 | Elementhood in a set of re... |
| elrfirn 42958 | Elementhood in a set of re... |
| elrfirn2 42959 | Elementhood in a set of re... |
| cmpfiiin 42960 | In a compact topology, a s... |
| ismrcd1 42961 | Any function from the subs... |
| ismrcd2 42962 | Second half of ~ ismrcd1 .... |
| istopclsd 42963 | A closure function which s... |
| ismrc 42964 | A function is a Moore clos... |
| isnacs 42967 | Expand definition of Noeth... |
| nacsfg 42968 | In a Noetherian-type closu... |
| isnacs2 42969 | Express Noetherian-type cl... |
| mrefg2 42970 | Slight variation on finite... |
| mrefg3 42971 | Slight variation on finite... |
| nacsacs 42972 | A closure system of Noethe... |
| isnacs3 42973 | A choice-free order equiva... |
| incssnn0 42974 | Transitivity induction of ... |
| nacsfix 42975 | An increasing sequence of ... |
| constmap 42976 | A constant (represented wi... |
| mapco2g 42977 | Renaming indices in a tupl... |
| mapco2 42978 | Post-composition (renaming... |
| mapfzcons 42979 | Extending a one-based mapp... |
| mapfzcons1 42980 | Recover prefix mapping fro... |
| mapfzcons1cl 42981 | A nonempty mapping has a p... |
| mapfzcons2 42982 | Recover added element from... |
| mptfcl 42983 | Interpret range of a maps-... |
| mzpclval 42988 | Substitution lemma for ` m... |
| elmzpcl 42989 | Double substitution lemma ... |
| mzpclall 42990 | The set of all functions w... |
| mzpcln0 42991 | Corollary of ~ mzpclall : ... |
| mzpcl1 42992 | Defining property 1 of a p... |
| mzpcl2 42993 | Defining property 2 of a p... |
| mzpcl34 42994 | Defining properties 3 and ... |
| mzpval 42995 | Value of the ` mzPoly ` fu... |
| dmmzp 42996 | ` mzPoly ` is defined for ... |
| mzpincl 42997 | Polynomial closedness is a... |
| mzpconst 42998 | Constant functions are pol... |
| mzpf 42999 | A polynomial function is a... |
| mzpproj 43000 | A projection function is p... |
| mzpadd 43001 | The pointwise sum of two p... |
| mzpmul 43002 | The pointwise product of t... |
| mzpconstmpt 43003 | A constant function expres... |
| mzpaddmpt 43004 | Sum of polynomial function... |
| mzpmulmpt 43005 | Product of polynomial func... |
| mzpsubmpt 43006 | The difference of two poly... |
| mzpnegmpt 43007 | Negation of a polynomial f... |
| mzpexpmpt 43008 | Raise a polynomial functio... |
| mzpindd 43009 | "Structural" induction to ... |
| mzpmfp 43010 | Relationship between multi... |
| mzpsubst 43011 | Substituting polynomials f... |
| mzprename 43012 | Simplified version of ~ mz... |
| mzpresrename 43013 | A polynomial is a polynomi... |
| mzpcompact2lem 43014 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 43015 | Polynomials are finitary o... |
| coeq0i 43016 | ~ coeq0 but without explic... |
| fzsplit1nn0 43017 | Split a finite 1-based set... |
| eldiophb 43020 | Initial expression of Diop... |
| eldioph 43021 | Condition for a set to be ... |
| diophrw 43022 | Renaming and adding unused... |
| eldioph2lem1 43023 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 43024 | Lemma for ~ eldioph2 . Co... |
| eldioph2 43025 | Construct a Diophantine se... |
| eldioph2b 43026 | While Diophantine sets wer... |
| eldiophelnn0 43027 | Remove antecedent on ` B `... |
| eldioph3b 43028 | Define Diophantine sets in... |
| eldioph3 43029 | Inference version of ~ eld... |
| ellz1 43030 | Membership in a lower set ... |
| lzunuz 43031 | The union of a lower set o... |
| fz1eqin 43032 | Express a one-based finite... |
| lzenom 43033 | Lower integers are countab... |
| elmapresaunres2 43034 | ~ fresaunres2 transposed t... |
| diophin 43035 | If two sets are Diophantin... |
| diophun 43036 | If two sets are Diophantin... |
| eldiophss 43037 | Diophantine sets are sets ... |
| diophrex 43038 | Projecting a Diophantine s... |
| eq0rabdioph 43039 | This is the first of a num... |
| eqrabdioph 43040 | Diophantine set builder fo... |
| 0dioph 43041 | The null set is Diophantin... |
| vdioph 43042 | The "universal" set (as la... |
| anrabdioph 43043 | Diophantine set builder fo... |
| orrabdioph 43044 | Diophantine set builder fo... |
| 3anrabdioph 43045 | Diophantine set builder fo... |
| 3orrabdioph 43046 | Diophantine set builder fo... |
| 2sbcrex 43047 | Exchange an existential qu... |
| sbcrexgOLD 43048 | Interchange class substitu... |
| 2sbcrexOLD 43049 | Exchange an existential qu... |
| sbc2rex 43050 | Exchange a substitution wi... |
| sbc2rexgOLD 43051 | Exchange a substitution wi... |
| sbc4rex 43052 | Exchange a substitution wi... |
| sbc4rexgOLD 43053 | Exchange a substitution wi... |
| sbcrot3 43054 | Rotate a sequence of three... |
| sbcrot5 43055 | Rotate a sequence of five ... |
| sbccomieg 43056 | Commute two explicit subst... |
| rexrabdioph 43057 | Diophantine set builder fo... |
| rexfrabdioph 43058 | Diophantine set builder fo... |
| 2rexfrabdioph 43059 | Diophantine set builder fo... |
| 3rexfrabdioph 43060 | Diophantine set builder fo... |
| 4rexfrabdioph 43061 | Diophantine set builder fo... |
| 6rexfrabdioph 43062 | Diophantine set builder fo... |
| 7rexfrabdioph 43063 | Diophantine set builder fo... |
| rabdiophlem1 43064 | Lemma for arithmetic dioph... |
| rabdiophlem2 43065 | Lemma for arithmetic dioph... |
| elnn0rabdioph 43066 | Diophantine set builder fo... |
| rexzrexnn0 43067 | Rewrite an existential qua... |
| lerabdioph 43068 | Diophantine set builder fo... |
| eluzrabdioph 43069 | Diophantine set builder fo... |
| elnnrabdioph 43070 | Diophantine set builder fo... |
| ltrabdioph 43071 | Diophantine set builder fo... |
| nerabdioph 43072 | Diophantine set builder fo... |
| dvdsrabdioph 43073 | Divisibility is a Diophant... |
| eldioph4b 43074 | Membership in ` Dioph ` ex... |
| eldioph4i 43075 | Forward-only version of ~ ... |
| diophren 43076 | Change variables in a Diop... |
| rabrenfdioph 43077 | Change variable numbers in... |
| rabren3dioph 43078 | Change variable numbers in... |
| fphpd 43079 | Pigeonhole principle expre... |
| fphpdo 43080 | Pigeonhole principle for s... |
| ctbnfien 43081 | An infinite subset of a co... |
| fiphp3d 43082 | Infinite pigeonhole princi... |
| rencldnfilem 43083 | Lemma for ~ rencldnfi . (... |
| rencldnfi 43084 | A set of real numbers whic... |
| irrapxlem1 43085 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 43086 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 43087 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 43088 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 43089 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 43090 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 43091 | Dirichlet's approximation ... |
| pellexlem1 43092 | Lemma for ~ pellex . Arit... |
| pellexlem2 43093 | Lemma for ~ pellex . Arit... |
| pellexlem3 43094 | Lemma for ~ pellex . To e... |
| pellexlem4 43095 | Lemma for ~ pellex . Invo... |
| pellexlem5 43096 | Lemma for ~ pellex . Invo... |
| pellexlem6 43097 | Lemma for ~ pellex . Doin... |
| pellex 43098 | Every Pell equation has a ... |
| pell1qrval 43109 | Value of the set of first-... |
| elpell1qr 43110 | Membership in a first-quad... |
| pell14qrval 43111 | Value of the set of positi... |
| elpell14qr 43112 | Membership in the set of p... |
| pell1234qrval 43113 | Value of the set of genera... |
| elpell1234qr 43114 | Membership in the set of g... |
| pell1234qrre 43115 | General Pell solutions are... |
| pell1234qrne0 43116 | No solution to a Pell equa... |
| pell1234qrreccl 43117 | General solutions of the P... |
| pell1234qrmulcl 43118 | General solutions of the P... |
| pell14qrss1234 43119 | A positive Pell solution i... |
| pell14qrre 43120 | A positive Pell solution i... |
| pell14qrne0 43121 | A positive Pell solution i... |
| pell14qrgt0 43122 | A positive Pell solution i... |
| pell14qrrp 43123 | A positive Pell solution i... |
| pell1234qrdich 43124 | A general Pell solution is... |
| elpell14qr2 43125 | A number is a positive Pel... |
| pell14qrmulcl 43126 | Positive Pell solutions ar... |
| pell14qrreccl 43127 | Positive Pell solutions ar... |
| pell14qrdivcl 43128 | Positive Pell solutions ar... |
| pell14qrexpclnn0 43129 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 43130 | Positive Pell solutions ar... |
| pell1qrss14 43131 | First-quadrant Pell soluti... |
| pell14qrdich 43132 | A positive Pell solution i... |
| pell1qrge1 43133 | A Pell solution in the fir... |
| pell1qr1 43134 | 1 is a Pell solution and i... |
| elpell1qr2 43135 | The first quadrant solutio... |
| pell1qrgaplem 43136 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 43137 | First-quadrant Pell soluti... |
| pell14qrgap 43138 | Positive Pell solutions ar... |
| pell14qrgapw 43139 | Positive Pell solutions ar... |
| pellqrexplicit 43140 | Condition for a calculated... |
| infmrgelbi 43141 | Any lower bound of a nonem... |
| pellqrex 43142 | There is a nontrivial solu... |
| pellfundval 43143 | Value of the fundamental s... |
| pellfundre 43144 | The fundamental solution o... |
| pellfundge 43145 | Lower bound on the fundame... |
| pellfundgt1 43146 | Weak lower bound on the Pe... |
| pellfundlb 43147 | A nontrivial first quadran... |
| pellfundglb 43148 | If a real is larger than t... |
| pellfundex 43149 | The fundamental solution a... |
| pellfund14gap 43150 | There are no solutions bet... |
| pellfundrp 43151 | The fundamental Pell solut... |
| pellfundne1 43152 | The fundamental Pell solut... |
| reglogcl 43153 | General logarithm is a rea... |
| reglogltb 43154 | General logarithm preserve... |
| reglogleb 43155 | General logarithm preserve... |
| reglogmul 43156 | Multiplication law for gen... |
| reglogexp 43157 | Power law for general log.... |
| reglogbas 43158 | General log of the base is... |
| reglog1 43159 | General log of 1 is 0. (C... |
| reglogexpbas 43160 | General log of a power of ... |
| pellfund14 43161 | Every positive Pell soluti... |
| pellfund14b 43162 | The positive Pell solution... |
| rmxfval 43167 | Value of the X sequence. ... |
| rmyfval 43168 | Value of the Y sequence. ... |
| rmspecsqrtnq 43169 | The discriminant used to d... |
| rmspecnonsq 43170 | The discriminant used to d... |
| qirropth 43171 | This lemma implements the ... |
| rmspecfund 43172 | The base of exponent used ... |
| rmxyelqirr 43173 | The solutions used to cons... |
| rmxypairf1o 43174 | The function used to extra... |
| rmxyelxp 43175 | Lemma for ~ frmx and ~ frm... |
| frmx 43176 | The X sequence is a nonneg... |
| frmy 43177 | The Y sequence is an integ... |
| rmxyval 43178 | Main definition of the X a... |
| rmspecpos 43179 | The discriminant used to d... |
| rmxycomplete 43180 | The X and Y sequences take... |
| rmxynorm 43181 | The X and Y sequences defi... |
| rmbaserp 43182 | The base of exponentiation... |
| rmxyneg 43183 | Negation law for X and Y s... |
| rmxyadd 43184 | Addition formula for X and... |
| rmxy1 43185 | Value of the X and Y seque... |
| rmxy0 43186 | Value of the X and Y seque... |
| rmxneg 43187 | Negation law (even functio... |
| rmx0 43188 | Value of X sequence at 0. ... |
| rmx1 43189 | Value of X sequence at 1. ... |
| rmxadd 43190 | Addition formula for X seq... |
| rmyneg 43191 | Negation formula for Y seq... |
| rmy0 43192 | Value of Y sequence at 0. ... |
| rmy1 43193 | Value of Y sequence at 1. ... |
| rmyadd 43194 | Addition formula for Y seq... |
| rmxp1 43195 | Special addition-of-1 form... |
| rmyp1 43196 | Special addition of 1 form... |
| rmxm1 43197 | Subtraction of 1 formula f... |
| rmym1 43198 | Subtraction of 1 formula f... |
| rmxluc 43199 | The X sequence is a Lucas ... |
| rmyluc 43200 | The Y sequence is a Lucas ... |
| rmyluc2 43201 | Lucas sequence property of... |
| rmxdbl 43202 | "Double-angle formula" for... |
| rmydbl 43203 | "Double-angle formula" for... |
| monotuz 43204 | A function defined on an u... |
| monotoddzzfi 43205 | A function which is odd an... |
| monotoddzz 43206 | A function (given implicit... |
| oddcomabszz 43207 | An odd function which take... |
| 2nn0ind 43208 | Induction on nonnegative i... |
| zindbi 43209 | Inductively transfer a pro... |
| rmxypos 43210 | For all nonnegative indice... |
| ltrmynn0 43211 | The Y-sequence is strictly... |
| ltrmxnn0 43212 | The X-sequence is strictly... |
| lermxnn0 43213 | The X-sequence is monotoni... |
| rmxnn 43214 | The X-sequence is defined ... |
| ltrmy 43215 | The Y-sequence is strictly... |
| rmyeq0 43216 | Y is zero only at zero. (... |
| rmyeq 43217 | Y is one-to-one. (Contrib... |
| lermy 43218 | Y is monotonic (non-strict... |
| rmynn 43219 | ` rmY ` is positive for po... |
| rmynn0 43220 | ` rmY ` is nonnegative for... |
| rmyabs 43221 | ` rmY ` commutes with ` ab... |
| jm2.24nn 43222 | X(n) is strictly greater t... |
| jm2.17a 43223 | First half of lemma 2.17 o... |
| jm2.17b 43224 | Weak form of the second ha... |
| jm2.17c 43225 | Second half of lemma 2.17 ... |
| jm2.24 43226 | Lemma 2.24 of [JonesMatija... |
| rmygeid 43227 | Y(n) increases faster than... |
| congtr 43228 | A wff of the form ` A || (... |
| congadd 43229 | If two pairs of numbers ar... |
| congmul 43230 | If two pairs of numbers ar... |
| congsym 43231 | Congruence mod ` A ` is a ... |
| congneg 43232 | If two integers are congru... |
| congsub 43233 | If two pairs of numbers ar... |
| congid 43234 | Every integer is congruent... |
| mzpcong 43235 | Polynomials commute with c... |
| congrep 43236 | Every integer is congruent... |
| congabseq 43237 | If two integers are congru... |
| acongid 43238 | A wff like that in this th... |
| acongsym 43239 | Symmetry of alternating co... |
| acongneg2 43240 | Negate right side of alter... |
| acongtr 43241 | Transitivity of alternatin... |
| acongeq12d 43242 | Substitution deduction for... |
| acongrep 43243 | Every integer is alternati... |
| fzmaxdif 43244 | Bound on the difference be... |
| fzneg 43245 | Reflection of a finite ran... |
| acongeq 43246 | Two numbers in the fundame... |
| dvdsacongtr 43247 | Alternating congruence pas... |
| coprmdvdsb 43248 | Multiplication by a coprim... |
| modabsdifz 43249 | Divisibility in terms of m... |
| dvdsabsmod0 43250 | Divisibility in terms of m... |
| jm2.18 43251 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 43252 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 43253 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 43254 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 43255 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 43256 | Lemma 2.19 of [JonesMatija... |
| jm2.21 43257 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 43258 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 43259 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 43260 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 43261 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 43262 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 43263 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 43264 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 43265 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 43266 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 43267 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 43268 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 43269 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 43270 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 43271 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 43272 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 43273 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 43274 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 43275 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 43276 | Lemma for ~ rmydioph . Us... |
| rmydioph 43277 | ~ jm2.27 restated in terms... |
| rmxdiophlem 43278 | X can be expressed in term... |
| rmxdioph 43279 | X is a Diophantine functio... |
| jm3.1lem1 43280 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 43281 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 43282 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 43283 | Diophantine expression for... |
| expdiophlem1 43284 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 43285 | Lemma for ~ expdioph . Ex... |
| expdioph 43286 | The exponential function i... |
| setindtr 43287 | Set induction for sets con... |
| setindtrs 43288 | Set induction scheme witho... |
| dford3lem1 43289 | Lemma for ~ dford3 . (Con... |
| dford3lem2 43290 | Lemma for ~ dford3 . (Con... |
| dford3 43291 | Ordinals are precisely the... |
| dford4 43292 | ~ dford3 expressed in prim... |
| wopprc 43293 | Unrelated: Wiener pairs t... |
| rpnnen3lem 43294 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 43295 | Dedekind cut injection of ... |
| axac10 43296 | Characterization of choice... |
| harinf 43297 | The Hartogs number of an i... |
| wdom2d2 43298 | Deduction for weak dominan... |
| ttac 43299 | Tarski's theorem about cho... |
| pw2f1ocnv 43300 | Define a bijection between... |
| pw2f1o2 43301 | Define a bijection between... |
| pw2f1o2val 43302 | Function value of the ~ pw... |
| pw2f1o2val2 43303 | Membership in a mapped set... |
| limsuc2 43304 | Limit ordinals in the sens... |
| wepwsolem 43305 | Transfer an ordering on ch... |
| wepwso 43306 | A well-ordering induces a ... |
| dnnumch1 43307 | Define an enumeration of a... |
| dnnumch2 43308 | Define an enumeration (wea... |
| dnnumch3lem 43309 | Value of the ordinal injec... |
| dnnumch3 43310 | Define an injection from a... |
| dnwech 43311 | Define a well-ordering fro... |
| fnwe2val 43312 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 43313 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 43314 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 43315 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 43316 | A well-ordering can be con... |
| aomclem1 43317 | Lemma for ~ dfac11 . This... |
| aomclem2 43318 | Lemma for ~ dfac11 . Succ... |
| aomclem3 43319 | Lemma for ~ dfac11 . Succ... |
| aomclem4 43320 | Lemma for ~ dfac11 . Limi... |
| aomclem5 43321 | Lemma for ~ dfac11 . Comb... |
| aomclem6 43322 | Lemma for ~ dfac11 . Tran... |
| aomclem7 43323 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 43324 | Lemma for ~ dfac11 . Perf... |
| dfac11 43325 | The right-hand side of thi... |
| kelac1 43326 | Kelley's choice, basic for... |
| kelac2lem 43327 | Lemma for ~ kelac2 and ~ d... |
| kelac2 43328 | Kelley's choice, most comm... |
| dfac21 43329 | Tychonoff's theorem is a c... |
| islmodfg 43332 | Property of a finitely gen... |
| islssfg 43333 | Property of a finitely gen... |
| islssfg2 43334 | Property of a finitely gen... |
| islssfgi 43335 | Finitely spanned subspaces... |
| fglmod 43336 | Finitely generated left mo... |
| lsmfgcl 43337 | The sum of two finitely ge... |
| islnm 43340 | Property of being a Noethe... |
| islnm2 43341 | Property of being a Noethe... |
| lnmlmod 43342 | A Noetherian left module i... |
| lnmlssfg 43343 | A submodule of Noetherian ... |
| lnmlsslnm 43344 | All submodules of a Noethe... |
| lnmfg 43345 | A Noetherian left module i... |
| kercvrlsm 43346 | The domain of a linear fun... |
| lmhmfgima 43347 | A homomorphism maps finite... |
| lnmepi 43348 | Epimorphic images of Noeth... |
| lmhmfgsplit 43349 | If the kernel and range of... |
| lmhmlnmsplit 43350 | If the kernel and range of... |
| lnmlmic 43351 | Noetherian is an invariant... |
| pwssplit4 43352 | Splitting for structure po... |
| filnm 43353 | Finite left modules are No... |
| pwslnmlem0 43354 | Zeroeth powers are Noether... |
| pwslnmlem1 43355 | First powers are Noetheria... |
| pwslnmlem2 43356 | A sum of powers is Noether... |
| pwslnm 43357 | Finite powers of Noetheria... |
| unxpwdom3 43358 | Weaker version of ~ unxpwd... |
| pwfi2f1o 43359 | The ~ pw2f1o bijection rel... |
| pwfi2en 43360 | Finitely supported indicat... |
| frlmpwfi 43361 | Formal linear combinations... |
| gicabl 43362 | Being Abelian is a group i... |
| imasgim 43363 | A relabeling of the elemen... |
| isnumbasgrplem1 43364 | A set which is equipollent... |
| harn0 43365 | The Hartogs number of a se... |
| numinfctb 43366 | A numerable infinite set c... |
| isnumbasgrplem2 43367 | If the (to be thought of a... |
| isnumbasgrplem3 43368 | Every nonempty numerable s... |
| isnumbasabl 43369 | A set is numerable iff it ... |
| isnumbasgrp 43370 | A set is numerable iff it ... |
| dfacbasgrp 43371 | A choice equivalent in abs... |
| islnr 43374 | Property of a left-Noether... |
| lnrring 43375 | Left-Noetherian rings are ... |
| lnrlnm 43376 | Left-Noetherian rings have... |
| islnr2 43377 | Property of being a left-N... |
| islnr3 43378 | Relate left-Noetherian rin... |
| lnr2i 43379 | Given an ideal in a left-N... |
| lpirlnr 43380 | Left principal ideal rings... |
| lnrfrlm 43381 | Finite-dimensional free mo... |
| lnrfg 43382 | Finitely-generated modules... |
| lnrfgtr 43383 | A submodule of a finitely ... |
| hbtlem1 43386 | Value of the leading coeff... |
| hbtlem2 43387 | Leading coefficient ideals... |
| hbtlem7 43388 | Functionality of leading c... |
| hbtlem4 43389 | The leading ideal function... |
| hbtlem3 43390 | The leading ideal function... |
| hbtlem5 43391 | The leading ideal function... |
| hbtlem6 43392 | There is a finite set of p... |
| hbt 43393 | The Hilbert Basis Theorem ... |
| dgrsub2 43398 | Subtracting two polynomial... |
| elmnc 43399 | Property of a monic polyno... |
| mncply 43400 | A monic polynomial is a po... |
| mnccoe 43401 | A monic polynomial has lea... |
| mncn0 43402 | A monic polynomial is not ... |
| dgraaval 43407 | Value of the degree functi... |
| dgraalem 43408 | Properties of the degree o... |
| dgraacl 43409 | Closure of the degree func... |
| dgraaf 43410 | Degree function on algebra... |
| dgraaub 43411 | Upper bound on degree of a... |
| dgraa0p 43412 | A rational polynomial of d... |
| mpaaeu 43413 | An algebraic number has ex... |
| mpaaval 43414 | Value of the minimal polyn... |
| mpaalem 43415 | Properties of the minimal ... |
| mpaacl 43416 | Minimal polynomial is a po... |
| mpaadgr 43417 | Minimal polynomial has deg... |
| mpaaroot 43418 | The minimal polynomial of ... |
| mpaamn 43419 | Minimal polynomial is moni... |
| itgoval 43424 | Value of the integral-over... |
| aaitgo 43425 | The standard algebraic num... |
| itgoss 43426 | An integral element is int... |
| itgocn 43427 | All integral elements are ... |
| cnsrexpcl 43428 | Exponentiation is closed i... |
| fsumcnsrcl 43429 | Finite sums are closed in ... |
| cnsrplycl 43430 | Polynomials are closed in ... |
| rgspnid 43431 | The span of a subring is i... |
| rngunsnply 43432 | Adjoining one element to a... |
| flcidc 43433 | Finite linear combinations... |
| algstr 43436 | Lemma to shorten proofs of... |
| algbase 43437 | The base set of a construc... |
| algaddg 43438 | The additive operation of ... |
| algmulr 43439 | The multiplicative operati... |
| algsca 43440 | The set of scalars of a co... |
| algvsca 43441 | The scalar product operati... |
| mendval 43442 | Value of the module endomo... |
| mendbas 43443 | Base set of the module end... |
| mendplusgfval 43444 | Addition in the module end... |
| mendplusg 43445 | A specific addition in the... |
| mendmulrfval 43446 | Multiplication in the modu... |
| mendmulr 43447 | A specific multiplication ... |
| mendsca 43448 | The module endomorphism al... |
| mendvscafval 43449 | Scalar multiplication in t... |
| mendvsca 43450 | A specific scalar multipli... |
| mendring 43451 | The module endomorphism al... |
| mendlmod 43452 | The module endomorphism al... |
| mendassa 43453 | The module endomorphism al... |
| idomodle 43454 | Limit on the number of ` N... |
| fiuneneq 43455 | Two finite sets of equal s... |
| idomsubgmo 43456 | The units of an integral d... |
| proot1mul 43457 | Any primitive ` N ` -th ro... |
| proot1hash 43458 | If an integral domain has ... |
| proot1ex 43459 | The complex field has prim... |
| mon1psubm 43462 | Monic polynomials are a mu... |
| deg1mhm 43463 | Homomorphic property of th... |
| cytpfn 43464 | Functionality of the cyclo... |
| cytpval 43465 | Substitutions for the Nth ... |
| fgraphopab 43466 | Express a function as a su... |
| fgraphxp 43467 | Express a function as a su... |
| hausgraph 43468 | The graph of a continuous ... |
| r1sssucd 43473 | Deductive form of ~ r1sssu... |
| iocunico 43474 | Split an open interval int... |
| iocinico 43475 | The intersection of two se... |
| iocmbl 43476 | An open-below, closed-abov... |
| cnioobibld 43477 | A bounded, continuous func... |
| arearect 43478 | The area of a rectangle wh... |
| areaquad 43479 | The area of a quadrilatera... |
| uniel 43480 | Two ways to say a union is... |
| unielss 43481 | Two ways to say the union ... |
| unielid 43482 | Two ways to say the union ... |
| ssunib 43483 | Two ways to say a class is... |
| rp-intrabeq 43484 | Equality theorem for supre... |
| rp-unirabeq 43485 | Equality theorem for infim... |
| onmaxnelsup 43486 | Two ways to say the maximu... |
| onsupneqmaxlim0 43487 | If the supremum of a class... |
| onsupcl2 43488 | The supremum of a set of o... |
| onuniintrab 43489 | The union of a set of ordi... |
| onintunirab 43490 | The intersection of a non-... |
| onsupnmax 43491 | If the union of a class of... |
| onsupuni 43492 | The supremum of a set of o... |
| onsupuni2 43493 | The supremum of a set of o... |
| onsupintrab 43494 | The supremum of a set of o... |
| onsupintrab2 43495 | The supremum of a set of o... |
| onsupcl3 43496 | The supremum of a set of o... |
| onsupex3 43497 | The supremum of a set of o... |
| onuniintrab2 43498 | The union of a set of ordi... |
| oninfint 43499 | The infimum of a non-empty... |
| oninfunirab 43500 | The infimum of a non-empty... |
| oninfcl2 43501 | The infimum of a non-empty... |
| onsupmaxb 43502 | The union of a class of or... |
| onexgt 43503 | For any ordinal, there is ... |
| onexomgt 43504 | For any ordinal, there is ... |
| omlimcl2 43505 | The product of a limit ord... |
| onexlimgt 43506 | For any ordinal, there is ... |
| onexoegt 43507 | For any ordinal, there is ... |
| oninfex2 43508 | The infimum of a non-empty... |
| onsupeqmax 43509 | Condition when the supremu... |
| onsupeqnmax 43510 | Condition when the supremu... |
| onsuplub 43511 | The supremum of a set of o... |
| onsupnub 43512 | An upper bound of a set of... |
| onfisupcl 43513 | Sufficient condition when ... |
| onelord 43514 | Every element of a ordinal... |
| onepsuc 43515 | Every ordinal is less than... |
| epsoon 43516 | The ordinals are strictly ... |
| epirron 43517 | The strict order on the or... |
| oneptr 43518 | The strict order on the or... |
| oneltr 43519 | The elementhood relation o... |
| oneptri 43520 | The strict, complete (line... |
| ordeldif 43521 | Membership in the differen... |
| ordeldifsucon 43522 | Membership in the differen... |
| ordeldif1o 43523 | Membership in the differen... |
| ordne0gt0 43524 | Ordinal zero is less than ... |
| ondif1i 43525 | Ordinal zero is less than ... |
| onsucelab 43526 | The successor of every ord... |
| dflim6 43527 | A limit ordinal is a non-z... |
| limnsuc 43528 | A limit ordinal is not an ... |
| onsucss 43529 | If one ordinal is less tha... |
| ordnexbtwnsuc 43530 | For any distinct pair of o... |
| orddif0suc 43531 | For any distinct pair of o... |
| onsucf1lem 43532 | For ordinals, the successo... |
| onsucf1olem 43533 | The successor operation is... |
| onsucrn 43534 | The successor operation is... |
| onsucf1o 43535 | The successor operation is... |
| dflim7 43536 | A limit ordinal is a non-z... |
| onov0suclim 43537 | Compactly express rules fo... |
| oa0suclim 43538 | Closed form expression of ... |
| om0suclim 43539 | Closed form expression of ... |
| oe0suclim 43540 | Closed form expression of ... |
| oaomoecl 43541 | The operations of addition... |
| onsupsucismax 43542 | If the union of a set of o... |
| onsssupeqcond 43543 | If for every element of a ... |
| limexissup 43544 | An ordinal which is a limi... |
| limiun 43545 | A limit ordinal is the uni... |
| limexissupab 43546 | An ordinal which is a limi... |
| om1om1r 43547 | Ordinal one is both a left... |
| oe0rif 43548 | Ordinal zero raised to any... |
| oasubex 43549 | While subtraction can't be... |
| nnamecl 43550 | Natural numbers are closed... |
| onsucwordi 43551 | The successor operation pr... |
| oalim2cl 43552 | The ordinal sum of any ord... |
| oaltublim 43553 | Given ` C ` is a limit ord... |
| oaordi3 43554 | Ordinal addition of the sa... |
| oaord3 43555 | When the same ordinal is a... |
| 1oaomeqom 43556 | Ordinal one plus omega is ... |
| oaabsb 43557 | The right addend absorbs t... |
| oaordnrex 43558 | When omega is added on the... |
| oaordnr 43559 | When the same ordinal is a... |
| omge1 43560 | Any non-zero ordinal produ... |
| omge2 43561 | Any non-zero ordinal produ... |
| omlim2 43562 | The non-zero product with ... |
| omord2lim 43563 | Given a limit ordinal, the... |
| omord2i 43564 | Ordinal multiplication of ... |
| omord2com 43565 | When the same non-zero ord... |
| 2omomeqom 43566 | Ordinal two times omega is... |
| omnord1ex 43567 | When omega is multiplied o... |
| omnord1 43568 | When the same non-zero ord... |
| oege1 43569 | Any non-zero ordinal power... |
| oege2 43570 | Any power of an ordinal at... |
| rp-oelim2 43571 | The power of an ordinal at... |
| oeord2lim 43572 | Given a limit ordinal, the... |
| oeord2i 43573 | Ordinal exponentiation of ... |
| oeord2com 43574 | When the same base at leas... |
| nnoeomeqom 43575 | Any natural number at leas... |
| df3o2 43576 | Ordinal 3 is the unordered... |
| df3o3 43577 | Ordinal 3, fully expanded.... |
| oenord1ex 43578 | When ordinals two and thre... |
| oenord1 43579 | When two ordinals (both at... |
| oaomoencom 43580 | Ordinal addition, multipli... |
| oenassex 43581 | Ordinal two raised to two ... |
| oenass 43582 | Ordinal exponentiation is ... |
| cantnftermord 43583 | For terms of the form of a... |
| cantnfub 43584 | Given a finite number of t... |
| cantnfub2 43585 | Given a finite number of t... |
| bropabg 43586 | Equivalence for two classe... |
| cantnfresb 43587 | A Cantor normal form which... |
| cantnf2 43588 | For every ordinal, ` A ` ,... |
| oawordex2 43589 | If ` C ` is between ` A ` ... |
| nnawordexg 43590 | If an ordinal, ` B ` , is ... |
| succlg 43591 | Closure law for ordinal su... |
| dflim5 43592 | A limit ordinal is either ... |
| oacl2g 43593 | Closure law for ordinal ad... |
| onmcl 43594 | If an ordinal is less than... |
| omabs2 43595 | Ordinal multiplication by ... |
| omcl2 43596 | Closure law for ordinal mu... |
| omcl3g 43597 | Closure law for ordinal mu... |
| ordsssucb 43598 | An ordinal number is less ... |
| tfsconcatlem 43599 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 43600 | The concatenation of two t... |
| tfsconcatfn 43601 | The concatenation of two t... |
| tfsconcatfv1 43602 | An early value of the conc... |
| tfsconcatfv2 43603 | A latter value of the conc... |
| tfsconcatfv 43604 | The value of the concatena... |
| tfsconcatrn 43605 | The range of the concatena... |
| tfsconcatfo 43606 | The concatenation of two t... |
| tfsconcatb0 43607 | The concatentation with th... |
| tfsconcat0i 43608 | The concatentation with th... |
| tfsconcat0b 43609 | The concatentation with th... |
| tfsconcat00 43610 | The concatentation of two ... |
| tfsconcatrev 43611 | If the domain of a transfi... |
| tfsconcatrnss12 43612 | The range of the concatena... |
| tfsconcatrnss 43613 | The concatenation of trans... |
| tfsconcatrnsson 43614 | The concatenation of trans... |
| tfsnfin 43615 | A transfinite sequence is ... |
| rp-tfslim 43616 | The limit of a sequence of... |
| ofoafg 43617 | Addition operator for func... |
| ofoaf 43618 | Addition operator for func... |
| ofoafo 43619 | Addition operator for func... |
| ofoacl 43620 | Closure law for component ... |
| ofoaid1 43621 | Identity law for component... |
| ofoaid2 43622 | Identity law for component... |
| ofoaass 43623 | Component-wise addition of... |
| ofoacom 43624 | Component-wise addition of... |
| naddcnff 43625 | Addition operator for Cant... |
| naddcnffn 43626 | Addition operator for Cant... |
| naddcnffo 43627 | Addition of Cantor normal ... |
| naddcnfcl 43628 | Closure law for component-... |
| naddcnfcom 43629 | Component-wise ordinal add... |
| naddcnfid1 43630 | Identity law for component... |
| naddcnfid2 43631 | Identity law for component... |
| naddcnfass 43632 | Component-wise addition of... |
| onsucunifi 43633 | The successor to the union... |
| sucunisn 43634 | The successor to the union... |
| onsucunipr 43635 | The successor to the union... |
| onsucunitp 43636 | The successor to the union... |
| oaun3lem1 43637 | The class of all ordinal s... |
| oaun3lem2 43638 | The class of all ordinal s... |
| oaun3lem3 43639 | The class of all ordinal s... |
| oaun3lem4 43640 | The class of all ordinal s... |
| rp-abid 43641 | Two ways to express a clas... |
| oadif1lem 43642 | Express the set difference... |
| oadif1 43643 | Express the set difference... |
| oaun2 43644 | Ordinal addition as a unio... |
| oaun3 43645 | Ordinal addition as a unio... |
| naddov4 43646 | Alternate expression for n... |
| nadd2rabtr 43647 | The set of ordinals which ... |
| nadd2rabord 43648 | The set of ordinals which ... |
| nadd2rabex 43649 | The class of ordinals whic... |
| nadd2rabon 43650 | The set of ordinals which ... |
| nadd1rabtr 43651 | The set of ordinals which ... |
| nadd1rabord 43652 | The set of ordinals which ... |
| nadd1rabex 43653 | The class of ordinals whic... |
| nadd1rabon 43654 | The set of ordinals which ... |
| nadd1suc 43655 | Natural addition with 1 is... |
| naddass1 43656 | Natural addition of ordina... |
| naddgeoa 43657 | Natural addition results i... |
| naddonnn 43658 | Natural addition with a na... |
| naddwordnexlem0 43659 | When ` A ` is the sum of a... |
| naddwordnexlem1 43660 | When ` A ` is the sum of a... |
| naddwordnexlem2 43661 | When ` A ` is the sum of a... |
| naddwordnexlem3 43662 | When ` A ` is the sum of a... |
| oawordex3 43663 | When ` A ` is the sum of a... |
| naddwordnexlem4 43664 | When ` A ` is the sum of a... |
| ordsssucim 43665 | If an ordinal is less than... |
| insucid 43666 | The intersection of a clas... |
| oaltom 43667 | Multiplication eventually ... |
| oe2 43668 | Two ways to square an ordi... |
| omltoe 43669 | Exponentiation eventually ... |
| abeqabi 43670 | Generalized condition for ... |
| abpr 43671 | Condition for a class abst... |
| abtp 43672 | Condition for a class abst... |
| ralopabb 43673 | Restricted universal quant... |
| fpwfvss 43674 | Functions into a powerset ... |
| sdomne0 43675 | A class that strictly domi... |
| sdomne0d 43676 | A class that strictly domi... |
| safesnsupfiss 43677 | If ` B ` is a finite subse... |
| safesnsupfiub 43678 | If ` B ` is a finite subse... |
| safesnsupfidom1o 43679 | If ` B ` is a finite subse... |
| safesnsupfilb 43680 | If ` B ` is a finite subse... |
| isoeq145d 43681 | Equality deduction for iso... |
| resisoeq45d 43682 | Equality deduction for equ... |
| negslem1 43683 | An equivalence between ide... |
| nvocnvb 43684 | Equivalence to saying the ... |
| rp-brsslt 43685 | Binary relation form of a ... |
| nla0002 43686 | Extending a linear order t... |
| nla0003 43687 | Extending a linear order t... |
| nla0001 43688 | Extending a linear order t... |
| faosnf0.11b 43689 | ` B ` is called a non-limi... |
| dfno2 43690 | A surreal number, in the f... |
| onnoxpg 43691 | Every ordinal maps to a su... |
| onnobdayg 43692 | Every ordinal maps to a su... |
| bdaybndex 43693 | Bounds formed from the bir... |
| bdaybndbday 43694 | Bounds formed from the bir... |
| onnoxp 43695 | Every ordinal maps to a su... |
| onnoxpi 43696 | Every ordinal maps to a su... |
| 0fno 43697 | Ordinal zero maps to a sur... |
| 1fno 43698 | Ordinal one maps to a surr... |
| 2fno 43699 | Ordinal two maps to a surr... |
| 3fno 43700 | Ordinal three maps to a su... |
| 4fno 43701 | Ordinal four maps to a sur... |
| fnimafnex 43702 | The functional image of a ... |
| nlimsuc 43703 | A successor is not a limit... |
| nlim1NEW 43704 | 1 is not a limit ordinal. ... |
| nlim2NEW 43705 | 2 is not a limit ordinal. ... |
| nlim3 43706 | 3 is not a limit ordinal. ... |
| nlim4 43707 | 4 is not a limit ordinal. ... |
| oa1un 43708 | Given ` A e. On ` , let ` ... |
| oa1cl 43709 | ` A +o 1o ` is in ` On ` .... |
| 0finon 43710 | 0 is a finite ordinal. Se... |
| 1finon 43711 | 1 is a finite ordinal. Se... |
| 2finon 43712 | 2 is a finite ordinal. Se... |
| 3finon 43713 | 3 is a finite ordinal. Se... |
| 4finon 43714 | 4 is a finite ordinal. Se... |
| finona1cl 43715 | The finite ordinals are cl... |
| finonex 43716 | The finite ordinals are a ... |
| fzunt 43717 | Union of two adjacent fini... |
| fzuntd 43718 | Union of two adjacent fini... |
| fzunt1d 43719 | Union of two overlapping f... |
| fzuntgd 43720 | Union of two adjacent or o... |
| ifpan123g 43721 | Conjunction of conditional... |
| ifpan23 43722 | Conjunction of conditional... |
| ifpdfor2 43723 | Define or in terms of cond... |
| ifporcor 43724 | Corollary of commutation o... |
| ifpdfan2 43725 | Define and with conditiona... |
| ifpancor 43726 | Corollary of commutation o... |
| ifpdfor 43727 | Define or in terms of cond... |
| ifpdfan 43728 | Define and with conditiona... |
| ifpbi2 43729 | Equivalence theorem for co... |
| ifpbi3 43730 | Equivalence theorem for co... |
| ifpim1 43731 | Restate implication as con... |
| ifpnot 43732 | Restate negated wff as con... |
| ifpid2 43733 | Restate wff as conditional... |
| ifpim2 43734 | Restate implication as con... |
| ifpbi23 43735 | Equivalence theorem for co... |
| ifpbiidcor 43736 | Restatement of ~ biid . (... |
| ifpbicor 43737 | Corollary of commutation o... |
| ifpxorcor 43738 | Corollary of commutation o... |
| ifpbi1 43739 | Equivalence theorem for co... |
| ifpnot23 43740 | Negation of conditional lo... |
| ifpnotnotb 43741 | Factor conditional logic o... |
| ifpnorcor 43742 | Corollary of commutation o... |
| ifpnancor 43743 | Corollary of commutation o... |
| ifpnot23b 43744 | Negation of conditional lo... |
| ifpbiidcor2 43745 | Restatement of ~ biid . (... |
| ifpnot23c 43746 | Negation of conditional lo... |
| ifpnot23d 43747 | Negation of conditional lo... |
| ifpdfnan 43748 | Define nand as conditional... |
| ifpdfxor 43749 | Define xor as conditional ... |
| ifpbi12 43750 | Equivalence theorem for co... |
| ifpbi13 43751 | Equivalence theorem for co... |
| ifpbi123 43752 | Equivalence theorem for co... |
| ifpidg 43753 | Restate wff as conditional... |
| ifpid3g 43754 | Restate wff as conditional... |
| ifpid2g 43755 | Restate wff as conditional... |
| ifpid1g 43756 | Restate wff as conditional... |
| ifpim23g 43757 | Restate implication as con... |
| ifpim3 43758 | Restate implication as con... |
| ifpnim1 43759 | Restate negated implicatio... |
| ifpim4 43760 | Restate implication as con... |
| ifpnim2 43761 | Restate negated implicatio... |
| ifpim123g 43762 | Implication of conditional... |
| ifpim1g 43763 | Implication of conditional... |
| ifp1bi 43764 | Substitute the first eleme... |
| ifpbi1b 43765 | When the first variable is... |
| ifpimimb 43766 | Factor conditional logic o... |
| ifpororb 43767 | Factor conditional logic o... |
| ifpananb 43768 | Factor conditional logic o... |
| ifpnannanb 43769 | Factor conditional logic o... |
| ifpor123g 43770 | Disjunction of conditional... |
| ifpimim 43771 | Consequnce of implication.... |
| ifpbibib 43772 | Factor conditional logic o... |
| ifpxorxorb 43773 | Factor conditional logic o... |
| rp-fakeimass 43774 | A special case where impli... |
| rp-fakeanorass 43775 | A special case where a mix... |
| rp-fakeoranass 43776 | A special case where a mix... |
| rp-fakeinunass 43777 | A special case where a mix... |
| rp-fakeuninass 43778 | A special case where a mix... |
| rp-isfinite5 43779 | A set is said to be finite... |
| rp-isfinite6 43780 | A set is said to be finite... |
| intabssd 43781 | When for each element ` y ... |
| eu0 43782 | There is only one empty se... |
| epelon2 43783 | Over the ordinal numbers, ... |
| ontric3g 43784 | For all ` x , y e. On ` , ... |
| dfsucon 43785 | ` A ` is called a successo... |
| snen1g 43786 | A singleton is equinumerou... |
| snen1el 43787 | A singleton is equinumerou... |
| sn1dom 43788 | A singleton is dominated b... |
| pr2dom 43789 | An unordered pair is domin... |
| tr3dom 43790 | An unordered triple is dom... |
| ensucne0 43791 | A class equinumerous to a ... |
| ensucne0OLD 43792 | A class equinumerous to a ... |
| dfom6 43793 | Let ` _om ` be defined to ... |
| infordmin 43794 | ` _om ` is the smallest in... |
| iscard4 43795 | Two ways to express the pr... |
| minregex 43796 | Given any cardinal number ... |
| minregex2 43797 | Given any cardinal number ... |
| iscard5 43798 | Two ways to express the pr... |
| elrncard 43799 | Let us define a cardinal n... |
| harval3 43800 | ` ( har `` A ) ` is the le... |
| harval3on 43801 | For any ordinal number ` A... |
| omssrncard 43802 | All natural numbers are ca... |
| 0iscard 43803 | 0 is a cardinal number. (... |
| 1iscard 43804 | 1 is a cardinal number. (... |
| omiscard 43805 | ` _om ` is a cardinal numb... |
| sucomisnotcard 43806 | ` _om +o 1o ` is not a car... |
| nna1iscard 43807 | For any natural number, th... |
| har2o 43808 | The least cardinal greater... |
| en2pr 43809 | A class is equinumerous to... |
| pr2cv 43810 | If an unordered pair is eq... |
| pr2el1 43811 | If an unordered pair is eq... |
| pr2cv1 43812 | If an unordered pair is eq... |
| pr2el2 43813 | If an unordered pair is eq... |
| pr2cv2 43814 | If an unordered pair is eq... |
| pren2 43815 | An unordered pair is equin... |
| pr2eldif1 43816 | If an unordered pair is eq... |
| pr2eldif2 43817 | If an unordered pair is eq... |
| pren2d 43818 | A pair of two distinct set... |
| aleph1min 43819 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 43820 | ` aleph ` is a strictly or... |
| alephiso3 43821 | ` aleph ` is a strictly or... |
| pwelg 43822 | The powerclass is an eleme... |
| pwinfig 43823 | The powerclass of an infin... |
| pwinfi2 43824 | The powerclass of an infin... |
| pwinfi3 43825 | The powerclass of an infin... |
| pwinfi 43826 | The powerclass of an infin... |
| fipjust 43827 | A definition of the finite... |
| cllem0 43828 | The class of all sets with... |
| superficl 43829 | The class of all supersets... |
| superuncl 43830 | The class of all supersets... |
| ssficl 43831 | The class of all subsets o... |
| ssuncl 43832 | The class of all subsets o... |
| ssdifcl 43833 | The class of all subsets o... |
| sssymdifcl 43834 | The class of all subsets o... |
| fiinfi 43835 | If two classes have the fi... |
| rababg 43836 | Condition when restricted ... |
| elinintab 43837 | Two ways of saying a set i... |
| elmapintrab 43838 | Two ways to say a set is a... |
| elinintrab 43839 | Two ways of saying a set i... |
| inintabss 43840 | Upper bound on intersectio... |
| inintabd 43841 | Value of the intersection ... |
| xpinintabd 43842 | Value of the intersection ... |
| relintabex 43843 | If the intersection of a c... |
| elcnvcnvintab 43844 | Two ways of saying a set i... |
| relintab 43845 | Value of the intersection ... |
| nonrel 43846 | A non-relation is equal to... |
| elnonrel 43847 | Only an ordered pair where... |
| cnvssb 43848 | Subclass theorem for conve... |
| relnonrel 43849 | The non-relation part of a... |
| cnvnonrel 43850 | The converse of the non-re... |
| brnonrel 43851 | A non-relation cannot rela... |
| dmnonrel 43852 | The domain of the non-rela... |
| rnnonrel 43853 | The range of the non-relat... |
| resnonrel 43854 | A restriction of the non-r... |
| imanonrel 43855 | An image under the non-rel... |
| cononrel1 43856 | Composition with the non-r... |
| cononrel2 43857 | Composition with the non-r... |
| elmapintab 43858 | Two ways to say a set is a... |
| fvnonrel 43859 | The function value of any ... |
| elinlem 43860 | Two ways to say a set is a... |
| elcnvcnvlem 43861 | Two ways to say a set is a... |
| cnvcnvintabd 43862 | Value of the relationship ... |
| elcnvlem 43863 | Two ways to say a set is a... |
| elcnvintab 43864 | Two ways of saying a set i... |
| cnvintabd 43865 | Value of the converse of t... |
| undmrnresiss 43866 | Two ways of saying the ide... |
| reflexg 43867 | Two ways of saying a relat... |
| cnvssco 43868 | A condition weaker than re... |
| refimssco 43869 | Reflexive relations are su... |
| cleq2lem 43870 | Equality implies bijection... |
| cbvcllem 43871 | Change of bound variable i... |
| clublem 43872 | If a superset ` Y ` of ` X... |
| clss2lem 43873 | The closure of a property ... |
| dfid7 43874 | Definition of identity rel... |
| mptrcllem 43875 | Show two versions of a clo... |
| cotrintab 43876 | The intersection of a clas... |
| rclexi 43877 | The reflexive closure of a... |
| rtrclexlem 43878 | Existence of relation impl... |
| rtrclex 43879 | The reflexive-transitive c... |
| trclubgNEW 43880 | If a relation exists then ... |
| trclubNEW 43881 | If a relation exists then ... |
| trclexi 43882 | The transitive closure of ... |
| rtrclexi 43883 | The reflexive-transitive c... |
| clrellem 43884 | When the property ` ps ` h... |
| clcnvlem 43885 | When ` A ` , an upper boun... |
| cnvtrucl0 43886 | The converse of the trivia... |
| cnvrcl0 43887 | The converse of the reflex... |
| cnvtrcl0 43888 | The converse of the transi... |
| dmtrcl 43889 | The domain of the transiti... |
| rntrcl 43890 | The range of the transitiv... |
| dfrtrcl5 43891 | Definition of reflexive-tr... |
| trcleq2lemRP 43892 | Equality implies bijection... |
| sqrtcvallem1 43893 | Two ways of saying a compl... |
| reabsifneg 43894 | Alternate expression for t... |
| reabsifnpos 43895 | Alternate expression for t... |
| reabsifpos 43896 | Alternate expression for t... |
| reabsifnneg 43897 | Alternate expression for t... |
| reabssgn 43898 | Alternate expression for t... |
| sqrtcvallem2 43899 | Equivalent to saying that ... |
| sqrtcvallem3 43900 | Equivalent to saying that ... |
| sqrtcvallem4 43901 | Equivalent to saying that ... |
| sqrtcvallem5 43902 | Equivalent to saying that ... |
| sqrtcval 43903 | Explicit formula for the c... |
| sqrtcval2 43904 | Explicit formula for the c... |
| resqrtval 43905 | Real part of the complex s... |
| imsqrtval 43906 | Imaginary part of the comp... |
| resqrtvalex 43907 | Example for ~ resqrtval . ... |
| imsqrtvalex 43908 | Example for ~ imsqrtval . ... |
| al3im 43909 | Version of ~ ax-4 for a ne... |
| intima0 43910 | Two ways of expressing the... |
| elimaint 43911 | Element of image of inters... |
| cnviun 43912 | Converse of indexed union.... |
| imaiun1 43913 | The image of an indexed un... |
| coiun1 43914 | Composition with an indexe... |
| elintima 43915 | Element of intersection of... |
| intimass 43916 | The image under the inters... |
| intimass2 43917 | The image under the inters... |
| intimag 43918 | Requirement for the image ... |
| intimasn 43919 | Two ways to express the im... |
| intimasn2 43920 | Two ways to express the im... |
| ss2iundf 43921 | Subclass theorem for index... |
| ss2iundv 43922 | Subclass theorem for index... |
| cbviuneq12df 43923 | Rule used to change the bo... |
| cbviuneq12dv 43924 | Rule used to change the bo... |
| conrel1d 43925 | Deduction about compositio... |
| conrel2d 43926 | Deduction about compositio... |
| trrelind 43927 | The intersection of transi... |
| xpintrreld 43928 | The intersection of a tran... |
| restrreld 43929 | The restriction of a trans... |
| trrelsuperreldg 43930 | Concrete construction of a... |
| trficl 43931 | The class of all transitiv... |
| cnvtrrel 43932 | The converse of a transiti... |
| trrelsuperrel2dg 43933 | Concrete construction of a... |
| dfrcl2 43936 | Reflexive closure of a rel... |
| dfrcl3 43937 | Reflexive closure of a rel... |
| dfrcl4 43938 | Reflexive closure of a rel... |
| relexp2 43939 | A set operated on by the r... |
| relexpnul 43940 | If the domain and range of... |
| eliunov2 43941 | Membership in the indexed ... |
| eltrclrec 43942 | Membership in the indexed ... |
| elrtrclrec 43943 | Membership in the indexed ... |
| briunov2 43944 | Two classes related by the... |
| brmptiunrelexpd 43945 | If two elements are connec... |
| fvmptiunrelexplb0d 43946 | If the indexed union range... |
| fvmptiunrelexplb0da 43947 | If the indexed union range... |
| fvmptiunrelexplb1d 43948 | If the indexed union range... |
| brfvid 43949 | If two elements are connec... |
| brfvidRP 43950 | If two elements are connec... |
| fvilbd 43951 | A set is a subset of its i... |
| fvilbdRP 43952 | A set is a subset of its i... |
| brfvrcld 43953 | If two elements are connec... |
| brfvrcld2 43954 | If two elements are connec... |
| fvrcllb0d 43955 | A restriction of the ident... |
| fvrcllb0da 43956 | A restriction of the ident... |
| fvrcllb1d 43957 | A set is a subset of its i... |
| brtrclrec 43958 | Two classes related by the... |
| brrtrclrec 43959 | Two classes related by the... |
| briunov2uz 43960 | Two classes related by the... |
| eliunov2uz 43961 | Membership in the indexed ... |
| ov2ssiunov2 43962 | Any particular operator va... |
| relexp0eq 43963 | The zeroth power of relati... |
| iunrelexp0 43964 | Simplification of zeroth p... |
| relexpxpnnidm 43965 | Any positive power of a Ca... |
| relexpiidm 43966 | Any power of any restricti... |
| relexpss1d 43967 | The relational power of a ... |
| comptiunov2i 43968 | The composition two indexe... |
| corclrcl 43969 | The reflexive closure is i... |
| iunrelexpmin1 43970 | The indexed union of relat... |
| relexpmulnn 43971 | With exponents limited to ... |
| relexpmulg 43972 | With ordered exponents, th... |
| trclrelexplem 43973 | The union of relational po... |
| iunrelexpmin2 43974 | The indexed union of relat... |
| relexp01min 43975 | With exponents limited to ... |
| relexp1idm 43976 | Repeated raising a relatio... |
| relexp0idm 43977 | Repeated raising a relatio... |
| relexp0a 43978 | Absorption law for zeroth ... |
| relexpxpmin 43979 | The composition of powers ... |
| relexpaddss 43980 | The composition of two pow... |
| iunrelexpuztr 43981 | The indexed union of relat... |
| dftrcl3 43982 | Transitive closure of a re... |
| brfvtrcld 43983 | If two elements are connec... |
| fvtrcllb1d 43984 | A set is a subset of its i... |
| trclfvcom 43985 | The transitive closure of ... |
| cnvtrclfv 43986 | The converse of the transi... |
| cotrcltrcl 43987 | The transitive closure is ... |
| trclimalb2 43988 | Lower bound for image unde... |
| brtrclfv2 43989 | Two ways to indicate two e... |
| trclfvdecomr 43990 | The transitive closure of ... |
| trclfvdecoml 43991 | The transitive closure of ... |
| dmtrclfvRP 43992 | The domain of the transiti... |
| rntrclfvRP 43993 | The range of the transitiv... |
| rntrclfv 43994 | The range of the transitiv... |
| dfrtrcl3 43995 | Reflexive-transitive closu... |
| brfvrtrcld 43996 | If two elements are connec... |
| fvrtrcllb0d 43997 | A restriction of the ident... |
| fvrtrcllb0da 43998 | A restriction of the ident... |
| fvrtrcllb1d 43999 | A set is a subset of its i... |
| dfrtrcl4 44000 | Reflexive-transitive closu... |
| corcltrcl 44001 | The composition of the ref... |
| cortrcltrcl 44002 | Composition with the refle... |
| corclrtrcl 44003 | Composition with the refle... |
| cotrclrcl 44004 | The composition of the ref... |
| cortrclrcl 44005 | Composition with the refle... |
| cotrclrtrcl 44006 | Composition with the refle... |
| cortrclrtrcl 44007 | The reflexive-transitive c... |
| frege77d 44008 | If the images of both ` { ... |
| frege81d 44009 | If the image of ` U ` is a... |
| frege83d 44010 | If the image of the union ... |
| frege96d 44011 | If ` C ` follows ` A ` in ... |
| frege87d 44012 | If the images of both ` { ... |
| frege91d 44013 | If ` B ` follows ` A ` in ... |
| frege97d 44014 | If ` A ` contains all elem... |
| frege98d 44015 | If ` C ` follows ` A ` and... |
| frege102d 44016 | If either ` A ` and ` C ` ... |
| frege106d 44017 | If ` B ` follows ` A ` in ... |
| frege108d 44018 | If either ` A ` and ` C ` ... |
| frege109d 44019 | If ` A ` contains all elem... |
| frege114d 44020 | If either ` R ` relates ` ... |
| frege111d 44021 | If either ` A ` and ` C ` ... |
| frege122d 44022 | If ` F ` is a function, ` ... |
| frege124d 44023 | If ` F ` is a function, ` ... |
| frege126d 44024 | If ` F ` is a function, ` ... |
| frege129d 44025 | If ` F ` is a function and... |
| frege131d 44026 | If ` F ` is a function and... |
| frege133d 44027 | If ` F ` is a function and... |
| dfxor4 44028 | Express exclusive-or in te... |
| dfxor5 44029 | Express exclusive-or in te... |
| df3or2 44030 | Express triple-or in terms... |
| df3an2 44031 | Express triple-and in term... |
| nev 44032 | Express that not every set... |
| 0pssin 44033 | Express that an intersecti... |
| dfhe2 44036 | The property of relation `... |
| dfhe3 44037 | The property of relation `... |
| heeq12 44038 | Equality law for relations... |
| heeq1 44039 | Equality law for relations... |
| heeq2 44040 | Equality law for relations... |
| sbcheg 44041 | Distribute proper substitu... |
| hess 44042 | Subclass law for relations... |
| xphe 44043 | Any Cartesian product is h... |
| 0he 44044 | The empty relation is here... |
| 0heALT 44045 | The empty relation is here... |
| he0 44046 | Any relation is hereditary... |
| unhe1 44047 | The union of two relations... |
| snhesn 44048 | Any singleton is hereditar... |
| idhe 44049 | The identity relation is h... |
| psshepw 44050 | The relation between sets ... |
| sshepw 44051 | The relation between sets ... |
| rp-simp2-frege 44054 | Simplification of triple c... |
| rp-simp2 44055 | Simplification of triple c... |
| rp-frege3g 44056 | Add antecedent to ~ ax-fre... |
| frege3 44057 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 44058 | Double-use of ~ ax-frege2 ... |
| rp-frege24 44059 | Introducing an embedded an... |
| rp-frege4g 44060 | Deduction related to distr... |
| frege4 44061 | Special case of closed for... |
| frege5 44062 | A closed form of ~ syl . ... |
| rp-7frege 44063 | Distribute antecedent and ... |
| rp-4frege 44064 | Elimination of a nested an... |
| rp-6frege 44065 | Elimination of a nested an... |
| rp-8frege 44066 | Eliminate antecedent when ... |
| rp-frege25 44067 | Closed form for ~ a1dd . ... |
| frege6 44068 | A closed form of ~ imim2d ... |
| axfrege8 44069 | Swap antecedents. Identic... |
| frege7 44070 | A closed form of ~ syl6 . ... |
| frege26 44072 | Identical to ~ idd . Prop... |
| frege27 44073 | We cannot (at the same tim... |
| frege9 44074 | Closed form of ~ syl with ... |
| frege12 44075 | A closed form of ~ com23 .... |
| frege11 44076 | Elimination of a nested an... |
| frege24 44077 | Closed form for ~ a1d . D... |
| frege16 44078 | A closed form of ~ com34 .... |
| frege25 44079 | Closed form for ~ a1dd . ... |
| frege18 44080 | Closed form of a syllogism... |
| frege22 44081 | A closed form of ~ com45 .... |
| frege10 44082 | Result commuting anteceden... |
| frege17 44083 | A closed form of ~ com3l .... |
| frege13 44084 | A closed form of ~ com3r .... |
| frege14 44085 | Closed form of a deduction... |
| frege19 44086 | A closed form of ~ syl6 . ... |
| frege23 44087 | Syllogism followed by rota... |
| frege15 44088 | A closed form of ~ com4r .... |
| frege21 44089 | Replace antecedent in ante... |
| frege20 44090 | A closed form of ~ syl8 . ... |
| axfrege28 44091 | Contraposition. Identical... |
| frege29 44093 | Closed form of ~ con3d . ... |
| frege30 44094 | Commuted, closed form of ~... |
| axfrege31 44095 | Identical to ~ notnotr . ... |
| frege32 44097 | Deduce ~ con1 from ~ con3 ... |
| frege33 44098 | If ` ph ` or ` ps ` takes ... |
| frege34 44099 | If as a consequence of the... |
| frege35 44100 | Commuted, closed form of ~... |
| frege36 44101 | The case in which ` ps ` i... |
| frege37 44102 | If ` ch ` is a necessary c... |
| frege38 44103 | Identical to ~ pm2.21 . P... |
| frege39 44104 | Syllogism between ~ pm2.18... |
| frege40 44105 | Anything implies ~ pm2.18 ... |
| axfrege41 44106 | Identical to ~ notnot . A... |
| frege42 44108 | Not not ~ id . Propositio... |
| frege43 44109 | If there is a choice only ... |
| frege44 44110 | Similar to a commuted ~ pm... |
| frege45 44111 | Deduce ~ pm2.6 from ~ con1... |
| frege46 44112 | If ` ps ` holds when ` ph ... |
| frege47 44113 | Deduce consequence follows... |
| frege48 44114 | Closed form of syllogism w... |
| frege49 44115 | Closed form of deduction w... |
| frege50 44116 | Closed form of ~ jaoi . P... |
| frege51 44117 | Compare with ~ jaod . Pro... |
| axfrege52a 44118 | Justification for ~ ax-fre... |
| frege52aid 44120 | The case when the content ... |
| frege53aid 44121 | Specialization of ~ frege5... |
| frege53a 44122 | Lemma for ~ frege55a . Pr... |
| axfrege54a 44123 | Justification for ~ ax-fre... |
| frege54cor0a 44125 | Synonym for logical equiva... |
| frege54cor1a 44126 | Reflexive equality. (Cont... |
| frege55aid 44127 | Lemma for ~ frege57aid . ... |
| frege55lem1a 44128 | Necessary deduction regard... |
| frege55lem2a 44129 | Core proof of Proposition ... |
| frege55a 44130 | Proposition 55 of [Frege18... |
| frege55cor1a 44131 | Proposition 55 of [Frege18... |
| frege56aid 44132 | Lemma for ~ frege57aid . ... |
| frege56a 44133 | Proposition 56 of [Frege18... |
| frege57aid 44134 | This is the all important ... |
| frege57a 44135 | Analogue of ~ frege57aid .... |
| axfrege58a 44136 | Identical to ~ anifp . Ju... |
| frege58acor 44138 | Lemma for ~ frege59a . (C... |
| frege59a 44139 | A kind of Aristotelian inf... |
| frege60a 44140 | Swap antecedents of ~ ax-f... |
| frege61a 44141 | Lemma for ~ frege65a . Pr... |
| frege62a 44142 | A kind of Aristotelian inf... |
| frege63a 44143 | Proposition 63 of [Frege18... |
| frege64a 44144 | Lemma for ~ frege65a . Pr... |
| frege65a 44145 | A kind of Aristotelian inf... |
| frege66a 44146 | Swap antecedents of ~ freg... |
| frege67a 44147 | Lemma for ~ frege68a . Pr... |
| frege68a 44148 | Combination of applying a ... |
| axfrege52c 44149 | Justification for ~ ax-fre... |
| frege52b 44151 | The case when the content ... |
| frege53b 44152 | Lemma for frege102 (via ~ ... |
| axfrege54c 44153 | Reflexive equality of clas... |
| frege54b 44155 | Reflexive equality of sets... |
| frege54cor1b 44156 | Reflexive equality. (Cont... |
| frege55lem1b 44157 | Necessary deduction regard... |
| frege55lem2b 44158 | Lemma for ~ frege55b . Co... |
| frege55b 44159 | Lemma for ~ frege57b . Pr... |
| frege56b 44160 | Lemma for ~ frege57b . Pr... |
| frege57b 44161 | Analogue of ~ frege57aid .... |
| axfrege58b 44162 | If ` A. x ph ` is affirmed... |
| frege58bid 44164 | If ` A. x ph ` is affirmed... |
| frege58bcor 44165 | Lemma for ~ frege59b . (C... |
| frege59b 44166 | A kind of Aristotelian inf... |
| frege60b 44167 | Swap antecedents of ~ ax-f... |
| frege61b 44168 | Lemma for ~ frege65b . Pr... |
| frege62b 44169 | A kind of Aristotelian inf... |
| frege63b 44170 | Lemma for ~ frege91 . Pro... |
| frege64b 44171 | Lemma for ~ frege65b . Pr... |
| frege65b 44172 | A kind of Aristotelian inf... |
| frege66b 44173 | Swap antecedents of ~ freg... |
| frege67b 44174 | Lemma for ~ frege68b . Pr... |
| frege68b 44175 | Combination of applying a ... |
| frege53c 44176 | Proposition 53 of [Frege18... |
| frege54cor1c 44177 | Reflexive equality. (Cont... |
| frege55lem1c 44178 | Necessary deduction regard... |
| frege55lem2c 44179 | Core proof of Proposition ... |
| frege55c 44180 | Proposition 55 of [Frege18... |
| frege56c 44181 | Lemma for ~ frege57c . Pr... |
| frege57c 44182 | Swap order of implication ... |
| frege58c 44183 | Principle related to ~ sp ... |
| frege59c 44184 | A kind of Aristotelian inf... |
| frege60c 44185 | Swap antecedents of ~ freg... |
| frege61c 44186 | Lemma for ~ frege65c . Pr... |
| frege62c 44187 | A kind of Aristotelian inf... |
| frege63c 44188 | Analogue of ~ frege63b . ... |
| frege64c 44189 | Lemma for ~ frege65c . Pr... |
| frege65c 44190 | A kind of Aristotelian inf... |
| frege66c 44191 | Swap antecedents of ~ freg... |
| frege67c 44192 | Lemma for ~ frege68c . Pr... |
| frege68c 44193 | Combination of applying a ... |
| dffrege69 44194 | If from the proposition th... |
| frege70 44195 | Lemma for ~ frege72 . Pro... |
| frege71 44196 | Lemma for ~ frege72 . Pro... |
| frege72 44197 | If property ` A ` is hered... |
| frege73 44198 | Lemma for ~ frege87 . Pro... |
| frege74 44199 | If ` X ` has a property ` ... |
| frege75 44200 | If from the proposition th... |
| dffrege76 44201 | If from the two propositio... |
| frege77 44202 | If ` Y ` follows ` X ` in ... |
| frege78 44203 | Commuted form of ~ frege77... |
| frege79 44204 | Distributed form of ~ freg... |
| frege80 44205 | Add additional condition t... |
| frege81 44206 | If ` X ` has a property ` ... |
| frege82 44207 | Closed-form deduction base... |
| frege83 44208 | Apply commuted form of ~ f... |
| frege84 44209 | Commuted form of ~ frege81... |
| frege85 44210 | Commuted form of ~ frege77... |
| frege86 44211 | Conclusion about element o... |
| frege87 44212 | If ` Z ` is a result of an... |
| frege88 44213 | Commuted form of ~ frege87... |
| frege89 44214 | One direction of ~ dffrege... |
| frege90 44215 | Add antecedent to ~ frege8... |
| frege91 44216 | Every result of an applica... |
| frege92 44217 | Inference from ~ frege91 .... |
| frege93 44218 | Necessary condition for tw... |
| frege94 44219 | Looking one past a pair re... |
| frege95 44220 | Looking one past a pair re... |
| frege96 44221 | Every result of an applica... |
| frege97 44222 | The property of following ... |
| frege98 44223 | If ` Y ` follows ` X ` and... |
| dffrege99 44224 | If ` Z ` is identical with... |
| frege100 44225 | One direction of ~ dffrege... |
| frege101 44226 | Lemma for ~ frege102 . Pr... |
| frege102 44227 | If ` Z ` belongs to the ` ... |
| frege103 44228 | Proposition 103 of [Frege1... |
| frege104 44229 | Proposition 104 of [Frege1... |
| frege105 44230 | Proposition 105 of [Frege1... |
| frege106 44231 | Whatever follows ` X ` in ... |
| frege107 44232 | Proposition 107 of [Frege1... |
| frege108 44233 | If ` Y ` belongs to the ` ... |
| frege109 44234 | The property of belonging ... |
| frege110 44235 | Proposition 110 of [Frege1... |
| frege111 44236 | If ` Y ` belongs to the ` ... |
| frege112 44237 | Identity implies belonging... |
| frege113 44238 | Proposition 113 of [Frege1... |
| frege114 44239 | If ` X ` belongs to the ` ... |
| dffrege115 44240 | If from the circumstance t... |
| frege116 44241 | One direction of ~ dffrege... |
| frege117 44242 | Lemma for ~ frege118 . Pr... |
| frege118 44243 | Simplified application of ... |
| frege119 44244 | Lemma for ~ frege120 . Pr... |
| frege120 44245 | Simplified application of ... |
| frege121 44246 | Lemma for ~ frege122 . Pr... |
| frege122 44247 | If ` X ` is a result of an... |
| frege123 44248 | Lemma for ~ frege124 . Pr... |
| frege124 44249 | If ` X ` is a result of an... |
| frege125 44250 | Lemma for ~ frege126 . Pr... |
| frege126 44251 | If ` M ` follows ` Y ` in ... |
| frege127 44252 | Communte antecedents of ~ ... |
| frege128 44253 | Lemma for ~ frege129 . Pr... |
| frege129 44254 | If the procedure ` R ` is ... |
| frege130 44255 | Lemma for ~ frege131 . Pr... |
| frege131 44256 | If the procedure ` R ` is ... |
| frege132 44257 | Lemma for ~ frege133 . Pr... |
| frege133 44258 | If the procedure ` R ` is ... |
| enrelmap 44259 | The set of all possible re... |
| enrelmapr 44260 | The set of all possible re... |
| enmappw 44261 | The set of all mappings fr... |
| enmappwid 44262 | The set of all mappings fr... |
| rfovd 44263 | Value of the operator, ` (... |
| rfovfvd 44264 | Value of the operator, ` (... |
| rfovfvfvd 44265 | Value of the operator, ` (... |
| rfovcnvf1od 44266 | Properties of the operator... |
| rfovcnvd 44267 | Value of the converse of t... |
| rfovf1od 44268 | The value of the operator,... |
| rfovcnvfvd 44269 | Value of the converse of t... |
| fsovd 44270 | Value of the operator, ` (... |
| fsovrfovd 44271 | The operator which gives a... |
| fsovfvd 44272 | Value of the operator, ` (... |
| fsovfvfvd 44273 | Value of the operator, ` (... |
| fsovfd 44274 | The operator, ` ( A O B ) ... |
| fsovcnvlem 44275 | The ` O ` operator, which ... |
| fsovcnvd 44276 | The value of the converse ... |
| fsovcnvfvd 44277 | The value of the converse ... |
| fsovf1od 44278 | The value of ` ( A O B ) `... |
| dssmapfvd 44279 | Value of the duality opera... |
| dssmapfv2d 44280 | Value of the duality opera... |
| dssmapfv3d 44281 | Value of the duality opera... |
| dssmapnvod 44282 | For any base set ` B ` the... |
| dssmapf1od 44283 | For any base set ` B ` the... |
| dssmap2d 44284 | For any base set ` B ` the... |
| or3or 44285 | Decompose disjunction into... |
| andi3or 44286 | Distribute over triple dis... |
| uneqsn 44287 | If a union of classes is e... |
| brfvimex 44288 | If a binary relation holds... |
| brovmptimex 44289 | If a binary relation holds... |
| brovmptimex1 44290 | If a binary relation holds... |
| brovmptimex2 44291 | If a binary relation holds... |
| brcoffn 44292 | Conditions allowing the de... |
| brcofffn 44293 | Conditions allowing the de... |
| brco2f1o 44294 | Conditions allowing the de... |
| brco3f1o 44295 | Conditions allowing the de... |
| ntrclsbex 44296 | If (pseudo-)interior and (... |
| ntrclsrcomplex 44297 | The relative complement of... |
| neik0imk0p 44298 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 44299 | If an interior function is... |
| ntrkbimka 44300 | If the interiors of disjoi... |
| ntrk0kbimka 44301 | If the interiors of disjoi... |
| clsk3nimkb 44302 | If the base set is not emp... |
| clsk1indlem0 44303 | The ansatz closure functio... |
| clsk1indlem2 44304 | The ansatz closure functio... |
| clsk1indlem3 44305 | The ansatz closure functio... |
| clsk1indlem4 44306 | The ansatz closure functio... |
| clsk1indlem1 44307 | The ansatz closure functio... |
| clsk1independent 44308 | For generalized closure fu... |
| neik0pk1imk0 44309 | Kuratowski's K0' and K1 ax... |
| isotone1 44310 | Two different ways to say ... |
| isotone2 44311 | Two different ways to say ... |
| ntrk1k3eqk13 44312 | An interior function is bo... |
| ntrclsf1o 44313 | If (pseudo-)interior and (... |
| ntrclsnvobr 44314 | If (pseudo-)interior and (... |
| ntrclsiex 44315 | If (pseudo-)interior and (... |
| ntrclskex 44316 | If (pseudo-)interior and (... |
| ntrclsfv1 44317 | If (pseudo-)interior and (... |
| ntrclsfv2 44318 | If (pseudo-)interior and (... |
| ntrclselnel1 44319 | If (pseudo-)interior and (... |
| ntrclselnel2 44320 | If (pseudo-)interior and (... |
| ntrclsfv 44321 | The value of the interior ... |
| ntrclsfveq1 44322 | If interior and closure fu... |
| ntrclsfveq2 44323 | If interior and closure fu... |
| ntrclsfveq 44324 | If interior and closure fu... |
| ntrclsss 44325 | If interior and closure fu... |
| ntrclsneine0lem 44326 | If (pseudo-)interior and (... |
| ntrclsneine0 44327 | If (pseudo-)interior and (... |
| ntrclscls00 44328 | If (pseudo-)interior and (... |
| ntrclsiso 44329 | If (pseudo-)interior and (... |
| ntrclsk2 44330 | An interior function is co... |
| ntrclskb 44331 | The interiors of disjoint ... |
| ntrclsk3 44332 | The intersection of interi... |
| ntrclsk13 44333 | The interior of the inters... |
| ntrclsk4 44334 | Idempotence of the interio... |
| ntrneibex 44335 | If (pseudo-)interior and (... |
| ntrneircomplex 44336 | The relative complement of... |
| ntrneif1o 44337 | If (pseudo-)interior and (... |
| ntrneiiex 44338 | If (pseudo-)interior and (... |
| ntrneinex 44339 | If (pseudo-)interior and (... |
| ntrneicnv 44340 | If (pseudo-)interior and (... |
| ntrneifv1 44341 | If (pseudo-)interior and (... |
| ntrneifv2 44342 | If (pseudo-)interior and (... |
| ntrneiel 44343 | If (pseudo-)interior and (... |
| ntrneifv3 44344 | The value of the neighbors... |
| ntrneineine0lem 44345 | If (pseudo-)interior and (... |
| ntrneineine1lem 44346 | If (pseudo-)interior and (... |
| ntrneifv4 44347 | The value of the interior ... |
| ntrneiel2 44348 | Membership in iterated int... |
| ntrneineine0 44349 | If (pseudo-)interior and (... |
| ntrneineine1 44350 | If (pseudo-)interior and (... |
| ntrneicls00 44351 | If (pseudo-)interior and (... |
| ntrneicls11 44352 | If (pseudo-)interior and (... |
| ntrneiiso 44353 | If (pseudo-)interior and (... |
| ntrneik2 44354 | An interior function is co... |
| ntrneix2 44355 | An interior (closure) func... |
| ntrneikb 44356 | The interiors of disjoint ... |
| ntrneixb 44357 | The interiors (closures) o... |
| ntrneik3 44358 | The intersection of interi... |
| ntrneix3 44359 | The closure of the union o... |
| ntrneik13 44360 | The interior of the inters... |
| ntrneix13 44361 | The closure of the union o... |
| ntrneik4w 44362 | Idempotence of the interio... |
| ntrneik4 44363 | Idempotence of the interio... |
| clsneibex 44364 | If (pseudo-)closure and (p... |
| clsneircomplex 44365 | The relative complement of... |
| clsneif1o 44366 | If a (pseudo-)closure func... |
| clsneicnv 44367 | If a (pseudo-)closure func... |
| clsneikex 44368 | If closure and neighborhoo... |
| clsneinex 44369 | If closure and neighborhoo... |
| clsneiel1 44370 | If a (pseudo-)closure func... |
| clsneiel2 44371 | If a (pseudo-)closure func... |
| clsneifv3 44372 | Value of the neighborhoods... |
| clsneifv4 44373 | Value of the closure (inte... |
| neicvgbex 44374 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 44375 | The relative complement of... |
| neicvgf1o 44376 | If neighborhood and conver... |
| neicvgnvo 44377 | If neighborhood and conver... |
| neicvgnvor 44378 | If neighborhood and conver... |
| neicvgmex 44379 | If the neighborhoods and c... |
| neicvgnex 44380 | If the neighborhoods and c... |
| neicvgel1 44381 | A subset being an element ... |
| neicvgel2 44382 | The complement of a subset... |
| neicvgfv 44383 | The value of the neighborh... |
| ntrrn 44384 | The range of the interior ... |
| ntrf 44385 | The interior function of a... |
| ntrf2 44386 | The interior function is a... |
| ntrelmap 44387 | The interior function is a... |
| clsf2 44388 | The closure function is a ... |
| clselmap 44389 | The closure function is a ... |
| dssmapntrcls 44390 | The interior and closure o... |
| dssmapclsntr 44391 | The closure and interior o... |
| gneispa 44392 | Each point ` p ` of the ne... |
| gneispb 44393 | Given a neighborhood ` N `... |
| gneispace2 44394 | The predicate that ` F ` i... |
| gneispace3 44395 | The predicate that ` F ` i... |
| gneispace 44396 | The predicate that ` F ` i... |
| gneispacef 44397 | A generic neighborhood spa... |
| gneispacef2 44398 | A generic neighborhood spa... |
| gneispacefun 44399 | A generic neighborhood spa... |
| gneispacern 44400 | A generic neighborhood spa... |
| gneispacern2 44401 | A generic neighborhood spa... |
| gneispace0nelrn 44402 | A generic neighborhood spa... |
| gneispace0nelrn2 44403 | A generic neighborhood spa... |
| gneispace0nelrn3 44404 | A generic neighborhood spa... |
| gneispaceel 44405 | Every neighborhood of a po... |
| gneispaceel2 44406 | Every neighborhood of a po... |
| gneispacess 44407 | All supersets of a neighbo... |
| gneispacess2 44408 | All supersets of a neighbo... |
| k0004lem1 44409 | Application of ~ ssin to r... |
| k0004lem2 44410 | A mapping with a particula... |
| k0004lem3 44411 | When the value of a mappin... |
| k0004val 44412 | The topological simplex of... |
| k0004ss1 44413 | The topological simplex of... |
| k0004ss2 44414 | The topological simplex of... |
| k0004ss3 44415 | The topological simplex of... |
| k0004val0 44416 | The topological simplex of... |
| inductionexd 44417 | Simple induction example. ... |
| wwlemuld 44418 | Natural deduction form of ... |
| leeq1d 44419 | Specialization of ~ breq1d... |
| leeq2d 44420 | Specialization of ~ breq2d... |
| absmulrposd 44421 | Specialization of absmuld ... |
| imadisjld 44422 | Natural dduction form of o... |
| wnefimgd 44423 | The image of a mapping fro... |
| fco2d 44424 | Natural deduction form of ... |
| wfximgfd 44425 | The value of a function on... |
| extoimad 44426 | If |f(x)| <= C for all x t... |
| imo72b2lem0 44427 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 44428 | Natural deduction form of ... |
| imo72b2lem2 44429 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 44430 | Natural deduction form of ... |
| imo72b2lem1 44431 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 44432 | 'Less than or equal to' re... |
| lemuldiv4d 44433 | 'Less than or equal to' re... |
| imo72b2 44434 | IMO 1972 B2. (14th Intern... |
| int-addcomd 44435 | AdditionCommutativity gene... |
| int-addassocd 44436 | AdditionAssociativity gene... |
| int-addsimpd 44437 | AdditionSimplification gen... |
| int-mulcomd 44438 | MultiplicationCommutativit... |
| int-mulassocd 44439 | MultiplicationAssociativit... |
| int-mulsimpd 44440 | MultiplicationSimplificati... |
| int-leftdistd 44441 | AdditionMultiplicationLeft... |
| int-rightdistd 44442 | AdditionMultiplicationRigh... |
| int-sqdefd 44443 | SquareDefinition generator... |
| int-mul11d 44444 | First MultiplicationOne ge... |
| int-mul12d 44445 | Second MultiplicationOne g... |
| int-add01d 44446 | First AdditionZero generat... |
| int-add02d 44447 | Second AdditionZero genera... |
| int-sqgeq0d 44448 | SquareGEQZero generator ru... |
| int-eqprincd 44449 | PrincipleOfEquality genera... |
| int-eqtransd 44450 | EqualityTransitivity gener... |
| int-eqmvtd 44451 | EquMoveTerm generator rule... |
| int-eqineqd 44452 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 44453 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 44454 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 44455 | SecondPrincipleOfInequalit... |
| int-ineqtransd 44456 | InequalityTransitivity gen... |
| unitadd 44457 | Theorem used in conjunctio... |
| gsumws3 44458 | Valuation of a length 3 wo... |
| gsumws4 44459 | Valuation of a length 4 wo... |
| amgm2d 44460 | Arithmetic-geometric mean ... |
| amgm3d 44461 | Arithmetic-geometric mean ... |
| amgm4d 44462 | Arithmetic-geometric mean ... |
| spALT 44463 | ~ sp can be proven from th... |
| elnelneqd 44464 | Two classes are not equal ... |
| elnelneq2d 44465 | Two classes are not equal ... |
| rr-spce 44466 | Prove an existential. (Co... |
| rexlimdvaacbv 44467 | Unpack a restricted existe... |
| rexlimddvcbvw 44468 | Unpack a restricted existe... |
| rexlimddvcbv 44469 | Unpack a restricted existe... |
| rr-elrnmpt3d 44470 | Elementhood in an image se... |
| rr-phpd 44471 | Equivalent of ~ php withou... |
| tfindsd 44472 | Deduction associated with ... |
| mnringvald 44475 | Value of the monoid ring f... |
| mnringnmulrd 44476 | Components of a monoid rin... |
| mnringbased 44477 | The base set of a monoid r... |
| mnringbaserd 44478 | The base set of a monoid r... |
| mnringelbased 44479 | Membership in the base set... |
| mnringbasefd 44480 | Elements of a monoid ring ... |
| mnringbasefsuppd 44481 | Elements of a monoid ring ... |
| mnringaddgd 44482 | The additive operation of ... |
| mnring0gd 44483 | The additive identity of a... |
| mnring0g2d 44484 | The additive identity of a... |
| mnringmulrd 44485 | The ring product of a mono... |
| mnringscad 44486 | The scalar ring of a monoi... |
| mnringvscad 44487 | The scalar product of a mo... |
| mnringlmodd 44488 | Monoid rings are left modu... |
| mnringmulrvald 44489 | Value of multiplication in... |
| mnringmulrcld 44490 | Monoid rings are closed un... |
| gru0eld 44491 | A nonempty Grothendieck un... |
| grusucd 44492 | Grothendieck universes are... |
| r1rankcld 44493 | Any rank of the cumulative... |
| grur1cld 44494 | Grothendieck universes are... |
| grurankcld 44495 | Grothendieck universes are... |
| grurankrcld 44496 | If a Grothendieck universe... |
| scotteqd 44499 | Equality theorem for the S... |
| scotteq 44500 | Closed form of ~ scotteqd ... |
| nfscott 44501 | Bound-variable hypothesis ... |
| scottabf 44502 | Value of the Scott operati... |
| scottab 44503 | Value of the Scott operati... |
| scottabes 44504 | Value of the Scott operati... |
| scottss 44505 | Scott's trick produces a s... |
| elscottab 44506 | An element of the output o... |
| scottex2 44507 | ~ scottex expressed using ... |
| scotteld 44508 | The Scott operation sends ... |
| scottelrankd 44509 | Property of a Scott's tric... |
| scottrankd 44510 | Rank of a nonempty Scott's... |
| gruscottcld 44511 | If a Grothendieck universe... |
| dfcoll2 44514 | Alternate definition of th... |
| colleq12d 44515 | Equality theorem for the c... |
| colleq1 44516 | Equality theorem for the c... |
| colleq2 44517 | Equality theorem for the c... |
| nfcoll 44518 | Bound-variable hypothesis ... |
| collexd 44519 | The output of the collecti... |
| cpcolld 44520 | Property of the collection... |
| cpcoll2d 44521 | ~ cpcolld with an extra ex... |
| grucollcld 44522 | A Grothendieck universe co... |
| ismnu 44523 | The hypothesis of this the... |
| mnuop123d 44524 | Operations of a minimal un... |
| mnussd 44525 | Minimal universes are clos... |
| mnuss2d 44526 | ~ mnussd with arguments pr... |
| mnu0eld 44527 | A nonempty minimal univers... |
| mnuop23d 44528 | Second and third operation... |
| mnupwd 44529 | Minimal universes are clos... |
| mnusnd 44530 | Minimal universes are clos... |
| mnuprssd 44531 | A minimal universe contain... |
| mnuprss2d 44532 | Special case of ~ mnuprssd... |
| mnuop3d 44533 | Third operation of a minim... |
| mnuprdlem1 44534 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 44535 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 44536 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 44537 | Lemma for ~ mnuprd . Gene... |
| mnuprd 44538 | Minimal universes are clos... |
| mnuunid 44539 | Minimal universes are clos... |
| mnuund 44540 | Minimal universes are clos... |
| mnutrcld 44541 | Minimal universes contain ... |
| mnutrd 44542 | Minimal universes are tran... |
| mnurndlem1 44543 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 44544 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 44545 | Minimal universes contain ... |
| mnugrud 44546 | Minimal universes are Grot... |
| grumnudlem 44547 | Lemma for ~ grumnud . (Co... |
| grumnud 44548 | Grothendieck universes are... |
| grumnueq 44549 | The class of Grothendieck ... |
| expandan 44550 | Expand conjunction to prim... |
| expandexn 44551 | Expand an existential quan... |
| expandral 44552 | Expand a restricted univer... |
| expandrexn 44553 | Expand a restricted existe... |
| expandrex 44554 | Expand a restricted existe... |
| expanduniss 44555 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 44556 | Express the predicate on `... |
| rr-grothprimbi 44557 | Express "every set is cont... |
| inagrud 44558 | Inaccessible levels of the... |
| inaex 44559 | Assuming the Tarski-Grothe... |
| gruex 44560 | Assuming the Tarski-Grothe... |
| rr-groth 44561 | An equivalent of ~ ax-grot... |
| rr-grothprim 44562 | An equivalent of ~ ax-grot... |
| ismnushort 44563 | Express the predicate on `... |
| dfuniv2 44564 | Alternative definition of ... |
| rr-grothshortbi 44565 | Express "every set is cont... |
| rr-grothshort 44566 | A shorter equivalent of ~ ... |
| nanorxor 44567 | 'nand' is equivalent to th... |
| undisjrab 44568 | Union of two disjoint rest... |
| iso0 44569 | The empty set is an ` R , ... |
| ssrecnpr 44570 | ` RR ` is a subset of both... |
| seff 44571 | Let set ` S ` be the real ... |
| sblpnf 44572 | The infinity ball in the a... |
| prmunb2 44573 | The primes are unbounded. ... |
| dvgrat 44574 | Ratio test for divergence ... |
| cvgdvgrat 44575 | Ratio test for convergence... |
| radcnvrat 44576 | Let ` L ` be the limit, if... |
| reldvds 44577 | The divides relation is in... |
| nznngen 44578 | All positive integers in t... |
| nzss 44579 | The set of multiples of _m... |
| nzin 44580 | The intersection of the se... |
| nzprmdif 44581 | Subtract one prime's multi... |
| hashnzfz 44582 | Special case of ~ hashdvds... |
| hashnzfz2 44583 | Special case of ~ hashnzfz... |
| hashnzfzclim 44584 | As the upper bound ` K ` o... |
| caofcan 44585 | Transfer a cancellation la... |
| ofsubid 44586 | Function analogue of ~ sub... |
| ofmul12 44587 | Function analogue of ~ mul... |
| ofdivrec 44588 | Function analogue of ~ div... |
| ofdivcan4 44589 | Function analogue of ~ div... |
| ofdivdiv2 44590 | Function analogue of ~ div... |
| lhe4.4ex1a 44591 | Example of the Fundamental... |
| dvsconst 44592 | Derivative of a constant f... |
| dvsid 44593 | Derivative of the identity... |
| dvsef 44594 | Derivative of the exponent... |
| expgrowthi 44595 | Exponential growth and dec... |
| dvconstbi 44596 | The derivative of a functi... |
| expgrowth 44597 | Exponential growth and dec... |
| bccval 44600 | Value of the generalized b... |
| bcccl 44601 | Closure of the generalized... |
| bcc0 44602 | The generalized binomial c... |
| bccp1k 44603 | Generalized binomial coeff... |
| bccm1k 44604 | Generalized binomial coeff... |
| bccn0 44605 | Generalized binomial coeff... |
| bccn1 44606 | Generalized binomial coeff... |
| bccbc 44607 | The binomial coefficient a... |
| uzmptshftfval 44608 | When ` F ` is a maps-to fu... |
| dvradcnv2 44609 | The radius of convergence ... |
| binomcxplemwb 44610 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 44611 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 44612 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 44613 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 44614 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 44615 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 44616 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 44617 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 44618 | Lemma for ~ binomcxp . Wh... |
| binomcxp 44619 | Generalize the binomial th... |
| pm10.12 44620 | Theorem *10.12 in [Whitehe... |
| pm10.14 44621 | Theorem *10.14 in [Whitehe... |
| pm10.251 44622 | Theorem *10.251 in [Whiteh... |
| pm10.252 44623 | Theorem *10.252 in [Whiteh... |
| pm10.253 44624 | Theorem *10.253 in [Whiteh... |
| albitr 44625 | Theorem *10.301 in [Whiteh... |
| pm10.42 44626 | Theorem *10.42 in [Whitehe... |
| pm10.52 44627 | Theorem *10.52 in [Whitehe... |
| pm10.53 44628 | Theorem *10.53 in [Whitehe... |
| pm10.541 44629 | Theorem *10.541 in [Whiteh... |
| pm10.542 44630 | Theorem *10.542 in [Whiteh... |
| pm10.55 44631 | Theorem *10.55 in [Whitehe... |
| pm10.56 44632 | Theorem *10.56 in [Whitehe... |
| pm10.57 44633 | Theorem *10.57 in [Whitehe... |
| 2alanimi 44634 | Removes two universal quan... |
| 2al2imi 44635 | Removes two universal quan... |
| pm11.11 44636 | Theorem *11.11 in [Whitehe... |
| pm11.12 44637 | Theorem *11.12 in [Whitehe... |
| 19.21vv 44638 | Compare Theorem *11.3 in [... |
| 2alim 44639 | Theorem *11.32 in [Whitehe... |
| 2albi 44640 | Theorem *11.33 in [Whitehe... |
| 2exim 44641 | Theorem *11.34 in [Whitehe... |
| 2exbi 44642 | Theorem *11.341 in [Whiteh... |
| spsbce-2 44643 | Theorem *11.36 in [Whitehe... |
| 19.33-2 44644 | Theorem *11.421 in [Whiteh... |
| 19.36vv 44645 | Theorem *11.43 in [Whitehe... |
| 19.31vv 44646 | Theorem *11.44 in [Whitehe... |
| 19.37vv 44647 | Theorem *11.46 in [Whitehe... |
| 19.28vv 44648 | Theorem *11.47 in [Whitehe... |
| pm11.52 44649 | Theorem *11.52 in [Whitehe... |
| aaanv 44650 | Theorem *11.56 in [Whitehe... |
| pm11.57 44651 | Theorem *11.57 in [Whitehe... |
| pm11.58 44652 | Theorem *11.58 in [Whitehe... |
| pm11.59 44653 | Theorem *11.59 in [Whitehe... |
| pm11.6 44654 | Theorem *11.6 in [Whitehea... |
| pm11.61 44655 | Theorem *11.61 in [Whitehe... |
| pm11.62 44656 | Theorem *11.62 in [Whitehe... |
| pm11.63 44657 | Theorem *11.63 in [Whitehe... |
| pm11.7 44658 | Theorem *11.7 in [Whitehea... |
| pm11.71 44659 | Theorem *11.71 in [Whitehe... |
| sbeqal1 44660 | If ` x = y ` always implie... |
| sbeqal1i 44661 | Suppose you know ` x = y `... |
| sbeqal2i 44662 | If ` x = y ` implies ` x =... |
| axc5c4c711 44663 | Proof of a theorem that ca... |
| axc5c4c711toc5 44664 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 44665 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 44666 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 44667 | Rederivation of ~ ax-11 fr... |
| axc11next 44668 | This theorem shows that, g... |
| pm13.13a 44669 | One result of theorem *13.... |
| pm13.13b 44670 | Theorem *13.13 in [Whitehe... |
| pm13.14 44671 | Theorem *13.14 in [Whitehe... |
| pm13.192 44672 | Theorem *13.192 in [Whiteh... |
| pm13.193 44673 | Theorem *13.193 in [Whiteh... |
| pm13.194 44674 | Theorem *13.194 in [Whiteh... |
| pm13.195 44675 | Theorem *13.195 in [Whiteh... |
| pm13.196a 44676 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 44677 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 44678 | Theorem *13.22 in [Whitehe... |
| iotain 44679 | Equivalence between two di... |
| iotaexeu 44680 | The iota class exists. Th... |
| iotasbc 44681 | Definition *14.01 in [Whit... |
| iotasbc2 44682 | Theorem *14.111 in [Whiteh... |
| pm14.12 44683 | Theorem *14.12 in [Whitehe... |
| pm14.122a 44684 | Theorem *14.122 in [Whiteh... |
| pm14.122b 44685 | Theorem *14.122 in [Whiteh... |
| pm14.122c 44686 | Theorem *14.122 in [Whiteh... |
| pm14.123a 44687 | Theorem *14.123 in [Whiteh... |
| pm14.123b 44688 | Theorem *14.123 in [Whiteh... |
| pm14.123c 44689 | Theorem *14.123 in [Whiteh... |
| pm14.18 44690 | Theorem *14.18 in [Whitehe... |
| iotaequ 44691 | Theorem *14.2 in [Whitehea... |
| iotavalb 44692 | Theorem *14.202 in [Whiteh... |
| iotasbc5 44693 | Theorem *14.205 in [Whiteh... |
| pm14.24 44694 | Theorem *14.24 in [Whitehe... |
| iotavalsb 44695 | Theorem *14.242 in [Whiteh... |
| sbiota1 44696 | Theorem *14.25 in [Whitehe... |
| sbaniota 44697 | Theorem *14.26 in [Whitehe... |
| iotasbcq 44698 | Theorem *14.272 in [Whiteh... |
| elnev 44699 | Any set that contains one ... |
| rusbcALT 44700 | A version of Russell's par... |
| compeq 44701 | Equality between two ways ... |
| compne 44702 | The complement of ` A ` is... |
| compab 44703 | Two ways of saying "the co... |
| conss2 44704 | Contrapositive law for sub... |
| conss1 44705 | Contrapositive law for sub... |
| ralbidar 44706 | More general form of ~ ral... |
| rexbidar 44707 | More general form of ~ rex... |
| dropab1 44708 | Theorem to aid use of the ... |
| dropab2 44709 | Theorem to aid use of the ... |
| ipo0 44710 | If the identity relation p... |
| ifr0 44711 | A class that is founded by... |
| ordpss 44712 | ~ ordelpss with an anteced... |
| fvsb 44713 | Explicit substitution of a... |
| fveqsb 44714 | Implicit substitution of a... |
| xpexb 44715 | A Cartesian product exists... |
| trelpss 44716 | An element of a transitive... |
| addcomgi 44717 | Generalization of commutat... |
| addrval 44727 | Value of the operation of ... |
| subrval 44728 | Value of the operation of ... |
| mulvval 44729 | Value of the operation of ... |
| addrfv 44730 | Vector addition at a value... |
| subrfv 44731 | Vector subtraction at a va... |
| mulvfv 44732 | Scalar multiplication at a... |
| addrfn 44733 | Vector addition produces a... |
| subrfn 44734 | Vector subtraction produce... |
| mulvfn 44735 | Scalar multiplication prod... |
| addrcom 44736 | Vector addition is commuta... |
| idiALT 44740 | Placeholder for ~ idi . T... |
| exbir 44741 | Exportation implication al... |
| 3impexpbicom 44742 | Version of ~ 3impexp where... |
| 3impexpbicomi 44743 | Inference associated with ... |
| bi1imp 44744 | Importation inference simi... |
| bi2imp 44745 | Importation inference simi... |
| bi3impb 44746 | Similar to ~ 3impb with im... |
| bi3impa 44747 | Similar to ~ 3impa with im... |
| bi23impib 44748 | ~ 3impib with the inner im... |
| bi13impib 44749 | ~ 3impib with the outer im... |
| bi123impib 44750 | ~ 3impib with the implicat... |
| bi13impia 44751 | ~ 3impia with the outer im... |
| bi123impia 44752 | ~ 3impia with the implicat... |
| bi33imp12 44753 | ~ 3imp with innermost impl... |
| bi13imp23 44754 | ~ 3imp with outermost impl... |
| bi13imp2 44755 | Similar to ~ 3imp except t... |
| bi12imp3 44756 | Similar to ~ 3imp except a... |
| bi23imp1 44757 | Similar to ~ 3imp except a... |
| bi123imp0 44758 | Similar to ~ 3imp except a... |
| 4animp1 44759 | A single hypothesis unific... |
| 4an31 44760 | A rearrangement of conjunc... |
| 4an4132 44761 | A rearrangement of conjunc... |
| expcomdg 44762 | Biconditional form of ~ ex... |
| iidn3 44763 | ~ idn3 without virtual ded... |
| ee222 44764 | ~ e222 without virtual ded... |
| ee3bir 44765 | Right-biconditional form o... |
| ee13 44766 | ~ e13 without virtual dedu... |
| ee121 44767 | ~ e121 without virtual ded... |
| ee122 44768 | ~ e122 without virtual ded... |
| ee333 44769 | ~ e333 without virtual ded... |
| ee323 44770 | ~ e323 without virtual ded... |
| 3ornot23 44771 | If the second and third di... |
| orbi1r 44772 | ~ orbi1 with order of disj... |
| 3orbi123 44773 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 44774 | Closed form of ~ syl5 . D... |
| impexpd 44775 | The following User's Proof... |
| com3rgbi 44776 | The following User's Proof... |
| impexpdcom 44777 | The following User's Proof... |
| ee1111 44778 | Non-virtual deduction form... |
| pm2.43bgbi 44779 | Logical equivalence of a 2... |
| pm2.43cbi 44780 | Logical equivalence of a 3... |
| ee233 44781 | Non-virtual deduction form... |
| imbi13 44782 | Join three logical equival... |
| ee33 44783 | Non-virtual deduction form... |
| con5 44784 | Biconditional contrapositi... |
| con5i 44785 | Inference form of ~ con5 .... |
| exlimexi 44786 | Inference similar to Theor... |
| sb5ALT 44787 | Equivalence for substituti... |
| eexinst01 44788 | ~ exinst01 without virtual... |
| eexinst11 44789 | ~ exinst11 without virtual... |
| vk15.4j 44790 | Excercise 4j of Unit 15 of... |
| notnotrALT 44791 | Converse of double negatio... |
| con3ALT2 44792 | Contraposition. Alternate... |
| ssralv2 44793 | Quantification restricted ... |
| sbc3or 44794 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 44795 | Closed form of ~ alrimi wi... |
| rspsbc2 44796 | ~ rspsbc with two quantify... |
| sbcoreleleq 44797 | Substitution of a setvar v... |
| tratrb 44798 | If a class is transitive a... |
| ordelordALT 44799 | An element of an ordinal c... |
| sbcim2g 44800 | Distribution of class subs... |
| sbcbi 44801 | Implication form of ~ sbcb... |
| trsbc 44802 | Formula-building inference... |
| truniALT 44803 | The union of a class of tr... |
| onfrALTlem5 44804 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 44805 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 44806 | Lemma for ~ onfrALT . (Co... |
| ggen31 44807 | ~ gen31 without virtual de... |
| onfrALTlem2 44808 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 44809 | A theorem pertaining to th... |
| onfrALTlem1 44810 | Lemma for ~ onfrALT . (Co... |
| onfrALT 44811 | The membership relation is... |
| 19.41rg 44812 | Closed form of right-to-le... |
| opelopab4 44813 | Ordered pair membership in... |
| 2pm13.193 44814 | ~ pm13.193 for two variabl... |
| hbntal 44815 | A closed form of ~ hbn . ~... |
| hbimpg 44816 | A closed form of ~ hbim . ... |
| hbalg 44817 | Closed form of ~ hbal . D... |
| hbexg 44818 | Closed form of ~ nfex . D... |
| ax6e2eq 44819 | Alternate form of ~ ax6e f... |
| ax6e2nd 44820 | If at least two sets exist... |
| ax6e2ndeq 44821 | "At least two sets exist" ... |
| 2sb5nd 44822 | Equivalence for double sub... |
| 2uasbanh 44823 | Distribute the unabbreviat... |
| 2uasban 44824 | Distribute the unabbreviat... |
| e2ebind 44825 | Absorption of an existenti... |
| elpwgded 44826 | ~ elpwgdedVD in convention... |
| trelded 44827 | Deduction form of ~ trel .... |
| jaoded 44828 | Deduction form of ~ jao . ... |
| sbtT 44829 | A substitution into a theo... |
| not12an2impnot1 44830 | If a double conjunction is... |
| in1 44833 | Inference form of ~ df-vd1... |
| iin1 44834 | ~ in1 without virtual dedu... |
| dfvd1ir 44835 | Inference form of ~ df-vd1... |
| idn1 44836 | Virtual deduction identity... |
| dfvd1imp 44837 | Left-to-right part of defi... |
| dfvd1impr 44838 | Right-to-left part of defi... |
| dfvd2 44841 | Definition of a 2-hypothes... |
| dfvd2an 44844 | Definition of a 2-hypothes... |
| dfvd2ani 44845 | Inference form of ~ dfvd2a... |
| dfvd2anir 44846 | Right-to-left inference fo... |
| dfvd2i 44847 | Inference form of ~ dfvd2 ... |
| dfvd2ir 44848 | Right-to-left inference fo... |
| dfvd3 44853 | Definition of a 3-hypothes... |
| dfvd3i 44854 | Inference form of ~ dfvd3 ... |
| dfvd3ir 44855 | Right-to-left inference fo... |
| dfvd3an 44856 | Definition of a 3-hypothes... |
| dfvd3ani 44857 | Inference form of ~ dfvd3a... |
| dfvd3anir 44858 | Right-to-left inference fo... |
| vd01 44859 | A virtual hypothesis virtu... |
| vd02 44860 | Two virtual hypotheses vir... |
| vd03 44861 | A theorem is virtually inf... |
| vd12 44862 | A virtual deduction with 1... |
| vd13 44863 | A virtual deduction with 1... |
| vd23 44864 | A virtual deduction with 2... |
| dfvd2imp 44865 | The virtual deduction form... |
| dfvd2impr 44866 | A 2-antecedent nested impl... |
| in2 44867 | The virtual deduction intr... |
| int2 44868 | The virtual deduction intr... |
| iin2 44869 | ~ in2 without virtual dedu... |
| in2an 44870 | The virtual deduction intr... |
| in3 44871 | The virtual deduction intr... |
| iin3 44872 | ~ in3 without virtual dedu... |
| in3an 44873 | The virtual deduction intr... |
| int3 44874 | The virtual deduction intr... |
| idn2 44875 | Virtual deduction identity... |
| iden2 44876 | Virtual deduction identity... |
| idn3 44877 | Virtual deduction identity... |
| gen11 44878 | Virtual deduction generali... |
| gen11nv 44879 | Virtual deduction generali... |
| gen12 44880 | Virtual deduction generali... |
| gen21 44881 | Virtual deduction generali... |
| gen21nv 44882 | Virtual deduction form of ... |
| gen31 44883 | Virtual deduction generali... |
| gen22 44884 | Virtual deduction generali... |
| ggen22 44885 | ~ gen22 without virtual de... |
| exinst 44886 | Existential Instantiation.... |
| exinst01 44887 | Existential Instantiation.... |
| exinst11 44888 | Existential Instantiation.... |
| e1a 44889 | A Virtual deduction elimin... |
| el1 44890 | A Virtual deduction elimin... |
| e1bi 44891 | Biconditional form of ~ e1... |
| e1bir 44892 | Right biconditional form o... |
| e2 44893 | A virtual deduction elimin... |
| e2bi 44894 | Biconditional form of ~ e2... |
| e2bir 44895 | Right biconditional form o... |
| ee223 44896 | ~ e223 without virtual ded... |
| e223 44897 | A virtual deduction elimin... |
| e222 44898 | A virtual deduction elimin... |
| e220 44899 | A virtual deduction elimin... |
| ee220 44900 | ~ e220 without virtual ded... |
| e202 44901 | A virtual deduction elimin... |
| ee202 44902 | ~ e202 without virtual ded... |
| e022 44903 | A virtual deduction elimin... |
| ee022 44904 | ~ e022 without virtual ded... |
| e002 44905 | A virtual deduction elimin... |
| ee002 44906 | ~ e002 without virtual ded... |
| e020 44907 | A virtual deduction elimin... |
| ee020 44908 | ~ e020 without virtual ded... |
| e200 44909 | A virtual deduction elimin... |
| ee200 44910 | ~ e200 without virtual ded... |
| e221 44911 | A virtual deduction elimin... |
| ee221 44912 | ~ e221 without virtual ded... |
| e212 44913 | A virtual deduction elimin... |
| ee212 44914 | ~ e212 without virtual ded... |
| e122 44915 | A virtual deduction elimin... |
| e112 44916 | A virtual deduction elimin... |
| ee112 44917 | ~ e112 without virtual ded... |
| e121 44918 | A virtual deduction elimin... |
| e211 44919 | A virtual deduction elimin... |
| ee211 44920 | ~ e211 without virtual ded... |
| e210 44921 | A virtual deduction elimin... |
| ee210 44922 | ~ e210 without virtual ded... |
| e201 44923 | A virtual deduction elimin... |
| ee201 44924 | ~ e201 without virtual ded... |
| e120 44925 | A virtual deduction elimin... |
| ee120 44926 | Virtual deduction rule ~ e... |
| e021 44927 | A virtual deduction elimin... |
| ee021 44928 | ~ e021 without virtual ded... |
| e012 44929 | A virtual deduction elimin... |
| ee012 44930 | ~ e012 without virtual ded... |
| e102 44931 | A virtual deduction elimin... |
| ee102 44932 | ~ e102 without virtual ded... |
| e22 44933 | A virtual deduction elimin... |
| e22an 44934 | Conjunction form of ~ e22 ... |
| ee22an 44935 | ~ e22an without virtual de... |
| e111 44936 | A virtual deduction elimin... |
| e1111 44937 | A virtual deduction elimin... |
| e110 44938 | A virtual deduction elimin... |
| ee110 44939 | ~ e110 without virtual ded... |
| e101 44940 | A virtual deduction elimin... |
| ee101 44941 | ~ e101 without virtual ded... |
| e011 44942 | A virtual deduction elimin... |
| ee011 44943 | ~ e011 without virtual ded... |
| e100 44944 | A virtual deduction elimin... |
| ee100 44945 | ~ e100 without virtual ded... |
| e010 44946 | A virtual deduction elimin... |
| ee010 44947 | ~ e010 without virtual ded... |
| e001 44948 | A virtual deduction elimin... |
| ee001 44949 | ~ e001 without virtual ded... |
| e11 44950 | A virtual deduction elimin... |
| e11an 44951 | Conjunction form of ~ e11 ... |
| ee11an 44952 | ~ e11an without virtual de... |
| e01 44953 | A virtual deduction elimin... |
| e01an 44954 | Conjunction form of ~ e01 ... |
| ee01an 44955 | ~ e01an without virtual de... |
| e10 44956 | A virtual deduction elimin... |
| e10an 44957 | Conjunction form of ~ e10 ... |
| ee10an 44958 | ~ e10an without virtual de... |
| e02 44959 | A virtual deduction elimin... |
| e02an 44960 | Conjunction form of ~ e02 ... |
| ee02an 44961 | ~ e02an without virtual de... |
| eel021old 44962 | ~ el021old without virtual... |
| el021old 44963 | A virtual deduction elimin... |
| eel000cT 44964 | An elimination deduction. ... |
| eel0TT 44965 | An elimination deduction. ... |
| eelT00 44966 | An elimination deduction. ... |
| eelTTT 44967 | An elimination deduction. ... |
| eelT11 44968 | An elimination deduction. ... |
| eelT1 44969 | Syllogism inference combin... |
| eelT12 44970 | An elimination deduction. ... |
| eelTT1 44971 | An elimination deduction. ... |
| eelT01 44972 | An elimination deduction. ... |
| eel0T1 44973 | An elimination deduction. ... |
| eel12131 44974 | An elimination deduction. ... |
| eel2131 44975 | ~ syl2an with antecedents ... |
| eel3132 44976 | ~ syl2an with antecedents ... |
| eel0321old 44977 | ~ el0321old without virtua... |
| el0321old 44978 | A virtual deduction elimin... |
| eel2122old 44979 | ~ el2122old without virtua... |
| el2122old 44980 | A virtual deduction elimin... |
| eel0000 44981 | Elimination rule similar t... |
| eel00001 44982 | An elimination deduction. ... |
| eel00000 44983 | Elimination rule similar ~... |
| eel11111 44984 | Five-hypothesis eliminatio... |
| e12 44985 | A virtual deduction elimin... |
| e12an 44986 | Conjunction form of ~ e12 ... |
| el12 44987 | Virtual deduction form of ... |
| e20 44988 | A virtual deduction elimin... |
| e20an 44989 | Conjunction form of ~ e20 ... |
| ee20an 44990 | ~ e20an without virtual de... |
| e21 44991 | A virtual deduction elimin... |
| e21an 44992 | Conjunction form of ~ e21 ... |
| ee21an 44993 | ~ e21an without virtual de... |
| e333 44994 | A virtual deduction elimin... |
| e33 44995 | A virtual deduction elimin... |
| e33an 44996 | Conjunction form of ~ e33 ... |
| ee33an 44997 | ~ e33an without virtual de... |
| e3 44998 | Meta-connective form of ~ ... |
| e3bi 44999 | Biconditional form of ~ e3... |
| e3bir 45000 | Right biconditional form o... |
| e03 45001 | A virtual deduction elimin... |
| ee03 45002 | ~ e03 without virtual dedu... |
| e03an 45003 | Conjunction form of ~ e03 ... |
| ee03an 45004 | Conjunction form of ~ ee03... |
| e30 45005 | A virtual deduction elimin... |
| ee30 45006 | ~ e30 without virtual dedu... |
| e30an 45007 | A virtual deduction elimin... |
| ee30an 45008 | Conjunction form of ~ ee30... |
| e13 45009 | A virtual deduction elimin... |
| e13an 45010 | A virtual deduction elimin... |
| ee13an 45011 | ~ e13an without virtual de... |
| e31 45012 | A virtual deduction elimin... |
| ee31 45013 | ~ e31 without virtual dedu... |
| e31an 45014 | A virtual deduction elimin... |
| ee31an 45015 | ~ e31an without virtual de... |
| e23 45016 | A virtual deduction elimin... |
| e23an 45017 | A virtual deduction elimin... |
| ee23an 45018 | ~ e23an without virtual de... |
| e32 45019 | A virtual deduction elimin... |
| ee32 45020 | ~ e32 without virtual dedu... |
| e32an 45021 | A virtual deduction elimin... |
| ee32an 45022 | ~ e33an without virtual de... |
| e123 45023 | A virtual deduction elimin... |
| ee123 45024 | ~ e123 without virtual ded... |
| el123 45025 | A virtual deduction elimin... |
| e233 45026 | A virtual deduction elimin... |
| e323 45027 | A virtual deduction elimin... |
| e000 45028 | A virtual deduction elimin... |
| e00 45029 | Elimination rule identical... |
| e00an 45030 | Elimination rule identical... |
| eel00cT 45031 | An elimination deduction. ... |
| eelTT 45032 | An elimination deduction. ... |
| e0a 45033 | Elimination rule identical... |
| eelT 45034 | An elimination deduction. ... |
| eel0cT 45035 | An elimination deduction. ... |
| eelT0 45036 | An elimination deduction. ... |
| e0bi 45037 | Elimination rule identical... |
| e0bir 45038 | Elimination rule identical... |
| uun0.1 45039 | Convention notation form o... |
| un0.1 45040 | ` T. ` is the constant tru... |
| uunT1 45041 | A deduction unionizing a n... |
| uunT1p1 45042 | A deduction unionizing a n... |
| uunT21 45043 | A deduction unionizing a n... |
| uun121 45044 | A deduction unionizing a n... |
| uun121p1 45045 | A deduction unionizing a n... |
| uun132 45046 | A deduction unionizing a n... |
| uun132p1 45047 | A deduction unionizing a n... |
| anabss7p1 45048 | A deduction unionizing a n... |
| un10 45049 | A unionizing deduction. (... |
| un01 45050 | A unionizing deduction. (... |
| un2122 45051 | A deduction unionizing a n... |
| uun2131 45052 | A deduction unionizing a n... |
| uun2131p1 45053 | A deduction unionizing a n... |
| uunTT1 45054 | A deduction unionizing a n... |
| uunTT1p1 45055 | A deduction unionizing a n... |
| uunTT1p2 45056 | A deduction unionizing a n... |
| uunT11 45057 | A deduction unionizing a n... |
| uunT11p1 45058 | A deduction unionizing a n... |
| uunT11p2 45059 | A deduction unionizing a n... |
| uunT12 45060 | A deduction unionizing a n... |
| uunT12p1 45061 | A deduction unionizing a n... |
| uunT12p2 45062 | A deduction unionizing a n... |
| uunT12p3 45063 | A deduction unionizing a n... |
| uunT12p4 45064 | A deduction unionizing a n... |
| uunT12p5 45065 | A deduction unionizing a n... |
| uun111 45066 | A deduction unionizing a n... |
| 3anidm12p1 45067 | A deduction unionizing a n... |
| 3anidm12p2 45068 | A deduction unionizing a n... |
| uun123 45069 | A deduction unionizing a n... |
| uun123p1 45070 | A deduction unionizing a n... |
| uun123p2 45071 | A deduction unionizing a n... |
| uun123p3 45072 | A deduction unionizing a n... |
| uun123p4 45073 | A deduction unionizing a n... |
| uun2221 45074 | A deduction unionizing a n... |
| uun2221p1 45075 | A deduction unionizing a n... |
| uun2221p2 45076 | A deduction unionizing a n... |
| 3impdirp1 45077 | A deduction unionizing a n... |
| 3impcombi 45078 | A 1-hypothesis proposition... |
| trsspwALT 45079 | Virtual deduction proof of... |
| trsspwALT2 45080 | Virtual deduction proof of... |
| trsspwALT3 45081 | Short predicate calculus p... |
| sspwtr 45082 | Virtual deduction proof of... |
| sspwtrALT 45083 | Virtual deduction proof of... |
| sspwtrALT2 45084 | Short predicate calculus p... |
| pwtrVD 45085 | Virtual deduction proof of... |
| pwtrrVD 45086 | Virtual deduction proof of... |
| suctrALT 45087 | The successor of a transit... |
| snssiALTVD 45088 | Virtual deduction proof of... |
| snssiALT 45089 | If a class is an element o... |
| snsslVD 45090 | Virtual deduction proof of... |
| snssl 45091 | If a singleton is a subcla... |
| snelpwrVD 45092 | Virtual deduction proof of... |
| unipwrVD 45093 | Virtual deduction proof of... |
| unipwr 45094 | A class is a subclass of t... |
| sstrALT2VD 45095 | Virtual deduction proof of... |
| sstrALT2 45096 | Virtual deduction proof of... |
| suctrALT2VD 45097 | Virtual deduction proof of... |
| suctrALT2 45098 | Virtual deduction proof of... |
| elex2VD 45099 | Virtual deduction proof of... |
| elex22VD 45100 | Virtual deduction proof of... |
| eqsbc2VD 45101 | Virtual deduction proof of... |
| zfregs2VD 45102 | Virtual deduction proof of... |
| tpid3gVD 45103 | Virtual deduction proof of... |
| en3lplem1VD 45104 | Virtual deduction proof of... |
| en3lplem2VD 45105 | Virtual deduction proof of... |
| en3lpVD 45106 | Virtual deduction proof of... |
| simplbi2VD 45107 | Virtual deduction proof of... |
| 3ornot23VD 45108 | Virtual deduction proof of... |
| orbi1rVD 45109 | Virtual deduction proof of... |
| bitr3VD 45110 | Virtual deduction proof of... |
| 3orbi123VD 45111 | Virtual deduction proof of... |
| sbc3orgVD 45112 | Virtual deduction proof of... |
| 19.21a3con13vVD 45113 | Virtual deduction proof of... |
| exbirVD 45114 | Virtual deduction proof of... |
| exbiriVD 45115 | Virtual deduction proof of... |
| rspsbc2VD 45116 | Virtual deduction proof of... |
| 3impexpVD 45117 | Virtual deduction proof of... |
| 3impexpbicomVD 45118 | Virtual deduction proof of... |
| 3impexpbicomiVD 45119 | Virtual deduction proof of... |
| sbcoreleleqVD 45120 | Virtual deduction proof of... |
| hbra2VD 45121 | Virtual deduction proof of... |
| tratrbVD 45122 | Virtual deduction proof of... |
| al2imVD 45123 | Virtual deduction proof of... |
| syl5impVD 45124 | Virtual deduction proof of... |
| idiVD 45125 | Virtual deduction proof of... |
| ancomstVD 45126 | Closed form of ~ ancoms . ... |
| ssralv2VD 45127 | Quantification restricted ... |
| ordelordALTVD 45128 | An element of an ordinal c... |
| equncomVD 45129 | If a class equals the unio... |
| equncomiVD 45130 | Inference form of ~ equnco... |
| sucidALTVD 45131 | A set belongs to its succe... |
| sucidALT 45132 | A set belongs to its succe... |
| sucidVD 45133 | A set belongs to its succe... |
| imbi12VD 45134 | Implication form of ~ imbi... |
| imbi13VD 45135 | Join three logical equival... |
| sbcim2gVD 45136 | Distribution of class subs... |
| sbcbiVD 45137 | Implication form of ~ sbcb... |
| trsbcVD 45138 | Formula-building inference... |
| truniALTVD 45139 | The union of a class of tr... |
| ee33VD 45140 | Non-virtual deduction form... |
| trintALTVD 45141 | The intersection of a clas... |
| trintALT 45142 | The intersection of a clas... |
| undif3VD 45143 | The first equality of Exer... |
| sbcssgVD 45144 | Virtual deduction proof of... |
| csbingVD 45145 | Virtual deduction proof of... |
| onfrALTlem5VD 45146 | Virtual deduction proof of... |
| onfrALTlem4VD 45147 | Virtual deduction proof of... |
| onfrALTlem3VD 45148 | Virtual deduction proof of... |
| simplbi2comtVD 45149 | Virtual deduction proof of... |
| onfrALTlem2VD 45150 | Virtual deduction proof of... |
| onfrALTlem1VD 45151 | Virtual deduction proof of... |
| onfrALTVD 45152 | Virtual deduction proof of... |
| csbeq2gVD 45153 | Virtual deduction proof of... |
| csbsngVD 45154 | Virtual deduction proof of... |
| csbxpgVD 45155 | Virtual deduction proof of... |
| csbresgVD 45156 | Virtual deduction proof of... |
| csbrngVD 45157 | Virtual deduction proof of... |
| csbima12gALTVD 45158 | Virtual deduction proof of... |
| csbunigVD 45159 | Virtual deduction proof of... |
| csbfv12gALTVD 45160 | Virtual deduction proof of... |
| con5VD 45161 | Virtual deduction proof of... |
| relopabVD 45162 | Virtual deduction proof of... |
| 19.41rgVD 45163 | Virtual deduction proof of... |
| 2pm13.193VD 45164 | Virtual deduction proof of... |
| hbimpgVD 45165 | Virtual deduction proof of... |
| hbalgVD 45166 | Virtual deduction proof of... |
| hbexgVD 45167 | Virtual deduction proof of... |
| ax6e2eqVD 45168 | The following User's Proof... |
| ax6e2ndVD 45169 | The following User's Proof... |
| ax6e2ndeqVD 45170 | The following User's Proof... |
| 2sb5ndVD 45171 | The following User's Proof... |
| 2uasbanhVD 45172 | The following User's Proof... |
| e2ebindVD 45173 | The following User's Proof... |
| sb5ALTVD 45174 | The following User's Proof... |
| vk15.4jVD 45175 | The following User's Proof... |
| notnotrALTVD 45176 | The following User's Proof... |
| con3ALTVD 45177 | The following User's Proof... |
| elpwgdedVD 45178 | Membership in a power clas... |
| sspwimp 45179 | If a class is a subclass o... |
| sspwimpVD 45180 | The following User's Proof... |
| sspwimpcf 45181 | If a class is a subclass o... |
| sspwimpcfVD 45182 | The following User's Proof... |
| suctrALTcf 45183 | The successor of a transit... |
| suctrALTcfVD 45184 | The following User's Proof... |
| suctrALT3 45185 | The successor of a transit... |
| sspwimpALT 45186 | If a class is a subclass o... |
| unisnALT 45187 | A set equals the union of ... |
| notnotrALT2 45188 | Converse of double negatio... |
| sspwimpALT2 45189 | If a class is a subclass o... |
| e2ebindALT 45190 | Absorption of an existenti... |
| ax6e2ndALT 45191 | If at least two sets exist... |
| ax6e2ndeqALT 45192 | "At least two sets exist" ... |
| 2sb5ndALT 45193 | Equivalence for double sub... |
| chordthmALT 45194 | The intersecting chords th... |
| isosctrlem1ALT 45195 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 45196 | The indexed union of conne... |
| iunconnALT 45197 | The indexed union of conne... |
| sineq0ALT 45198 | A complex number whose sin... |
| rspesbcd 45199 | Restricted quantifier vers... |
| rext0 45200 | Nonempty existential quant... |
| dfbi1ALTa 45201 | Version of ~ dfbi1ALT usin... |
| simprimi 45202 | Inference associated with ... |
| dfbi1ALTb 45203 | Further shorten ~ dfbi1ALT... |
| relpeq1 45206 | Equality theorem for relat... |
| relpeq2 45207 | Equality theorem for relat... |
| relpeq3 45208 | Equality theorem for relat... |
| relpeq4 45209 | Equality theorem for relat... |
| relpeq5 45210 | Equality theorem for relat... |
| nfrelp 45211 | Bound-variable hypothesis ... |
| relpf 45212 | A relation-preserving func... |
| relprel 45213 | A relation-preserving func... |
| relpmin 45214 | A preimage of a minimal el... |
| relpfrlem 45215 | Lemma for ~ relpfr . Prov... |
| relpfr 45216 | If the image of a set unde... |
| orbitex 45217 | Orbits exist. Given a set... |
| orbitinit 45218 | A set is contained in its ... |
| orbitcl 45219 | The orbit under a function... |
| orbitclmpt 45220 | Version of ~ orbitcl using... |
| trwf 45221 | The class of well-founded ... |
| rankrelp 45222 | The rank function preserve... |
| wffr 45223 | The class of well-founded ... |
| trfr 45224 | A transitive class well-fo... |
| tcfr 45225 | A set is well-founded if a... |
| xpwf 45226 | The Cartesian product of t... |
| dmwf 45227 | The domain of a well-found... |
| rnwf 45228 | The range of a well-founde... |
| relwf 45229 | A relation is a well-found... |
| ralabso 45230 | Simplification of restrict... |
| rexabso 45231 | Simplification of restrict... |
| ralabsod 45232 | Deduction form of ~ ralabs... |
| rexabsod 45233 | Deduction form of ~ rexabs... |
| ralabsobidv 45234 | Formula-building lemma for... |
| rexabsobidv 45235 | Formula-building lemma for... |
| ssabso 45236 | The notion " ` x ` is a su... |
| disjabso 45237 | Disjointness is absolute f... |
| n0abso 45238 | Nonemptiness is absolute f... |
| traxext 45239 | A transitive class models ... |
| modelaxreplem1 45240 | Lemma for ~ modelaxrep . ... |
| modelaxreplem2 45241 | Lemma for ~ modelaxrep . ... |
| modelaxreplem3 45242 | Lemma for ~ modelaxrep . ... |
| modelaxrep 45243 | Conditions which guarantee... |
| ssclaxsep 45244 | A class that is closed und... |
| 0elaxnul 45245 | A class that contains the ... |
| pwclaxpow 45246 | Suppose ` M ` is a transit... |
| prclaxpr 45247 | A class that is closed und... |
| uniclaxun 45248 | A class that is closed und... |
| sswfaxreg 45249 | A subclass of the class of... |
| omssaxinf2 45250 | A class that contains all ... |
| omelaxinf2 45251 | A transitive class that co... |
| dfac5prim 45252 | ~ dfac5 expanded into prim... |
| ac8prim 45253 | ~ ac8 expanded into primit... |
| modelac8prim 45254 | If ` M ` is a transitive c... |
| wfaxext 45255 | The class of well-founded ... |
| wfaxrep 45256 | The class of well-founded ... |
| wfaxsep 45257 | The class of well-founded ... |
| wfaxnul 45258 | The class of well-founded ... |
| wfaxpow 45259 | The class of well-founded ... |
| wfaxpr 45260 | The class of well-founded ... |
| wfaxun 45261 | The class of well-founded ... |
| wfaxreg 45262 | The class of well-founded ... |
| wfaxinf2 45263 | The class of well-founded ... |
| wfac8prim 45264 | The class of well-founded ... |
| brpermmodel 45265 | The membership relation in... |
| brpermmodelcnv 45266 | Ordinary membership expres... |
| permaxext 45267 | The Axiom of Extensionalit... |
| permaxrep 45268 | The Axiom of Replacement ~... |
| permaxsep 45269 | The Axiom of Separation ~ ... |
| permaxnul 45270 | The Null Set Axiom ~ ax-nu... |
| permaxpow 45271 | The Axiom of Power Sets ~ ... |
| permaxpr 45272 | The Axiom of Pairing ~ ax-... |
| permaxun 45273 | The Axiom of Union ~ ax-un... |
| permaxinf2lem 45274 | Lemma for ~ permaxinf2 . ... |
| permaxinf2 45275 | The Axiom of Infinity ~ ax... |
| permac8prim 45276 | The Axiom of Choice ~ ac8p... |
| nregmodelf1o 45277 | Define a permutation ` F `... |
| nregmodellem 45278 | Lemma for ~ nregmodel . (... |
| nregmodel 45279 | The Axiom of Regularity ~ ... |
| nregmodelaxext 45280 | The Axiom of Extensionalit... |
| evth2f 45281 | A version of ~ evth2 using... |
| elunif 45282 | A version of ~ eluni using... |
| rzalf 45283 | A version of ~ rzal using ... |
| fvelrnbf 45284 | A version of ~ fvelrnb usi... |
| rfcnpre1 45285 | If F is a continuous funct... |
| ubelsupr 45286 | If U belongs to A and U is... |
| fsumcnf 45287 | A finite sum of functions ... |
| mulltgt0 45288 | The product of a negative ... |
| rspcegf 45289 | A version of ~ rspcev usin... |
| rabexgf 45290 | A version of ~ rabexg usin... |
| fcnre 45291 | A function continuous with... |
| sumsnd 45292 | A sum of a singleton is th... |
| evthf 45293 | A version of ~ evth using ... |
| cnfex 45294 | The class of continuous fu... |
| fnchoice 45295 | For a finite set, a choice... |
| refsumcn 45296 | A finite sum of continuous... |
| rfcnpre2 45297 | If ` F ` is a continuous f... |
| cncmpmax 45298 | When the hypothesis for th... |
| rfcnpre3 45299 | If F is a continuous funct... |
| rfcnpre4 45300 | If F is a continuous funct... |
| sumpair 45301 | Sum of two distinct comple... |
| rfcnnnub 45302 | Given a real continuous fu... |
| refsum2cnlem1 45303 | This is the core Lemma for... |
| refsum2cn 45304 | The sum of two continuus r... |
| adantlllr 45305 | Deduction adding a conjunc... |
| 3adantlr3 45306 | Deduction adding a conjunc... |
| 3adantll2 45307 | Deduction adding a conjunc... |
| 3adantll3 45308 | Deduction adding a conjunc... |
| ssnel 45309 | If not element of a set, t... |
| sncldre 45310 | A singleton is closed w.r.... |
| n0p 45311 | A polynomial with a nonzer... |
| pm2.65ni 45312 | Inference rule for proof b... |
| iuneq2df 45313 | Equality deduction for ind... |
| nnfoctb 45314 | There exists a mapping fro... |
| elpwinss 45315 | An element of the powerset... |
| unidmex 45316 | If ` F ` is a set, then ` ... |
| ndisj2 45317 | A non-disjointness conditi... |
| zenom 45318 | The set of integer numbers... |
| uzwo4 45319 | Well-ordering principle: a... |
| unisn0 45320 | The union of the singleton... |
| ssin0 45321 | If two classes are disjoin... |
| inabs3 45322 | Absorption law for interse... |
| pwpwuni 45323 | Relationship between power... |
| disjiun2 45324 | In a disjoint collection, ... |
| 0pwfi 45325 | The empty set is in any po... |
| ssinss2d 45326 | Intersection preserves sub... |
| zct 45327 | The set of integer numbers... |
| pwfin0 45328 | A finite set always belong... |
| uzct 45329 | An upper integer set is co... |
| iunxsnf 45330 | A singleton index picks ou... |
| fiiuncl 45331 | If a set is closed under t... |
| iunp1 45332 | The addition of the next s... |
| fiunicl 45333 | If a set is closed under t... |
| ixpeq2d 45334 | Equality theorem for infin... |
| disjxp1 45335 | The sets of a cartesian pr... |
| disjsnxp 45336 | The sets in the cartesian ... |
| eliind 45337 | Membership in indexed inte... |
| rspcef 45338 | Restricted existential spe... |
| ixpssmapc 45339 | An infinite Cartesian prod... |
| elintd 45340 | Membership in class inters... |
| ssdf 45341 | A sufficient condition for... |
| brneqtrd 45342 | Substitution of equal clas... |
| ssnct 45343 | A set containing an uncoun... |
| ssuniint 45344 | Sufficient condition for b... |
| elintdv 45345 | Membership in class inters... |
| ssd 45346 | A sufficient condition for... |
| ralimralim 45347 | Introducing any antecedent... |
| snelmap 45348 | Membership of the element ... |
| xrnmnfpnf 45349 | An extended real that is n... |
| iuneq1i 45350 | Equality theorem for index... |
| nssrex 45351 | Negation of subclass relat... |
| ssinc 45352 | Inclusion relation for a m... |
| ssdec 45353 | Inclusion relation for a m... |
| elixpconstg 45354 | Membership in an infinite ... |
| iineq1d 45355 | Equality theorem for index... |
| metpsmet 45356 | A metric is a pseudometric... |
| ixpssixp 45357 | Subclass theorem for infin... |
| ballss3 45358 | A sufficient condition for... |
| iunincfi 45359 | Given a sequence of increa... |
| nsstr 45360 | If it's not a subclass, it... |
| rexanuz3 45361 | Combine two different uppe... |
| cbvmpo2 45362 | Rule to change the second ... |
| cbvmpo1 45363 | Rule to change the first b... |
| eliuniin 45364 | Indexed union of indexed i... |
| ssabf 45365 | Subclass of a class abstra... |
| pssnssi 45366 | A proper subclass does not... |
| rabidim2 45367 | Membership in a restricted... |
| eluni2f 45368 | Membership in class union.... |
| eliin2f 45369 | Membership in indexed inte... |
| nssd 45370 | Negation of subclass relat... |
| iineq12dv 45371 | Equality deduction for ind... |
| supxrcld 45372 | The supremum of an arbitra... |
| elrestd 45373 | A sufficient condition for... |
| eliuniincex 45374 | Counterexample to show tha... |
| eliincex 45375 | Counterexample to show tha... |
| eliinid 45376 | Membership in an indexed i... |
| abssf 45377 | Class abstraction in a sub... |
| supxrubd 45378 | A member of a set of exten... |
| ssrabf 45379 | Subclass of a restricted c... |
| ssrabdf 45380 | Subclass of a restricted c... |
| eliin2 45381 | Membership in indexed inte... |
| ssrab2f 45382 | Subclass relation for a re... |
| restuni3 45383 | The underlying set of a su... |
| rabssf 45384 | Restricted class abstracti... |
| eliuniin2 45385 | Indexed union of indexed i... |
| restuni4 45386 | The underlying set of a su... |
| restuni6 45387 | The underlying set of a su... |
| restuni5 45388 | The underlying set of a su... |
| unirestss 45389 | The union of an elementwis... |
| iniin1 45390 | Indexed intersection of in... |
| iniin2 45391 | Indexed intersection of in... |
| cbvrabv2 45392 | A more general version of ... |
| cbvrabv2w 45393 | A more general version of ... |
| iinssiin 45394 | Subset implication for an ... |
| eliind2 45395 | Membership in indexed inte... |
| iinssd 45396 | Subset implication for an ... |
| rabbida2 45397 | Equivalent wff's yield equ... |
| iinexd 45398 | The existence of an indexe... |
| rabexf 45399 | Separation Scheme in terms... |
| rabbida3 45400 | Equivalent wff's yield equ... |
| r19.36vf 45401 | Restricted quantifier vers... |
| raleqd 45402 | Equality deduction for res... |
| iinssf 45403 | Subset implication for an ... |
| iinssdf 45404 | Subset implication for an ... |
| resabs2i 45405 | Absorption law for restric... |
| ssdf2 45406 | A sufficient condition for... |
| rabssd 45407 | Restricted class abstracti... |
| rexnegd 45408 | Minus a real number. (Con... |
| rexlimd3 45409 | * Inference from Theorem 1... |
| nel1nelini 45410 | Membership in an intersect... |
| nel2nelini 45411 | Membership in an intersect... |
| eliunid 45412 | Membership in indexed unio... |
| reximdd 45413 | Deduction from Theorem 19.... |
| inopnd 45414 | The intersection of two op... |
| ss2rabdf 45415 | Deduction of restricted ab... |
| restopn3 45416 | If ` A ` is open, then ` A... |
| restopnssd 45417 | A topology restricted to a... |
| restsubel 45418 | A subset belongs in the sp... |
| toprestsubel 45419 | A subset is open in the to... |
| rabidd 45420 | An "identity" law of concr... |
| iunssdf 45421 | Subset theorem for an inde... |
| iinss2d 45422 | Subset implication for an ... |
| r19.3rzf 45423 | Restricted quantification ... |
| r19.28zf 45424 | Restricted quantifier vers... |
| iindif2f 45425 | Indexed intersection of cl... |
| ralfal 45426 | Two ways of expressing emp... |
| archd 45427 | Archimedean property of re... |
| nimnbi 45428 | If an implication is false... |
| nimnbi2 45429 | If an implication is false... |
| notbicom 45430 | Commutative law for the ne... |
| rexeqif 45431 | Equality inference for res... |
| rspced 45432 | Restricted existential spe... |
| fnresdmss 45433 | A function does not change... |
| fmptsnxp 45434 | Maps-to notation and Carte... |
| fvmpt2bd 45435 | Value of a function given ... |
| rnmptfi 45436 | The range of a function wi... |
| fresin2 45437 | Restriction of a function ... |
| ffi 45438 | A function with finite dom... |
| suprnmpt 45439 | An explicit bound for the ... |
| rnffi 45440 | The range of a function wi... |
| mptelpm 45441 | A function in maps-to nota... |
| rnmptpr 45442 | Range of a function define... |
| resmpti 45443 | Restriction of the mapping... |
| founiiun 45444 | Union expressed as an inde... |
| rnresun 45445 | Distribution law for range... |
| elrnmptf 45446 | The range of a function in... |
| rnmptssrn 45447 | Inclusion relation for two... |
| disjf1 45448 | A 1 to 1 mapping built fro... |
| rnsnf 45449 | The range of a function wh... |
| wessf1ornlem 45450 | Given a function ` F ` on ... |
| wessf1orn 45451 | Given a function ` F ` on ... |
| nelrnres 45452 | If ` A ` is not in the ran... |
| disjrnmpt2 45453 | Disjointness of the range ... |
| elrnmpt1sf 45454 | Elementhood in an image se... |
| founiiun0 45455 | Union expressed as an inde... |
| disjf1o 45456 | A bijection built from dis... |
| disjinfi 45457 | Only a finite number of di... |
| fvovco 45458 | Value of the composition o... |
| ssnnf1octb 45459 | There exists a bijection b... |
| nnf1oxpnn 45460 | There is a bijection betwe... |
| rnmptssd 45461 | The range of a function gi... |
| projf1o 45462 | A biijection from a set to... |
| fvmap 45463 | Function value for a membe... |
| fvixp2 45464 | Projection of a factor of ... |
| choicefi 45465 | For a finite set, a choice... |
| mpct 45466 | The exponentiation of a co... |
| cnmetcoval 45467 | Value of the distance func... |
| fcomptss 45468 | Express composition of two... |
| elmapsnd 45469 | Membership in a set expone... |
| mapss2 45470 | Subset inheritance for set... |
| fsneq 45471 | Equality condition for two... |
| difmap 45472 | Difference of two sets exp... |
| unirnmap 45473 | Given a subset of a set ex... |
| inmap 45474 | Intersection of two sets e... |
| fcoss 45475 | Composition of two mapping... |
| fsneqrn 45476 | Equality condition for two... |
| difmapsn 45477 | Difference of two sets exp... |
| mapssbi 45478 | Subset inheritance for set... |
| unirnmapsn 45479 | Equality theorem for a sub... |
| iunmapss 45480 | The indexed union of set e... |
| ssmapsn 45481 | A subset ` C ` of a set ex... |
| iunmapsn 45482 | The indexed union of set e... |
| absfico 45483 | Mapping domain and codomai... |
| icof 45484 | The set of left-closed rig... |
| elpmrn 45485 | The range of a partial fun... |
| imaexi 45486 | The image of a set is a se... |
| axccdom 45487 | Relax the constraint on ax... |
| dmmptdff 45488 | The domain of the mapping ... |
| dmmptdf 45489 | The domain of the mapping ... |
| elpmi2 45490 | The domain of a partial fu... |
| dmrelrnrel 45491 | A relation preserving func... |
| fvcod 45492 | Value of a function compos... |
| elrnmpoid 45493 | Membership in the range of... |
| axccd 45494 | An alternative version of ... |
| axccd2 45495 | An alternative version of ... |
| feqresmptf 45496 | Express a restricted funct... |
| dmmptssf 45497 | The domain of a mapping is... |
| dmmptdf2 45498 | The domain of the mapping ... |
| dmuz 45499 | Domain of the upper intege... |
| fmptd2f 45500 | Domain and codomain of the... |
| mpteq1df 45501 | An equality theorem for th... |
| mptexf 45502 | If the domain of a functio... |
| fvmpt4 45503 | Value of a function given ... |
| fmptf 45504 | Functionality of the mappi... |
| resimass 45505 | The image of a restriction... |
| mptssid 45506 | The mapping operation expr... |
| mptfnd 45507 | The maps-to notation defin... |
| rnmptlb 45508 | Boundness below of the ran... |
| rnmptbddlem 45509 | Boundness of the range of ... |
| rnmptbdd 45510 | Boundness of the range of ... |
| funimaeq 45511 | Membership relation for th... |
| rnmptssf 45512 | The range of a function gi... |
| rnmptbd2lem 45513 | Boundness below of the ran... |
| rnmptbd2 45514 | Boundness below of the ran... |
| infnsuprnmpt 45515 | The indexed infimum of rea... |
| suprclrnmpt 45516 | Closure of the indexed sup... |
| suprubrnmpt2 45517 | A member of a nonempty ind... |
| suprubrnmpt 45518 | A member of a nonempty ind... |
| rnmptssdf 45519 | The range of a function gi... |
| rnmptbdlem 45520 | Boundness above of the ran... |
| rnmptbd 45521 | Boundness above of the ran... |
| rnmptss2 45522 | The range of a function gi... |
| elmptima 45523 | The image of a function in... |
| ralrnmpt3 45524 | A restricted quantifier ov... |
| rnmptssbi 45525 | The range of a function gi... |
| imass2d 45526 | Subset theorem for image. ... |
| imassmpt 45527 | Membership relation for th... |
| fpmd 45528 | A total function is a part... |
| fconst7 45529 | An alternative way to expr... |
| fnmptif 45530 | Functionality and domain o... |
| dmmptif 45531 | Domain of the mapping oper... |
| mpteq2dfa 45532 | Slightly more general equa... |
| dmmpt1 45533 | The domain of the mapping ... |
| fmptff 45534 | Functionality of the mappi... |
| fvmptelcdmf 45535 | The value of a function at... |
| fmptdff 45536 | A version of ~ fmptd using... |
| fvmpt2df 45537 | Deduction version of ~ fvm... |
| rn1st 45538 | The range of a function wi... |
| rnmptssff 45539 | The range of a function gi... |
| rnmptssdff 45540 | The range of a function gi... |
| fvmpt4d 45541 | Value of a function given ... |
| sub2times 45542 | Subtracting from a number,... |
| nnxrd 45543 | A natural number is an ext... |
| nnxr 45544 | A natural number is an ext... |
| abssubrp 45545 | The distance of two distin... |
| elfzfzo 45546 | Relationship between membe... |
| oddfl 45547 | Odd number representation ... |
| abscosbd 45548 | Bound for the absolute val... |
| mul13d 45549 | Commutative/associative la... |
| negpilt0 45550 | Negative ` _pi ` is negati... |
| dstregt0 45551 | A complex number ` A ` tha... |
| subadd4b 45552 | Rearrangement of 4 terms i... |
| xrlttri5d 45553 | Not equal and not larger i... |
| zltlesub 45554 | If an integer ` N ` is les... |
| divlt0gt0d 45555 | The ratio of a negative nu... |
| subsub23d 45556 | Swap subtrahend and result... |
| 2timesgt 45557 | Double of a positive real ... |
| reopn 45558 | The reals are open with re... |
| sub31 45559 | Swap the first and third t... |
| nnne1ge2 45560 | A positive integer which i... |
| lefldiveq 45561 | A closed enough, smaller r... |
| negsubdi3d 45562 | Distribution of negative o... |
| ltdiv2dd 45563 | Division of a positive num... |
| abssinbd 45564 | Bound for the absolute val... |
| halffl 45565 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 45566 | Ordering relation for a st... |
| hashssle 45567 | The size of a subset of a ... |
| lttri5d 45568 | Not equal and not larger i... |
| fzisoeu 45569 | A finite ordered set has a... |
| lt3addmuld 45570 | If three real numbers are ... |
| absnpncan2d 45571 | Triangular inequality, com... |
| fperiodmullem 45572 | A function with period ` T... |
| fperiodmul 45573 | A function with period T i... |
| upbdrech 45574 | Choice of an upper bound f... |
| lt4addmuld 45575 | If four real numbers are l... |
| absnpncan3d 45576 | Triangular inequality, com... |
| upbdrech2 45577 | Choice of an upper bound f... |
| ssfiunibd 45578 | A finite union of bounded ... |
| fzdifsuc2 45579 | Remove a successor from th... |
| fzsscn 45580 | A finite sequence of integ... |
| divcan8d 45581 | A cancellation law for div... |
| dmmcand 45582 | Cancellation law for divis... |
| fzssre 45583 | A finite sequence of integ... |
| bccld 45584 | A binomial coefficient, in... |
| fzssnn0 45585 | A finite set of sequential... |
| xreqle 45586 | Equality implies 'less tha... |
| xaddlidd 45587 | ` 0 ` is a left identity f... |
| xadd0ge 45588 | A number is less than or e... |
| xrleneltd 45589 | 'Less than or equal to' an... |
| xaddcomd 45590 | The extended real addition... |
| supxrre3 45591 | The supremum of a nonempty... |
| uzfissfz 45592 | For any finite subset of t... |
| xleadd2d 45593 | Addition of extended reals... |
| suprltrp 45594 | The supremum of a nonempty... |
| xleadd1d 45595 | Addition of extended reals... |
| xreqled 45596 | Equality implies 'less tha... |
| xrgepnfd 45597 | An extended real greater t... |
| xrge0nemnfd 45598 | A nonnegative extended rea... |
| supxrgere 45599 | If a real number can be ap... |
| iuneqfzuzlem 45600 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 45601 | If two unions indexed by u... |
| xle2addd 45602 | Adding both side of two in... |
| supxrgelem 45603 | If an extended real number... |
| supxrge 45604 | If an extended real number... |
| suplesup 45605 | If any element of ` A ` ca... |
| infxrglb 45606 | The infimum of a set of ex... |
| xadd0ge2 45607 | A number is less than or e... |
| nepnfltpnf 45608 | An extended real that is n... |
| ltadd12dd 45609 | Addition to both sides of ... |
| nemnftgtmnft 45610 | An extended real that is n... |
| xrgtso 45611 | 'Greater than' is a strict... |
| rpex 45612 | The positive reals form a ... |
| xrge0ge0 45613 | A nonnegative extended rea... |
| xrssre 45614 | A subset of extended reals... |
| ssuzfz 45615 | A finite subset of the upp... |
| absfun 45616 | The absolute value is a fu... |
| infrpge 45617 | The infimum of a nonempty,... |
| xrlexaddrp 45618 | If an extended real number... |
| supsubc 45619 | The supremum function dist... |
| xralrple2 45620 | Show that ` A ` is less th... |
| nnuzdisj 45621 | The first ` N ` elements o... |
| ltdivgt1 45622 | Divsion by a number greate... |
| xrltned 45623 | 'Less than' implies not eq... |
| nnsplit 45624 | Express the set of positiv... |
| divdiv3d 45625 | Division into a fraction. ... |
| abslt2sqd 45626 | Comparison of the square o... |
| qenom 45627 | The set of rational number... |
| qct 45628 | The set of rational number... |
| lenlteq 45629 | 'less than or equal to' bu... |
| xrred 45630 | An extended real that is n... |
| rr2sscn2 45631 | The cartesian square of ` ... |
| infxr 45632 | The infimum of a set of ex... |
| infxrunb2 45633 | The infimum of an unbounde... |
| infxrbnd2 45634 | The infimum of a bounded-b... |
| infleinflem1 45635 | Lemma for ~ infleinf , cas... |
| infleinflem2 45636 | Lemma for ~ infleinf , whe... |
| infleinf 45637 | If any element of ` B ` ca... |
| xralrple4 45638 | Show that ` A ` is less th... |
| xralrple3 45639 | Show that ` A ` is less th... |
| eluzelzd 45640 | A member of an upper set o... |
| suplesup2 45641 | If any element of ` A ` is... |
| recnnltrp 45642 | ` N ` is a natural number ... |
| nnn0 45643 | The set of positive intege... |
| fzct 45644 | A finite set of sequential... |
| rpgtrecnn 45645 | Any positive real number i... |
| fzossuz 45646 | A half-open integer interv... |
| infxrrefi 45647 | The real and extended real... |
| xrralrecnnle 45648 | Show that ` A ` is less th... |
| fzoct 45649 | A finite set of sequential... |
| frexr 45650 | A function taking real val... |
| nnrecrp 45651 | The reciprocal of a positi... |
| reclt0d 45652 | The reciprocal of a negati... |
| lt0neg1dd 45653 | If a number is negative, i... |
| infxrcld 45654 | The infimum of an arbitrar... |
| xrralrecnnge 45655 | Show that ` A ` is less th... |
| reclt0 45656 | The reciprocal of a negati... |
| ltmulneg 45657 | Multiplying by a negative ... |
| allbutfi 45658 | For all but finitely many.... |
| ltdiv23neg 45659 | Swap denominator with othe... |
| xreqnltd 45660 | A consequence of trichotom... |
| mnfnre2 45661 | Minus infinity is not a re... |
| zssxr 45662 | The integers are a subset ... |
| fisupclrnmpt 45663 | A nonempty finite indexed ... |
| supxrunb3 45664 | The supremum of an unbound... |
| elfzod 45665 | Membership in a half-open ... |
| fimaxre4 45666 | A nonempty finite set of r... |
| ren0 45667 | The set of reals is nonemp... |
| eluzelz2 45668 | A member of an upper set o... |
| resabs2d 45669 | Absorption law for restric... |
| uzid2 45670 | Membership of the least me... |
| supxrleubrnmpt 45671 | The supremum of a nonempty... |
| uzssre2 45672 | An upper set of integers i... |
| uzssd 45673 | Subset relationship for tw... |
| eluzd 45674 | Membership in an upper set... |
| infxrlbrnmpt2 45675 | A member of a nonempty ind... |
| xrre4 45676 | An extended real is real i... |
| uz0 45677 | The upper integers functio... |
| eluzelz2d 45678 | A member of an upper set o... |
| infleinf2 45679 | If any element in ` B ` is... |
| unb2ltle 45680 | "Unbounded below" expresse... |
| uzidd2 45681 | Membership of the least me... |
| uzssd2 45682 | Subset relationship for tw... |
| rexabslelem 45683 | An indexed set of absolute... |
| rexabsle 45684 | An indexed set of absolute... |
| allbutfiinf 45685 | Given a "for all but finit... |
| supxrrernmpt 45686 | The real and extended real... |
| suprleubrnmpt 45687 | The supremum of a nonempty... |
| infrnmptle 45688 | An indexed infimum of exte... |
| infxrunb3 45689 | The infimum of an unbounde... |
| uzn0d 45690 | The upper integers are all... |
| uzssd3 45691 | Subset relationship for tw... |
| rexabsle2 45692 | An indexed set of absolute... |
| infxrunb3rnmpt 45693 | The infimum of an unbounde... |
| supxrre3rnmpt 45694 | The indexed supremum of a ... |
| uzublem 45695 | A set of reals, indexed by... |
| uzub 45696 | A set of reals, indexed by... |
| ssrexr 45697 | A subset of the reals is a... |
| supxrmnf2 45698 | Removing minus infinity fr... |
| supxrcli 45699 | The supremum of an arbitra... |
| uzid3 45700 | Membership of the least me... |
| infxrlesupxr 45701 | The supremum of a nonempty... |
| xnegeqd 45702 | Equality of two extended n... |
| xnegrecl 45703 | The extended real negative... |
| xnegnegi 45704 | Extended real version of ~... |
| xnegeqi 45705 | Equality of two extended n... |
| nfxnegd 45706 | Deduction version of ~ nfx... |
| xnegnegd 45707 | Extended real version of ~... |
| uzred 45708 | An upper integer is a real... |
| xnegcli 45709 | Closure of extended real n... |
| supminfrnmpt 45710 | The indexed supremum of a ... |
| infxrpnf 45711 | Adding plus infinity to a ... |
| infxrrnmptcl 45712 | The infimum of an arbitrar... |
| leneg2d 45713 | Negative of one side of 'l... |
| supxrltinfxr 45714 | The supremum of the empty ... |
| max1d 45715 | A number is less than or e... |
| supxrleubrnmptf 45716 | The supremum of a nonempty... |
| nleltd 45717 | 'Not less than or equal to... |
| zxrd 45718 | An integer is an extended ... |
| infxrgelbrnmpt 45719 | The infimum of an indexed ... |
| rphalfltd 45720 | Half of a positive real is... |
| uzssz2 45721 | An upper set of integers i... |
| leneg3d 45722 | Negative of one side of 'l... |
| max2d 45723 | A number is less than or e... |
| uzn0bi 45724 | The upper integers functio... |
| xnegrecl2 45725 | If the extended real negat... |
| nfxneg 45726 | Bound-variable hypothesis ... |
| uzxrd 45727 | An upper integer is an ext... |
| infxrpnf2 45728 | Removing plus infinity fro... |
| supminfxr 45729 | The extended real suprema ... |
| infrpgernmpt 45730 | The infimum of a nonempty,... |
| xnegre 45731 | An extended real is real i... |
| xnegrecl2d 45732 | If the extended real negat... |
| uzxr 45733 | An upper integer is an ext... |
| supminfxr2 45734 | The extended real suprema ... |
| xnegred 45735 | An extended real is real i... |
| supminfxrrnmpt 45736 | The indexed supremum of a ... |
| min1d 45737 | The minimum of two numbers... |
| min2d 45738 | The minimum of two numbers... |
| xrnpnfmnf 45739 | An extended real that is n... |
| uzsscn 45740 | An upper set of integers i... |
| absimnre 45741 | The absolute value of the ... |
| uzsscn2 45742 | An upper set of integers i... |
| xrtgcntopre 45743 | The standard topologies on... |
| absimlere 45744 | The absolute value of the ... |
| rpssxr 45745 | The positive reals are a s... |
| monoordxrv 45746 | Ordering relation for a mo... |
| monoordxr 45747 | Ordering relation for a mo... |
| monoord2xrv 45748 | Ordering relation for a mo... |
| monoord2xr 45749 | Ordering relation for a mo... |
| xrpnf 45750 | An extended real is plus i... |
| xlenegcon1 45751 | Extended real version of ~... |
| xlenegcon2 45752 | Extended real version of ~... |
| pimxrneun 45753 | The preimage of a set of e... |
| caucvgbf 45754 | A function is convergent i... |
| cvgcau 45755 | A convergent function is C... |
| cvgcaule 45756 | A convergent function is C... |
| rexanuz2nf 45757 | A simple counterexample re... |
| gtnelioc 45758 | A real number larger than ... |
| ioossioc 45759 | An open interval is a subs... |
| ioondisj2 45760 | A condition for two open i... |
| ioondisj1 45761 | A condition for two open i... |
| ioogtlb 45762 | An element of a closed int... |
| evthiccabs 45763 | Extreme Value Theorem on y... |
| ltnelicc 45764 | A real number smaller than... |
| eliood 45765 | Membership in an open real... |
| iooabslt 45766 | An upper bound for the dis... |
| gtnelicc 45767 | A real number greater than... |
| iooinlbub 45768 | An open interval has empty... |
| iocgtlb 45769 | An element of a left-open ... |
| iocleub 45770 | An element of a left-open ... |
| eliccd 45771 | Membership in a closed rea... |
| eliccre 45772 | A member of a closed inter... |
| eliooshift 45773 | Element of an open interva... |
| eliocd 45774 | Membership in a left-open ... |
| icoltub 45775 | An element of a left-close... |
| eliocre 45776 | A member of a left-open ri... |
| iooltub 45777 | An element of an open inte... |
| ioontr 45778 | The interior of an interva... |
| snunioo1 45779 | The closure of one end of ... |
| lbioc 45780 | A left-open right-closed i... |
| ioomidp 45781 | The midpoint is an element... |
| iccdifioo 45782 | If the open inverval is re... |
| iccdifprioo 45783 | An open interval is the cl... |
| ioossioobi 45784 | Biconditional form of ~ io... |
| iccshift 45785 | A closed interval shifted ... |
| iccsuble 45786 | An upper bound to the dist... |
| iocopn 45787 | A left-open right-closed i... |
| eliccelioc 45788 | Membership in a closed int... |
| iooshift 45789 | An open interval shifted b... |
| iccintsng 45790 | Intersection of two adiace... |
| icoiccdif 45791 | Left-closed right-open int... |
| icoopn 45792 | A left-closed right-open i... |
| icoub 45793 | A left-closed, right-open ... |
| eliccxrd 45794 | Membership in a closed rea... |
| pnfel0pnf 45795 | ` +oo ` is a nonnegative e... |
| eliccnelico 45796 | An element of a closed int... |
| eliccelicod 45797 | A member of a closed inter... |
| ge0xrre 45798 | A nonnegative extended rea... |
| ge0lere 45799 | A nonnegative extended Rea... |
| elicores 45800 | Membership in a left-close... |
| inficc 45801 | The infimum of a nonempty ... |
| qinioo 45802 | The rational numbers are d... |
| lenelioc 45803 | A real number smaller than... |
| ioonct 45804 | A nonempty open interval i... |
| xrgtnelicc 45805 | A real number greater than... |
| iccdificc 45806 | The difference of two clos... |
| iocnct 45807 | A nonempty left-open, righ... |
| iccnct 45808 | A closed interval, with mo... |
| iooiinicc 45809 | A closed interval expresse... |
| iccgelbd 45810 | An element of a closed int... |
| iooltubd 45811 | An element of an open inte... |
| icoltubd 45812 | An element of a left-close... |
| qelioo 45813 | The rational numbers are d... |
| tgqioo2 45814 | Every open set of reals is... |
| iccleubd 45815 | An element of a closed int... |
| elioored 45816 | A member of an open interv... |
| ioogtlbd 45817 | An element of a closed int... |
| ioofun 45818 | ` (,) ` is a function. (C... |
| icomnfinre 45819 | A left-closed, right-open,... |
| sqrlearg 45820 | The square compared with i... |
| ressiocsup 45821 | If the supremum belongs to... |
| ressioosup 45822 | If the supremum does not b... |
| iooiinioc 45823 | A left-open, right-closed ... |
| ressiooinf 45824 | If the infimum does not be... |
| iocleubd 45825 | An element of a left-open ... |
| uzinico 45826 | An upper interval of integ... |
| preimaiocmnf 45827 | Preimage of a right-closed... |
| uzinico2 45828 | An upper interval of integ... |
| uzinico3 45829 | An upper interval of integ... |
| dmico 45830 | The domain of the closed-b... |
| ndmico 45831 | The closed-below, open-abo... |
| uzubioo 45832 | The upper integers are unb... |
| uzubico 45833 | The upper integers are unb... |
| uzubioo2 45834 | The upper integers are unb... |
| uzubico2 45835 | The upper integers are unb... |
| iocgtlbd 45836 | An element of a left-open ... |
| xrtgioo2 45837 | The topology on the extend... |
| fsummulc1f 45838 | Closure of a finite sum of... |
| fsumnncl 45839 | Closure of a nonempty, fin... |
| fsumge0cl 45840 | The finite sum of nonnegat... |
| fsumf1of 45841 | Re-index a finite sum usin... |
| fsumiunss 45842 | Sum over a disjoint indexe... |
| fsumreclf 45843 | Closure of a finite sum of... |
| fsumlessf 45844 | A shorter sum of nonnegati... |
| fsumsupp0 45845 | Finite sum of function val... |
| fsumsermpt 45846 | A finite sum expressed in ... |
| fmul01 45847 | Multiplying a finite numbe... |
| fmulcl 45848 | If ' Y ' is closed under t... |
| fmuldfeqlem1 45849 | induction step for the pro... |
| fmuldfeq 45850 | X and Z are two equivalent... |
| fmul01lt1lem1 45851 | Given a finite multiplicat... |
| fmul01lt1lem2 45852 | Given a finite multiplicat... |
| fmul01lt1 45853 | Given a finite multiplicat... |
| cncfmptss 45854 | A continuous complex funct... |
| rrpsscn 45855 | The positive reals are a s... |
| mulc1cncfg 45856 | A version of ~ mulc1cncf u... |
| infrglb 45857 | The infimum of a nonempty ... |
| expcnfg 45858 | If ` F ` is a complex cont... |
| prodeq2ad 45859 | Equality deduction for pro... |
| fprodsplit1 45860 | Separate out a term in a f... |
| fprodexp 45861 | Positive integer exponenti... |
| fprodabs2 45862 | The absolute value of a fi... |
| fprod0 45863 | A finite product with a ze... |
| mccllem 45864 | * Induction step for ~ mcc... |
| mccl 45865 | A multinomial coefficient,... |
| fprodcnlem 45866 | A finite product of functi... |
| fprodcn 45867 | A finite product of functi... |
| clim1fr1 45868 | A class of sequences of fr... |
| isumneg 45869 | Negation of a converging s... |
| climrec 45870 | Limit of the reciprocal of... |
| climmulf 45871 | A version of ~ climmul usi... |
| climexp 45872 | The limit of natural power... |
| climinf 45873 | A bounded monotonic noninc... |
| climsuselem1 45874 | The subsequence index ` I ... |
| climsuse 45875 | A subsequence ` G ` of a c... |
| climrecf 45876 | A version of ~ climrec usi... |
| climneg 45877 | Complex limit of the negat... |
| climinff 45878 | A version of ~ climinf usi... |
| climdivf 45879 | Limit of the ratio of two ... |
| climreeq 45880 | If ` F ` is a real functio... |
| ellimciota 45881 | An explicit value for the ... |
| climaddf 45882 | A version of ~ climadd usi... |
| mullimc 45883 | Limit of the product of tw... |
| ellimcabssub0 45884 | An equivalent condition fo... |
| limcdm0 45885 | If a function has empty do... |
| islptre 45886 | An equivalence condition f... |
| limccog 45887 | Limit of the composition o... |
| limciccioolb 45888 | The limit of a function at... |
| climf 45889 | Express the predicate: Th... |
| mullimcf 45890 | Limit of the multiplicatio... |
| constlimc 45891 | Limit of constant function... |
| rexlim2d 45892 | Inference removing two res... |
| idlimc 45893 | Limit of the identity func... |
| divcnvg 45894 | The sequence of reciprocal... |
| limcperiod 45895 | If ` F ` is a periodic fun... |
| limcrecl 45896 | If ` F ` is a real-valued ... |
| sumnnodd 45897 | A series indexed by ` NN `... |
| lptioo2 45898 | The upper bound of an open... |
| lptioo1 45899 | The lower bound of an open... |
| limcmptdm 45900 | The domain of a maps-to fu... |
| clim2f 45901 | Express the predicate: Th... |
| limcicciooub 45902 | The limit of a function at... |
| ltmod 45903 | A sufficient condition for... |
| islpcn 45904 | A characterization for a l... |
| lptre2pt 45905 | If a set in the real line ... |
| limsupre 45906 | If a sequence is bounded, ... |
| limcresiooub 45907 | The left limit doesn't cha... |
| limcresioolb 45908 | The right limit doesn't ch... |
| limcleqr 45909 | If the left and the right ... |
| lptioo2cn 45910 | The upper bound of an open... |
| lptioo1cn 45911 | The lower bound of an open... |
| neglimc 45912 | Limit of the negative func... |
| addlimc 45913 | Sum of two limits. (Contr... |
| 0ellimcdiv 45914 | If the numerator converges... |
| clim2cf 45915 | Express the predicate ` F ... |
| limclner 45916 | For a limit point, both fr... |
| sublimc 45917 | Subtraction of two limits.... |
| reclimc 45918 | Limit of the reciprocal of... |
| clim0cf 45919 | Express the predicate ` F ... |
| limclr 45920 | For a limit point, both fr... |
| divlimc 45921 | Limit of the quotient of t... |
| expfac 45922 | Factorial grows faster tha... |
| climconstmpt 45923 | A constant sequence conver... |
| climresmpt 45924 | A function restricted to u... |
| climsubmpt 45925 | Limit of the difference of... |
| climsubc2mpt 45926 | Limit of the difference of... |
| climsubc1mpt 45927 | Limit of the difference of... |
| fnlimfv 45928 | The value of the limit fun... |
| climreclf 45929 | The limit of a convergent ... |
| climeldmeq 45930 | Two functions that are eve... |
| climf2 45931 | Express the predicate: Th... |
| fnlimcnv 45932 | The sequence of function v... |
| climeldmeqmpt 45933 | Two functions that are eve... |
| climfveq 45934 | Two functions that are eve... |
| clim2f2 45935 | Express the predicate: Th... |
| climfveqmpt 45936 | Two functions that are eve... |
| climd 45937 | Express the predicate: Th... |
| clim2d 45938 | The limit of complex numbe... |
| fnlimfvre 45939 | The limit function of real... |
| allbutfifvre 45940 | Given a sequence of real-v... |
| climleltrp 45941 | The limit of complex numbe... |
| fnlimfvre2 45942 | The limit function of real... |
| fnlimf 45943 | The limit function of real... |
| fnlimabslt 45944 | A sequence of function val... |
| climfveqf 45945 | Two functions that are eve... |
| climmptf 45946 | Exhibit a function ` G ` w... |
| climfveqmpt3 45947 | Two functions that are eve... |
| climeldmeqf 45948 | Two functions that are eve... |
| climreclmpt 45949 | The limit of B convergent ... |
| limsupref 45950 | If a sequence is bounded, ... |
| limsupbnd1f 45951 | If a sequence is eventuall... |
| climbddf 45952 | A converging sequence of c... |
| climeqf 45953 | Two functions that are eve... |
| climeldmeqmpt3 45954 | Two functions that are eve... |
| limsupcld 45955 | Closure of the superior li... |
| climfv 45956 | The limit of a convergent ... |
| limsupval3 45957 | The superior limit of an i... |
| climfveqmpt2 45958 | Two functions that are eve... |
| limsup0 45959 | The superior limit of the ... |
| climeldmeqmpt2 45960 | Two functions that are eve... |
| limsupresre 45961 | The supremum limit of a fu... |
| climeqmpt 45962 | Two functions that are eve... |
| climfvd 45963 | The limit of a convergent ... |
| limsuplesup 45964 | An upper bound for the sup... |
| limsupresico 45965 | The superior limit doesn't... |
| limsuppnfdlem 45966 | If the restriction of a fu... |
| limsuppnfd 45967 | If the restriction of a fu... |
| limsupresuz 45968 | If the real part of the do... |
| limsupub 45969 | If the limsup is not ` +oo... |
| limsupres 45970 | The superior limit of a re... |
| climinf2lem 45971 | A convergent, nonincreasin... |
| climinf2 45972 | A convergent, nonincreasin... |
| limsupvaluz 45973 | The superior limit, when t... |
| limsupresuz2 45974 | If the domain of a functio... |
| limsuppnflem 45975 | If the restriction of a fu... |
| limsuppnf 45976 | If the restriction of a fu... |
| limsupubuzlem 45977 | If the limsup is not ` +oo... |
| limsupubuz 45978 | For a real-valued function... |
| climinf2mpt 45979 | A bounded below, monotonic... |
| climinfmpt 45980 | A bounded below, monotonic... |
| climinf3 45981 | A convergent, nonincreasin... |
| limsupvaluzmpt 45982 | The superior limit, when t... |
| limsupequzmpt2 45983 | Two functions that are eve... |
| limsupubuzmpt 45984 | If the limsup is not ` +oo... |
| limsupmnflem 45985 | The superior limit of a fu... |
| limsupmnf 45986 | The superior limit of a fu... |
| limsupequzlem 45987 | Two functions that are eve... |
| limsupequz 45988 | Two functions that are eve... |
| limsupre2lem 45989 | Given a function on the ex... |
| limsupre2 45990 | Given a function on the ex... |
| limsupmnfuzlem 45991 | The superior limit of a fu... |
| limsupmnfuz 45992 | The superior limit of a fu... |
| limsupequzmptlem 45993 | Two functions that are eve... |
| limsupequzmpt 45994 | Two functions that are eve... |
| limsupre2mpt 45995 | Given a function on the ex... |
| limsupequzmptf 45996 | Two functions that are eve... |
| limsupre3lem 45997 | Given a function on the ex... |
| limsupre3 45998 | Given a function on the ex... |
| limsupre3mpt 45999 | Given a function on the ex... |
| limsupre3uzlem 46000 | Given a function on the ex... |
| limsupre3uz 46001 | Given a function on the ex... |
| limsupreuz 46002 | Given a function on the re... |
| limsupvaluz2 46003 | The superior limit, when t... |
| limsupreuzmpt 46004 | Given a function on the re... |
| supcnvlimsup 46005 | If a function on a set of ... |
| supcnvlimsupmpt 46006 | If a function on a set of ... |
| 0cnv 46007 | If ` (/) ` is a complex nu... |
| climuzlem 46008 | Express the predicate: Th... |
| climuz 46009 | Express the predicate: Th... |
| lmbr3v 46010 | Express the binary relatio... |
| climisp 46011 | If a sequence converges to... |
| lmbr3 46012 | Express the binary relatio... |
| climrescn 46013 | A sequence converging w.r.... |
| climxrrelem 46014 | If a sequence ranging over... |
| climxrre 46015 | If a sequence ranging over... |
| limsuplt2 46018 | The defining property of t... |
| liminfgord 46019 | Ordering property of the i... |
| limsupvald 46020 | The superior limit of a se... |
| limsupresicompt 46021 | The superior limit doesn't... |
| limsupcli 46022 | Closure of the superior li... |
| liminfgf 46023 | Closure of the inferior li... |
| liminfval 46024 | The inferior limit of a se... |
| climlimsup 46025 | A sequence of real numbers... |
| limsupge 46026 | The defining property of t... |
| liminfgval 46027 | Value of the inferior limi... |
| liminfcl 46028 | Closure of the inferior li... |
| liminfvald 46029 | The inferior limit of a se... |
| liminfval5 46030 | The inferior limit of an i... |
| limsupresxr 46031 | The superior limit of a fu... |
| liminfresxr 46032 | The inferior limit of a fu... |
| liminfval2 46033 | The superior limit, relati... |
| climlimsupcex 46034 | Counterexample for ~ climl... |
| liminfcld 46035 | Closure of the inferior li... |
| liminfresico 46036 | The inferior limit doesn't... |
| limsup10exlem 46037 | The range of the given fun... |
| limsup10ex 46038 | The superior limit of a fu... |
| liminf10ex 46039 | The inferior limit of a fu... |
| liminflelimsuplem 46040 | The superior limit is grea... |
| liminflelimsup 46041 | The superior limit is grea... |
| limsupgtlem 46042 | For any positive real, the... |
| limsupgt 46043 | Given a sequence of real n... |
| liminfresre 46044 | The inferior limit of a fu... |
| liminfresicompt 46045 | The inferior limit doesn't... |
| liminfltlimsupex 46046 | An example where the ` lim... |
| liminfgelimsup 46047 | The inferior limit is grea... |
| liminfvalxr 46048 | Alternate definition of ` ... |
| liminfresuz 46049 | If the real part of the do... |
| liminflelimsupuz 46050 | The superior limit is grea... |
| liminfvalxrmpt 46051 | Alternate definition of ` ... |
| liminfresuz2 46052 | If the domain of a functio... |
| liminfgelimsupuz 46053 | The inferior limit is grea... |
| liminfval4 46054 | Alternate definition of ` ... |
| liminfval3 46055 | Alternate definition of ` ... |
| liminfequzmpt2 46056 | Two functions that are eve... |
| liminfvaluz 46057 | Alternate definition of ` ... |
| liminf0 46058 | The inferior limit of the ... |
| limsupval4 46059 | Alternate definition of ` ... |
| liminfvaluz2 46060 | Alternate definition of ` ... |
| liminfvaluz3 46061 | Alternate definition of ` ... |
| liminflelimsupcex 46062 | A counterexample for ~ lim... |
| limsupvaluz3 46063 | Alternate definition of ` ... |
| liminfvaluz4 46064 | Alternate definition of ` ... |
| limsupvaluz4 46065 | Alternate definition of ` ... |
| climliminflimsupd 46066 | If a sequence of real numb... |
| liminfreuzlem 46067 | Given a function on the re... |
| liminfreuz 46068 | Given a function on the re... |
| liminfltlem 46069 | Given a sequence of real n... |
| liminflt 46070 | Given a sequence of real n... |
| climliminf 46071 | A sequence of real numbers... |
| liminflimsupclim 46072 | A sequence of real numbers... |
| climliminflimsup 46073 | A sequence of real numbers... |
| climliminflimsup2 46074 | A sequence of real numbers... |
| climliminflimsup3 46075 | A sequence of real numbers... |
| climliminflimsup4 46076 | A sequence of real numbers... |
| limsupub2 46077 | A extended real valued fun... |
| limsupubuz2 46078 | A sequence with values in ... |
| xlimpnfxnegmnf 46079 | A sequence converges to ` ... |
| liminflbuz2 46080 | A sequence with values in ... |
| liminfpnfuz 46081 | The inferior limit of a fu... |
| liminflimsupxrre 46082 | A sequence with values in ... |
| xlimrel 46085 | The limit on extended real... |
| xlimres 46086 | A function converges iff i... |
| xlimcl 46087 | The limit of a sequence of... |
| rexlimddv2 46088 | Restricted existential eli... |
| xlimclim 46089 | Given a sequence of reals,... |
| xlimconst 46090 | A constant sequence conver... |
| climxlim 46091 | A converging sequence in t... |
| xlimbr 46092 | Express the binary relatio... |
| fuzxrpmcn 46093 | A function mapping from an... |
| cnrefiisplem 46094 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 46095 | A non-real, complex number... |
| xlimxrre 46096 | If a sequence ranging over... |
| xlimmnfvlem1 46097 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 46098 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 46099 | A function converges to mi... |
| xlimconst2 46100 | A sequence that eventually... |
| xlimpnfvlem1 46101 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 46102 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 46103 | A function converges to pl... |
| xlimclim2lem 46104 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 46105 | Given a sequence of extend... |
| xlimmnf 46106 | A function converges to mi... |
| xlimpnf 46107 | A function converges to pl... |
| xlimmnfmpt 46108 | A function converges to pl... |
| xlimpnfmpt 46109 | A function converges to pl... |
| climxlim2lem 46110 | In this lemma for ~ climxl... |
| climxlim2 46111 | A sequence of extended rea... |
| dfxlim2v 46112 | An alternative definition ... |
| dfxlim2 46113 | An alternative definition ... |
| climresd 46114 | A function restricted to u... |
| climresdm 46115 | A real function converges ... |
| dmclimxlim 46116 | A real valued sequence tha... |
| xlimmnflimsup2 46117 | A sequence of extended rea... |
| xlimuni 46118 | An infinite sequence conve... |
| xlimclimdm 46119 | A sequence of extended rea... |
| xlimfun 46120 | The convergence relation o... |
| xlimmnflimsup 46121 | If a sequence of extended ... |
| xlimdm 46122 | Two ways to express that a... |
| xlimpnfxnegmnf2 46123 | A sequence converges to ` ... |
| xlimresdm 46124 | A function converges in th... |
| xlimpnfliminf 46125 | If a sequence of extended ... |
| xlimpnfliminf2 46126 | A sequence of extended rea... |
| xlimliminflimsup 46127 | A sequence of extended rea... |
| xlimlimsupleliminf 46128 | A sequence of extended rea... |
| coseq0 46129 | A complex number whose cos... |
| sinmulcos 46130 | Multiplication formula for... |
| coskpi2 46131 | The cosine of an integer m... |
| cosnegpi 46132 | The cosine of negative ` _... |
| sinaover2ne0 46133 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 46134 | The cosine of an integer m... |
| mulcncff 46135 | The multiplication of two ... |
| cncfmptssg 46136 | A continuous complex funct... |
| constcncfg 46137 | A constant function is a c... |
| idcncfg 46138 | The identity function is a... |
| cncfshift 46139 | A periodic continuous func... |
| resincncf 46140 | ` sin ` restricted to real... |
| addccncf2 46141 | Adding a constant is a con... |
| 0cnf 46142 | The empty set is a continu... |
| fsumcncf 46143 | The finite sum of continuo... |
| cncfperiod 46144 | A periodic continuous func... |
| subcncff 46145 | The subtraction of two con... |
| negcncfg 46146 | The opposite of a continuo... |
| cnfdmsn 46147 | A function with a singleto... |
| cncfcompt 46148 | Composition of continuous ... |
| addcncff 46149 | The sum of two continuous ... |
| ioccncflimc 46150 | Limit at the upper bound o... |
| cncfuni 46151 | A complex function on a su... |
| icccncfext 46152 | A continuous function on a... |
| cncficcgt0 46153 | A the absolute value of a ... |
| icocncflimc 46154 | Limit at the lower bound, ... |
| cncfdmsn 46155 | A complex function with a ... |
| divcncff 46156 | The quotient of two contin... |
| cncfshiftioo 46157 | A periodic continuous func... |
| cncfiooicclem1 46158 | A continuous function ` F ... |
| cncfiooicc 46159 | A continuous function ` F ... |
| cncfiooiccre 46160 | A continuous function ` F ... |
| cncfioobdlem 46161 | ` G ` actually extends ` F... |
| cncfioobd 46162 | A continuous function ` F ... |
| jumpncnp 46163 | Jump discontinuity or disc... |
| cxpcncf2 46164 | The complex power function... |
| fprodcncf 46165 | The finite product of cont... |
| add1cncf 46166 | Addition to a constant is ... |
| add2cncf 46167 | Addition to a constant is ... |
| sub1cncfd 46168 | Subtracting a constant is ... |
| sub2cncfd 46169 | Subtraction from a constan... |
| fprodsub2cncf 46170 | ` F ` is continuous. (Con... |
| fprodadd2cncf 46171 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 46172 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 46173 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 46174 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 46175 | The sequence ` S ` of fini... |
| dvsinexp 46176 | The derivative of sin^N . ... |
| dvcosre 46177 | The real derivative of the... |
| dvsinax 46178 | Derivative exercise: the d... |
| dvsubf 46179 | The subtraction rule for e... |
| dvmptconst 46180 | Function-builder for deriv... |
| dvcnre 46181 | From complex differentiati... |
| dvmptidg 46182 | Function-builder for deriv... |
| dvresntr 46183 | Function-builder for deriv... |
| fperdvper 46184 | The derivative of a period... |
| dvasinbx 46185 | Derivative exercise: the d... |
| dvresioo 46186 | Restriction of a derivativ... |
| dvdivf 46187 | The quotient rule for ever... |
| dvdivbd 46188 | A sufficient condition for... |
| dvsubcncf 46189 | A sufficient condition for... |
| dvmulcncf 46190 | A sufficient condition for... |
| dvcosax 46191 | Derivative exercise: the d... |
| dvdivcncf 46192 | A sufficient condition for... |
| dvbdfbdioolem1 46193 | Given a function with boun... |
| dvbdfbdioolem2 46194 | A function on an open inte... |
| dvbdfbdioo 46195 | A function on an open inte... |
| ioodvbdlimc1lem1 46196 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 46197 | Limit at the lower bound o... |
| ioodvbdlimc1 46198 | A real function with bound... |
| ioodvbdlimc2lem 46199 | Limit at the upper bound o... |
| ioodvbdlimc2 46200 | A real function with bound... |
| dvdmsscn 46201 | ` X ` is a subset of ` CC ... |
| dvmptmulf 46202 | Function-builder for deriv... |
| dvnmptdivc 46203 | Function-builder for itera... |
| dvdsn1add 46204 | If ` K ` divides ` N ` but... |
| dvxpaek 46205 | Derivative of the polynomi... |
| dvnmptconst 46206 | The ` N ` -th derivative o... |
| dvnxpaek 46207 | The ` n ` -th derivative o... |
| dvnmul 46208 | Function-builder for the `... |
| dvmptfprodlem 46209 | Induction step for ~ dvmpt... |
| dvmptfprod 46210 | Function-builder for deriv... |
| dvnprodlem1 46211 | ` D ` is bijective. (Cont... |
| dvnprodlem2 46212 | Induction step for ~ dvnpr... |
| dvnprodlem3 46213 | The multinomial formula fo... |
| dvnprod 46214 | The multinomial formula fo... |
| itgsin0pilem1 46215 | Calculation of the integra... |
| ibliccsinexp 46216 | sin^n on a closed interval... |
| itgsin0pi 46217 | Calculation of the integra... |
| iblioosinexp 46218 | sin^n on an open integral ... |
| itgsinexplem1 46219 | Integration by parts is ap... |
| itgsinexp 46220 | A recursive formula for th... |
| iblconstmpt 46221 | A constant function is int... |
| itgeq1d 46222 | Equality theorem for an in... |
| mbfres2cn 46223 | Measurability of a piecewi... |
| vol0 46224 | The measure of the empty s... |
| ditgeqiooicc 46225 | A function ` F ` on an ope... |
| volge0 46226 | The volume of a set is alw... |
| cnbdibl 46227 | A continuous bounded funct... |
| snmbl 46228 | A singleton is measurable.... |
| ditgeq3d 46229 | Equality theorem for the d... |
| iblempty 46230 | The empty function is inte... |
| iblsplit 46231 | The union of two integrabl... |
| volsn 46232 | A singleton has 0 Lebesgue... |
| itgvol0 46233 | If the domani is negligibl... |
| itgcoscmulx 46234 | Exercise: the integral of ... |
| iblsplitf 46235 | A version of ~ iblsplit us... |
| ibliooicc 46236 | If a function is integrabl... |
| volioc 46237 | The measure of a left-open... |
| iblspltprt 46238 | If a function is integrabl... |
| itgsincmulx 46239 | Exercise: the integral of ... |
| itgsubsticclem 46240 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 46241 | Integration by u-substitut... |
| itgioocnicc 46242 | The integral of a piecewis... |
| iblcncfioo 46243 | A continuous function ` F ... |
| itgspltprt 46244 | The ` S. ` integral splits... |
| itgiccshift 46245 | The integral of a function... |
| itgperiod 46246 | The integral of a periodic... |
| itgsbtaddcnst 46247 | Integral substitution, add... |
| volico 46248 | The measure of left-closed... |
| sublevolico 46249 | The Lebesgue measure of a ... |
| dmvolss 46250 | Lebesgue measurable sets a... |
| ismbl3 46251 | The predicate " ` A ` is L... |
| volioof 46252 | The function that assigns ... |
| ovolsplit 46253 | The Lebesgue outer measure... |
| fvvolioof 46254 | The function value of the ... |
| volioore 46255 | The measure of an open int... |
| fvvolicof 46256 | The function value of the ... |
| voliooico 46257 | An open interval and a lef... |
| ismbl4 46258 | The predicate " ` A ` is L... |
| volioofmpt 46259 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 46260 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 46261 | The Lebesgue measure of op... |
| volicofmpt 46262 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 46263 | The Lebesgue measure of a ... |
| voliccico 46264 | A closed interval and a le... |
| mbfdmssre 46265 | The domain of a measurable... |
| stoweidlem1 46266 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 46267 | lemma for ~ stoweid : here... |
| stoweidlem3 46268 | Lemma for ~ stoweid : if `... |
| stoweidlem4 46269 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 46270 | There exists a δ as ... |
| stoweidlem6 46271 | Lemma for ~ stoweid : two ... |
| stoweidlem7 46272 | This lemma is used to prov... |
| stoweidlem8 46273 | Lemma for ~ stoweid : two ... |
| stoweidlem9 46274 | Lemma for ~ stoweid : here... |
| stoweidlem10 46275 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 46276 | This lemma is used to prov... |
| stoweidlem12 46277 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 46278 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 46279 | There exists a ` k ` as in... |
| stoweidlem15 46280 | This lemma is used to prov... |
| stoweidlem16 46281 | Lemma for ~ stoweid . The... |
| stoweidlem17 46282 | This lemma proves that the... |
| stoweidlem18 46283 | This theorem proves Lemma ... |
| stoweidlem19 46284 | If a set of real functions... |
| stoweidlem20 46285 | If a set A of real functio... |
| stoweidlem21 46286 | Once the Stone Weierstrass... |
| stoweidlem22 46287 | If a set of real functions... |
| stoweidlem23 46288 | This lemma is used to prov... |
| stoweidlem24 46289 | This lemma proves that for... |
| stoweidlem25 46290 | This lemma proves that for... |
| stoweidlem26 46291 | This lemma is used to prov... |
| stoweidlem27 46292 | This lemma is used to prov... |
| stoweidlem28 46293 | There exists a δ as ... |
| stoweidlem29 46294 | When the hypothesis for th... |
| stoweidlem30 46295 | This lemma is used to prov... |
| stoweidlem31 46296 | This lemma is used to prov... |
| stoweidlem32 46297 | If a set A of real functio... |
| stoweidlem33 46298 | If a set of real functions... |
| stoweidlem34 46299 | This lemma proves that for... |
| stoweidlem35 46300 | This lemma is used to prov... |
| stoweidlem36 46301 | This lemma is used to prov... |
| stoweidlem37 46302 | This lemma is used to prov... |
| stoweidlem38 46303 | This lemma is used to prov... |
| stoweidlem39 46304 | This lemma is used to prov... |
| stoweidlem40 46305 | This lemma proves that q_n... |
| stoweidlem41 46306 | This lemma is used to prov... |
| stoweidlem42 46307 | This lemma is used to prov... |
| stoweidlem43 46308 | This lemma is used to prov... |
| stoweidlem44 46309 | This lemma is used to prov... |
| stoweidlem45 46310 | This lemma proves that, gi... |
| stoweidlem46 46311 | This lemma proves that set... |
| stoweidlem47 46312 | Subtracting a constant fro... |
| stoweidlem48 46313 | This lemma is used to prov... |
| stoweidlem49 46314 | There exists a function q_... |
| stoweidlem50 46315 | This lemma proves that set... |
| stoweidlem51 46316 | There exists a function x ... |
| stoweidlem52 46317 | There exists a neighborhoo... |
| stoweidlem53 46318 | This lemma is used to prov... |
| stoweidlem54 46319 | There exists a function ` ... |
| stoweidlem55 46320 | This lemma proves the exis... |
| stoweidlem56 46321 | This theorem proves Lemma ... |
| stoweidlem57 46322 | There exists a function x ... |
| stoweidlem58 46323 | This theorem proves Lemma ... |
| stoweidlem59 46324 | This lemma proves that the... |
| stoweidlem60 46325 | This lemma proves that the... |
| stoweidlem61 46326 | This lemma proves that the... |
| stoweidlem62 46327 | This theorem proves the St... |
| stoweid 46328 | This theorem proves the St... |
| stowei 46329 | This theorem proves the St... |
| wallispilem1 46330 | ` I ` is monotone: increas... |
| wallispilem2 46331 | A first set of properties ... |
| wallispilem3 46332 | I maps to real values. (C... |
| wallispilem4 46333 | ` F ` maps to explicit exp... |
| wallispilem5 46334 | The sequence ` H ` converg... |
| wallispi 46335 | Wallis' formula for π :... |
| wallispi2lem1 46336 | An intermediate step betwe... |
| wallispi2lem2 46337 | Two expressions are proven... |
| wallispi2 46338 | An alternative version of ... |
| stirlinglem1 46339 | A simple limit of fraction... |
| stirlinglem2 46340 | ` A ` maps to positive rea... |
| stirlinglem3 46341 | Long but simple algebraic ... |
| stirlinglem4 46342 | Algebraic manipulation of ... |
| stirlinglem5 46343 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 46344 | A series that converges to... |
| stirlinglem7 46345 | Algebraic manipulation of ... |
| stirlinglem8 46346 | If ` A ` converges to ` C ... |
| stirlinglem9 46347 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 46348 | A bound for any B(N)-B(N +... |
| stirlinglem11 46349 | ` B ` is decreasing. (Con... |
| stirlinglem12 46350 | The sequence ` B ` is boun... |
| stirlinglem13 46351 | ` B ` is decreasing and ha... |
| stirlinglem14 46352 | The sequence ` A ` converg... |
| stirlinglem15 46353 | The Stirling's formula is ... |
| stirling 46354 | Stirling's approximation f... |
| stirlingr 46355 | Stirling's approximation f... |
| dirkerval 46356 | The N_th Dirichlet Kernel.... |
| dirker2re 46357 | The Dirichlet Kernel value... |
| dirkerdenne0 46358 | The Dirichlet Kernel denom... |
| dirkerval2 46359 | The N_th Dirichlet Kernel ... |
| dirkerre 46360 | The Dirichlet Kernel at an... |
| dirkerper 46361 | the Dirichlet Kernel has p... |
| dirkerf 46362 | For any natural number ` N... |
| dirkertrigeqlem1 46363 | Sum of an even number of a... |
| dirkertrigeqlem2 46364 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 46365 | Trigonometric equality lem... |
| dirkertrigeq 46366 | Trigonometric equality for... |
| dirkeritg 46367 | The definite integral of t... |
| dirkercncflem1 46368 | If ` Y ` is a multiple of ... |
| dirkercncflem2 46369 | Lemma used to prove that t... |
| dirkercncflem3 46370 | The Dirichlet Kernel is co... |
| dirkercncflem4 46371 | The Dirichlet Kernel is co... |
| dirkercncf 46372 | For any natural number ` N... |
| fourierdlem1 46373 | A partition interval is a ... |
| fourierdlem2 46374 | Membership in a partition.... |
| fourierdlem3 46375 | Membership in a partition.... |
| fourierdlem4 46376 | ` E ` is a function that m... |
| fourierdlem5 46377 | ` S ` is a function. (Con... |
| fourierdlem6 46378 | ` X ` is in the periodic p... |
| fourierdlem7 46379 | The difference between the... |
| fourierdlem8 46380 | A partition interval is a ... |
| fourierdlem9 46381 | ` H ` is a complex functio... |
| fourierdlem10 46382 | Condition on the bounds of... |
| fourierdlem11 46383 | If there is a partition, t... |
| fourierdlem12 46384 | A point of a partition is ... |
| fourierdlem13 46385 | Value of ` V ` in terms of... |
| fourierdlem14 46386 | Given the partition ` V ` ... |
| fourierdlem15 46387 | The range of the partition... |
| fourierdlem16 46388 | The coefficients of the fo... |
| fourierdlem17 46389 | The defined ` L ` is actua... |
| fourierdlem18 46390 | The function ` S ` is cont... |
| fourierdlem19 46391 | If two elements of ` D ` h... |
| fourierdlem20 46392 | Every interval in the part... |
| fourierdlem21 46393 | The coefficients of the fo... |
| fourierdlem22 46394 | The coefficients of the fo... |
| fourierdlem23 46395 | If ` F ` is continuous and... |
| fourierdlem24 46396 | A sufficient condition for... |
| fourierdlem25 46397 | If ` C ` is not in the ran... |
| fourierdlem26 46398 | Periodic image of a point ... |
| fourierdlem27 46399 | A partition open interval ... |
| fourierdlem28 46400 | Derivative of ` ( F `` ( X... |
| fourierdlem29 46401 | Explicit function value fo... |
| fourierdlem30 46402 | Sum of three small pieces ... |
| fourierdlem31 46403 | If ` A ` is finite and for... |
| fourierdlem32 46404 | Limit of a continuous func... |
| fourierdlem33 46405 | Limit of a continuous func... |
| fourierdlem34 46406 | A partition is one to one.... |
| fourierdlem35 46407 | There is a single point in... |
| fourierdlem36 46408 | ` F ` is an isomorphism. ... |
| fourierdlem37 46409 | ` I ` is a function that m... |
| fourierdlem38 46410 | The function ` F ` is cont... |
| fourierdlem39 46411 | Integration by parts of ... |
| fourierdlem40 46412 | ` H ` is a continuous func... |
| fourierdlem41 46413 | Lemma used to prove that e... |
| fourierdlem42 46414 | The set of points in a mov... |
| fourierdlem43 46415 | ` K ` is a real function. ... |
| fourierdlem44 46416 | A condition for having ` (... |
| fourierdlem46 46417 | The function ` F ` has a l... |
| fourierdlem47 46418 | For ` r ` large enough, th... |
| fourierdlem48 46419 | The given periodic functio... |
| fourierdlem49 46420 | The given periodic functio... |
| fourierdlem50 46421 | Continuity of ` O ` and it... |
| fourierdlem51 46422 | ` X ` is in the periodic p... |
| fourierdlem52 46423 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 46424 | The limit of ` F ( s ) ` a... |
| fourierdlem54 46425 | Given a partition ` Q ` an... |
| fourierdlem55 46426 | ` U ` is a real function. ... |
| fourierdlem56 46427 | Derivative of the ` K ` fu... |
| fourierdlem57 46428 | The derivative of ` O ` . ... |
| fourierdlem58 46429 | The derivative of ` K ` is... |
| fourierdlem59 46430 | The derivative of ` H ` is... |
| fourierdlem60 46431 | Given a differentiable fun... |
| fourierdlem61 46432 | Given a differentiable fun... |
| fourierdlem62 46433 | The function ` K ` is cont... |
| fourierdlem63 46434 | The upper bound of interva... |
| fourierdlem64 46435 | The partition ` V ` is fin... |
| fourierdlem65 46436 | The distance of two adjace... |
| fourierdlem66 46437 | Value of the ` G ` functio... |
| fourierdlem67 46438 | ` G ` is a function. (Con... |
| fourierdlem68 46439 | The derivative of ` O ` is... |
| fourierdlem69 46440 | A piecewise continuous fun... |
| fourierdlem70 46441 | A piecewise continuous fun... |
| fourierdlem71 46442 | A periodic piecewise conti... |
| fourierdlem72 46443 | The derivative of ` O ` is... |
| fourierdlem73 46444 | A version of the Riemann L... |
| fourierdlem74 46445 | Given a piecewise smooth f... |
| fourierdlem75 46446 | Given a piecewise smooth f... |
| fourierdlem76 46447 | Continuity of ` O ` and it... |
| fourierdlem77 46448 | If ` H ` is bounded, then ... |
| fourierdlem78 46449 | ` G ` is continuous when r... |
| fourierdlem79 46450 | ` E ` projects every inter... |
| fourierdlem80 46451 | The derivative of ` O ` is... |
| fourierdlem81 46452 | The integral of a piecewis... |
| fourierdlem82 46453 | Integral by substitution, ... |
| fourierdlem83 46454 | The fourier partial sum fo... |
| fourierdlem84 46455 | If ` F ` is piecewise cont... |
| fourierdlem85 46456 | Limit of the function ` G ... |
| fourierdlem86 46457 | Continuity of ` O ` and it... |
| fourierdlem87 46458 | The integral of ` G ` goes... |
| fourierdlem88 46459 | Given a piecewise continuo... |
| fourierdlem89 46460 | Given a piecewise continuo... |
| fourierdlem90 46461 | Given a piecewise continuo... |
| fourierdlem91 46462 | Given a piecewise continuo... |
| fourierdlem92 46463 | The integral of a piecewis... |
| fourierdlem93 46464 | Integral by substitution (... |
| fourierdlem94 46465 | For a piecewise smooth fun... |
| fourierdlem95 46466 | Algebraic manipulation of ... |
| fourierdlem96 46467 | limit for ` F ` at the low... |
| fourierdlem97 46468 | ` F ` is continuous on the... |
| fourierdlem98 46469 | ` F ` is continuous on the... |
| fourierdlem99 46470 | limit for ` F ` at the upp... |
| fourierdlem100 46471 | A piecewise continuous fun... |
| fourierdlem101 46472 | Integral by substitution f... |
| fourierdlem102 46473 | For a piecewise smooth fun... |
| fourierdlem103 46474 | The half lower part of the... |
| fourierdlem104 46475 | The half upper part of the... |
| fourierdlem105 46476 | A piecewise continuous fun... |
| fourierdlem106 46477 | For a piecewise smooth fun... |
| fourierdlem107 46478 | The integral of a piecewis... |
| fourierdlem108 46479 | The integral of a piecewis... |
| fourierdlem109 46480 | The integral of a piecewis... |
| fourierdlem110 46481 | The integral of a piecewis... |
| fourierdlem111 46482 | The fourier partial sum fo... |
| fourierdlem112 46483 | Here abbreviations (local ... |
| fourierdlem113 46484 | Fourier series convergence... |
| fourierdlem114 46485 | Fourier series convergence... |
| fourierdlem115 46486 | Fourier serier convergence... |
| fourierd 46487 | Fourier series convergence... |
| fourierclimd 46488 | Fourier series convergence... |
| fourierclim 46489 | Fourier series convergence... |
| fourier 46490 | Fourier series convergence... |
| fouriercnp 46491 | If ` F ` is continuous at ... |
| fourier2 46492 | Fourier series convergence... |
| sqwvfoura 46493 | Fourier coefficients for t... |
| sqwvfourb 46494 | Fourier series ` B ` coeff... |
| fourierswlem 46495 | The Fourier series for the... |
| fouriersw 46496 | Fourier series convergence... |
| fouriercn 46497 | If the derivative of ` F `... |
| elaa2lem 46498 | Elementhood in the set of ... |
| elaa2 46499 | Elementhood in the set of ... |
| etransclem1 46500 | ` H ` is a function. (Con... |
| etransclem2 46501 | Derivative of ` G ` . (Co... |
| etransclem3 46502 | The given ` if ` term is a... |
| etransclem4 46503 | ` F ` expressed as a finit... |
| etransclem5 46504 | A change of bound variable... |
| etransclem6 46505 | A change of bound variable... |
| etransclem7 46506 | The given product is an in... |
| etransclem8 46507 | ` F ` is a function. (Con... |
| etransclem9 46508 | If ` K ` divides ` N ` but... |
| etransclem10 46509 | The given ` if ` term is a... |
| etransclem11 46510 | A change of bound variable... |
| etransclem12 46511 | ` C ` applied to ` N ` . ... |
| etransclem13 46512 | ` F ` applied to ` Y ` . ... |
| etransclem14 46513 | Value of the term ` T ` , ... |
| etransclem15 46514 | Value of the term ` T ` , ... |
| etransclem16 46515 | Every element in the range... |
| etransclem17 46516 | The ` N ` -th derivative o... |
| etransclem18 46517 | The given function is inte... |
| etransclem19 46518 | The ` N ` -th derivative o... |
| etransclem20 46519 | ` H ` is smooth. (Contrib... |
| etransclem21 46520 | The ` N ` -th derivative o... |
| etransclem22 46521 | The ` N ` -th derivative o... |
| etransclem23 46522 | This is the claim proof in... |
| etransclem24 46523 | ` P ` divides the I -th de... |
| etransclem25 46524 | ` P ` factorial divides th... |
| etransclem26 46525 | Every term in the sum of t... |
| etransclem27 46526 | The ` N ` -th derivative o... |
| etransclem28 46527 | ` ( P - 1 ) ` factorial di... |
| etransclem29 46528 | The ` N ` -th derivative o... |
| etransclem30 46529 | The ` N ` -th derivative o... |
| etransclem31 46530 | The ` N ` -th derivative o... |
| etransclem32 46531 | This is the proof for the ... |
| etransclem33 46532 | ` F ` is smooth. (Contrib... |
| etransclem34 46533 | The ` N ` -th derivative o... |
| etransclem35 46534 | ` P ` does not divide the ... |
| etransclem36 46535 | The ` N ` -th derivative o... |
| etransclem37 46536 | ` ( P - 1 ) ` factorial di... |
| etransclem38 46537 | ` P ` divides the I -th de... |
| etransclem39 46538 | ` G ` is a function. (Con... |
| etransclem40 46539 | The ` N ` -th derivative o... |
| etransclem41 46540 | ` P ` does not divide the ... |
| etransclem42 46541 | The ` N ` -th derivative o... |
| etransclem43 46542 | ` G ` is a continuous func... |
| etransclem44 46543 | The given finite sum is no... |
| etransclem45 46544 | ` K ` is an integer. (Con... |
| etransclem46 46545 | This is the proof for equa... |
| etransclem47 46546 | ` _e ` is transcendental. ... |
| etransclem48 46547 | ` _e ` is transcendental. ... |
| etransc 46548 | ` _e ` is transcendental. ... |
| rrxtopn 46549 | The topology of the genera... |
| rrxngp 46550 | Generalized Euclidean real... |
| rrxtps 46551 | Generalized Euclidean real... |
| rrxtopnfi 46552 | The topology of the n-dime... |
| rrxtopon 46553 | The topology on generalize... |
| rrxtop 46554 | The topology on generalize... |
| rrndistlt 46555 | Given two points in the sp... |
| rrxtoponfi 46556 | The topology on n-dimensio... |
| rrxunitopnfi 46557 | The base set of the standa... |
| rrxtopn0 46558 | The topology of the zero-d... |
| qndenserrnbllem 46559 | n-dimensional rational num... |
| qndenserrnbl 46560 | n-dimensional rational num... |
| rrxtopn0b 46561 | The topology of the zero-d... |
| qndenserrnopnlem 46562 | n-dimensional rational num... |
| qndenserrnopn 46563 | n-dimensional rational num... |
| qndenserrn 46564 | n-dimensional rational num... |
| rrxsnicc 46565 | A multidimensional singlet... |
| rrnprjdstle 46566 | The distance between two p... |
| rrndsmet 46567 | ` D ` is a metric for the ... |
| rrndsxmet 46568 | ` D ` is an extended metri... |
| ioorrnopnlem 46569 | The a point in an indexed ... |
| ioorrnopn 46570 | The indexed product of ope... |
| ioorrnopnxrlem 46571 | Given a point ` F ` that b... |
| ioorrnopnxr 46572 | The indexed product of ope... |
| issal 46579 | Express the predicate " ` ... |
| pwsal 46580 | The power set of a given s... |
| salunicl 46581 | SAlg sigma-algebra is clos... |
| saluncl 46582 | The union of two sets in a... |
| prsal 46583 | The pair of the empty set ... |
| saldifcl 46584 | The complement of an eleme... |
| 0sal 46585 | The empty set belongs to e... |
| salgenval 46586 | The sigma-algebra generate... |
| saliunclf 46587 | SAlg sigma-algebra is clos... |
| saliuncl 46588 | SAlg sigma-algebra is clos... |
| salincl 46589 | The intersection of two se... |
| saluni 46590 | A set is an element of any... |
| saliinclf 46591 | SAlg sigma-algebra is clos... |
| saliincl 46592 | SAlg sigma-algebra is clos... |
| saldifcl2 46593 | The difference of two elem... |
| intsaluni 46594 | The union of an arbitrary ... |
| intsal 46595 | The arbitrary intersection... |
| salgenn0 46596 | The set used in the defini... |
| salgencl 46597 | ` SalGen ` actually genera... |
| issald 46598 | Sufficient condition to pr... |
| salexct 46599 | An example of nontrivial s... |
| sssalgen 46600 | A set is a subset of the s... |
| salgenss 46601 | The sigma-algebra generate... |
| salgenuni 46602 | The base set of the sigma-... |
| issalgend 46603 | One side of ~ dfsalgen2 . ... |
| salexct2 46604 | An example of a subset tha... |
| unisalgen 46605 | The union of a set belongs... |
| dfsalgen2 46606 | Alternate characterization... |
| salexct3 46607 | An example of a sigma-alge... |
| salgencntex 46608 | This counterexample shows ... |
| salgensscntex 46609 | This counterexample shows ... |
| issalnnd 46610 | Sufficient condition to pr... |
| dmvolsal 46611 | Lebesgue measurable sets f... |
| saldifcld 46612 | The complement of an eleme... |
| saluncld 46613 | The union of two sets in a... |
| salgencld 46614 | ` SalGen ` actually genera... |
| 0sald 46615 | The empty set belongs to e... |
| iooborel 46616 | An open interval is a Bore... |
| salincld 46617 | The intersection of two se... |
| salunid 46618 | A set is an element of any... |
| unisalgen2 46619 | The union of a set belongs... |
| bor1sal 46620 | The Borel sigma-algebra on... |
| iocborel 46621 | A left-open, right-closed ... |
| subsaliuncllem 46622 | A subspace sigma-algebra i... |
| subsaliuncl 46623 | A subspace sigma-algebra i... |
| subsalsal 46624 | A subspace sigma-algebra i... |
| subsaluni 46625 | A set belongs to the subsp... |
| salrestss 46626 | A sigma-algebra restricted... |
| sge0rnre 46629 | When ` sum^ ` is applied t... |
| fge0icoicc 46630 | If ` F ` maps to nonnegati... |
| sge0val 46631 | The value of the sum of no... |
| fge0npnf 46632 | If ` F ` maps to nonnegati... |
| sge0rnn0 46633 | The range used in the defi... |
| sge0vald 46634 | The value of the sum of no... |
| fge0iccico 46635 | A range of nonnegative ext... |
| gsumge0cl 46636 | Closure of group sum, for ... |
| sge0reval 46637 | Value of the sum of nonneg... |
| sge0pnfval 46638 | If a term in the sum of no... |
| fge0iccre 46639 | A range of nonnegative ext... |
| sge0z 46640 | Any nonnegative extended s... |
| sge00 46641 | The sum of nonnegative ext... |
| fsumlesge0 46642 | Every finite subsum of non... |
| sge0revalmpt 46643 | Value of the sum of nonneg... |
| sge0sn 46644 | A sum of a nonnegative ext... |
| sge0tsms 46645 | ` sum^ ` applied to a nonn... |
| sge0cl 46646 | The arbitrary sum of nonne... |
| sge0f1o 46647 | Re-index a nonnegative ext... |
| sge0snmpt 46648 | A sum of a nonnegative ext... |
| sge0ge0 46649 | The sum of nonnegative ext... |
| sge0xrcl 46650 | The arbitrary sum of nonne... |
| sge0repnf 46651 | The of nonnegative extende... |
| sge0fsum 46652 | The arbitrary sum of a fin... |
| sge0rern 46653 | If the sum of nonnegative ... |
| sge0supre 46654 | If the arbitrary sum of no... |
| sge0fsummpt 46655 | The arbitrary sum of a fin... |
| sge0sup 46656 | The arbitrary sum of nonne... |
| sge0less 46657 | A shorter sum of nonnegati... |
| sge0rnbnd 46658 | The range used in the defi... |
| sge0pr 46659 | Sum of a pair of nonnegati... |
| sge0gerp 46660 | The arbitrary sum of nonne... |
| sge0pnffigt 46661 | If the sum of nonnegative ... |
| sge0ssre 46662 | If a sum of nonnegative ex... |
| sge0lefi 46663 | A sum of nonnegative exten... |
| sge0lessmpt 46664 | A shorter sum of nonnegati... |
| sge0ltfirp 46665 | If the sum of nonnegative ... |
| sge0prle 46666 | The sum of a pair of nonne... |
| sge0gerpmpt 46667 | The arbitrary sum of nonne... |
| sge0resrnlem 46668 | The sum of nonnegative ext... |
| sge0resrn 46669 | The sum of nonnegative ext... |
| sge0ssrempt 46670 | If a sum of nonnegative ex... |
| sge0resplit 46671 | ` sum^ ` splits into two p... |
| sge0le 46672 | If all of the terms of sum... |
| sge0ltfirpmpt 46673 | If the extended sum of non... |
| sge0split 46674 | Split a sum of nonnegative... |
| sge0lempt 46675 | If all of the terms of sum... |
| sge0splitmpt 46676 | Split a sum of nonnegative... |
| sge0ss 46677 | Change the index set to a ... |
| sge0iunmptlemfi 46678 | Sum of nonnegative extende... |
| sge0p1 46679 | The addition of the next t... |
| sge0iunmptlemre 46680 | Sum of nonnegative extende... |
| sge0fodjrnlem 46681 | Re-index a nonnegative ext... |
| sge0fodjrn 46682 | Re-index a nonnegative ext... |
| sge0iunmpt 46683 | Sum of nonnegative extende... |
| sge0iun 46684 | Sum of nonnegative extende... |
| sge0nemnf 46685 | The generalized sum of non... |
| sge0rpcpnf 46686 | The sum of an infinite num... |
| sge0rernmpt 46687 | If the sum of nonnegative ... |
| sge0lefimpt 46688 | A sum of nonnegative exten... |
| nn0ssge0 46689 | Nonnegative integers are n... |
| sge0clmpt 46690 | The generalized sum of non... |
| sge0ltfirpmpt2 46691 | If the extended sum of non... |
| sge0isum 46692 | If a series of nonnegative... |
| sge0xrclmpt 46693 | The generalized sum of non... |
| sge0xp 46694 | Combine two generalized su... |
| sge0isummpt 46695 | If a series of nonnegative... |
| sge0ad2en 46696 | The value of the infinite ... |
| sge0isummpt2 46697 | If a series of nonnegative... |
| sge0xaddlem1 46698 | The extended addition of t... |
| sge0xaddlem2 46699 | The extended addition of t... |
| sge0xadd 46700 | The extended addition of t... |
| sge0fsummptf 46701 | The generalized sum of a f... |
| sge0snmptf 46702 | A sum of a nonnegative ext... |
| sge0ge0mpt 46703 | The sum of nonnegative ext... |
| sge0repnfmpt 46704 | The of nonnegative extende... |
| sge0pnffigtmpt 46705 | If the generalized sum of ... |
| sge0splitsn 46706 | Separate out a term in a g... |
| sge0pnffsumgt 46707 | If the sum of nonnegative ... |
| sge0gtfsumgt 46708 | If the generalized sum of ... |
| sge0uzfsumgt 46709 | If a real number is smalle... |
| sge0pnfmpt 46710 | If a term in the sum of no... |
| sge0seq 46711 | A series of nonnegative re... |
| sge0reuz 46712 | Value of the generalized s... |
| sge0reuzb 46713 | Value of the generalized s... |
| ismea 46716 | Express the predicate " ` ... |
| dmmeasal 46717 | The domain of a measure is... |
| meaf 46718 | A measure is a function th... |
| mea0 46719 | The measure of the empty s... |
| nnfoctbdjlem 46720 | There exists a mapping fro... |
| nnfoctbdj 46721 | There exists a mapping fro... |
| meadjuni 46722 | The measure of the disjoin... |
| meacl 46723 | The measure of a set is a ... |
| iundjiunlem 46724 | The sets in the sequence `... |
| iundjiun 46725 | Given a sequence ` E ` of ... |
| meaxrcl 46726 | The measure of a set is an... |
| meadjun 46727 | The measure of the union o... |
| meassle 46728 | The measure of a set is gr... |
| meaunle 46729 | The measure of the union o... |
| meadjiunlem 46730 | The sum of nonnegative ext... |
| meadjiun 46731 | The measure of the disjoin... |
| ismeannd 46732 | Sufficient condition to pr... |
| meaiunlelem 46733 | The measure of the union o... |
| meaiunle 46734 | The measure of the union o... |
| psmeasurelem 46735 | ` M ` applied to a disjoin... |
| psmeasure 46736 | Point supported measure, R... |
| voliunsge0lem 46737 | The Lebesgue measure funct... |
| voliunsge0 46738 | The Lebesgue measure funct... |
| volmea 46739 | The Lebesgue measure on th... |
| meage0 46740 | If the measure of a measur... |
| meadjunre 46741 | The measure of the union o... |
| meassre 46742 | If the measure of a measur... |
| meale0eq0 46743 | A measure that is less tha... |
| meadif 46744 | The measure of the differe... |
| meaiuninclem 46745 | Measures are continuous fr... |
| meaiuninc 46746 | Measures are continuous fr... |
| meaiuninc2 46747 | Measures are continuous fr... |
| meaiunincf 46748 | Measures are continuous fr... |
| meaiuninc3v 46749 | Measures are continuous fr... |
| meaiuninc3 46750 | Measures are continuous fr... |
| meaiininclem 46751 | Measures are continuous fr... |
| meaiininc 46752 | Measures are continuous fr... |
| meaiininc2 46753 | Measures are continuous fr... |
| caragenval 46758 | The sigma-algebra generate... |
| isome 46759 | Express the predicate " ` ... |
| caragenel 46760 | Membership in the Caratheo... |
| omef 46761 | An outer measure is a func... |
| ome0 46762 | The outer measure of the e... |
| omessle 46763 | The outer measure of a set... |
| omedm 46764 | The domain of an outer mea... |
| caragensplit 46765 | If ` E ` is in the set gen... |
| caragenelss 46766 | An element of the Caratheo... |
| carageneld 46767 | Membership in the Caratheo... |
| omecl 46768 | The outer measure of a set... |
| caragenss 46769 | The sigma-algebra generate... |
| omeunile 46770 | The outer measure of the u... |
| caragen0 46771 | The empty set belongs to a... |
| omexrcl 46772 | The outer measure of a set... |
| caragenunidm 46773 | The base set of an outer m... |
| caragensspw 46774 | The sigma-algebra generate... |
| omessre 46775 | If the outer measure of a ... |
| caragenuni 46776 | The base set of the sigma-... |
| caragenuncllem 46777 | The Caratheodory's constru... |
| caragenuncl 46778 | The Caratheodory's constru... |
| caragendifcl 46779 | The Caratheodory's constru... |
| caragenfiiuncl 46780 | The Caratheodory's constru... |
| omeunle 46781 | The outer measure of the u... |
| omeiunle 46782 | The outer measure of the i... |
| omelesplit 46783 | The outer measure of a set... |
| omeiunltfirp 46784 | If the outer measure of a ... |
| omeiunlempt 46785 | The outer measure of the i... |
| carageniuncllem1 46786 | The outer measure of ` A i... |
| carageniuncllem2 46787 | The Caratheodory's constru... |
| carageniuncl 46788 | The Caratheodory's constru... |
| caragenunicl 46789 | The Caratheodory's constru... |
| caragensal 46790 | Caratheodory's method gene... |
| caratheodorylem1 46791 | Lemma used to prove that C... |
| caratheodorylem2 46792 | Caratheodory's constructio... |
| caratheodory 46793 | Caratheodory's constructio... |
| 0ome 46794 | The map that assigns 0 to ... |
| isomenndlem 46795 | ` O ` is sub-additive w.r.... |
| isomennd 46796 | Sufficient condition to pr... |
| caragenel2d 46797 | Membership in the Caratheo... |
| omege0 46798 | If the outer measure of a ... |
| omess0 46799 | If the outer measure of a ... |
| caragencmpl 46800 | A measure built with the C... |
| vonval 46805 | Value of the Lebesgue meas... |
| ovnval 46806 | Value of the Lebesgue oute... |
| elhoi 46807 | Membership in a multidimen... |
| icoresmbl 46808 | A closed-below, open-above... |
| hoissre 46809 | The projection of a half-o... |
| ovnval2 46810 | Value of the Lebesgue oute... |
| volicorecl 46811 | The Lebesgue measure of a ... |
| hoiprodcl 46812 | The pre-measure of half-op... |
| hoicvr 46813 | ` I ` is a countable set o... |
| hoissrrn 46814 | A half-open interval is a ... |
| ovn0val 46815 | The Lebesgue outer measure... |
| ovnn0val 46816 | The value of a (multidimen... |
| ovnval2b 46817 | Value of the Lebesgue oute... |
| volicorescl 46818 | The Lebesgue measure of a ... |
| ovnprodcl 46819 | The product used in the de... |
| hoiprodcl2 46820 | The pre-measure of half-op... |
| hoicvrrex 46821 | Any subset of the multidim... |
| ovnsupge0 46822 | The set used in the defini... |
| ovnlecvr 46823 | Given a subset of multidim... |
| ovnpnfelsup 46824 | ` +oo ` is an element of t... |
| ovnsslelem 46825 | The (multidimensional, non... |
| ovnssle 46826 | The (multidimensional) Leb... |
| ovnlerp 46827 | The Lebesgue outer measure... |
| ovnf 46828 | The Lebesgue outer measure... |
| ovncvrrp 46829 | The Lebesgue outer measure... |
| ovn0lem 46830 | For any finite dimension, ... |
| ovn0 46831 | For any finite dimension, ... |
| ovncl 46832 | The Lebesgue outer measure... |
| ovn02 46833 | For the zero-dimensional s... |
| ovnxrcl 46834 | The Lebesgue outer measure... |
| ovnsubaddlem1 46835 | The Lebesgue outer measure... |
| ovnsubaddlem2 46836 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 46837 | ` ( voln* `` X ) ` is suba... |
| ovnome 46838 | ` ( voln* `` X ) ` is an o... |
| vonmea 46839 | ` ( voln `` X ) ` is a mea... |
| volicon0 46840 | The measure of a nonempty ... |
| hsphoif 46841 | ` H ` is a function (that ... |
| hoidmvval 46842 | The dimensional volume of ... |
| hoissrrn2 46843 | A half-open interval is a ... |
| hsphoival 46844 | ` H ` is a function (that ... |
| hoiprodcl3 46845 | The pre-measure of half-op... |
| volicore 46846 | The Lebesgue measure of a ... |
| hoidmvcl 46847 | The dimensional volume of ... |
| hoidmv0val 46848 | The dimensional volume of ... |
| hoidmvn0val 46849 | The dimensional volume of ... |
| hsphoidmvle2 46850 | The dimensional volume of ... |
| hsphoidmvle 46851 | The dimensional volume of ... |
| hoidmvval0 46852 | The dimensional volume of ... |
| hoiprodp1 46853 | The dimensional volume of ... |
| sge0hsphoire 46854 | If the generalized sum of ... |
| hoidmvval0b 46855 | The dimensional volume of ... |
| hoidmv1lelem1 46856 | The supremum of ` U ` belo... |
| hoidmv1lelem2 46857 | This is the contradiction ... |
| hoidmv1lelem3 46858 | The dimensional volume of ... |
| hoidmv1le 46859 | The dimensional volume of ... |
| hoidmvlelem1 46860 | The supremum of ` U ` belo... |
| hoidmvlelem2 46861 | This is the contradiction ... |
| hoidmvlelem3 46862 | This is the contradiction ... |
| hoidmvlelem4 46863 | The dimensional volume of ... |
| hoidmvlelem5 46864 | The dimensional volume of ... |
| hoidmvle 46865 | The dimensional volume of ... |
| ovnhoilem1 46866 | The Lebesgue outer measure... |
| ovnhoilem2 46867 | The Lebesgue outer measure... |
| ovnhoi 46868 | The Lebesgue outer measure... |
| dmovn 46869 | The domain of the Lebesgue... |
| hoicoto2 46870 | The half-open interval exp... |
| dmvon 46871 | Lebesgue measurable n-dime... |
| hoi2toco 46872 | The half-open interval exp... |
| hoidifhspval 46873 | ` D ` is a function that r... |
| hspval 46874 | The value of the half-spac... |
| ovnlecvr2 46875 | Given a subset of multidim... |
| ovncvr2 46876 | ` B ` and ` T ` are the le... |
| dmovnsal 46877 | The domain of the Lebesgue... |
| unidmovn 46878 | Base set of the n-dimensio... |
| rrnmbl 46879 | The set of n-dimensional R... |
| hoidifhspval2 46880 | ` D ` is a function that r... |
| hspdifhsp 46881 | A n-dimensional half-open ... |
| unidmvon 46882 | Base set of the n-dimensio... |
| hoidifhspf 46883 | ` D ` is a function that r... |
| hoidifhspval3 46884 | ` D ` is a function that r... |
| hoidifhspdmvle 46885 | The dimensional volume of ... |
| voncmpl 46886 | The Lebesgue measure is co... |
| hoiqssbllem1 46887 | The center of the n-dimens... |
| hoiqssbllem2 46888 | The center of the n-dimens... |
| hoiqssbllem3 46889 | A n-dimensional ball conta... |
| hoiqssbl 46890 | A n-dimensional ball conta... |
| hspmbllem1 46891 | Any half-space of the n-di... |
| hspmbllem2 46892 | Any half-space of the n-di... |
| hspmbllem3 46893 | Any half-space of the n-di... |
| hspmbl 46894 | Any half-space of the n-di... |
| hoimbllem 46895 | Any n-dimensional half-ope... |
| hoimbl 46896 | Any n-dimensional half-ope... |
| opnvonmbllem1 46897 | The half-open interval exp... |
| opnvonmbllem2 46898 | An open subset of the n-di... |
| opnvonmbl 46899 | An open subset of the n-di... |
| opnssborel 46900 | Open sets of a generalized... |
| borelmbl 46901 | All Borel subsets of the n... |
| volicorege0 46902 | The Lebesgue measure of a ... |
| isvonmbl 46903 | The predicate " ` A ` is m... |
| mblvon 46904 | The n-dimensional Lebesgue... |
| vonmblss 46905 | n-dimensional Lebesgue mea... |
| volico2 46906 | The measure of left-closed... |
| vonmblss2 46907 | n-dimensional Lebesgue mea... |
| ovolval2lem 46908 | The value of the Lebesgue ... |
| ovolval2 46909 | The value of the Lebesgue ... |
| ovnsubadd2lem 46910 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 46911 | ` ( voln* `` X ) ` is suba... |
| ovolval3 46912 | The value of the Lebesgue ... |
| ovnsplit 46913 | The n-dimensional Lebesgue... |
| ovolval4lem1 46914 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 46915 | The value of the Lebesgue ... |
| ovolval4 46916 | The value of the Lebesgue ... |
| ovolval5lem1 46917 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 46918 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 46919 | The value of the Lebesgue ... |
| ovolval5 46920 | The value of the Lebesgue ... |
| ovnovollem1 46921 | if ` F ` is a cover of ` B... |
| ovnovollem2 46922 | if ` I ` is a cover of ` (... |
| ovnovollem3 46923 | The 1-dimensional Lebesgue... |
| ovnovol 46924 | The 1-dimensional Lebesgue... |
| vonvolmbllem 46925 | If a subset ` B ` of real ... |
| vonvolmbl 46926 | A subset of Real numbers i... |
| vonvol 46927 | The 1-dimensional Lebesgue... |
| vonvolmbl2 46928 | A subset ` X ` of the spac... |
| vonvol2 46929 | The 1-dimensional Lebesgue... |
| hoimbl2 46930 | Any n-dimensional half-ope... |
| voncl 46931 | The Lebesgue measure of a ... |
| vonhoi 46932 | The Lebesgue outer measure... |
| vonxrcl 46933 | The Lebesgue measure of a ... |
| ioosshoi 46934 | A n-dimensional open inter... |
| vonn0hoi 46935 | The Lebesgue outer measure... |
| von0val 46936 | The Lebesgue measure (for ... |
| vonhoire 46937 | The Lebesgue measure of a ... |
| iinhoiicclem 46938 | A n-dimensional closed int... |
| iinhoiicc 46939 | A n-dimensional closed int... |
| iunhoiioolem 46940 | A n-dimensional open inter... |
| iunhoiioo 46941 | A n-dimensional open inter... |
| ioovonmbl 46942 | Any n-dimensional open int... |
| iccvonmbllem 46943 | Any n-dimensional closed i... |
| iccvonmbl 46944 | Any n-dimensional closed i... |
| vonioolem1 46945 | The sequence of the measur... |
| vonioolem2 46946 | The n-dimensional Lebesgue... |
| vonioo 46947 | The n-dimensional Lebesgue... |
| vonicclem1 46948 | The sequence of the measur... |
| vonicclem2 46949 | The n-dimensional Lebesgue... |
| vonicc 46950 | The n-dimensional Lebesgue... |
| snvonmbl 46951 | A n-dimensional singleton ... |
| vonn0ioo 46952 | The n-dimensional Lebesgue... |
| vonn0icc 46953 | The n-dimensional Lebesgue... |
| ctvonmbl 46954 | Any n-dimensional countabl... |
| vonn0ioo2 46955 | The n-dimensional Lebesgue... |
| vonsn 46956 | The n-dimensional Lebesgue... |
| vonn0icc2 46957 | The n-dimensional Lebesgue... |
| vonct 46958 | The n-dimensional Lebesgue... |
| vitali2 46959 | There are non-measurable s... |
| pimltmnf2f 46962 | Given a real-valued functi... |
| pimltmnf2 46963 | Given a real-valued functi... |
| preimagelt 46964 | The preimage of a right-op... |
| preimalegt 46965 | The preimage of a left-ope... |
| pimconstlt0 46966 | Given a constant function,... |
| pimconstlt1 46967 | Given a constant function,... |
| pimltpnff 46968 | Given a real-valued functi... |
| pimltpnf 46969 | Given a real-valued functi... |
| pimgtpnf2f 46970 | Given a real-valued functi... |
| pimgtpnf2 46971 | Given a real-valued functi... |
| salpreimagelt 46972 | If all the preimages of le... |
| pimrecltpos 46973 | The preimage of an unbound... |
| salpreimalegt 46974 | If all the preimages of ri... |
| pimiooltgt 46975 | The preimage of an open in... |
| preimaicomnf 46976 | Preimage of an open interv... |
| pimltpnf2f 46977 | Given a real-valued functi... |
| pimltpnf2 46978 | Given a real-valued functi... |
| pimgtmnf2 46979 | Given a real-valued functi... |
| pimdecfgtioc 46980 | Given a nonincreasing func... |
| pimincfltioc 46981 | Given a nondecreasing func... |
| pimdecfgtioo 46982 | Given a nondecreasing func... |
| pimincfltioo 46983 | Given a nondecreasing func... |
| preimaioomnf 46984 | Preimage of an open interv... |
| preimageiingt 46985 | A preimage of a left-close... |
| preimaleiinlt 46986 | A preimage of a left-open,... |
| pimgtmnff 46987 | Given a real-valued functi... |
| pimgtmnf 46988 | Given a real-valued functi... |
| pimrecltneg 46989 | The preimage of an unbound... |
| salpreimagtge 46990 | If all the preimages of le... |
| salpreimaltle 46991 | If all the preimages of ri... |
| issmflem 46992 | The predicate " ` F ` is a... |
| issmf 46993 | The predicate " ` F ` is a... |
| salpreimalelt 46994 | If all the preimages of ri... |
| salpreimagtlt 46995 | If all the preimages of le... |
| smfpreimalt 46996 | Given a function measurabl... |
| smff 46997 | A function measurable w.r.... |
| smfdmss 46998 | The domain of a function m... |
| issmff 46999 | The predicate " ` F ` is a... |
| issmfd 47000 | A sufficient condition for... |
| smfpreimaltf 47001 | Given a function measurabl... |
| issmfdf 47002 | A sufficient condition for... |
| sssmf 47003 | The restriction of a sigma... |
| mbfresmf 47004 | A real-valued measurable f... |
| cnfsmf 47005 | A continuous function is m... |
| incsmflem 47006 | A nondecreasing function i... |
| incsmf 47007 | A real-valued, nondecreasi... |
| smfsssmf 47008 | If a function is measurabl... |
| issmflelem 47009 | The predicate " ` F ` is a... |
| issmfle 47010 | The predicate " ` F ` is a... |
| smfpimltmpt 47011 | Given a function measurabl... |
| smfpimltxr 47012 | Given a function measurabl... |
| issmfdmpt 47013 | A sufficient condition for... |
| smfconst 47014 | Given a sigma-algebra over... |
| sssmfmpt 47015 | The restriction of a sigma... |
| cnfrrnsmf 47016 | A function, continuous fro... |
| smfid 47017 | The identity function is B... |
| bormflebmf 47018 | A Borel measurable functio... |
| smfpreimale 47019 | Given a function measurabl... |
| issmfgtlem 47020 | The predicate " ` F ` is a... |
| issmfgt 47021 | The predicate " ` F ` is a... |
| issmfled 47022 | A sufficient condition for... |
| smfpimltxrmptf 47023 | Given a function measurabl... |
| smfpimltxrmpt 47024 | Given a function measurabl... |
| smfmbfcex 47025 | A constant function, with ... |
| issmfgtd 47026 | A sufficient condition for... |
| smfpreimagt 47027 | Given a function measurabl... |
| smfaddlem1 47028 | Given the sum of two funct... |
| smfaddlem2 47029 | The sum of two sigma-measu... |
| smfadd 47030 | The sum of two sigma-measu... |
| decsmflem 47031 | A nonincreasing function i... |
| decsmf 47032 | A real-valued, nonincreasi... |
| smfpreimagtf 47033 | Given a function measurabl... |
| issmfgelem 47034 | The predicate " ` F ` is a... |
| issmfge 47035 | The predicate " ` F ` is a... |
| smflimlem1 47036 | Lemma for the proof that t... |
| smflimlem2 47037 | Lemma for the proof that t... |
| smflimlem3 47038 | The limit of sigma-measura... |
| smflimlem4 47039 | Lemma for the proof that t... |
| smflimlem5 47040 | Lemma for the proof that t... |
| smflimlem6 47041 | Lemma for the proof that t... |
| smflim 47042 | The limit of sigma-measura... |
| nsssmfmbflem 47043 | The sigma-measurable funct... |
| nsssmfmbf 47044 | The sigma-measurable funct... |
| smfpimgtxr 47045 | Given a function measurabl... |
| smfpimgtmpt 47046 | Given a function measurabl... |
| smfpreimage 47047 | Given a function measurabl... |
| mbfpsssmf 47048 | Real-valued measurable fun... |
| smfpimgtxrmptf 47049 | Given a function measurabl... |
| smfpimgtxrmpt 47050 | Given a function measurabl... |
| smfpimioompt 47051 | Given a function measurabl... |
| smfpimioo 47052 | Given a function measurabl... |
| smfresal 47053 | Given a sigma-measurable f... |
| smfrec 47054 | The reciprocal of a sigma-... |
| smfres 47055 | The restriction of sigma-m... |
| smfmullem1 47056 | The multiplication of two ... |
| smfmullem2 47057 | The multiplication of two ... |
| smfmullem3 47058 | The multiplication of two ... |
| smfmullem4 47059 | The multiplication of two ... |
| smfmul 47060 | The multiplication of two ... |
| smfmulc1 47061 | A sigma-measurable functio... |
| smfdiv 47062 | The fraction of two sigma-... |
| smfpimbor1lem1 47063 | Every open set belongs to ... |
| smfpimbor1lem2 47064 | Given a sigma-measurable f... |
| smfpimbor1 47065 | Given a sigma-measurable f... |
| smf2id 47066 | Twice the identity functio... |
| smfco 47067 | The composition of a Borel... |
| smfneg 47068 | The negative of a sigma-me... |
| smffmptf 47069 | A function measurable w.r.... |
| smffmpt 47070 | A function measurable w.r.... |
| smflim2 47071 | The limit of a sequence of... |
| smfpimcclem 47072 | Lemma for ~ smfpimcc given... |
| smfpimcc 47073 | Given a countable set of s... |
| issmfle2d 47074 | A sufficient condition for... |
| smflimmpt 47075 | The limit of a sequence of... |
| smfsuplem1 47076 | The supremum of a countabl... |
| smfsuplem2 47077 | The supremum of a countabl... |
| smfsuplem3 47078 | The supremum of a countabl... |
| smfsup 47079 | The supremum of a countabl... |
| smfsupmpt 47080 | The supremum of a countabl... |
| smfsupxr 47081 | The supremum of a countabl... |
| smfinflem 47082 | The infimum of a countable... |
| smfinf 47083 | The infimum of a countable... |
| smfinfmpt 47084 | The infimum of a countable... |
| smflimsuplem1 47085 | If ` H ` converges, the ` ... |
| smflimsuplem2 47086 | The superior limit of a se... |
| smflimsuplem3 47087 | The limit of the ` ( H `` ... |
| smflimsuplem4 47088 | If ` H ` converges, the ` ... |
| smflimsuplem5 47089 | ` H ` converges to the sup... |
| smflimsuplem6 47090 | The superior limit of a se... |
| smflimsuplem7 47091 | The superior limit of a se... |
| smflimsuplem8 47092 | The superior limit of a se... |
| smflimsup 47093 | The superior limit of a se... |
| smflimsupmpt 47094 | The superior limit of a se... |
| smfliminflem 47095 | The inferior limit of a co... |
| smfliminf 47096 | The inferior limit of a co... |
| smfliminfmpt 47097 | The inferior limit of a co... |
| adddmmbl 47098 | If two functions have doma... |
| adddmmbl2 47099 | If two functions have doma... |
| muldmmbl 47100 | If two functions have doma... |
| muldmmbl2 47101 | If two functions have doma... |
| smfdmmblpimne 47102 | If a measurable function w... |
| smfdivdmmbl 47103 | If a functions and a sigma... |
| smfpimne 47104 | Given a function measurabl... |
| smfpimne2 47105 | Given a function measurabl... |
| smfdivdmmbl2 47106 | If a functions and a sigma... |
| fsupdm 47107 | The domain of the sup func... |
| fsupdm2 47108 | The domain of the sup func... |
| smfsupdmmbllem 47109 | If a countable set of sigm... |
| smfsupdmmbl 47110 | If a countable set of sigm... |
| finfdm 47111 | The domain of the inf func... |
| finfdm2 47112 | The domain of the inf func... |
| smfinfdmmbllem 47113 | If a countable set of sigm... |
| smfinfdmmbl 47114 | If a countable set of sigm... |
| sigarval 47115 | Define the signed area by ... |
| sigarim 47116 | Signed area takes value in... |
| sigarac 47117 | Signed area is anticommuta... |
| sigaraf 47118 | Signed area is additive by... |
| sigarmf 47119 | Signed area is additive (w... |
| sigaras 47120 | Signed area is additive by... |
| sigarms 47121 | Signed area is additive (w... |
| sigarls 47122 | Signed area is linear by t... |
| sigarid 47123 | Signed area of a flat para... |
| sigarexp 47124 | Expand the signed area for... |
| sigarperm 47125 | Signed area ` ( A - C ) G ... |
| sigardiv 47126 | If signed area between vec... |
| sigarimcd 47127 | Signed area takes value in... |
| sigariz 47128 | If signed area is zero, th... |
| sigarcol 47129 | Given three points ` A ` ,... |
| sharhght 47130 | Let ` A B C ` be a triangl... |
| sigaradd 47131 | Subtracting (double) area ... |
| cevathlem1 47132 | Ceva's theorem first lemma... |
| cevathlem2 47133 | Ceva's theorem second lemm... |
| cevath 47134 | Ceva's theorem. Let ` A B... |
| simpcntrab 47135 | The center of a simple gro... |
| et-ltneverrefl 47136 | Less-than class is never r... |
| et-equeucl 47137 | Alternative proof that equ... |
| et-sqrtnegnre 47138 | The square root of a negat... |
| ormklocald 47139 | If elements of a certain s... |
| ormkglobd 47140 | If all adjacent elements o... |
| natlocalincr 47141 | Global monotonicity on hal... |
| natglobalincr 47142 | Local monotonicity on half... |
| chnsubseqword 47143 | A subsequence of a chain i... |
| chnsubseqwl 47144 | A subsequence of a chain h... |
| chnsubseq 47145 | An order-preserving subseq... |
| chnsuslle 47146 | Length of a subsequence is... |
| chnerlem1 47147 | In a chain constructed on ... |
| chnerlem2 47148 | Lemma for ~ chner where th... |
| chnerlem3 47149 | Lemma for ~ chner - tricho... |
| chner 47150 | Any two elements are equiv... |
| nthrucw 47151 | Some number sets form a ch... |
| evenwodadd 47152 | If an integer is multiplie... |
| squeezedltsq 47153 | If a real value is squeeze... |
| lambert0 47154 | A value of Lambert W (prod... |
| lamberte 47155 | A value of Lambert W (prod... |
| cjnpoly 47156 | Complex conjugation operat... |
| tannpoly 47157 | The tangent function is no... |
| sinnpoly 47158 | Sine function is not a pol... |
| hirstL-ax3 47159 | The third axiom of a syste... |
| ax3h 47160 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 47161 | A closed form showing (a i... |
| aibandbiaiaiffb 47162 | A closed form showing (a i... |
| notatnand 47163 | Do not use. Use intnanr i... |
| aistia 47164 | Given a is equivalent to `... |
| aisfina 47165 | Given a is equivalent to `... |
| bothtbothsame 47166 | Given both a, b are equiva... |
| bothfbothsame 47167 | Given both a, b are equiva... |
| aiffbbtat 47168 | Given a is equivalent to b... |
| aisbbisfaisf 47169 | Given a is equivalent to b... |
| axorbtnotaiffb 47170 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 47171 | Given a is equivalent to (... |
| axorbciffatcxorb 47172 | Given a is equivalent to (... |
| aibnbna 47173 | Given a implies b, (not b)... |
| aibnbaif 47174 | Given a implies b, not b, ... |
| aiffbtbat 47175 | Given a is equivalent to b... |
| astbstanbst 47176 | Given a is equivalent to T... |
| aistbistaandb 47177 | Given a is equivalent to T... |
| aisbnaxb 47178 | Given a is equivalent to b... |
| atbiffatnnb 47179 | If a implies b, then a imp... |
| bisaiaisb 47180 | Application of bicom1 with... |
| atbiffatnnbalt 47181 | If a implies b, then a imp... |
| abnotbtaxb 47182 | Assuming a, not b, there e... |
| abnotataxb 47183 | Assuming not a, b, there e... |
| conimpf 47184 | Assuming a, not b, and a i... |
| conimpfalt 47185 | Assuming a, not b, and a i... |
| aistbisfiaxb 47186 | Given a is equivalent to T... |
| aisfbistiaxb 47187 | Given a is equivalent to F... |
| aifftbifffaibif 47188 | Given a is equivalent to T... |
| aifftbifffaibifff 47189 | Given a is equivalent to T... |
| atnaiana 47190 | Given a, it is not the cas... |
| ainaiaandna 47191 | Given a, a implies it is n... |
| abcdta 47192 | Given (((a and b) and c) a... |
| abcdtb 47193 | Given (((a and b) and c) a... |
| abcdtc 47194 | Given (((a and b) and c) a... |
| abcdtd 47195 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 47196 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 47197 | Operands in a biconditiona... |
| nabctnabc 47198 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 47199 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 47200 | From one negated implicati... |
| twonotinotbothi 47201 | From these two negated imp... |
| clifte 47202 | show d is the same as an i... |
| cliftet 47203 | show d is the same as an i... |
| clifteta 47204 | show d is the same as an i... |
| cliftetb 47205 | show d is the same as an i... |
| confun 47206 | Given the hypotheses there... |
| confun2 47207 | Confun simplified to two p... |
| confun3 47208 | Confun's more complex form... |
| confun4 47209 | An attempt at derivative. ... |
| confun5 47210 | An attempt at derivative. ... |
| plcofph 47211 | Given, a,b and a "definiti... |
| pldofph 47212 | Given, a,b c, d, "definiti... |
| plvcofph 47213 | Given, a,b,d, and "definit... |
| plvcofphax 47214 | Given, a,b,d, and "definit... |
| plvofpos 47215 | rh is derivable because ON... |
| mdandyv0 47216 | Given the equivalences set... |
| mdandyv1 47217 | Given the equivalences set... |
| mdandyv2 47218 | Given the equivalences set... |
| mdandyv3 47219 | Given the equivalences set... |
| mdandyv4 47220 | Given the equivalences set... |
| mdandyv5 47221 | Given the equivalences set... |
| mdandyv6 47222 | Given the equivalences set... |
| mdandyv7 47223 | Given the equivalences set... |
| mdandyv8 47224 | Given the equivalences set... |
| mdandyv9 47225 | Given the equivalences set... |
| mdandyv10 47226 | Given the equivalences set... |
| mdandyv11 47227 | Given the equivalences set... |
| mdandyv12 47228 | Given the equivalences set... |
| mdandyv13 47229 | Given the equivalences set... |
| mdandyv14 47230 | Given the equivalences set... |
| mdandyv15 47231 | Given the equivalences set... |
| mdandyvr0 47232 | Given the equivalences set... |
| mdandyvr1 47233 | Given the equivalences set... |
| mdandyvr2 47234 | Given the equivalences set... |
| mdandyvr3 47235 | Given the equivalences set... |
| mdandyvr4 47236 | Given the equivalences set... |
| mdandyvr5 47237 | Given the equivalences set... |
| mdandyvr6 47238 | Given the equivalences set... |
| mdandyvr7 47239 | Given the equivalences set... |
| mdandyvr8 47240 | Given the equivalences set... |
| mdandyvr9 47241 | Given the equivalences set... |
| mdandyvr10 47242 | Given the equivalences set... |
| mdandyvr11 47243 | Given the equivalences set... |
| mdandyvr12 47244 | Given the equivalences set... |
| mdandyvr13 47245 | Given the equivalences set... |
| mdandyvr14 47246 | Given the equivalences set... |
| mdandyvr15 47247 | Given the equivalences set... |
| mdandyvrx0 47248 | Given the exclusivities se... |
| mdandyvrx1 47249 | Given the exclusivities se... |
| mdandyvrx2 47250 | Given the exclusivities se... |
| mdandyvrx3 47251 | Given the exclusivities se... |
| mdandyvrx4 47252 | Given the exclusivities se... |
| mdandyvrx5 47253 | Given the exclusivities se... |
| mdandyvrx6 47254 | Given the exclusivities se... |
| mdandyvrx7 47255 | Given the exclusivities se... |
| mdandyvrx8 47256 | Given the exclusivities se... |
| mdandyvrx9 47257 | Given the exclusivities se... |
| mdandyvrx10 47258 | Given the exclusivities se... |
| mdandyvrx11 47259 | Given the exclusivities se... |
| mdandyvrx12 47260 | Given the exclusivities se... |
| mdandyvrx13 47261 | Given the exclusivities se... |
| mdandyvrx14 47262 | Given the exclusivities se... |
| mdandyvrx15 47263 | Given the exclusivities se... |
| H15NH16TH15IH16 47264 | Given 15 hypotheses and a ... |
| dandysum2p2e4 47265 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 47266 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 47267 | Replacement of a nested an... |
| adh-minim 47268 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 47269 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 47270 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 47271 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 47272 | Fourth lemma for the deriv... |
| adh-minim-ax1 47273 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 47274 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 47275 | Sixth lemma for the deriva... |
| adh-minim-ax2c 47276 | Derivation of a commuted f... |
| adh-minim-ax2 47277 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 47278 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 47279 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 47280 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 47281 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 47282 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 47283 | Third lemma for the deriva... |
| adh-minimp-sylsimp 47284 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 47285 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 47286 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 47287 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 47288 | Fourth lemma for the deriv... |
| adh-minimp-ax2 47289 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 47290 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 47291 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 47292 | If a class with one elemen... |
| eusnsn 47293 | There is a unique element ... |
| absnsb 47294 | If the class abstraction `... |
| euabsneu 47295 | Another way to express exi... |
| elprneb 47296 | An element of a proper uno... |
| oppr 47297 | Equality for ordered pairs... |
| opprb 47298 | Equality for unordered pai... |
| or2expropbilem1 47299 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 47300 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 47301 | If two classes are strictl... |
| eubrv 47302 | If there is a unique set w... |
| eubrdm 47303 | If there is a unique set w... |
| eldmressn 47304 | Element of the domain of a... |
| iota0def 47305 | Example for a defined iota... |
| iota0ndef 47306 | Example for an undefined i... |
| fveqvfvv 47307 | If a function's value at a... |
| fnresfnco 47308 | Composition of two functio... |
| funcoressn 47309 | A composition restricted t... |
| funressnfv 47310 | A restriction to a singlet... |
| funressndmfvrn 47311 | The value of a function ` ... |
| funressnvmo 47312 | A function restricted to a... |
| funressnmo 47313 | A function restricted to a... |
| funressneu 47314 | There is exactly one value... |
| fresfo 47315 | Conditions for a restricti... |
| fsetsniunop 47316 | The class of all functions... |
| fsetabsnop 47317 | The class of all functions... |
| fsetsnf 47318 | The mapping of an element ... |
| fsetsnf1 47319 | The mapping of an element ... |
| fsetsnfo 47320 | The mapping of an element ... |
| fsetsnf1o 47321 | The mapping of an element ... |
| fsetsnprcnex 47322 | The class of all functions... |
| cfsetssfset 47323 | The class of constant func... |
| cfsetsnfsetfv 47324 | The function value of the ... |
| cfsetsnfsetf 47325 | The mapping of the class o... |
| cfsetsnfsetf1 47326 | The mapping of the class o... |
| cfsetsnfsetfo 47327 | The mapping of the class o... |
| cfsetsnfsetf1o 47328 | The mapping of the class o... |
| fsetprcnexALT 47329 | First version of proof for... |
| fcoreslem1 47330 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 47331 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 47332 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 47333 | Lemma 4 for ~ fcores . (C... |
| fcores 47334 | Every composite function `... |
| fcoresf1lem 47335 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 47336 | If a composition is inject... |
| fcoresf1b 47337 | A composition is injective... |
| fcoresfo 47338 | If a composition is surjec... |
| fcoresfob 47339 | A composition is surjectiv... |
| fcoresf1ob 47340 | A composition is bijective... |
| f1cof1blem 47341 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 47342 | The composition of three b... |
| 3f1oss2 47343 | The composition of three b... |
| f1cof1b 47344 | If the range of ` F ` equa... |
| funfocofob 47345 | If the domain of a functio... |
| fnfocofob 47346 | If the domain of a functio... |
| focofob 47347 | If the domain of a functio... |
| f1ocof1ob 47348 | If the range of ` F ` equa... |
| f1ocof1ob2 47349 | If the range of ` F ` equa... |
| aiotajust 47351 | Soundness justification th... |
| dfaiota2 47353 | Alternate definition of th... |
| reuabaiotaiota 47354 | The iota and the alternate... |
| reuaiotaiota 47355 | The iota and the alternate... |
| aiotaexb 47356 | The alternate iota over a ... |
| aiotavb 47357 | The alternate iota over a ... |
| aiotaint 47358 | This is to ~ df-aiota what... |
| dfaiota3 47359 | Alternate definition of ` ... |
| iotan0aiotaex 47360 | If the iota over a wff ` p... |
| aiotaexaiotaiota 47361 | The alternate iota over a ... |
| aiotaval 47362 | Theorem 8.19 in [Quine] p.... |
| aiota0def 47363 | Example for a defined alte... |
| aiota0ndef 47364 | Example for an undefined a... |
| r19.32 47365 | Theorem 19.32 of [Margaris... |
| rexsb 47366 | An equivalent expression f... |
| rexrsb 47367 | An equivalent expression f... |
| 2rexsb 47368 | An equivalent expression f... |
| 2rexrsb 47369 | An equivalent expression f... |
| cbvral2 47370 | Change bound variables of ... |
| cbvrex2 47371 | Change bound variables of ... |
| ralndv1 47372 | Example for a theorem abou... |
| ralndv2 47373 | Second example for a theor... |
| reuf1odnf 47374 | There is exactly one eleme... |
| reuf1od 47375 | There is exactly one eleme... |
| euoreqb 47376 | There is a set which is eq... |
| 2reu3 47377 | Double restricted existent... |
| 2reu7 47378 | Two equivalent expressions... |
| 2reu8 47379 | Two equivalent expressions... |
| 2reu8i 47380 | Implication of a double re... |
| 2reuimp0 47381 | Implication of a double re... |
| 2reuimp 47382 | Implication of a double re... |
| ralbinrald 47389 | Elemination of a restricte... |
| nvelim 47390 | If a class is the universa... |
| alneu 47391 | If a statement holds for a... |
| eu2ndop1stv 47392 | If there is a unique secon... |
| dfateq12d 47393 | Equality deduction for "de... |
| nfdfat 47394 | Bound-variable hypothesis ... |
| dfdfat2 47395 | Alternate definition of th... |
| fundmdfat 47396 | A function is defined at a... |
| dfatprc 47397 | A function is not defined ... |
| dfatelrn 47398 | The value of a function ` ... |
| dfafv2 47399 | Alternative definition of ... |
| afveq12d 47400 | Equality deduction for fun... |
| afveq1 47401 | Equality theorem for funct... |
| afveq2 47402 | Equality theorem for funct... |
| nfafv 47403 | Bound-variable hypothesis ... |
| csbafv12g 47404 | Move class substitution in... |
| afvfundmfveq 47405 | If a class is a function r... |
| afvnfundmuv 47406 | If a set is not in the dom... |
| ndmafv 47407 | The value of a class outsi... |
| afvvdm 47408 | If the function value of a... |
| nfunsnafv 47409 | If the restriction of a cl... |
| afvvfunressn 47410 | If the function value of a... |
| afvprc 47411 | A function's value at a pr... |
| afvvv 47412 | If a function's value at a... |
| afvpcfv0 47413 | If the value of the altern... |
| afvnufveq 47414 | The value of the alternati... |
| afvvfveq 47415 | The value of the alternati... |
| afv0fv0 47416 | If the value of the altern... |
| afvfvn0fveq 47417 | If the function's value at... |
| afv0nbfvbi 47418 | The function's value at an... |
| afvfv0bi 47419 | The function's value at an... |
| afveu 47420 | The value of a function at... |
| fnbrafvb 47421 | Equivalence of function va... |
| fnopafvb 47422 | Equivalence of function va... |
| funbrafvb 47423 | Equivalence of function va... |
| funopafvb 47424 | Equivalence of function va... |
| funbrafv 47425 | The second argument of a b... |
| funbrafv2b 47426 | Function value in terms of... |
| dfafn5a 47427 | Representation of a functi... |
| dfafn5b 47428 | Representation of a functi... |
| fnrnafv 47429 | The range of a function ex... |
| afvelrnb 47430 | A member of a function's r... |
| afvelrnb0 47431 | A member of a function's r... |
| dfaimafn 47432 | Alternate definition of th... |
| dfaimafn2 47433 | Alternate definition of th... |
| afvelima 47434 | Function value in an image... |
| afvelrn 47435 | A function's value belongs... |
| fnafvelrn 47436 | A function's value belongs... |
| fafvelcdm 47437 | A function's value belongs... |
| ffnafv 47438 | A function maps to a class... |
| afvres 47439 | The value of a restricted ... |
| tz6.12-afv 47440 | Function value. Theorem 6... |
| tz6.12-1-afv 47441 | Function value (Theorem 6.... |
| dmfcoafv 47442 | Domains of a function comp... |
| afvco2 47443 | Value of a function compos... |
| rlimdmafv 47444 | Two ways to express that a... |
| aoveq123d 47445 | Equality deduction for ope... |
| nfaov 47446 | Bound-variable hypothesis ... |
| csbaovg 47447 | Move class substitution in... |
| aovfundmoveq 47448 | If a class is a function r... |
| aovnfundmuv 47449 | If an ordered pair is not ... |
| ndmaov 47450 | The value of an operation ... |
| ndmaovg 47451 | The value of an operation ... |
| aovvdm 47452 | If the operation value of ... |
| nfunsnaov 47453 | If the restriction of a cl... |
| aovvfunressn 47454 | If the operation value of ... |
| aovprc 47455 | The value of an operation ... |
| aovrcl 47456 | Reverse closure for an ope... |
| aovpcov0 47457 | If the alternative value o... |
| aovnuoveq 47458 | The alternative value of t... |
| aovvoveq 47459 | The alternative value of t... |
| aov0ov0 47460 | If the alternative value o... |
| aovovn0oveq 47461 | If the operation's value a... |
| aov0nbovbi 47462 | The operation's value on a... |
| aovov0bi 47463 | The operation's value on a... |
| rspceaov 47464 | A frequently used special ... |
| fnotaovb 47465 | Equivalence of operation v... |
| ffnaov 47466 | An operation maps to a cla... |
| faovcl 47467 | Closure law for an operati... |
| aovmpt4g 47468 | Value of a function given ... |
| aoprssdm 47469 | Domain of closure of an op... |
| ndmaovcl 47470 | The "closure" of an operat... |
| ndmaovrcl 47471 | Reverse closure law, in co... |
| ndmaovcom 47472 | Any operation is commutati... |
| ndmaovass 47473 | Any operation is associati... |
| ndmaovdistr 47474 | Any operation is distribut... |
| dfatafv2iota 47477 | If a function is defined a... |
| ndfatafv2 47478 | The alternate function val... |
| ndfatafv2undef 47479 | The alternate function val... |
| dfatafv2ex 47480 | The alternate function val... |
| afv2ex 47481 | The alternate function val... |
| afv2eq12d 47482 | Equality deduction for fun... |
| afv2eq1 47483 | Equality theorem for funct... |
| afv2eq2 47484 | Equality theorem for funct... |
| nfafv2 47485 | Bound-variable hypothesis ... |
| csbafv212g 47486 | Move class substitution in... |
| fexafv2ex 47487 | The alternate function val... |
| ndfatafv2nrn 47488 | The alternate function val... |
| ndmafv2nrn 47489 | The value of a class outsi... |
| funressndmafv2rn 47490 | The alternate function val... |
| afv2ndefb 47491 | Two ways to say that an al... |
| nfunsnafv2 47492 | If the restriction of a cl... |
| afv2prc 47493 | A function's value at a pr... |
| dfatafv2rnb 47494 | The alternate function val... |
| afv2orxorb 47495 | If a set is in the range o... |
| dmafv2rnb 47496 | The alternate function val... |
| fundmafv2rnb 47497 | The alternate function val... |
| afv2elrn 47498 | An alternate function valu... |
| afv20defat 47499 | If the alternate function ... |
| fnafv2elrn 47500 | An alternate function valu... |
| fafv2elcdm 47501 | An alternate function valu... |
| fafv2elrnb 47502 | An alternate function valu... |
| fcdmvafv2v 47503 | If the codomain of a funct... |
| tz6.12-2-afv2 47504 | Function value when ` F ` ... |
| afv2eu 47505 | The value of a function at... |
| afv2res 47506 | The value of a restricted ... |
| tz6.12-afv2 47507 | Function value (Theorem 6.... |
| tz6.12-1-afv2 47508 | Function value (Theorem 6.... |
| tz6.12c-afv2 47509 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 47510 | Corollary of Theorem 6.12(... |
| funressnbrafv2 47511 | The second argument of a b... |
| dfatbrafv2b 47512 | Equivalence of function va... |
| dfatopafv2b 47513 | Equivalence of function va... |
| funbrafv2 47514 | The second argument of a b... |
| fnbrafv2b 47515 | Equivalence of function va... |
| fnopafv2b 47516 | Equivalence of function va... |
| funbrafv22b 47517 | Equivalence of function va... |
| funopafv2b 47518 | Equivalence of function va... |
| dfatsnafv2 47519 | Singleton of function valu... |
| dfafv23 47520 | A definition of function v... |
| dfatdmfcoafv2 47521 | Domain of a function compo... |
| dfatcolem 47522 | Lemma for ~ dfatco . (Con... |
| dfatco 47523 | The predicate "defined at"... |
| afv2co2 47524 | Value of a function compos... |
| rlimdmafv2 47525 | Two ways to express that a... |
| dfafv22 47526 | Alternate definition of ` ... |
| afv2ndeffv0 47527 | If the alternate function ... |
| dfatafv2eqfv 47528 | If a function is defined a... |
| afv2rnfveq 47529 | If the alternate function ... |
| afv20fv0 47530 | If the alternate function ... |
| afv2fvn0fveq 47531 | If the function's value at... |
| afv2fv0 47532 | If the function's value at... |
| afv2fv0b 47533 | The function's value at an... |
| afv2fv0xorb 47534 | If a set is in the range o... |
| an4com24 47535 | Rearrangement of 4 conjunc... |
| 3an4ancom24 47536 | Commutative law for a conj... |
| 4an21 47537 | Rearrangement of 4 conjunc... |
| dfnelbr2 47540 | Alternate definition of th... |
| nelbr 47541 | The binary relation of a s... |
| nelbrim 47542 | If a set is related to ano... |
| nelbrnel 47543 | A set is related to anothe... |
| nelbrnelim 47544 | If a set is related to ano... |
| ralralimp 47545 | Selecting one of two alter... |
| otiunsndisjX 47546 | The union of singletons co... |
| fvifeq 47547 | Equality of function value... |
| rnfdmpr 47548 | The range of a one-to-one ... |
| imarnf1pr 47549 | The image of the range of ... |
| funop1 47550 | A function is an ordered p... |
| fun2dmnopgexmpl 47551 | A function with a domain c... |
| opabresex0d 47552 | A collection of ordered pa... |
| opabbrfex0d 47553 | A collection of ordered pa... |
| opabresexd 47554 | A collection of ordered pa... |
| opabbrfexd 47555 | A collection of ordered pa... |
| f1oresf1orab 47556 | Build a bijection by restr... |
| f1oresf1o 47557 | Build a bijection by restr... |
| f1oresf1o2 47558 | Build a bijection by restr... |
| fvmptrab 47559 | Value of a function mappin... |
| fvmptrabdm 47560 | Value of a function mappin... |
| cnambpcma 47561 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 47562 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 47563 | The sum of two complex num... |
| leaddsuble 47564 | Addition and subtraction o... |
| 2leaddle2 47565 | If two real numbers are le... |
| ltnltne 47566 | Variant of trichotomy law ... |
| p1lep2 47567 | A real number increasd by ... |
| ltsubsubaddltsub 47568 | If the result of subtracti... |
| zm1nn 47569 | An integer minus 1 is posi... |
| readdcnnred 47570 | The sum of a real number a... |
| resubcnnred 47571 | The difference of a real n... |
| recnmulnred 47572 | The product of a real numb... |
| cndivrenred 47573 | The quotient of an imagina... |
| sqrtnegnre 47574 | The square root of a negat... |
| nn0resubcl 47575 | Closure law for subtractio... |
| zgeltp1eq 47576 | If an integer is between a... |
| 1t10e1p1e11 47577 | 11 is 1 times 10 to the po... |
| deccarry 47578 | Add 1 to a 2 digit number ... |
| eluzge0nn0 47579 | If an integer is greater t... |
| nltle2tri 47580 | Negated extended trichotom... |
| ssfz12 47581 | Subset relationship for fi... |
| elfz2z 47582 | Membership of an integer i... |
| 2elfz3nn0 47583 | If there are two elements ... |
| fz0addcom 47584 | The addition of two member... |
| 2elfz2melfz 47585 | If the sum of two integers... |
| fz0addge0 47586 | The sum of two integers in... |
| elfzlble 47587 | Membership of an integer i... |
| elfzelfzlble 47588 | Membership of an element o... |
| fzopred 47589 | Join a predecessor to the ... |
| fzopredsuc 47590 | Join a predecessor and a s... |
| 1fzopredsuc 47591 | Join 0 and a successor to ... |
| el1fzopredsuc 47592 | An element of an open inte... |
| subsubelfzo0 47593 | Subtracting a difference f... |
| 2ffzoeq 47594 | Two functions over a half-... |
| 2ltceilhalf 47595 | The ceiling of half of an ... |
| ceilhalfgt1 47596 | The ceiling of half of an ... |
| ceilhalfelfzo1 47597 | A positive integer less th... |
| gpgedgvtx1lem 47598 | Lemma for ~ gpgedgvtx1 . ... |
| 2tceilhalfelfzo1 47599 | Two times a positive integ... |
| ceilbi 47600 | A condition equivalent to ... |
| ceilhalf1 47601 | The ceiling of one half is... |
| rehalfge1 47602 | Half of a real number grea... |
| ceilhalfnn 47603 | The ceiling of half of a p... |
| 1elfzo1ceilhalf1 47604 | 1 is in the half-open inte... |
| fldivmod 47605 | Expressing the floor of a ... |
| ceildivmod 47606 | Expressing the ceiling of ... |
| ceil5half3 47607 | The ceiling of half of 5 i... |
| submodaddmod 47608 | Subtraction and addition m... |
| difltmodne 47609 | Two nonnegative integers a... |
| zplusmodne 47610 | A nonnegative integer is n... |
| addmodne 47611 | The sum of a nonnegative i... |
| plusmod5ne 47612 | A nonnegative integer is n... |
| zp1modne 47613 | An integer is not itself p... |
| p1modne 47614 | A nonnegative integer is n... |
| m1modne 47615 | A nonnegative integer is n... |
| minusmod5ne 47616 | A nonnegative integer is n... |
| submodlt 47617 | The difference of an eleme... |
| submodneaddmod 47618 | An integer minus ` B ` is ... |
| m1modnep2mod 47619 | A nonnegative integer minu... |
| minusmodnep2tmod 47620 | A nonnegative integer minu... |
| m1mod0mod1 47621 | An integer decreased by 1 ... |
| elmod2 47622 | An integer modulo 2 is eit... |
| mod0mul 47623 | If an integer is 0 modulo ... |
| modn0mul 47624 | If an integer is not 0 mod... |
| m1modmmod 47625 | An integer decreased by 1 ... |
| difmodm1lt 47626 | The difference between an ... |
| 8mod5e3 47627 | 8 modulo 5 is 3. (Contrib... |
| modmkpkne 47628 | If an integer minus a cons... |
| modmknepk 47629 | A nonnegative integer less... |
| modlt0b 47630 | An integer with an absolut... |
| mod2addne 47631 | The sums of a nonnegative ... |
| modm1nep1 47632 | A nonnegative integer less... |
| modm2nep1 47633 | A nonnegative integer less... |
| modp2nep1 47634 | A nonnegative integer less... |
| modm1nep2 47635 | A nonnegative integer less... |
| modm1nem2 47636 | A nonnegative integer less... |
| modm1p1ne 47637 | If an integer minus one eq... |
| smonoord 47638 | Ordering relation for a st... |
| fsummsndifre 47639 | A finite sum with one of i... |
| fsumsplitsndif 47640 | Separate out a term in a f... |
| fsummmodsndifre 47641 | A finite sum of summands m... |
| fsummmodsnunz 47642 | A finite sum of summands m... |
| setsidel 47643 | The injected slot is an el... |
| setsnidel 47644 | The injected slot is an el... |
| setsv 47645 | The value of the structure... |
| preimafvsnel 47646 | The preimage of a function... |
| preimafvn0 47647 | The preimage of a function... |
| uniimafveqt 47648 | The union of the image of ... |
| uniimaprimaeqfv 47649 | The union of the image of ... |
| setpreimafvex 47650 | The class ` P ` of all pre... |
| elsetpreimafvb 47651 | The characterization of an... |
| elsetpreimafv 47652 | An element of the class ` ... |
| elsetpreimafvssdm 47653 | An element of the class ` ... |
| fvelsetpreimafv 47654 | There is an element in a p... |
| preimafvelsetpreimafv 47655 | The preimage of a function... |
| preimafvsspwdm 47656 | The class ` P ` of all pre... |
| 0nelsetpreimafv 47657 | The empty set is not an el... |
| elsetpreimafvbi 47658 | An element of the preimage... |
| elsetpreimafveqfv 47659 | The elements of the preima... |
| eqfvelsetpreimafv 47660 | If an element of the domai... |
| elsetpreimafvrab 47661 | An element of the preimage... |
| imaelsetpreimafv 47662 | The image of an element of... |
| uniimaelsetpreimafv 47663 | The union of the image of ... |
| elsetpreimafveq 47664 | If two preimages of functi... |
| fundcmpsurinjlem1 47665 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 47666 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 47667 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 47668 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 47669 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 47670 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 47671 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 47672 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 47673 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 47674 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 47675 | Every function ` F : A -->... |
| fundcmpsurinj 47676 | Every function ` F : A -->... |
| fundcmpsurbijinj 47677 | Every function ` F : A -->... |
| fundcmpsurinjimaid 47678 | Every function ` F : A -->... |
| fundcmpsurinjALT 47679 | Alternate proof of ~ fundc... |
| iccpval 47682 | Partition consisting of a ... |
| iccpart 47683 | A special partition. Corr... |
| iccpartimp 47684 | Implications for a class b... |
| iccpartres 47685 | The restriction of a parti... |
| iccpartxr 47686 | If there is a partition, t... |
| iccpartgtprec 47687 | If there is a partition, t... |
| iccpartipre 47688 | If there is a partition, t... |
| iccpartiltu 47689 | If there is a partition, t... |
| iccpartigtl 47690 | If there is a partition, t... |
| iccpartlt 47691 | If there is a partition, t... |
| iccpartltu 47692 | If there is a partition, t... |
| iccpartgtl 47693 | If there is a partition, t... |
| iccpartgt 47694 | If there is a partition, t... |
| iccpartleu 47695 | If there is a partition, t... |
| iccpartgel 47696 | If there is a partition, t... |
| iccpartrn 47697 | If there is a partition, t... |
| iccpartf 47698 | The range of the partition... |
| iccpartel 47699 | If there is a partition, t... |
| iccelpart 47700 | An element of any partitio... |
| iccpartiun 47701 | A half-open interval of ex... |
| icceuelpartlem 47702 | Lemma for ~ icceuelpart . ... |
| icceuelpart 47703 | An element of a partitione... |
| iccpartdisj 47704 | The segments of a partitio... |
| iccpartnel 47705 | A point of a partition is ... |
| fargshiftfv 47706 | If a class is a function, ... |
| fargshiftf 47707 | If a class is a function, ... |
| fargshiftf1 47708 | If a function is 1-1, then... |
| fargshiftfo 47709 | If a function is onto, the... |
| fargshiftfva 47710 | The values of a shifted fu... |
| lswn0 47711 | The last symbol of a nonem... |
| nfich1 47714 | The first interchangeable ... |
| nfich2 47715 | The second interchangeable... |
| ichv 47716 | Setvar variables are inter... |
| ichf 47717 | Setvar variables are inter... |
| ichid 47718 | A setvar variable is alway... |
| icht 47719 | A theorem is interchangeab... |
| ichbidv 47720 | Formula building rule for ... |
| ichcircshi 47721 | The setvar variables are i... |
| ichan 47722 | If two setvar variables ar... |
| ichn 47723 | Negation does not affect i... |
| ichim 47724 | Formula building rule for ... |
| dfich2 47725 | Alternate definition of th... |
| ichcom 47726 | The interchangeability of ... |
| ichbi12i 47727 | Equivalence for interchang... |
| icheqid 47728 | In an equality for the sam... |
| icheq 47729 | In an equality of setvar v... |
| ichnfimlem 47730 | Lemma for ~ ichnfim : A s... |
| ichnfim 47731 | If in an interchangeabilit... |
| ichnfb 47732 | If ` x ` and ` y ` are int... |
| ichal 47733 | Move a universal quantifie... |
| ich2al 47734 | Two setvar variables are a... |
| ich2ex 47735 | Two setvar variables are a... |
| ichexmpl1 47736 | Example for interchangeabl... |
| ichexmpl2 47737 | Example for interchangeabl... |
| ich2exprop 47738 | If the setvar variables ar... |
| ichnreuop 47739 | If the setvar variables ar... |
| ichreuopeq 47740 | If the setvar variables ar... |
| sprid 47741 | Two identical representati... |
| elsprel 47742 | An unordered pair is an el... |
| spr0nelg 47743 | The empty set is not an el... |
| sprval 47746 | The set of all unordered p... |
| sprvalpw 47747 | The set of all unordered p... |
| sprssspr 47748 | The set of all unordered p... |
| spr0el 47749 | The empty set is not an un... |
| sprvalpwn0 47750 | The set of all unordered p... |
| sprel 47751 | An element of the set of a... |
| prssspr 47752 | An element of a subset of ... |
| prelspr 47753 | An unordered pair of eleme... |
| prsprel 47754 | The elements of a pair fro... |
| prsssprel 47755 | The elements of a pair fro... |
| sprvalpwle2 47756 | The set of all unordered p... |
| sprsymrelfvlem 47757 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 47758 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 47759 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 47760 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 47761 | The value of the function ... |
| sprsymrelf 47762 | The mapping ` F ` is a fun... |
| sprsymrelf1 47763 | The mapping ` F ` is a one... |
| sprsymrelfo 47764 | The mapping ` F ` is a fun... |
| sprsymrelf1o 47765 | The mapping ` F ` is a bij... |
| sprbisymrel 47766 | There is a bijection betwe... |
| sprsymrelen 47767 | The class ` P ` of subsets... |
| prpair 47768 | Characterization of a prop... |
| prproropf1olem0 47769 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 47770 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 47771 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 47772 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 47773 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 47774 | There is a bijection betwe... |
| prproropen 47775 | The set of proper pairs an... |
| prproropreud 47776 | There is exactly one order... |
| pairreueq 47777 | Two equivalent representat... |
| paireqne 47778 | Two sets are not equal iff... |
| prprval 47781 | The set of all proper unor... |
| prprvalpw 47782 | The set of all proper unor... |
| prprelb 47783 | An element of the set of a... |
| prprelprb 47784 | A set is an element of the... |
| prprspr2 47785 | The set of all proper unor... |
| prprsprreu 47786 | There is a unique proper u... |
| prprreueq 47787 | There is a unique proper u... |
| sbcpr 47788 | The proper substitution of... |
| reupr 47789 | There is a unique unordere... |
| reuprpr 47790 | There is a unique proper u... |
| poprelb 47791 | Equality for unordered pai... |
| 2exopprim 47792 | The existence of an ordere... |
| reuopreuprim 47793 | There is a unique unordere... |
| fmtno 47796 | The ` N ` th Fermat number... |
| fmtnoge3 47797 | Each Fermat number is grea... |
| fmtnonn 47798 | Each Fermat number is a po... |
| fmtnom1nn 47799 | A Fermat number minus one ... |
| fmtnoodd 47800 | Each Fermat number is odd.... |
| fmtnorn 47801 | A Fermat number is a funct... |
| fmtnof1 47802 | The enumeration of the Fer... |
| fmtnoinf 47803 | The set of Fermat numbers ... |
| fmtnorec1 47804 | The first recurrence relat... |
| sqrtpwpw2p 47805 | The floor of the square ro... |
| fmtnosqrt 47806 | The floor of the square ro... |
| fmtno0 47807 | The ` 0 ` th Fermat number... |
| fmtno1 47808 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 47809 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 47810 | The second recurrence rela... |
| fmtnodvds 47811 | Any Fermat number divides ... |
| goldbachthlem1 47812 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 47813 | Lemma 2 for ~ goldbachth .... |
| goldbachth 47814 | Goldbach's theorem: Two d... |
| fmtnorec3 47815 | The third recurrence relat... |
| fmtnorec4 47816 | The fourth recurrence rela... |
| fmtno2 47817 | The ` 2 ` nd Fermat number... |
| fmtno3 47818 | The ` 3 ` rd Fermat number... |
| fmtno4 47819 | The ` 4 ` th Fermat number... |
| fmtno5lem1 47820 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 47821 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 47822 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 47823 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 47824 | The ` 5 ` th Fermat number... |
| fmtno0prm 47825 | The ` 0 ` th Fermat number... |
| fmtno1prm 47826 | The ` 1 ` st Fermat number... |
| fmtno2prm 47827 | The ` 2 ` nd Fermat number... |
| 257prm 47828 | 257 is a prime number (the... |
| fmtno3prm 47829 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 47830 | Any power of two is coprim... |
| fmtnoprmfac1lem 47831 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 47832 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 47833 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 47834 | Divisor of Fermat number (... |
| fmtnofac2lem 47835 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 47836 | Divisor of Fermat number (... |
| fmtnofac1 47837 | Divisor of Fermat number (... |
| fmtno4sqrt 47838 | The floor of the square ro... |
| fmtno4prmfac 47839 | If P was a (prime) factor ... |
| fmtno4prmfac193 47840 | If P was a (prime) factor ... |
| fmtno4nprmfac193 47841 | 193 is not a (prime) facto... |
| fmtno4prm 47842 | The ` 4 `-th Fermat number... |
| 65537prm 47843 | 65537 is a prime number (t... |
| fmtnofz04prm 47844 | The first five Fermat numb... |
| fmtnole4prm 47845 | The first five Fermat numb... |
| fmtno5faclem1 47846 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 47847 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 47848 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 47849 | The factorization of the `... |
| fmtno5nprm 47850 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 47851 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 47852 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 47853 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 47854 | The mapping of a Fermat nu... |
| prminf2 47855 | The set of prime numbers i... |
| 2pwp1prm 47856 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 47857 | Every prime number of the ... |
| m2prm 47858 | The second Mersenne number... |
| m3prm 47859 | The third Mersenne number ... |
| flsqrt 47860 | A condition equivalent to ... |
| flsqrt5 47861 | The floor of the square ro... |
| 3ndvds4 47862 | 3 does not divide 4. (Con... |
| 139prmALT 47863 | 139 is a prime number. In... |
| 31prm 47864 | 31 is a prime number. In ... |
| m5prm 47865 | The fifth Mersenne number ... |
| 127prm 47866 | 127 is a prime number. (C... |
| m7prm 47867 | The seventh Mersenne numbe... |
| m11nprm 47868 | The eleventh Mersenne numb... |
| mod42tp1mod8 47869 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 47870 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 47871 | If ` P ` is a Sophie Germa... |
| lighneallem1 47872 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 47873 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 47874 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 47875 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 47876 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 47877 | Lemma 3 for ~ lighneal . ... |
| lighneal 47878 | If a power of a prime ` P ... |
| modexp2m1d 47879 | The square of an integer w... |
| proththdlem 47880 | Lemma for ~ proththd . (C... |
| proththd 47881 | Proth's theorem (1878). I... |
| 5tcu2e40 47882 | 5 times the cube of 2 is 4... |
| 3exp4mod41 47883 | 3 to the fourth power is -... |
| 41prothprmlem1 47884 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 47885 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 47886 | 41 is a _Proth prime_. (C... |
| quad1 47887 | A condition for a quadrati... |
| requad01 47888 | A condition for a quadrati... |
| requad1 47889 | A condition for a quadrati... |
| requad2 47890 | A condition for a quadrati... |
| iseven 47895 | The predicate "is an even ... |
| isodd 47896 | The predicate "is an odd n... |
| evenz 47897 | An even number is an integ... |
| oddz 47898 | An odd number is an intege... |
| evendiv2z 47899 | The result of dividing an ... |
| oddp1div2z 47900 | The result of dividing an ... |
| oddm1div2z 47901 | The result of dividing an ... |
| isodd2 47902 | The predicate "is an odd n... |
| dfodd2 47903 | Alternate definition for o... |
| dfodd6 47904 | Alternate definition for o... |
| dfeven4 47905 | Alternate definition for e... |
| evenm1odd 47906 | The predecessor of an even... |
| evenp1odd 47907 | The successor of an even n... |
| oddp1eveni 47908 | The successor of an odd nu... |
| oddm1eveni 47909 | The predecessor of an odd ... |
| evennodd 47910 | An even number is not an o... |
| oddneven 47911 | An odd number is not an ev... |
| enege 47912 | The negative of an even nu... |
| onego 47913 | The negative of an odd num... |
| m1expevenALTV 47914 | Exponentiation of -1 by an... |
| m1expoddALTV 47915 | Exponentiation of -1 by an... |
| dfeven2 47916 | Alternate definition for e... |
| dfodd3 47917 | Alternate definition for o... |
| iseven2 47918 | The predicate "is an even ... |
| isodd3 47919 | The predicate "is an odd n... |
| 2dvdseven 47920 | 2 divides an even number. ... |
| m2even 47921 | A multiple of 2 is an even... |
| 2ndvdsodd 47922 | 2 does not divide an odd n... |
| 2dvdsoddp1 47923 | 2 divides an odd number in... |
| 2dvdsoddm1 47924 | 2 divides an odd number de... |
| dfeven3 47925 | Alternate definition for e... |
| dfodd4 47926 | Alternate definition for o... |
| dfodd5 47927 | Alternate definition for o... |
| zefldiv2ALTV 47928 | The floor of an even numbe... |
| zofldiv2ALTV 47929 | The floor of an odd number... |
| oddflALTV 47930 | Odd number representation ... |
| iseven5 47931 | The predicate "is an even ... |
| isodd7 47932 | The predicate "is an odd n... |
| dfeven5 47933 | Alternate definition for e... |
| dfodd7 47934 | Alternate definition for o... |
| gcd2odd1 47935 | The greatest common diviso... |
| zneoALTV 47936 | No even integer equals an ... |
| zeoALTV 47937 | An integer is even or odd.... |
| zeo2ALTV 47938 | An integer is even or odd ... |
| nneoALTV 47939 | A positive integer is even... |
| nneoiALTV 47940 | A positive integer is even... |
| odd2np1ALTV 47941 | An integer is odd iff it i... |
| oddm1evenALTV 47942 | An integer is odd iff its ... |
| oddp1evenALTV 47943 | An integer is odd iff its ... |
| oexpnegALTV 47944 | The exponential of the neg... |
| oexpnegnz 47945 | The exponential of the neg... |
| bits0ALTV 47946 | Value of the zeroth bit. ... |
| bits0eALTV 47947 | The zeroth bit of an even ... |
| bits0oALTV 47948 | The zeroth bit of an odd n... |
| divgcdoddALTV 47949 | Either ` A / ( A gcd B ) `... |
| opoeALTV 47950 | The sum of two odds is eve... |
| opeoALTV 47951 | The sum of an odd and an e... |
| omoeALTV 47952 | The difference of two odds... |
| omeoALTV 47953 | The difference of an odd a... |
| oddprmALTV 47954 | A prime not equal to ` 2 `... |
| 0evenALTV 47955 | 0 is an even number. (Con... |
| 0noddALTV 47956 | 0 is not an odd number. (... |
| 1oddALTV 47957 | 1 is an odd number. (Cont... |
| 1nevenALTV 47958 | 1 is not an even number. ... |
| 2evenALTV 47959 | 2 is an even number. (Con... |
| 2noddALTV 47960 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 47961 | An odd nonnegative integer... |
| nnoALTV 47962 | An alternate characterizat... |
| nn0oALTV 47963 | An alternate characterizat... |
| nn0e 47964 | An alternate characterizat... |
| nneven 47965 | An alternate characterizat... |
| nn0onn0exALTV 47966 | For each odd nonnegative i... |
| nn0enn0exALTV 47967 | For each even nonnegative ... |
| nnennexALTV 47968 | For each even positive int... |
| nnpw2evenALTV 47969 | 2 to the power of a positi... |
| epoo 47970 | The sum of an even and an ... |
| emoo 47971 | The difference of an even ... |
| epee 47972 | The sum of two even number... |
| emee 47973 | The difference of two even... |
| evensumeven 47974 | If a summand is even, the ... |
| 3odd 47975 | 3 is an odd number. (Cont... |
| 4even 47976 | 4 is an even number. (Con... |
| 5odd 47977 | 5 is an odd number. (Cont... |
| 6even 47978 | 6 is an even number. (Con... |
| 7odd 47979 | 7 is an odd number. (Cont... |
| 8even 47980 | 8 is an even number. (Con... |
| evenprm2 47981 | A prime number is even iff... |
| oddprmne2 47982 | Every prime number not bei... |
| oddprmuzge3 47983 | A prime number which is od... |
| evenltle 47984 | If an even number is great... |
| odd2prm2 47985 | If an odd number is the su... |
| even3prm2 47986 | If an even number is the s... |
| mogoldbblem 47987 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 47988 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 47989 | Lemma for ~ perfectALTV . ... |
| perfectALTV 47990 | The Euclid-Euler theorem, ... |
| fppr 47993 | The set of Fermat pseudopr... |
| fpprmod 47994 | The set of Fermat pseudopr... |
| fpprel 47995 | A Fermat pseudoprime to th... |
| fpprbasnn 47996 | The base of a Fermat pseud... |
| fpprnn 47997 | A Fermat pseudoprime to th... |
| fppr2odd 47998 | A Fermat pseudoprime to th... |
| 11t31e341 47999 | 341 is the product of 11 a... |
| 2exp340mod341 48000 | Eight to the eighth power ... |
| 341fppr2 48001 | 341 is the (smallest) _Pou... |
| 4fppr1 48002 | 4 is the (smallest) Fermat... |
| 8exp8mod9 48003 | Eight to the eighth power ... |
| 9fppr8 48004 | 9 is the (smallest) Fermat... |
| dfwppr 48005 | Alternate definition of a ... |
| fpprwppr 48006 | A Fermat pseudoprime to th... |
| fpprwpprb 48007 | An integer ` X ` which is ... |
| fpprel2 48008 | An alternate definition fo... |
| nfermltl8rev 48009 | Fermat's little theorem wi... |
| nfermltl2rev 48010 | Fermat's little theorem wi... |
| nfermltlrev 48011 | Fermat's little theorem re... |
| isgbe 48018 | The predicate "is an even ... |
| isgbow 48019 | The predicate "is a weak o... |
| isgbo 48020 | The predicate "is an odd G... |
| gbeeven 48021 | An even Goldbach number is... |
| gbowodd 48022 | A weak odd Goldbach number... |
| gbogbow 48023 | A (strong) odd Goldbach nu... |
| gboodd 48024 | An odd Goldbach number is ... |
| gbepos 48025 | Any even Goldbach number i... |
| gbowpos 48026 | Any weak odd Goldbach numb... |
| gbopos 48027 | Any odd Goldbach number is... |
| gbegt5 48028 | Any even Goldbach number i... |
| gbowgt5 48029 | Any weak odd Goldbach numb... |
| gbowge7 48030 | Any weak odd Goldbach numb... |
| gboge9 48031 | Any odd Goldbach number is... |
| gbege6 48032 | Any even Goldbach number i... |
| gbpart6 48033 | The Goldbach partition of ... |
| gbpart7 48034 | The (weak) Goldbach partit... |
| gbpart8 48035 | The Goldbach partition of ... |
| gbpart9 48036 | The (strong) Goldbach part... |
| gbpart11 48037 | The (strong) Goldbach part... |
| 6gbe 48038 | 6 is an even Goldbach numb... |
| 7gbow 48039 | 7 is a weak odd Goldbach n... |
| 8gbe 48040 | 8 is an even Goldbach numb... |
| 9gbo 48041 | 9 is an odd Goldbach numbe... |
| 11gbo 48042 | 11 is an odd Goldbach numb... |
| stgoldbwt 48043 | If the strong ternary Gold... |
| sbgoldbwt 48044 | If the strong binary Goldb... |
| sbgoldbst 48045 | If the strong binary Goldb... |
| sbgoldbaltlem1 48046 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 48047 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 48048 | An alternate (related to t... |
| sbgoldbb 48049 | If the strong binary Goldb... |
| sgoldbeven3prm 48050 | If the binary Goldbach con... |
| sbgoldbm 48051 | If the strong binary Goldb... |
| mogoldbb 48052 | If the modern version of t... |
| sbgoldbmb 48053 | The strong binary Goldbach... |
| sbgoldbo 48054 | If the strong binary Goldb... |
| nnsum3primes4 48055 | 4 is the sum of at most 3 ... |
| nnsum4primes4 48056 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 48057 | Every prime is "the sum of... |
| nnsum4primesprm 48058 | Every prime is "the sum of... |
| nnsum3primesgbe 48059 | Any even Goldbach number i... |
| nnsum4primesgbe 48060 | Any even Goldbach number i... |
| nnsum3primesle9 48061 | Every integer greater than... |
| nnsum4primesle9 48062 | Every integer greater than... |
| nnsum4primesodd 48063 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 48064 | If the (strong) ternary Go... |
| evengpop3 48065 | If the (weak) ternary Gold... |
| evengpoap3 48066 | If the (strong) ternary Go... |
| nnsum4primeseven 48067 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 48068 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 48069 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 48070 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 48071 | If the binary Goldbach con... |
| bgoldbtbndlem1 48072 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 48073 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 48074 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 48075 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 48076 | If the binary Goldbach con... |
| tgoldbachgtALTV 48079 | Variant of Thierry Arnoux'... |
| bgoldbachlt 48080 | The binary Goldbach conjec... |
| tgblthelfgott 48082 | The ternary Goldbach conje... |
| tgoldbachlt 48083 | The ternary Goldbach conje... |
| tgoldbach 48084 | The ternary Goldbach conje... |
| clnbgrprc0 48087 | The closed neighborhood is... |
| clnbgrcl 48088 | If a class ` X ` has at le... |
| clnbgrval 48089 | The closed neighborhood of... |
| dfclnbgr2 48090 | Alternate definition of th... |
| dfclnbgr4 48091 | Alternate definition of th... |
| elclnbgrelnbgr 48092 | An element of the closed n... |
| dfclnbgr3 48093 | Alternate definition of th... |
| clnbgrnvtx0 48094 | If a class ` X ` is not a ... |
| clnbgrel 48095 | Characterization of a memb... |
| clnbgrvtxel 48096 | Every vertex ` K ` is a me... |
| clnbgrisvtx 48097 | Every member ` N ` of the ... |
| clnbgrssvtx 48098 | The closed neighborhood of... |
| clnbgrn0 48099 | The closed neighborhood of... |
| clnbupgr 48100 | The closed neighborhood of... |
| clnbupgrel 48101 | A member of the closed nei... |
| clnbupgreli 48102 | A member of the closed nei... |
| clnbgr0vtx 48103 | In a null graph (with no v... |
| clnbgr0edg 48104 | In an empty graph (with no... |
| clnbgrsym 48105 | In a graph, the closed nei... |
| predgclnbgrel 48106 | If a (not necessarily prop... |
| clnbgredg 48107 | A vertex connected by an e... |
| clnbgrssedg 48108 | The vertices connected by ... |
| edgusgrclnbfin 48109 | The size of the closed nei... |
| clnbusgrfi 48110 | The closed neighborhood of... |
| clnbfiusgrfi 48111 | The closed neighborhood of... |
| clnbgrlevtx 48112 | The size of the closed nei... |
| dfsclnbgr2 48113 | Alternate definition of th... |
| sclnbgrel 48114 | Characterization of a memb... |
| sclnbgrelself 48115 | A vertex ` N ` is a member... |
| sclnbgrisvtx 48116 | Every member ` X ` of the ... |
| dfclnbgr5 48117 | Alternate definition of th... |
| dfnbgr5 48118 | Alternate definition of th... |
| dfnbgrss 48119 | Subset chain for different... |
| dfvopnbgr2 48120 | Alternate definition of th... |
| vopnbgrel 48121 | Characterization of a memb... |
| vopnbgrelself 48122 | A vertex ` N ` is a member... |
| dfclnbgr6 48123 | Alternate definition of th... |
| dfnbgr6 48124 | Alternate definition of th... |
| dfsclnbgr6 48125 | Alternate definition of a ... |
| dfnbgrss2 48126 | Subset chain for different... |
| isisubgr 48129 | The subgraph induced by a ... |
| isubgriedg 48130 | The edges of an induced su... |
| isubgrvtxuhgr 48131 | The subgraph induced by th... |
| isubgredgss 48132 | The edges of an induced su... |
| isubgredg 48133 | An edge of an induced subg... |
| isubgrvtx 48134 | The vertices of an induced... |
| isubgruhgr 48135 | An induced subgraph of a h... |
| isubgrsubgr 48136 | An induced subgraph of a h... |
| isubgrupgr 48137 | An induced subgraph of a p... |
| isubgrumgr 48138 | An induced subgraph of a m... |
| isubgrusgr 48139 | An induced subgraph of a s... |
| isubgr0uhgr 48140 | The subgraph induced by an... |
| grimfn 48146 | The graph isomorphism func... |
| grimdmrel 48147 | The domain of the graph is... |
| isgrim 48149 | An isomorphism of graphs i... |
| grimprop 48150 | Properties of an isomorphi... |
| grimf1o 48151 | An isomorphism of graphs i... |
| grimidvtxedg 48152 | The identity relation rest... |
| grimid 48153 | The identity relation rest... |
| grimuhgr 48154 | If there is a graph isomor... |
| grimcnv 48155 | The converse of a graph is... |
| grimco 48156 | The composition of graph i... |
| uhgrimedgi 48157 | An isomorphism between gra... |
| uhgrimedg 48158 | An isomorphism between gra... |
| uhgrimprop 48159 | An isomorphism between hyp... |
| isuspgrim0lem 48160 | An isomorphism of simple p... |
| isuspgrim0 48161 | An isomorphism of simple p... |
| isuspgrimlem 48162 | Lemma for ~ isuspgrim . (... |
| isuspgrim 48163 | A class is an isomorphism ... |
| upgrimwlklem1 48164 | Lemma 1 for ~ upgrimwlk an... |
| upgrimwlklem2 48165 | Lemma 2 for ~ upgrimwlk . ... |
| upgrimwlklem3 48166 | Lemma 3 for ~ upgrimwlk . ... |
| upgrimwlklem4 48167 | Lemma 4 for ~ upgrimwlk . ... |
| upgrimwlklem5 48168 | Lemma 5 for ~ upgrimwlk . ... |
| upgrimwlk 48169 | Graph isomorphisms between... |
| upgrimwlklen 48170 | Graph isomorphisms between... |
| upgrimtrlslem1 48171 | Lemma 1 for ~ upgrimtrls .... |
| upgrimtrlslem2 48172 | Lemma 2 for ~ upgrimtrls .... |
| upgrimtrls 48173 | Graph isomorphisms between... |
| upgrimpthslem1 48174 | Lemma 1 for ~ upgrimpths .... |
| upgrimpthslem2 48175 | Lemma 2 for ~ upgrimpths .... |
| upgrimpths 48176 | Graph isomorphisms between... |
| upgrimspths 48177 | Graph isomorphisms between... |
| upgrimcycls 48178 | Graph isomorphisms between... |
| brgric 48179 | The relation "is isomorphi... |
| brgrici 48180 | Prove that two graphs are ... |
| gricrcl 48181 | Reverse closure of the "is... |
| dfgric2 48182 | Alternate, explicit defini... |
| gricbri 48183 | Implications of two graphs... |
| gricushgr 48184 | The "is isomorphic to" rel... |
| gricuspgr 48185 | The "is isomorphic to" rel... |
| gricrel 48186 | The "is isomorphic to" rel... |
| gricref 48187 | Graph isomorphism is refle... |
| gricsym 48188 | Graph isomorphism is symme... |
| gricsymb 48189 | Graph isomorphism is symme... |
| grictr 48190 | Graph isomorphism is trans... |
| gricer 48191 | Isomorphism is an equivale... |
| gricen 48192 | Isomorphic graphs have equ... |
| opstrgric 48193 | A graph represented as an ... |
| ushggricedg 48194 | A simple hypergraph (with ... |
| cycldlenngric 48195 | Two simple pseudographs ar... |
| isubgrgrim 48196 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 48197 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 48198 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 48199 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 48200 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 48201 | Graph isomorphisms between... |
| grimedg 48202 | For two isomorphic graphs,... |
| grimedgi 48203 | Graph isomorphisms map edg... |
| grtriproplem 48206 | Lemma for ~ grtriprop . (... |
| grtri 48207 | The triangles in a graph. ... |
| grtriprop 48208 | The properties of a triang... |
| grtrif1o 48209 | Any bijection onto a trian... |
| isgrtri 48210 | A triangle in a graph. (C... |
| grtrissvtx 48211 | A triangle is a subset of ... |
| grtriclwlk3 48212 | A triangle induces a close... |
| cycl3grtrilem 48213 | Lemma for ~ cycl3grtri . ... |
| cycl3grtri 48214 | The vertices of a cycle of... |
| grtrimap 48215 | Conditions for mapping tri... |
| grimgrtri 48216 | Graph isomorphisms map tri... |
| usgrgrtrirex 48217 | Conditions for a simple gr... |
| stgrfv 48220 | The star graph S_N. (Contr... |
| stgrvtx 48221 | The vertices of the star g... |
| stgriedg 48222 | The indexed edges of the s... |
| stgredg 48223 | The edges of the star grap... |
| stgredgel 48224 | An edge of the star graph ... |
| stgredgiun 48225 | The edges of the star grap... |
| stgrusgra 48226 | The star graph S_N is a si... |
| stgr0 48227 | The star graph S_0 consist... |
| stgr1 48228 | The star graph S_1 consist... |
| stgrvtx0 48229 | The center ("internal node... |
| stgrorder 48230 | The order of a star graph ... |
| stgrnbgr0 48231 | All vertices of a star gra... |
| stgrclnbgr0 48232 | All vertices of a star gra... |
| isubgr3stgrlem1 48233 | Lemma 1 for ~ isubgr3stgr ... |
| isubgr3stgrlem2 48234 | Lemma 2 for ~ isubgr3stgr ... |
| isubgr3stgrlem3 48235 | Lemma 3 for ~ isubgr3stgr ... |
| isubgr3stgrlem4 48236 | Lemma 4 for ~ isubgr3stgr ... |
| isubgr3stgrlem5 48237 | Lemma 5 for ~ isubgr3stgr ... |
| isubgr3stgrlem6 48238 | Lemma 6 for ~ isubgr3stgr ... |
| isubgr3stgrlem7 48239 | Lemma 7 for ~ isubgr3stgr ... |
| isubgr3stgrlem8 48240 | Lemma 8 for ~ isubgr3stgr ... |
| isubgr3stgrlem9 48241 | Lemma 9 for ~ isubgr3stgr ... |
| isubgr3stgr 48242 | If a vertex of a simple gr... |
| grlimfn 48246 | The graph local isomorphis... |
| grlimdmrel 48247 | The domain of the graph lo... |
| isgrlim 48249 | A local isomorphism of gra... |
| isgrlim2 48250 | A local isomorphism of gra... |
| grlimprop 48251 | Properties of a local isom... |
| grlimf1o 48252 | A local isomorphism of gra... |
| grlimprop2 48253 | Properties of a local isom... |
| uhgrimgrlim 48254 | An isomorphism of hypergra... |
| uspgrlimlem1 48255 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 48256 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 48257 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 48258 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 48259 | A local isomorphism of sim... |
| usgrlimprop 48260 | Properties of a local isom... |
| clnbgrvtxedg 48261 | An edge ` E ` containing a... |
| grlimedgclnbgr 48262 | For two locally isomorphic... |
| grlimprclnbgr 48263 | For two locally isomorphic... |
| grlimprclnbgredg 48264 | For two locally isomorphic... |
| grlimpredg 48265 | For two locally isomorphic... |
| grlimprclnbgrvtx 48266 | For two locally isomorphic... |
| grlimgredgex 48267 | Local isomorphisms between... |
| grlimgrtrilem1 48268 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 48269 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 48270 | If one of two locally isom... |
| brgrlic 48271 | The relation "is locally i... |
| brgrilci 48272 | Prove that two graphs are ... |
| grlicrel 48273 | The "is locally isomorphic... |
| grlicrcl 48274 | Reverse closure of the "is... |
| dfgrlic2 48275 | Alternate, explicit defini... |
| grilcbri 48276 | Implications of two graphs... |
| dfgrlic3 48277 | Alternate, explicit defini... |
| grilcbri2 48278 | Implications of two graphs... |
| grlicref 48279 | Graph local isomorphism is... |
| grlicsym 48280 | Graph local isomorphism is... |
| grlicsymb 48281 | Graph local isomorphism is... |
| grlictr 48282 | Graph local isomorphism is... |
| grlicer 48283 | Local isomorphism is an eq... |
| grlicen 48284 | Locally isomorphic graphs ... |
| gricgrlic 48285 | Isomorphic hypergraphs are... |
| clnbgr3stgrgrlim 48286 | If all (closed) neighborho... |
| clnbgr3stgrgrlic 48287 | If all (closed) neighborho... |
| usgrexmpl1lem 48288 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 48289 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 48290 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 48291 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 48292 | ` G ` contains a triangle ... |
| usgrexmpl2lem 48293 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 48294 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 48295 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 48296 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 48297 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 48298 | The neighborhood of the fi... |
| usgrexmpl2nb1 48299 | The neighborhood of the se... |
| usgrexmpl2nb2 48300 | The neighborhood of the th... |
| usgrexmpl2nb3 48301 | The neighborhood of the fo... |
| usgrexmpl2nb4 48302 | The neighborhood of the fi... |
| usgrexmpl2nb5 48303 | The neighborhood of the si... |
| usgrexmpl2trifr 48304 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 48305 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 48306 | The graphs ` H ` and ` G `... |
| gpgov 48309 | The generalized Petersen g... |
| gpgvtx 48310 | The vertices of the genera... |
| gpgiedg 48311 | The indexed edges of the g... |
| gpgedg 48312 | The edges of the generaliz... |
| gpgiedgdmellem 48313 | Lemma for ~ gpgiedgdmel an... |
| gpgvtxel 48314 | A vertex in a generalized ... |
| gpgvtxel2 48315 | The second component of a ... |
| gpgiedgdmel 48316 | An index of edges of the g... |
| gpgedgel 48317 | An edge in a generalized P... |
| gpgprismgriedgdmel 48318 | An index of edges of the g... |
| gpgprismgriedgdmss 48319 | A subset of the index of e... |
| gpgvtx0 48320 | The outside vertices in a ... |
| gpgvtx1 48321 | The inside vertices in a g... |
| opgpgvtx 48322 | A vertex in a generalized ... |
| gpgusgralem 48323 | Lemma for ~ gpgusgra . (C... |
| gpgusgra 48324 | The generalized Petersen g... |
| gpgprismgrusgra 48325 | The generalized Petersen g... |
| gpgorder 48326 | The order of the generaliz... |
| gpg5order 48327 | The order of a generalized... |
| gpgedgvtx0 48328 | The edges starting at an o... |
| gpgedgvtx1 48329 | The edges starting at an i... |
| gpgvtxedg0 48330 | The edges starting at an o... |
| gpgvtxedg1 48331 | The edges starting at an i... |
| gpgedgiov 48332 | The edges of the generaliz... |
| gpgedg2ov 48333 | The edges of the generaliz... |
| gpgedg2iv 48334 | The edges of the generaliz... |
| gpg5nbgrvtx03starlem1 48335 | Lemma 1 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem2 48336 | Lemma 2 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem3 48337 | Lemma 3 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx13starlem1 48338 | Lemma 1 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem2 48339 | Lemma 2 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem3 48340 | Lemma 3 for ~ gpg5nbgr3sta... |
| gpgnbgrvtx0 48341 | The (open) neighborhood of... |
| gpgnbgrvtx1 48342 | The (open) neighborhood of... |
| gpg3nbgrvtx0 48343 | In a generalized Petersen ... |
| gpg3nbgrvtx0ALT 48344 | In a generalized Petersen ... |
| gpg3nbgrvtx1 48345 | In a generalized Petersen ... |
| gpgcubic 48346 | Every generalized Petersen... |
| gpg5nbgrvtx03star 48347 | In a generalized Petersen ... |
| gpg5nbgr3star 48348 | In a generalized Petersen ... |
| gpgvtxdg3 48349 | Every vertex in a generali... |
| gpg3kgrtriexlem1 48350 | Lemma 1 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem2 48351 | Lemma 2 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem3 48352 | Lemma 3 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem4 48353 | Lemma 4 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem5 48354 | Lemma 5 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem6 48355 | Lemma 6 for ~ gpg3kgrtriex... |
| gpg3kgrtriex 48356 | All generalized Petersen g... |
| gpg5gricstgr3 48357 | Each closed neighborhood i... |
| pglem 48358 | Lemma for theorems about P... |
| pgjsgr 48359 | A Petersen graph is a simp... |
| gpg5grlim 48360 | A local isomorphism betwee... |
| gpg5grlic 48361 | The two generalized Peters... |
| gpgprismgr4cycllem1 48362 | Lemma 1 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem2 48363 | Lemma 2 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem3 48364 | Lemma 3 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem4 48365 | Lemma 4 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem5 48366 | Lemma 5 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem6 48367 | Lemma 6 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem7 48368 | Lemma 7 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem8 48369 | Lemma 8 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem9 48370 | Lemma 9 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem10 48371 | Lemma 10 for ~ gpgprismgr4... |
| gpgprismgr4cycllem11 48372 | Lemma 11 for ~ gpgprismgr4... |
| gpgprismgr4cycl0 48373 | The generalized Petersen g... |
| gpgprismgr4cyclex 48374 | The generalized Petersen g... |
| pgnioedg1 48375 | An inside and an outside v... |
| pgnioedg2 48376 | An inside and an outside v... |
| pgnioedg3 48377 | An inside and an outside v... |
| pgnioedg4 48378 | An inside and an outside v... |
| pgnioedg5 48379 | An inside and an outside v... |
| pgnbgreunbgrlem1 48380 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem1 48381 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem2 48382 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem3 48383 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2 48384 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem3 48385 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem4 48386 | Lemma 4 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem1 48387 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem2 48388 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem3 48389 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5 48390 | Lemma 5 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem6 48391 | Lemma 6 for ~ pgnbgreunbgr... |
| pgnbgreunbgr 48392 | In a Petersen graph, two d... |
| pgn4cyclex 48393 | A cycle in a Petersen grap... |
| pg4cyclnex 48394 | In the Petersen graph G(5,... |
| gpg5ngric 48395 | The two generalized Peters... |
| lgricngricex 48396 | There are two different lo... |
| gpg5edgnedg 48397 | Two consecutive (according... |
| grlimedgnedg 48398 | In general, the image of a... |
| 1hegrlfgr 48399 | A graph ` G ` with one hyp... |
| upwlksfval 48402 | The set of simple walks (i... |
| isupwlk 48403 | Properties of a pair of fu... |
| isupwlkg 48404 | Generalization of ~ isupwl... |
| upwlkbprop 48405 | Basic properties of a simp... |
| upwlkwlk 48406 | A simple walk is a walk. ... |
| upgrwlkupwlk 48407 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 48408 | In a pseudograph, the defi... |
| upgrisupwlkALT 48409 | Alternate proof of ~ upgri... |
| upgredgssspr 48410 | The set of edges of a pseu... |
| uspgropssxp 48411 | The set ` G ` of "simple p... |
| uspgrsprfv 48412 | The value of the function ... |
| uspgrsprf 48413 | The mapping ` F ` is a fun... |
| uspgrsprf1 48414 | The mapping ` F ` is a one... |
| uspgrsprfo 48415 | The mapping ` F ` is a fun... |
| uspgrsprf1o 48416 | The mapping ` F ` is a bij... |
| uspgrex 48417 | The class ` G ` of all "si... |
| uspgrbispr 48418 | There is a bijection betwe... |
| uspgrspren 48419 | The set ` G ` of the "simp... |
| uspgrymrelen 48420 | The set ` G ` of the "simp... |
| uspgrbisymrel 48421 | There is a bijection betwe... |
| uspgrbisymrelALT 48422 | Alternate proof of ~ uspgr... |
| ovn0dmfun 48423 | If a class operation value... |
| xpsnopab 48424 | A Cartesian product with a... |
| xpiun 48425 | A Cartesian product expres... |
| ovn0ssdmfun 48426 | If a class' operation valu... |
| fnxpdmdm 48427 | The domain of the domain o... |
| cnfldsrngbas 48428 | The base set of a subring ... |
| cnfldsrngadd 48429 | The group addition operati... |
| cnfldsrngmul 48430 | The ring multiplication op... |
| plusfreseq 48431 | If the empty set is not co... |
| mgmplusfreseq 48432 | If the empty set is not co... |
| 0mgm 48433 | A set with an empty base s... |
| opmpoismgm 48434 | A structure with a group a... |
| copissgrp 48435 | A structure with a constan... |
| copisnmnd 48436 | A structure with a constan... |
| 0nodd 48437 | 0 is not an odd integer. ... |
| 1odd 48438 | 1 is an odd integer. (Con... |
| 2nodd 48439 | 2 is not an odd integer. ... |
| oddibas 48440 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 48441 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 48442 | The structure of all odd i... |
| nnsgrpmgm 48443 | The structure of positive ... |
| nnsgrp 48444 | The structure of positive ... |
| nnsgrpnmnd 48445 | The structure of positive ... |
| nn0mnd 48446 | The set of nonnegative int... |
| gsumsplit2f 48447 | Split a group sum into two... |
| gsumdifsndf 48448 | Extract a summand from a f... |
| gsumfsupp 48449 | A group sum of a family ca... |
| iscllaw 48456 | The predicate "is a closed... |
| iscomlaw 48457 | The predicate "is a commut... |
| clcllaw 48458 | Closure of a closed operat... |
| isasslaw 48459 | The predicate "is an assoc... |
| asslawass 48460 | Associativity of an associ... |
| mgmplusgiopALT 48461 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 48462 | Slot 2 (group operation) o... |
| intopval 48469 | The internal (binary) oper... |
| intop 48470 | An internal (binary) opera... |
| clintopval 48471 | The closed (internal binar... |
| assintopval 48472 | The associative (closed in... |
| assintopmap 48473 | The associative (closed in... |
| isclintop 48474 | The predicate "is a closed... |
| clintop 48475 | A closed (internal binary)... |
| assintop 48476 | An associative (closed int... |
| isassintop 48477 | The predicate "is an assoc... |
| clintopcllaw 48478 | The closure law holds for ... |
| assintopcllaw 48479 | The closure low holds for ... |
| assintopasslaw 48480 | The associative low holds ... |
| assintopass 48481 | An associative (closed int... |
| ismgmALT 48490 | The predicate "is a magma"... |
| iscmgmALT 48491 | The predicate "is a commut... |
| issgrpALT 48492 | The predicate "is a semigr... |
| iscsgrpALT 48493 | The predicate "is a commut... |
| mgm2mgm 48494 | Equivalence of the two def... |
| sgrp2sgrp 48495 | Equivalence of the two def... |
| lmod0rng 48496 | If the scalar ring of a mo... |
| nzrneg1ne0 48497 | The additive inverse of th... |
| lidldomn1 48498 | If a (left) ideal (which i... |
| lidlabl 48499 | A (left) ideal of a ring i... |
| lidlrng 48500 | A (left) ideal of a ring i... |
| zlidlring 48501 | The zero (left) ideal of a... |
| uzlidlring 48502 | Only the zero (left) ideal... |
| lidldomnnring 48503 | A (left) ideal of a domain... |
| 0even 48504 | 0 is an even integer. (Co... |
| 1neven 48505 | 1 is not an even integer. ... |
| 2even 48506 | 2 is an even integer. (Co... |
| 2zlidl 48507 | The even integers are a (l... |
| 2zrng 48508 | The ring of integers restr... |
| 2zrngbas 48509 | The base set of R is the s... |
| 2zrngadd 48510 | The group addition operati... |
| 2zrng0 48511 | The additive identity of R... |
| 2zrngamgm 48512 | R is an (additive) magma. ... |
| 2zrngasgrp 48513 | R is an (additive) semigro... |
| 2zrngamnd 48514 | R is an (additive) monoid.... |
| 2zrngacmnd 48515 | R is a commutative (additi... |
| 2zrngagrp 48516 | R is an (additive) group. ... |
| 2zrngaabl 48517 | R is an (additive) abelian... |
| 2zrngmul 48518 | The ring multiplication op... |
| 2zrngmmgm 48519 | R is a (multiplicative) ma... |
| 2zrngmsgrp 48520 | R is a (multiplicative) se... |
| 2zrngALT 48521 | The ring of integers restr... |
| 2zrngnmlid 48522 | R has no multiplicative (l... |
| 2zrngnmrid 48523 | R has no multiplicative (r... |
| 2zrngnmlid2 48524 | R has no multiplicative (l... |
| 2zrngnring 48525 | R is not a unital ring. (... |
| cznrnglem 48526 | Lemma for ~ cznrng : The ... |
| cznabel 48527 | The ring constructed from ... |
| cznrng 48528 | The ring constructed from ... |
| cznnring 48529 | The ring constructed from ... |
| rngcvalALTV 48532 | Value of the category of n... |
| rngcbasALTV 48533 | Set of objects of the cate... |
| rngchomfvalALTV 48534 | Set of arrows of the categ... |
| rngchomALTV 48535 | Set of arrows of the categ... |
| elrngchomALTV 48536 | A morphism of non-unital r... |
| rngccofvalALTV 48537 | Composition in the categor... |
| rngccoALTV 48538 | Composition in the categor... |
| rngccatidALTV 48539 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 48540 | The category of non-unital... |
| rngcidALTV 48541 | The identity arrow in the ... |
| rngcsectALTV 48542 | A section in the category ... |
| rngcinvALTV 48543 | An inverse in the category... |
| rngcisoALTV 48544 | An isomorphism in the cate... |
| rngchomffvalALTV 48545 | The value of the functiona... |
| rngchomrnghmresALTV 48546 | The value of the functiona... |
| rngcrescrhmALTV 48547 | The category of non-unital... |
| rhmsubcALTVlem1 48548 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 48549 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 48550 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 48551 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 48552 | According to ~ df-subc , t... |
| rhmsubcALTVcat 48553 | The restriction of the cat... |
| ringcvalALTV 48556 | Value of the category of r... |
| funcringcsetcALTV2lem1 48557 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 48558 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 48559 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 48560 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 48561 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 48562 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 48563 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 48564 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 48565 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 48566 | The "natural forgetful fun... |
| ringcbasALTV 48567 | Set of objects of the cate... |
| ringchomfvalALTV 48568 | Set of arrows of the categ... |
| ringchomALTV 48569 | Set of arrows of the categ... |
| elringchomALTV 48570 | A morphism of rings is a f... |
| ringccofvalALTV 48571 | Composition in the categor... |
| ringccoALTV 48572 | Composition in the categor... |
| ringccatidALTV 48573 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 48574 | The category of rings is a... |
| ringcidALTV 48575 | The identity arrow in the ... |
| ringcsectALTV 48576 | A section in the category ... |
| ringcinvALTV 48577 | An inverse in the category... |
| ringcisoALTV 48578 | An isomorphism in the cate... |
| ringcbasbasALTV 48579 | An element of the base set... |
| funcringcsetclem1ALTV 48580 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 48581 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 48582 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 48583 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 48584 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 48585 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 48586 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 48587 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 48588 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 48589 | The "natural forgetful fun... |
| srhmsubcALTVlem1 48590 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 48591 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 48592 | According to ~ df-subc , t... |
| sringcatALTV 48593 | The restriction of the cat... |
| crhmsubcALTV 48594 | According to ~ df-subc , t... |
| cringcatALTV 48595 | The restriction of the cat... |
| drhmsubcALTV 48596 | According to ~ df-subc , t... |
| drngcatALTV 48597 | The restriction of the cat... |
| fldcatALTV 48598 | The restriction of the cat... |
| fldcALTV 48599 | The restriction of the cat... |
| fldhmsubcALTV 48600 | According to ~ df-subc , t... |
| eliunxp2 48601 | Membership in a union of C... |
| mpomptx2 48602 | Express a two-argument fun... |
| cbvmpox2 48603 | Rule to change the bound v... |
| dmmpossx2 48604 | The domain of a mapping is... |
| mpoexxg2 48605 | Existence of an operation ... |
| ovmpordxf 48606 | Value of an operation give... |
| ovmpordx 48607 | Value of an operation give... |
| ovmpox2 48608 | The value of an operation ... |
| fdmdifeqresdif 48609 | The restriction of a condi... |
| ofaddmndmap 48610 | The function operation app... |
| mapsnop 48611 | A singleton of an ordered ... |
| fprmappr 48612 | A function with a domain o... |
| mapprop 48613 | An unordered pair containi... |
| ztprmneprm 48614 | A prime is not an integer ... |
| 2t6m3t4e0 48615 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 48616 | For any finite subset of `... |
| nn0sumltlt 48617 | If the sum of two nonnegat... |
| bcpascm1 48618 | Pascal's rule for the bino... |
| altgsumbc 48619 | The sum of binomial coeffi... |
| altgsumbcALT 48620 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 48621 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 48622 | An element of the (base se... |
| zlmodzxz0 48623 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 48624 | The scalar multiplication ... |
| zlmodzxzadd 48625 | The addition of the ` ZZ `... |
| zlmodzxzsubm 48626 | The subtraction of the ` Z... |
| zlmodzxzsub 48627 | The subtraction of the ` Z... |
| mgpsumunsn 48628 | Extract a summand/factor f... |
| mgpsumz 48629 | If the group sum for the m... |
| mgpsumn 48630 | If the group sum for the m... |
| exple2lt6 48631 | A nonnegative integer to t... |
| pgrple2abl 48632 | Every symmetric group on a... |
| pgrpgt2nabl 48633 | Every symmetric group on a... |
| invginvrid 48634 | Identity for a multiplicat... |
| rmsupp0 48635 | The support of a mapping o... |
| domnmsuppn0 48636 | The support of a mapping o... |
| rmsuppss 48637 | The support of a mapping o... |
| scmsuppss 48638 | The support of a mapping o... |
| rmsuppfi 48639 | The support of a mapping o... |
| rmfsupp 48640 | A mapping of a multiplicat... |
| scmsuppfi 48641 | The support of a mapping o... |
| scmfsupp 48642 | A mapping of a scalar mult... |
| suppmptcfin 48643 | The support of a mapping w... |
| mptcfsupp 48644 | A mapping with value 0 exc... |
| fsuppmptdmf 48645 | A mapping with a finite do... |
| lmodvsmdi 48646 | Multiple distributive law ... |
| gsumlsscl 48647 | Closure of a group sum in ... |
| assaascl0 48648 | The scalar 0 embedded into... |
| assaascl1 48649 | The scalar 1 embedded into... |
| ply1vr1smo 48650 | The variable in a polynomi... |
| ply1sclrmsm 48651 | The ring multiplication of... |
| coe1sclmulval 48652 | The value of the coefficie... |
| ply1mulgsumlem1 48653 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 48654 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 48655 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 48656 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 48657 | The product of two polynom... |
| evl1at0 48658 | Polynomial evaluation for ... |
| evl1at1 48659 | Polynomial evaluation for ... |
| linply1 48660 | A term of the form ` x - C... |
| lineval 48661 | A term of the form ` x - C... |
| linevalexample 48662 | The polynomial ` x - 3 ` o... |
| dmatALTval 48667 | The algebra of ` N ` x ` N... |
| dmatALTbas 48668 | The base set of the algebr... |
| dmatALTbasel 48669 | An element of the base set... |
| dmatbas 48670 | The set of all ` N ` x ` N... |
| lincop 48675 | A linear combination as op... |
| lincval 48676 | The value of a linear comb... |
| dflinc2 48677 | Alternative definition of ... |
| lcoop 48678 | A linear combination as op... |
| lcoval 48679 | The value of a linear comb... |
| lincfsuppcl 48680 | A linear combination of ve... |
| linccl 48681 | A linear combination of ve... |
| lincval0 48682 | The value of an empty line... |
| lincvalsng 48683 | The linear combination ove... |
| lincvalsn 48684 | The linear combination ove... |
| lincvalpr 48685 | The linear combination ove... |
| lincval1 48686 | The linear combination ove... |
| lcosn0 48687 | Properties of a linear com... |
| lincvalsc0 48688 | The linear combination whe... |
| lcoc0 48689 | Properties of a linear com... |
| linc0scn0 48690 | If a set contains the zero... |
| lincdifsn 48691 | A vector is a linear combi... |
| linc1 48692 | A vector is a linear combi... |
| lincellss 48693 | A linear combination of a ... |
| lco0 48694 | The set of empty linear co... |
| lcoel0 48695 | The zero vector is always ... |
| lincsum 48696 | The sum of two linear comb... |
| lincscm 48697 | A linear combinations mult... |
| lincsumcl 48698 | The sum of two linear comb... |
| lincscmcl 48699 | The multiplication of a li... |
| lincsumscmcl 48700 | The sum of a linear combin... |
| lincolss 48701 | According to the statement... |
| ellcoellss 48702 | Every linear combination o... |
| lcoss 48703 | A set of vectors of a modu... |
| lspsslco 48704 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 48705 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 48706 | Equivalence of a _span_ of... |
| rellininds 48710 | The class defining the rel... |
| linindsv 48712 | The classes of the module ... |
| islininds 48713 | The property of being a li... |
| linindsi 48714 | The implications of being ... |
| linindslinci 48715 | The implications of being ... |
| islinindfis 48716 | The property of being a li... |
| islinindfiss 48717 | The property of being a li... |
| linindscl 48718 | A linearly independent set... |
| lindepsnlininds 48719 | A linearly dependent subse... |
| islindeps 48720 | The property of being a li... |
| lincext1 48721 | Property 1 of an extension... |
| lincext2 48722 | Property 2 of an extension... |
| lincext3 48723 | Property 3 of an extension... |
| lindslinindsimp1 48724 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 48725 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 48726 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 48727 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 48728 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 48729 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 48730 | Implication 2 for ~ lindsl... |
| lindslininds 48731 | Equivalence of definitions... |
| linds0 48732 | The empty set is always a ... |
| el0ldep 48733 | A set containing the zero ... |
| el0ldepsnzr 48734 | A set containing the zero ... |
| lindsrng01 48735 | Any subset of a module is ... |
| lindszr 48736 | Any subset of a module ove... |
| snlindsntorlem 48737 | Lemma for ~ snlindsntor . ... |
| snlindsntor 48738 | A singleton is linearly in... |
| ldepsprlem 48739 | Lemma for ~ ldepspr . (Co... |
| ldepspr 48740 | If a vector is a scalar mu... |
| lincresunit3lem3 48741 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 48742 | Lemma 1 for properties of ... |
| lincresunitlem2 48743 | Lemma for properties of a ... |
| lincresunit1 48744 | Property 1 of a specially ... |
| lincresunit2 48745 | Property 2 of a specially ... |
| lincresunit3lem1 48746 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 48747 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 48748 | Property 3 of a specially ... |
| lincreslvec3 48749 | Property 3 of a specially ... |
| islindeps2 48750 | Conditions for being a lin... |
| islininds2 48751 | Implication of being a lin... |
| isldepslvec2 48752 | Alternative definition of ... |
| lindssnlvec 48753 | A singleton not containing... |
| lmod1lem1 48754 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 48755 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 48756 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 48757 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 48758 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 48759 | The (smallest) structure r... |
| lmod1zr 48760 | The (smallest) structure r... |
| lmod1zrnlvec 48761 | There is a (left) module (... |
| lmodn0 48762 | Left modules exist. (Cont... |
| zlmodzxzequa 48763 | Example of an equation wit... |
| zlmodzxznm 48764 | Example of a linearly depe... |
| zlmodzxzldeplem 48765 | A and B are not equal. (C... |
| zlmodzxzequap 48766 | Example of an equation wit... |
| zlmodzxzldeplem1 48767 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 48768 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 48769 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 48770 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 48771 | { A , B } is a linearly de... |
| ldepsnlinclem1 48772 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 48773 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 48774 | The class of all (left) ve... |
| ldepsnlinc 48775 | The reverse implication of... |
| ldepslinc 48776 | For (left) vector spaces, ... |
| suppdm 48777 | If the range of a function... |
| eluz2cnn0n1 48778 | An integer greater than 1 ... |
| divge1b 48779 | The ratio of a real number... |
| divgt1b 48780 | The ratio of a real number... |
| ltsubaddb 48781 | Equivalence for the "less ... |
| ltsubsubb 48782 | Equivalence for the "less ... |
| ltsubadd2b 48783 | Equivalence for the "less ... |
| divsub1dir 48784 | Distribution of division o... |
| expnegico01 48785 | An integer greater than 1 ... |
| elfzolborelfzop1 48786 | An element of a half-open ... |
| pw2m1lepw2m1 48787 | 2 to the power of a positi... |
| zgtp1leeq 48788 | If an integer is between a... |
| flsubz 48789 | An integer can be moved in... |
| nn0onn0ex 48790 | For each odd nonnegative i... |
| nn0enn0ex 48791 | For each even nonnegative ... |
| nnennex 48792 | For each even positive int... |
| nneop 48793 | A positive integer is even... |
| nneom 48794 | A positive integer is even... |
| nn0eo 48795 | A nonnegative integer is e... |
| nnpw2even 48796 | 2 to the power of a positi... |
| zefldiv2 48797 | The floor of an even integ... |
| zofldiv2 48798 | The floor of an odd intege... |
| nn0ofldiv2 48799 | The floor of an odd nonneg... |
| flnn0div2ge 48800 | The floor of a positive in... |
| flnn0ohalf 48801 | The floor of the half of a... |
| logcxp0 48802 | Logarithm of a complex pow... |
| regt1loggt0 48803 | The natural logarithm for ... |
| fdivval 48806 | The quotient of two functi... |
| fdivmpt 48807 | The quotient of two functi... |
| fdivmptf 48808 | The quotient of two functi... |
| refdivmptf 48809 | The quotient of two functi... |
| fdivpm 48810 | The quotient of two functi... |
| refdivpm 48811 | The quotient of two functi... |
| fdivmptfv 48812 | The function value of a qu... |
| refdivmptfv 48813 | The function value of a qu... |
| bigoval 48816 | Set of functions of order ... |
| elbigofrcl 48817 | Reverse closure of the "bi... |
| elbigo 48818 | Properties of a function o... |
| elbigo2 48819 | Properties of a function o... |
| elbigo2r 48820 | Sufficient condition for a... |
| elbigof 48821 | A function of order G(x) i... |
| elbigodm 48822 | The domain of a function o... |
| elbigoimp 48823 | The defining property of a... |
| elbigolo1 48824 | A function (into the posit... |
| rege1logbrege0 48825 | The general logarithm, wit... |
| rege1logbzge0 48826 | The general logarithm, wit... |
| fllogbd 48827 | A real number is between t... |
| relogbmulbexp 48828 | The logarithm of the produ... |
| relogbdivb 48829 | The logarithm of the quoti... |
| logbge0b 48830 | The logarithm of a number ... |
| logblt1b 48831 | The logarithm of a number ... |
| fldivexpfllog2 48832 | The floor of a positive re... |
| nnlog2ge0lt1 48833 | A positive integer is 1 if... |
| logbpw2m1 48834 | The floor of the binary lo... |
| fllog2 48835 | The floor of the binary lo... |
| blenval 48838 | The binary length of an in... |
| blen0 48839 | The binary length of 0. (... |
| blenn0 48840 | The binary length of a "nu... |
| blenre 48841 | The binary length of a pos... |
| blennn 48842 | The binary length of a pos... |
| blennnelnn 48843 | The binary length of a pos... |
| blennn0elnn 48844 | The binary length of a non... |
| blenpw2 48845 | The binary length of a pow... |
| blenpw2m1 48846 | The binary length of a pow... |
| nnpw2blen 48847 | A positive integer is betw... |
| nnpw2blenfzo 48848 | A positive integer is betw... |
| nnpw2blenfzo2 48849 | A positive integer is eith... |
| nnpw2pmod 48850 | Every positive integer can... |
| blen1 48851 | The binary length of 1. (... |
| blen2 48852 | The binary length of 2. (... |
| nnpw2p 48853 | Every positive integer can... |
| nnpw2pb 48854 | A number is a positive int... |
| blen1b 48855 | The binary length of a non... |
| blennnt2 48856 | The binary length of a pos... |
| nnolog2flm1 48857 | The floor of the binary lo... |
| blennn0em1 48858 | The binary length of the h... |
| blennngt2o2 48859 | The binary length of an od... |
| blengt1fldiv2p1 48860 | The binary length of an in... |
| blennn0e2 48861 | The binary length of an ev... |
| digfval 48864 | Operation to obtain the ` ... |
| digval 48865 | The ` K ` th digit of a no... |
| digvalnn0 48866 | The ` K ` th digit of a no... |
| nn0digval 48867 | The ` K ` th digit of a no... |
| dignn0fr 48868 | The digits of the fraction... |
| dignn0ldlem 48869 | Lemma for ~ dignnld . (Co... |
| dignnld 48870 | The leading digits of a po... |
| dig2nn0ld 48871 | The leading digits of a po... |
| dig2nn1st 48872 | The first (relevant) digit... |
| dig0 48873 | All digits of 0 are 0. (C... |
| digexp 48874 | The ` K ` th digit of a po... |
| dig1 48875 | All but one digits of 1 ar... |
| 0dig1 48876 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 48877 | The integers 0 and 1 corre... |
| dig2nn0 48878 | A digit of a nonnegative i... |
| 0dig2nn0e 48879 | The last bit of an even in... |
| 0dig2nn0o 48880 | The last bit of an odd int... |
| dig2bits 48881 | The ` K ` th digit of a no... |
| dignn0flhalflem1 48882 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 48883 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 48884 | The digits of the half of ... |
| dignn0flhalf 48885 | The digits of the rounded ... |
| nn0sumshdiglemA 48886 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 48887 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 48888 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 48889 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 48890 | A nonnegative integer can ... |
| nn0mulfsum 48891 | Trivial algorithm to calcu... |
| nn0mullong 48892 | Standard algorithm (also k... |
| naryfval 48895 | The set of the n-ary (endo... |
| naryfvalixp 48896 | The set of the n-ary (endo... |
| naryfvalel 48897 | An n-ary (endo)function on... |
| naryrcl 48898 | Reverse closure for n-ary ... |
| naryfvalelfv 48899 | The value of an n-ary (end... |
| naryfvalelwrdf 48900 | An n-ary (endo)function on... |
| 0aryfvalel 48901 | A nullary (endo)function o... |
| 0aryfvalelfv 48902 | The value of a nullary (en... |
| 1aryfvalel 48903 | A unary (endo)function on ... |
| fv1arycl 48904 | Closure of a unary (endo)f... |
| 1arympt1 48905 | A unary (endo)function in ... |
| 1arympt1fv 48906 | The value of a unary (endo... |
| 1arymaptfv 48907 | The value of the mapping o... |
| 1arymaptf 48908 | The mapping of unary (endo... |
| 1arymaptf1 48909 | The mapping of unary (endo... |
| 1arymaptfo 48910 | The mapping of unary (endo... |
| 1arymaptf1o 48911 | The mapping of unary (endo... |
| 1aryenef 48912 | The set of unary (endo)fun... |
| 1aryenefmnd 48913 | The set of unary (endo)fun... |
| 2aryfvalel 48914 | A binary (endo)function on... |
| fv2arycl 48915 | Closure of a binary (endo)... |
| 2arympt 48916 | A binary (endo)function in... |
| 2arymptfv 48917 | The value of a binary (end... |
| 2arymaptfv 48918 | The value of the mapping o... |
| 2arymaptf 48919 | The mapping of binary (end... |
| 2arymaptf1 48920 | The mapping of binary (end... |
| 2arymaptfo 48921 | The mapping of binary (end... |
| 2arymaptf1o 48922 | The mapping of binary (end... |
| 2aryenef 48923 | The set of binary (endo)fu... |
| itcoval 48928 | The value of the function ... |
| itcoval0 48929 | A function iterated zero t... |
| itcoval1 48930 | A function iterated once. ... |
| itcoval2 48931 | A function iterated twice.... |
| itcoval3 48932 | A function iterated three ... |
| itcoval0mpt 48933 | A mapping iterated zero ti... |
| itcovalsuc 48934 | The value of the function ... |
| itcovalsucov 48935 | The value of the function ... |
| itcovalendof 48936 | The n-th iterate of an end... |
| itcovalpclem1 48937 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 48938 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 48939 | The value of the function ... |
| itcovalt2lem2lem1 48940 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 48941 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 48942 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 48943 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 48944 | The value of the function ... |
| ackvalsuc1mpt 48945 | The Ackermann function at ... |
| ackvalsuc1 48946 | The Ackermann function at ... |
| ackval0 48947 | The Ackermann function at ... |
| ackval1 48948 | The Ackermann function at ... |
| ackval2 48949 | The Ackermann function at ... |
| ackval3 48950 | The Ackermann function at ... |
| ackendofnn0 48951 | The Ackermann function at ... |
| ackfnnn0 48952 | The Ackermann function at ... |
| ackval0val 48953 | The Ackermann function at ... |
| ackvalsuc0val 48954 | The Ackermann function at ... |
| ackvalsucsucval 48955 | The Ackermann function at ... |
| ackval0012 48956 | The Ackermann function at ... |
| ackval1012 48957 | The Ackermann function at ... |
| ackval2012 48958 | The Ackermann function at ... |
| ackval3012 48959 | The Ackermann function at ... |
| ackval40 48960 | The Ackermann function at ... |
| ackval41a 48961 | The Ackermann function at ... |
| ackval41 48962 | The Ackermann function at ... |
| ackval42 48963 | The Ackermann function at ... |
| ackval42a 48964 | The Ackermann function at ... |
| ackval50 48965 | The Ackermann function at ... |
| fv1prop 48966 | The function value of unor... |
| fv2prop 48967 | The function value of unor... |
| submuladdmuld 48968 | Transformation of a sum of... |
| affinecomb1 48969 | Combination of two real af... |
| affinecomb2 48970 | Combination of two real af... |
| affineid 48971 | Identity of an affine comb... |
| 1subrec1sub 48972 | Subtract the reciprocal of... |
| resum2sqcl 48973 | The sum of two squares of ... |
| resum2sqgt0 48974 | The sum of the square of a... |
| resum2sqrp 48975 | The sum of the square of a... |
| resum2sqorgt0 48976 | The sum of the square of t... |
| reorelicc 48977 | Membership in and outside ... |
| rrx2pxel 48978 | The x-coordinate of a poin... |
| rrx2pyel 48979 | The y-coordinate of a poin... |
| prelrrx2 48980 | An unordered pair of order... |
| prelrrx2b 48981 | An unordered pair of order... |
| rrx2pnecoorneor 48982 | If two different points ` ... |
| rrx2pnedifcoorneor 48983 | If two different points ` ... |
| rrx2pnedifcoorneorr 48984 | If two different points ` ... |
| rrx2xpref1o 48985 | There is a bijection betwe... |
| rrx2xpreen 48986 | The set of points in the t... |
| rrx2plord 48987 | The lexicographical orderi... |
| rrx2plord1 48988 | The lexicographical orderi... |
| rrx2plord2 48989 | The lexicographical orderi... |
| rrx2plordisom 48990 | The set of points in the t... |
| rrx2plordso 48991 | The lexicographical orderi... |
| ehl2eudisval0 48992 | The Euclidean distance of ... |
| ehl2eudis0lt 48993 | An upper bound of the Eucl... |
| lines 48998 | The lines passing through ... |
| line 48999 | The line passing through t... |
| rrxlines 49000 | Definition of lines passin... |
| rrxline 49001 | The line passing through t... |
| rrxlinesc 49002 | Definition of lines passin... |
| rrxlinec 49003 | The line passing through t... |
| eenglngeehlnmlem1 49004 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 49005 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 49006 | The line definition in the... |
| rrx2line 49007 | The line passing through t... |
| rrx2vlinest 49008 | The vertical line passing ... |
| rrx2linest 49009 | The line passing through t... |
| rrx2linesl 49010 | The line passing through t... |
| rrx2linest2 49011 | The line passing through t... |
| elrrx2linest2 49012 | The line passing through t... |
| spheres 49013 | The spheres for given cent... |
| sphere 49014 | A sphere with center ` X `... |
| rrxsphere 49015 | The sphere with center ` M... |
| 2sphere 49016 | The sphere with center ` M... |
| 2sphere0 49017 | The sphere around the orig... |
| line2ylem 49018 | Lemma for ~ line2y . This... |
| line2 49019 | Example for a line ` G ` p... |
| line2xlem 49020 | Lemma for ~ line2x . This... |
| line2x 49021 | Example for a horizontal l... |
| line2y 49022 | Example for a vertical lin... |
| itsclc0lem1 49023 | Lemma for theorems about i... |
| itsclc0lem2 49024 | Lemma for theorems about i... |
| itsclc0lem3 49025 | Lemma for theorems about i... |
| itscnhlc0yqe 49026 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 49027 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 49028 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 49029 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 49030 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 49031 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 49032 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 49033 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 49034 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 49035 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 49036 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 49037 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 49038 | The intersection points of... |
| itsclc0b 49039 | The intersection points of... |
| itsclinecirc0 49040 | The intersection points of... |
| itsclinecirc0b 49041 | The intersection points of... |
| itsclinecirc0in 49042 | The intersection points of... |
| itsclquadb 49043 | Quadratic equation for the... |
| itsclquadeu 49044 | Quadratic equation for the... |
| 2itscplem1 49045 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 49046 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 49047 | Lemma D for ~ 2itscp . (C... |
| 2itscp 49048 | A condition for a quadrati... |
| itscnhlinecirc02plem1 49049 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 49050 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 49051 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 49052 | Intersection of a nonhoriz... |
| inlinecirc02plem 49053 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 49054 | Intersection of a line wit... |
| inlinecirc02preu 49055 | Intersection of a line wit... |
| pm4.71da 49056 | Deduction converting a bic... |
| logic1 49057 | Distribution of implicatio... |
| logic1a 49058 | Variant of ~ logic1 . (Co... |
| logic2 49059 | Variant of ~ logic1 . (Co... |
| pm5.32dav 49060 | Distribution of implicatio... |
| pm5.32dra 49061 | Reverse distribution of im... |
| exp12bd 49062 | The import-export theorem ... |
| mpbiran3d 49063 | Equivalence with a conjunc... |
| mpbiran4d 49064 | Equivalence with a conjunc... |
| dtrucor3 49065 | An example of how ~ ax-5 w... |
| ralbidb 49066 | Formula-building rule for ... |
| ralbidc 49067 | Formula-building rule for ... |
| r19.41dv 49068 | A complex deduction form o... |
| rmotru 49069 | Two ways of expressing "at... |
| reutru 49070 | Two ways of expressing "ex... |
| reutruALT 49071 | Alternate proof of ~ reutr... |
| reueqbidva 49072 | Formula-building rule for ... |
| reuxfr1dd 49073 | Transfer existential uniqu... |
| ssdisjd 49074 | Subset preserves disjointn... |
| ssdisjdr 49075 | Subset preserves disjointn... |
| disjdifb 49076 | Relative complement is ant... |
| predisj 49077 | Preimages of disjoint sets... |
| vsn 49078 | The singleton of the unive... |
| mosn 49079 | "At most one" element in a... |
| mo0 49080 | "At most one" element in a... |
| mosssn 49081 | "At most one" element in a... |
| mo0sn 49082 | Two ways of expressing "at... |
| mosssn2 49083 | Two ways of expressing "at... |
| unilbss 49084 | Superclass of the greatest... |
| iuneq0 49085 | An indexed union is empty ... |
| iineq0 49086 | An indexed intersection is... |
| iunlub 49087 | The indexed union is the t... |
| iinglb 49088 | The indexed intersection i... |
| iuneqconst2 49089 | Indexed union of identical... |
| iineqconst2 49090 | Indexed intersection of id... |
| inpw 49091 | Two ways of expressing a c... |
| opth1neg 49092 | Two ordered pairs are not ... |
| opth2neg 49093 | Two ordered pairs are not ... |
| brab2dd 49094 | Expressing that two sets a... |
| brab2ddw 49095 | Expressing that two sets a... |
| brab2ddw2 49096 | Expressing that two sets a... |
| iinxp 49097 | Indexed intersection of Ca... |
| intxp 49098 | Intersection of Cartesian ... |
| coxp 49099 | Composition with a Cartesi... |
| cosn 49100 | Composition with an ordere... |
| cosni 49101 | Composition with an ordere... |
| inisegn0a 49102 | The inverse image of a sin... |
| dmrnxp 49103 | A Cartesian product is the... |
| mof0 49104 | There is at most one funct... |
| mof02 49105 | A variant of ~ mof0 . (Co... |
| mof0ALT 49106 | Alternate proof of ~ mof0 ... |
| eufsnlem 49107 | There is exactly one funct... |
| eufsn 49108 | There is exactly one funct... |
| eufsn2 49109 | There is exactly one funct... |
| mofsn 49110 | There is at most one funct... |
| mofsn2 49111 | There is at most one funct... |
| mofsssn 49112 | There is at most one funct... |
| mofmo 49113 | There is at most one funct... |
| mofeu 49114 | The uniqueness of a functi... |
| elfvne0 49115 | If a function value has a ... |
| fdomne0 49116 | A function with non-empty ... |
| f1sn2g 49117 | A function that maps a sin... |
| f102g 49118 | A function that maps the e... |
| f1mo 49119 | A function that maps a set... |
| f002 49120 | A function with an empty c... |
| map0cor 49121 | A function exists iff an e... |
| ffvbr 49122 | Relation with function val... |
| xpco2 49123 | Composition of a Cartesian... |
| ovsng 49124 | The operation value of a s... |
| ovsng2 49125 | The operation value of a s... |
| ovsn 49126 | The operation value of a s... |
| ovsn2 49127 | The operation value of a s... |
| fvconstr 49128 | Two ways of expressing ` A... |
| fvconstrn0 49129 | Two ways of expressing ` A... |
| fvconstr2 49130 | Two ways of expressing ` A... |
| ovmpt4d 49131 | Deduction version of ~ ovm... |
| eqfnovd 49132 | Deduction for equality of ... |
| fonex 49133 | The domain of a surjection... |
| eloprab1st2nd 49134 | Reconstruction of a nested... |
| fmpodg 49135 | Domain and codomain of the... |
| fmpod 49136 | Domain and codomain of the... |
| resinsnlem 49137 | Lemma for ~ resinsnALT . ... |
| resinsn 49138 | Restriction to the interse... |
| resinsnALT 49139 | Restriction to the interse... |
| dftpos5 49140 | Alternate definition of ` ... |
| dftpos6 49141 | Alternate definition of ` ... |
| dmtposss 49142 | The domain of ` tpos F ` i... |
| tposres0 49143 | The transposition of a set... |
| tposresg 49144 | The transposition restrict... |
| tposrescnv 49145 | The transposition restrict... |
| tposres2 49146 | The transposition restrict... |
| tposres3 49147 | The transposition restrict... |
| tposres 49148 | The transposition restrict... |
| tposresxp 49149 | The transposition restrict... |
| tposf1o 49150 | Condition of a bijective t... |
| tposid 49151 | Swap an ordered pair. (Co... |
| tposidres 49152 | Swap an ordered pair. (Co... |
| tposidf1o 49153 | The swap function, or the ... |
| tposideq 49154 | Two ways of expressing the... |
| tposideq2 49155 | Two ways of expressing the... |
| ixpv 49156 | Infinite Cartesian product... |
| fvconst0ci 49157 | A constant function's valu... |
| fvconstdomi 49158 | A constant function's valu... |
| f1omo 49159 | There is at most one eleme... |
| f1omoOLD 49160 | Obsolete version of ~ f1om... |
| f1omoALT 49161 | There is at most one eleme... |
| iccin 49162 | Intersection of two closed... |
| iccdisj2 49163 | If the upper bound of one ... |
| iccdisj 49164 | If the upper bound of one ... |
| slotresfo 49165 | The condition of a structu... |
| mreuniss 49166 | The union of a collection ... |
| clduni 49167 | The union of closed sets i... |
| opncldeqv 49168 | Conditions on open sets ar... |
| opndisj 49169 | Two ways of saying that tw... |
| clddisj 49170 | Two ways of saying that tw... |
| neircl 49171 | Reverse closure of the nei... |
| opnneilem 49172 | Lemma factoring out common... |
| opnneir 49173 | If something is true for a... |
| opnneirv 49174 | A variant of ~ opnneir wit... |
| opnneilv 49175 | The converse of ~ opnneir ... |
| opnneil 49176 | A variant of ~ opnneilv . ... |
| opnneieqv 49177 | The equivalence between ne... |
| opnneieqvv 49178 | The equivalence between ne... |
| restcls2lem 49179 | A closed set in a subspace... |
| restcls2 49180 | A closed set in a subspace... |
| restclsseplem 49181 | Lemma for ~ restclssep . ... |
| restclssep 49182 | Two disjoint closed sets i... |
| cnneiima 49183 | Given a continuous functio... |
| iooii 49184 | Open intervals are open se... |
| icccldii 49185 | Closed intervals are close... |
| i0oii 49186 | ` ( 0 [,) A ) ` is open in... |
| io1ii 49187 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 49188 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 49189 | Open neighborhood and neig... |
| sepnsepo 49190 | Open neighborhood and neig... |
| sepdisj 49191 | Separated sets are disjoin... |
| seposep 49192 | If two sets are separated ... |
| sepcsepo 49193 | If two sets are separated ... |
| sepfsepc 49194 | If two sets are separated ... |
| seppsepf 49195 | If two sets are precisely ... |
| seppcld 49196 | If two sets are precisely ... |
| isnrm4 49197 | A topological space is nor... |
| dfnrm2 49198 | A topological space is nor... |
| dfnrm3 49199 | A topological space is nor... |
| iscnrm3lem1 49200 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 49201 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem4 49202 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 49203 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 49204 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 49205 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 49206 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 49207 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 49208 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 49209 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 49210 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 49211 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 49212 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 49213 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 49214 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 49215 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 49216 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 49217 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 49218 | A completely normal topolo... |
| iscnrm3v 49219 | A topology is completely n... |
| iscnrm4 49220 | A completely normal topolo... |
| isprsd 49221 | Property of being a preord... |
| lubeldm2 49222 | Member of the domain of th... |
| glbeldm2 49223 | Member of the domain of th... |
| lubeldm2d 49224 | Member of the domain of th... |
| glbeldm2d 49225 | Member of the domain of th... |
| lubsscl 49226 | If a subset of ` S ` conta... |
| glbsscl 49227 | If a subset of ` S ` conta... |
| lubprlem 49228 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 49229 | The set of two comparable ... |
| lubpr 49230 | The LUB of the set of two ... |
| glbprlem 49231 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 49232 | The set of two comparable ... |
| glbpr 49233 | The GLB of the set of two ... |
| joindm2 49234 | The join of any two elemen... |
| joindm3 49235 | The join of any two elemen... |
| meetdm2 49236 | The meet of any two elemen... |
| meetdm3 49237 | The meet of any two elemen... |
| posjidm 49238 | Poset join is idempotent. ... |
| posmidm 49239 | Poset meet is idempotent. ... |
| resiposbas 49240 | Construct a poset ( ~ resi... |
| resipos 49241 | A set equipped with an ord... |
| exbaspos 49242 | There exists a poset for a... |
| exbasprs 49243 | There exists a preordered ... |
| basresposfo 49244 | The base function restrict... |
| basresprsfo 49245 | The base function restrict... |
| posnex 49246 | The class of posets is a p... |
| prsnex 49247 | The class of preordered se... |
| toslat 49248 | A toset is a lattice. (Co... |
| isclatd 49249 | The predicate "is a comple... |
| intubeu 49250 | Existential uniqueness of ... |
| unilbeu 49251 | Existential uniqueness of ... |
| ipolublem 49252 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 49253 | The domain of the LUB of t... |
| ipolub 49254 | The LUB of the inclusion p... |
| ipoglblem 49255 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 49256 | The domain of the GLB of t... |
| ipoglb 49257 | The GLB of the inclusion p... |
| ipolub0 49258 | The LUB of the empty set i... |
| ipolub00 49259 | The LUB of the empty set i... |
| ipoglb0 49260 | The GLB of the empty set i... |
| mrelatlubALT 49261 | Least upper bounds in a Mo... |
| mrelatglbALT 49262 | Greatest lower bounds in a... |
| mreclat 49263 | A Moore space is a complet... |
| topclat 49264 | A topology is a complete l... |
| toplatglb0 49265 | The empty intersection in ... |
| toplatlub 49266 | Least upper bounds in a to... |
| toplatglb 49267 | Greatest lower bounds in a... |
| toplatjoin 49268 | Joins in a topology are re... |
| toplatmeet 49269 | Meets in a topology are re... |
| topdlat 49270 | A topology is a distributi... |
| elmgpcntrd 49271 | The center of a ring. (Co... |
| asclelbasALT 49272 | Alternate proof of ~ ascle... |
| asclcntr 49273 | The algebra scalar lifting... |
| asclcom 49274 | Scalars are commutative af... |
| homf0 49275 | The base is empty iff the ... |
| catprslem 49276 | Lemma for ~ catprs . (Con... |
| catprs 49277 | A preorder can be extracte... |
| catprs2 49278 | A category equipped with t... |
| catprsc 49279 | A construction of the preo... |
| catprsc2 49280 | An alternate construction ... |
| endmndlem 49281 | A diagonal hom-set in a ca... |
| oppccatb 49282 | An opposite category is a ... |
| oppcmndclem 49283 | Lemma for ~ oppcmndc . Ev... |
| oppcendc 49284 | The opposite category of a... |
| oppcmndc 49285 | The opposite category of a... |
| idmon 49286 | An identity arrow, or an i... |
| idepi 49287 | An identity arrow, or an i... |
| sectrcl 49288 | Reverse closure for sectio... |
| sectrcl2 49289 | Reverse closure for sectio... |
| invrcl 49290 | Reverse closure for invers... |
| invrcl2 49291 | Reverse closure for invers... |
| isinv2 49292 | The property " ` F ` is an... |
| isisod 49293 | The predicate "is an isomo... |
| upeu2lem 49294 | Lemma for ~ upeu2 . There... |
| sectfn 49295 | The function value of the ... |
| invfn 49296 | The function value of the ... |
| isofnALT 49297 | The function value of the ... |
| isofval2 49298 | Function value of the func... |
| isorcl 49299 | Reverse closure for isomor... |
| isorcl2 49300 | Reverse closure for isomor... |
| isoval2 49301 | The isomorphisms are the d... |
| sectpropdlem 49302 | Lemma for ~ sectpropd . (... |
| sectpropd 49303 | Two structures with the sa... |
| invpropdlem 49304 | Lemma for ~ invpropd . (C... |
| invpropd 49305 | Two structures with the sa... |
| isopropdlem 49306 | Lemma for ~ isopropd . (C... |
| isopropd 49307 | Two structures with the sa... |
| cicfn 49308 | ` ~=c ` is a function on `... |
| cicrcl2 49309 | Isomorphism implies the st... |
| oppccic 49310 | Isomorphic objects are iso... |
| relcic 49311 | The set of isomorphic obje... |
| cicerALT 49312 | Isomorphism is an equivale... |
| cic1st2nd 49313 | Reconstruction of a pair o... |
| cic1st2ndbr 49314 | Rewrite the predicate of i... |
| cicpropdlem 49315 | Lemma for ~ cicpropd . (C... |
| cicpropd 49316 | Two structures with the sa... |
| oppccicb 49317 | Isomorphic objects are iso... |
| oppcciceq 49318 | The opposite category has ... |
| dmdm 49319 | The double domain of a fun... |
| iinfssclem1 49320 | Lemma for ~ iinfssc . (Co... |
| iinfssclem2 49321 | Lemma for ~ iinfssc . (Co... |
| iinfssclem3 49322 | Lemma for ~ iinfssc . (Co... |
| iinfssc 49323 | Indexed intersection of su... |
| iinfsubc 49324 | Indexed intersection of su... |
| iinfprg 49325 | Indexed intersection of fu... |
| infsubc 49326 | The intersection of two su... |
| infsubc2 49327 | The intersection of two su... |
| infsubc2d 49328 | The intersection of two su... |
| discsubclem 49329 | Lemma for ~ discsubc . (C... |
| discsubc 49330 | A discrete category, whose... |
| iinfconstbaslem 49331 | Lemma for ~ iinfconstbas .... |
| iinfconstbas 49332 | The discrete category is t... |
| nelsubclem 49333 | Lemma for ~ nelsubc . (Co... |
| nelsubc 49334 | An empty "hom-set" for non... |
| nelsubc2 49335 | An empty "hom-set" for non... |
| nelsubc3lem 49336 | Lemma for ~ nelsubc3 . (C... |
| nelsubc3 49337 | Remark 4.2(2) of [Adamek] ... |
| ssccatid 49338 | A category ` C ` restricte... |
| resccatlem 49339 | Lemma for ~ resccat . (Co... |
| resccat 49340 | A class ` C ` restricted b... |
| reldmfunc 49341 | The domain of ` Func ` is ... |
| func1st2nd 49342 | Rewrite the functor predic... |
| func1st 49343 | Extract the first member o... |
| func2nd 49344 | Extract the second member ... |
| funcrcl2 49345 | Reverse closure for a func... |
| funcrcl3 49346 | Reverse closure for a func... |
| funcf2lem 49347 | A utility theorem for prov... |
| funcf2lem2 49348 | A utility theorem for prov... |
| 0funcglem 49349 | Lemma for ~ 0funcg . (Con... |
| 0funcg2 49350 | The functor from the empty... |
| 0funcg 49351 | The functor from the empty... |
| 0funclem 49352 | Lemma for ~ 0funcALT . (C... |
| 0func 49353 | The functor from the empty... |
| 0funcALT 49354 | Alternate proof of ~ 0func... |
| func0g 49355 | The source category of a f... |
| func0g2 49356 | The source category of a f... |
| initc 49357 | Sets with empty base are t... |
| cofu1st2nd 49358 | Rewrite the functor compos... |
| rescofuf 49359 | The restriction of functor... |
| cofu1a 49360 | Value of the object part o... |
| cofu2a 49361 | Value of the morphism part... |
| cofucla 49362 | The composition of two fun... |
| funchomf 49363 | Source categories of a fun... |
| idfurcl 49364 | Reverse closure for an ide... |
| idfu1stf1o 49365 | The identity functor/inclu... |
| idfu1stalem 49366 | Lemma for ~ idfu1sta . (C... |
| idfu1sta 49367 | Value of the object part o... |
| idfu1a 49368 | Value of the object part o... |
| idfu2nda 49369 | Value of the morphism part... |
| imasubclem1 49370 | Lemma for ~ imasubc . (Co... |
| imasubclem2 49371 | Lemma for ~ imasubc . (Co... |
| imasubclem3 49372 | Lemma for ~ imasubc . (Co... |
| imaf1homlem 49373 | Lemma for ~ imaf1hom and o... |
| imaf1hom 49374 | The hom-set of an image of... |
| imaidfu2lem 49375 | Lemma for ~ imaidfu2 . (C... |
| imaidfu 49376 | The image of the identity ... |
| imaidfu2 49377 | The image of the identity ... |
| cofid1a 49378 | Express the object part of... |
| cofid2a 49379 | Express the morphism part ... |
| cofid1 49380 | Express the object part of... |
| cofid2 49381 | Express the morphism part ... |
| cofidvala 49382 | The property " ` F ` is a ... |
| cofidf2a 49383 | If " ` F ` is a section of... |
| cofidf1a 49384 | If " ` F ` is a section of... |
| cofidval 49385 | The property " ` <. F , G ... |
| cofidf2 49386 | If " ` F ` is a section of... |
| cofidf1 49387 | If " ` <. F , G >. ` is a ... |
| oppffn 49390 | ` oppFunc ` is a function ... |
| reldmoppf 49391 | The domain of ` oppFunc ` ... |
| oppfvalg 49392 | Value of the opposite func... |
| oppfrcllem 49393 | Lemma for ~ oppfrcl . (Co... |
| oppfrcl 49394 | If an opposite functor of ... |
| oppfrcl2 49395 | If an opposite functor of ... |
| oppfrcl3 49396 | If an opposite functor of ... |
| oppf1st2nd 49397 | Rewrite the opposite funct... |
| 2oppf 49398 | The double opposite functo... |
| eloppf 49399 | The pre-image of a non-emp... |
| eloppf2 49400 | Both components of a pre-i... |
| oppfvallem 49401 | Lemma for ~ oppfval . (Co... |
| oppfval 49402 | Value of the opposite func... |
| oppfval2 49403 | Value of the opposite func... |
| oppfval3 49404 | Value of the opposite func... |
| oppf1 49405 | Value of the object part o... |
| oppf2 49406 | Value of the morphism part... |
| oppfoppc 49407 | The opposite functor is a ... |
| oppfoppc2 49408 | The opposite functor is a ... |
| funcoppc2 49409 | A functor on opposite cate... |
| funcoppc4 49410 | A functor on opposite cate... |
| funcoppc5 49411 | A functor on opposite cate... |
| 2oppffunc 49412 | The opposite functor of an... |
| funcoppc3 49413 | A functor on opposite cate... |
| oppff1 49414 | The operation generating o... |
| oppff1o 49415 | The operation generating o... |
| cofuoppf 49416 | Composition of opposite fu... |
| imasubc 49417 | An image of a full functor... |
| imasubc2 49418 | An image of a full functor... |
| imassc 49419 | An image of a functor sati... |
| imaid 49420 | An image of a functor pres... |
| imaf1co 49421 | An image of a functor whos... |
| imasubc3 49422 | An image of a functor inje... |
| fthcomf 49423 | Source categories of a fai... |
| idfth 49424 | The inclusion functor is a... |
| idemb 49425 | The inclusion functor is a... |
| idsubc 49426 | The source category of an ... |
| idfullsubc 49427 | The source category of an ... |
| cofidfth 49428 | If " ` F ` is a section of... |
| fulloppf 49429 | The opposite functor of a ... |
| fthoppf 49430 | The opposite functor of a ... |
| ffthoppf 49431 | The opposite functor of a ... |
| upciclem1 49432 | Lemma for ~ upcic , ~ upeu... |
| upciclem2 49433 | Lemma for ~ upciclem3 and ... |
| upciclem3 49434 | Lemma for ~ upciclem4 . (... |
| upciclem4 49435 | Lemma for ~ upcic and ~ up... |
| upcic 49436 | A universal property defin... |
| upeu 49437 | A universal property defin... |
| upeu2 49438 | Generate new universal mor... |
| reldmup 49441 | The domain of ` UP ` is a ... |
| upfval 49442 | Function value of the clas... |
| upfval2 49443 | Function value of the clas... |
| upfval3 49444 | Function value of the clas... |
| isuplem 49445 | Lemma for ~ isup and other... |
| isup 49446 | The predicate "is a univer... |
| uppropd 49447 | If two categories have the... |
| reldmup2 49448 | The domain of ` ( D UP E )... |
| relup 49449 | The set of universal pairs... |
| uprcl 49450 | Reverse closure for the cl... |
| up1st2nd 49451 | Rewrite the universal prop... |
| up1st2ndr 49452 | Combine separated parts in... |
| up1st2ndb 49453 | Combine/separate parts in ... |
| up1st2nd2 49454 | Rewrite the universal prop... |
| uprcl2 49455 | Reverse closure for the cl... |
| uprcl3 49456 | Reverse closure for the cl... |
| uprcl4 49457 | Reverse closure for the cl... |
| uprcl5 49458 | Reverse closure for the cl... |
| uobrcl 49459 | Reverse closure for univer... |
| isup2 49460 | The universal property of ... |
| upeu3 49461 | The universal pair ` <. X ... |
| upeu4 49462 | Generate a new universal m... |
| uptposlem 49463 | Lemma for ~ uptpos . (Con... |
| uptpos 49464 | Rewrite the predicate of u... |
| oppcuprcl4 49465 | Reverse closure for the cl... |
| oppcuprcl3 49466 | Reverse closure for the cl... |
| oppcuprcl5 49467 | Reverse closure for the cl... |
| oppcuprcl2 49468 | Reverse closure for the cl... |
| uprcl2a 49469 | Reverse closure for the cl... |
| oppfuprcl 49470 | Reverse closure for the cl... |
| oppfuprcl2 49471 | Reverse closure for the cl... |
| oppcup3lem 49472 | Lemma for ~ oppcup3 . (Co... |
| oppcup 49473 | The universal pair ` <. X ... |
| oppcup2 49474 | The universal property for... |
| oppcup3 49475 | The universal property for... |
| uptrlem1 49476 | Lemma for ~ uptr . (Contr... |
| uptrlem2 49477 | Lemma for ~ uptr . (Contr... |
| uptrlem3 49478 | Lemma for ~ uptr . (Contr... |
| uptr 49479 | Universal property and ful... |
| uptri 49480 | Universal property and ful... |
| uptra 49481 | Universal property and ful... |
| uptrar 49482 | Universal property and ful... |
| uptrai 49483 | Universal property and ful... |
| uobffth 49484 | A fully faithful functor g... |
| uobeqw 49485 | If a full functor (in fact... |
| uobeq 49486 | If a full functor (in fact... |
| uptr2 49487 | Universal property and ful... |
| uptr2a 49488 | Universal property and ful... |
| isnatd 49489 | Property of being a natura... |
| natrcl2 49490 | Reverse closure for a natu... |
| natrcl3 49491 | Reverse closure for a natu... |
| catbas 49492 | The base of the category s... |
| cathomfval 49493 | The hom-sets of the catego... |
| catcofval 49494 | Composition of the categor... |
| natoppf 49495 | A natural transformation i... |
| natoppf2 49496 | A natural transformation i... |
| natoppfb 49497 | A natural transformation i... |
| initoo2 49498 | An initial object is an ob... |
| termoo2 49499 | A terminal object is an ob... |
| zeroo2 49500 | A zero object is an object... |
| oppcinito 49501 | Initial objects are termin... |
| oppctermo 49502 | Terminal objects are initi... |
| oppczeroo 49503 | Zero objects are zero in t... |
| termoeu2 49504 | Terminal objects are essen... |
| initopropdlemlem 49505 | Lemma for ~ initopropdlem ... |
| initopropdlem 49506 | Lemma for ~ initopropd . ... |
| termopropdlem 49507 | Lemma for ~ termopropd . ... |
| zeroopropdlem 49508 | Lemma for ~ zeroopropd . ... |
| initopropd 49509 | Two structures with the sa... |
| termopropd 49510 | Two structures with the sa... |
| zeroopropd 49511 | Two structures with the sa... |
| reldmxpc 49512 | The binary product of cate... |
| reldmxpcALT 49513 | Alternate proof of ~ reldm... |
| elxpcbasex1 49514 | A non-empty base set of th... |
| elxpcbasex1ALT 49515 | Alternate proof of ~ elxpc... |
| elxpcbasex2 49516 | A non-empty base set of th... |
| elxpcbasex2ALT 49517 | Alternate proof of ~ elxpc... |
| xpcfucbas 49518 | The base set of the produc... |
| xpcfuchomfval 49519 | Set of morphisms of the bi... |
| xpcfuchom 49520 | Set of morphisms of the bi... |
| xpcfuchom2 49521 | Value of the set of morphi... |
| xpcfucco2 49522 | Value of composition in th... |
| xpcfuccocl 49523 | The composition of two nat... |
| xpcfucco3 49524 | Value of composition in th... |
| dfswapf2 49527 | Alternate definition of ` ... |
| swapfval 49528 | Value of the swap functor.... |
| swapfelvv 49529 | A swap functor is an order... |
| swapf2fvala 49530 | The morphism part of the s... |
| swapf2fval 49531 | The morphism part of the s... |
| swapf1vala 49532 | The object part of the swa... |
| swapf1val 49533 | The object part of the swa... |
| swapf2fn 49534 | The morphism part of the s... |
| swapf1a 49535 | The object part of the swa... |
| swapf2vala 49536 | The morphism part of the s... |
| swapf2a 49537 | The morphism part of the s... |
| swapf1 49538 | The object part of the swa... |
| swapf2val 49539 | The morphism part of the s... |
| swapf2 49540 | The morphism part of the s... |
| swapf1f1o 49541 | The object part of the swa... |
| swapf2f1o 49542 | The morphism part of the s... |
| swapf2f1oa 49543 | The morphism part of the s... |
| swapf2f1oaALT 49544 | Alternate proof of ~ swapf... |
| swapfid 49545 | Each identity morphism in ... |
| swapfida 49546 | Each identity morphism in ... |
| swapfcoa 49547 | Composition in the source ... |
| swapffunc 49548 | The swap functor is a func... |
| swapfffth 49549 | The swap functor is a full... |
| swapffunca 49550 | The swap functor is a func... |
| swapfiso 49551 | The swap functor is an iso... |
| swapciso 49552 | The product category is ca... |
| oppc1stflem 49553 | A utility theorem for prov... |
| oppc1stf 49554 | The opposite functor of th... |
| oppc2ndf 49555 | The opposite functor of th... |
| 1stfpropd 49556 | If two categories have the... |
| 2ndfpropd 49557 | If two categories have the... |
| diagpropd 49558 | If two categories have the... |
| cofuswapfcl 49559 | The bifunctor pre-composed... |
| cofuswapf1 49560 | The object part of a bifun... |
| cofuswapf2 49561 | The morphism part of a bif... |
| tposcurf1cl 49562 | The partially evaluated tr... |
| tposcurf11 49563 | Value of the double evalua... |
| tposcurf12 49564 | The partially evaluated tr... |
| tposcurf1 49565 | Value of the object part o... |
| tposcurf2 49566 | Value of the transposed cu... |
| tposcurf2val 49567 | Value of a component of th... |
| tposcurf2cl 49568 | The transposed curry funct... |
| tposcurfcl 49569 | The transposed curry funct... |
| diag1 49570 | The constant functor of ` ... |
| diag1a 49571 | The constant functor of ` ... |
| diag1f1lem 49572 | The object part of the dia... |
| diag1f1 49573 | The object part of the dia... |
| diag2f1lem 49574 | Lemma for ~ diag2f1 . The... |
| diag2f1 49575 | If ` B ` is non-empty, the... |
| fucofulem1 49576 | Lemma for proving functor ... |
| fucofulem2 49577 | Lemma for proving functor ... |
| fuco2el 49578 | Equivalence of product fun... |
| fuco2eld 49579 | Equivalence of product fun... |
| fuco2eld2 49580 | Equivalence of product fun... |
| fuco2eld3 49581 | Equivalence of product fun... |
| fucofvalg 49584 | Value of the function givi... |
| fucofval 49585 | Value of the function givi... |
| fucoelvv 49586 | A functor composition bifu... |
| fuco1 49587 | The object part of the fun... |
| fucof1 49588 | The object part of the fun... |
| fuco2 49589 | The morphism part of the f... |
| fucofn2 49590 | The morphism part of the f... |
| fucofvalne 49591 | Value of the function givi... |
| fuco11 49592 | The object part of the fun... |
| fuco11cl 49593 | The object part of the fun... |
| fuco11a 49594 | The object part of the fun... |
| fuco112 49595 | The object part of the fun... |
| fuco111 49596 | The object part of the fun... |
| fuco111x 49597 | The object part of the fun... |
| fuco112x 49598 | The object part of the fun... |
| fuco112xa 49599 | The object part of the fun... |
| fuco11id 49600 | The identity morphism of t... |
| fuco11idx 49601 | The identity morphism of t... |
| fuco21 49602 | The morphism part of the f... |
| fuco11b 49603 | The object part of the fun... |
| fuco11bALT 49604 | Alternate proof of ~ fuco1... |
| fuco22 49605 | The morphism part of the f... |
| fucofn22 49606 | The morphism part of the f... |
| fuco23 49607 | The morphism part of the f... |
| fuco22natlem1 49608 | Lemma for ~ fuco22nat . T... |
| fuco22natlem2 49609 | Lemma for ~ fuco22nat . T... |
| fuco22natlem3 49610 | Combine ~ fuco22natlem2 wi... |
| fuco22natlem 49611 | The composed natural trans... |
| fuco22nat 49612 | The composed natural trans... |
| fucof21 49613 | The morphism part of the f... |
| fucoid 49614 | Each identity morphism in ... |
| fucoid2 49615 | Each identity morphism in ... |
| fuco22a 49616 | The morphism part of the f... |
| fuco23alem 49617 | The naturality property ( ... |
| fuco23a 49618 | The morphism part of the f... |
| fucocolem1 49619 | Lemma for ~ fucoco . Asso... |
| fucocolem2 49620 | Lemma for ~ fucoco . The ... |
| fucocolem3 49621 | Lemma for ~ fucoco . The ... |
| fucocolem4 49622 | Lemma for ~ fucoco . The ... |
| fucoco 49623 | Composition in the source ... |
| fucoco2 49624 | Composition in the source ... |
| fucofunc 49625 | The functor composition bi... |
| fucofunca 49626 | The functor composition bi... |
| fucolid 49627 | Post-compose a natural tra... |
| fucorid 49628 | Pre-composing a natural tr... |
| fucorid2 49629 | Pre-composing a natural tr... |
| postcofval 49630 | Value of the post-composit... |
| postcofcl 49631 | The post-composition funct... |
| precofvallem 49632 | Lemma for ~ precofval to e... |
| precofval 49633 | Value of the pre-compositi... |
| precofvalALT 49634 | Alternate proof of ~ preco... |
| precofval2 49635 | Value of the pre-compositi... |
| precofcl 49636 | The pre-composition functo... |
| precofval3 49637 | Value of the pre-compositi... |
| precoffunc 49638 | The pre-composition functo... |
| reldmprcof 49641 | The domain of ` -o.F ` is ... |
| prcofvalg 49642 | Value of the pre-compositi... |
| prcofvala 49643 | Value of the pre-compositi... |
| prcofval 49644 | Value of the pre-compositi... |
| prcofpropd 49645 | If the categories have the... |
| prcofelvv 49646 | The pre-composition functo... |
| reldmprcof1 49647 | The domain of the object p... |
| reldmprcof2 49648 | The domain of the morphism... |
| prcoftposcurfuco 49649 | The pre-composition functo... |
| prcoftposcurfucoa 49650 | The pre-composition functo... |
| prcoffunc 49651 | The pre-composition functo... |
| prcoffunca 49652 | The pre-composition functo... |
| prcoffunca2 49653 | The pre-composition functo... |
| prcof1 49654 | The object part of the pre... |
| prcof2a 49655 | The morphism part of the p... |
| prcof2 49656 | The morphism part of the p... |
| prcof21a 49657 | The morphism part of the p... |
| prcof22a 49658 | The morphism part of the p... |
| prcofdiag1 49659 | A constant functor pre-com... |
| prcofdiag 49660 | A diagonal functor post-co... |
| catcrcl 49661 | Reverse closure for the ca... |
| catcrcl2 49662 | Reverse closure for the ca... |
| elcatchom 49663 | A morphism of the category... |
| catcsect 49664 | The property " ` F ` is a ... |
| catcinv 49665 | The property " ` F ` is an... |
| catcisoi 49666 | A functor is an isomorphis... |
| uobeq2 49667 | If a full functor (in fact... |
| uobeq3 49668 | An isomorphism between cat... |
| opf11 49669 | The object part of the op ... |
| opf12 49670 | The object part of the op ... |
| opf2fval 49671 | The morphism part of the o... |
| opf2 49672 | The morphism part of the o... |
| fucoppclem 49673 | Lemma for ~ fucoppc . (Co... |
| fucoppcid 49674 | The opposite category of f... |
| fucoppcco 49675 | The opposite category of f... |
| fucoppc 49676 | The isomorphism from the o... |
| fucoppcffth 49677 | A fully faithful functor f... |
| fucoppcfunc 49678 | A functor from the opposit... |
| fucoppccic 49679 | The opposite category of f... |
| oppfdiag1 49680 | A constant functor for opp... |
| oppfdiag1a 49681 | A constant functor for opp... |
| oppfdiag 49682 | A diagonal functor for opp... |
| isthinc 49685 | The predicate "is a thin c... |
| isthinc2 49686 | A thin category is a categ... |
| isthinc3 49687 | A thin category is a categ... |
| thincc 49688 | A thin category is a categ... |
| thinccd 49689 | A thin category is a categ... |
| thincssc 49690 | A thin category is a categ... |
| isthincd2lem1 49691 | Lemma for ~ isthincd2 and ... |
| thincmo2 49692 | Morphisms in the same hom-... |
| thinchom 49693 | A non-empty hom-set of a t... |
| thincmo 49694 | There is at most one morph... |
| thincmoALT 49695 | Alternate proof of ~ thinc... |
| thincmod 49696 | At most one morphism in ea... |
| thincn0eu 49697 | In a thin category, a hom-... |
| thincid 49698 | In a thin category, a morp... |
| thincmon 49699 | In a thin category, all mo... |
| thincepi 49700 | In a thin category, all mo... |
| isthincd2lem2 49701 | Lemma for ~ isthincd2 . (... |
| isthincd 49702 | The predicate "is a thin c... |
| isthincd2 49703 | The predicate " ` C ` is a... |
| oppcthin 49704 | The opposite category of a... |
| oppcthinco 49705 | If the opposite category o... |
| oppcthinendc 49706 | The opposite category of a... |
| oppcthinendcALT 49707 | Alternate proof of ~ oppct... |
| thincpropd 49708 | Two structures with the sa... |
| subthinc 49709 | A subcategory of a thin ca... |
| functhinclem1 49710 | Lemma for ~ functhinc . G... |
| functhinclem2 49711 | Lemma for ~ functhinc . (... |
| functhinclem3 49712 | Lemma for ~ functhinc . T... |
| functhinclem4 49713 | Lemma for ~ functhinc . O... |
| functhinc 49714 | A functor to a thin catego... |
| functhincfun 49715 | A functor to a thin catego... |
| fullthinc 49716 | A functor to a thin catego... |
| fullthinc2 49717 | A full functor to a thin c... |
| thincfth 49718 | A functor from a thin cate... |
| thincciso 49719 | Two thin categories are is... |
| thinccisod 49720 | Two thin categories are is... |
| thincciso2 49721 | Categories isomorphic to a... |
| thincciso3 49722 | Categories isomorphic to a... |
| thincciso4 49723 | Two isomorphic categories ... |
| 0thincg 49724 | Any structure with an empt... |
| 0thinc 49725 | The empty category (see ~ ... |
| indcthing 49726 | An indiscrete category, i.... |
| discthing 49727 | A discrete category, i.e.,... |
| indthinc 49728 | An indiscrete category in ... |
| indthincALT 49729 | An alternate proof of ~ in... |
| prsthinc 49730 | Preordered sets as categor... |
| setcthin 49731 | A category of sets all of ... |
| setc2othin 49732 | The category ` ( SetCat ``... |
| thincsect 49733 | In a thin category, one mo... |
| thincsect2 49734 | In a thin category, ` F ` ... |
| thincinv 49735 | In a thin category, ` F ` ... |
| thinciso 49736 | In a thin category, ` F : ... |
| thinccic 49737 | In a thin category, two ob... |
| istermc 49740 | The predicate "is a termin... |
| istermc2 49741 | The predicate "is a termin... |
| istermc3 49742 | The predicate "is a termin... |
| termcthin 49743 | A terminal category is a t... |
| termcthind 49744 | A terminal category is a t... |
| termccd 49745 | A terminal category is a c... |
| termcbas 49746 | The base of a terminal cat... |
| termco 49747 | The object of a terminal c... |
| termcbas2 49748 | The base of a terminal cat... |
| termcbasmo 49749 | Two objects in a terminal ... |
| termchomn0 49750 | All hom-sets of a terminal... |
| termchommo 49751 | All morphisms of a termina... |
| termcid 49752 | The morphism of a terminal... |
| termcid2 49753 | The morphism of a terminal... |
| termchom 49754 | The hom-set of a terminal ... |
| termchom2 49755 | The hom-set of a terminal ... |
| setcsnterm 49756 | The category of one set, e... |
| setc1oterm 49757 | The category ` ( SetCat ``... |
| setc1obas 49758 | The base of the trivial ca... |
| setc1ohomfval 49759 | Set of morphisms of the tr... |
| setc1ocofval 49760 | Composition in the trivial... |
| setc1oid 49761 | The identity morphism of t... |
| funcsetc1ocl 49762 | The functor to the trivial... |
| funcsetc1o 49763 | Value of the functor to th... |
| isinito2lem 49764 | The predicate "is an initi... |
| isinito2 49765 | The predicate "is an initi... |
| isinito3 49766 | The predicate "is an initi... |
| dfinito4 49767 | An alternate definition of... |
| dftermo4 49768 | An alternate definition of... |
| termcpropd 49769 | Two structures with the sa... |
| oppctermhom 49770 | The opposite category of a... |
| oppctermco 49771 | The opposite category of a... |
| oppcterm 49772 | The opposite category of a... |
| functermclem 49773 | Lemma for ~ functermc . (... |
| functermc 49774 | Functor to a terminal cate... |
| functermc2 49775 | Functor to a terminal cate... |
| functermceu 49776 | There exists a unique func... |
| fulltermc 49777 | A functor to a terminal ca... |
| fulltermc2 49778 | Given a full functor to a ... |
| termcterm 49779 | A terminal category is a t... |
| termcterm2 49780 | A terminal object of the c... |
| termcterm3 49781 | In the category of small c... |
| termcciso 49782 | A category is isomorphic t... |
| termccisoeu 49783 | The isomorphism between te... |
| termc2 49784 | If there exists a unique f... |
| termc 49785 | Alternate definition of ` ... |
| dftermc2 49786 | Alternate definition of ` ... |
| eufunclem 49787 | If there exists a unique f... |
| eufunc 49788 | If there exists a unique f... |
| idfudiag1lem 49789 | Lemma for ~ idfudiag1bas a... |
| idfudiag1bas 49790 | If the identity functor of... |
| idfudiag1 49791 | If the identity functor of... |
| euendfunc 49792 | If there exists a unique e... |
| euendfunc2 49793 | If there exists a unique e... |
| termcarweu 49794 | There exists a unique disj... |
| arweuthinc 49795 | If a structure has a uniqu... |
| arweutermc 49796 | If a structure has a uniqu... |
| dftermc3 49797 | Alternate definition of ` ... |
| termcfuncval 49798 | The value of a functor fro... |
| diag1f1olem 49799 | To any functor from a term... |
| diag1f1o 49800 | The object part of the dia... |
| termcnatval 49801 | Value of natural transform... |
| diag2f1olem 49802 | Lemma for ~ diag2f1o . (C... |
| diag2f1o 49803 | If ` D ` is terminal, the ... |
| diagffth 49804 | The diagonal functor is a ... |
| diagciso 49805 | The diagonal functor is an... |
| diagcic 49806 | Any category ` C ` is isom... |
| funcsn 49807 | The category of one functo... |
| fucterm 49808 | The category of functors t... |
| 0fucterm 49809 | The category of functors f... |
| termfucterm 49810 | All functors between two t... |
| cofuterm 49811 | Post-compose with a functo... |
| uobeqterm 49812 | Universal objects and term... |
| isinito4 49813 | The predicate "is an initi... |
| isinito4a 49814 | The predicate "is an initi... |
| prstcval 49817 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 49818 | Lemma for ~ prstcnid and ~... |
| prstcnid 49819 | Components other than ` Ho... |
| prstcbas 49820 | The base set is unchanged.... |
| prstcleval 49821 | Value of the less-than-or-... |
| prstcle 49822 | Value of the less-than-or-... |
| prstcocval 49823 | Orthocomplementation is un... |
| prstcoc 49824 | Orthocomplementation is un... |
| prstchomval 49825 | Hom-sets of the constructe... |
| prstcprs 49826 | The category is a preorder... |
| prstcthin 49827 | The preordered set is equi... |
| prstchom 49828 | Hom-sets of the constructe... |
| prstchom2 49829 | Hom-sets of the constructe... |
| prstchom2ALT 49830 | Hom-sets of the constructe... |
| oduoppcbas 49831 | The dual of a preordered s... |
| oduoppcciso 49832 | The dual of a preordered s... |
| postcpos 49833 | The converted category is ... |
| postcposALT 49834 | Alternate proof of ~ postc... |
| postc 49835 | The converted category is ... |
| discsntermlem 49836 | A singlegon is an element ... |
| basrestermcfolem 49837 | An element of the class of... |
| discbas 49838 | A discrete category (a cat... |
| discthin 49839 | A discrete category (a cat... |
| discsnterm 49840 | A discrete category (a cat... |
| basrestermcfo 49841 | The base function restrict... |
| termcnex 49842 | The class of all terminal ... |
| mndtcval 49845 | Value of the category buil... |
| mndtcbasval 49846 | The base set of the catego... |
| mndtcbas 49847 | The category built from a ... |
| mndtcob 49848 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 49849 | Two objects in a category ... |
| mndtchom 49850 | The only hom-set of the ca... |
| mndtcco 49851 | The composition of the cat... |
| mndtcco2 49852 | The composition of the cat... |
| mndtccatid 49853 | Lemma for ~ mndtccat and ~... |
| mndtccat 49854 | The function value is a ca... |
| mndtcid 49855 | The identity morphism, or ... |
| oppgoppchom 49856 | The converted opposite mon... |
| oppgoppcco 49857 | The converted opposite mon... |
| oppgoppcid 49858 | The converted opposite mon... |
| grptcmon 49859 | All morphisms in a categor... |
| grptcepi 49860 | All morphisms in a categor... |
| 2arwcatlem1 49861 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem2 49862 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem3 49863 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem4 49864 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem5 49865 | Lemma for ~ 2arwcat . (Co... |
| 2arwcat 49866 | The condition for a struct... |
| incat 49867 | Constructing a category wi... |
| setc1onsubc 49868 | Construct a category with ... |
| cnelsubclem 49869 | Lemma for ~ cnelsubc . (C... |
| cnelsubc 49870 | Remark 4.2(2) of [Adamek] ... |
| lanfn 49875 | ` Lan ` is a function on `... |
| ranfn 49876 | ` Ran ` is a function on `... |
| reldmlan 49877 | The domain of ` Lan ` is a... |
| reldmran 49878 | The domain of ` Ran ` is a... |
| lanfval 49879 | Value of the function gene... |
| ranfval 49880 | Value of the function gene... |
| lanpropd 49881 | If the categories have the... |
| ranpropd 49882 | If the categories have the... |
| reldmlan2 49883 | The domain of ` ( P Lan E ... |
| reldmran2 49884 | The domain of ` ( P Ran E ... |
| lanval 49885 | Value of the set of left K... |
| ranval 49886 | Value of the set of right ... |
| lanrcl 49887 | Reverse closure for left K... |
| ranrcl 49888 | Reverse closure for right ... |
| rellan 49889 | The set of left Kan extens... |
| relran 49890 | The set of right Kan exten... |
| islan 49891 | A left Kan extension is a ... |
| islan2 49892 | A left Kan extension is a ... |
| lanval2 49893 | The set of left Kan extens... |
| isran 49894 | A right Kan extension is a... |
| isran2 49895 | A right Kan extension is a... |
| ranval2 49896 | The set of right Kan exten... |
| ranval3 49897 | The set of right Kan exten... |
| lanrcl2 49898 | Reverse closure for left K... |
| lanrcl3 49899 | Reverse closure for left K... |
| lanrcl4 49900 | The first component of a l... |
| lanrcl5 49901 | The second component of a ... |
| ranrcl2 49902 | Reverse closure for right ... |
| ranrcl3 49903 | Reverse closure for right ... |
| ranrcl4lem 49904 | Lemma for ~ ranrcl4 and ~ ... |
| ranrcl4 49905 | The first component of a r... |
| ranrcl5 49906 | The second component of a ... |
| lanup 49907 | The universal property of ... |
| ranup 49908 | The universal property of ... |
| reldmlmd 49913 | The domain of ` Limit ` is... |
| reldmcmd 49914 | The domain of ` Colimit ` ... |
| lmdfval 49915 | Function value of ` Limit ... |
| cmdfval 49916 | Function value of ` Colimi... |
| lmdrcl 49917 | Reverse closure for a limi... |
| cmdrcl 49918 | Reverse closure for a coli... |
| reldmlmd2 49919 | The domain of ` ( C Limit ... |
| reldmcmd2 49920 | The domain of ` ( C Colimi... |
| lmdfval2 49921 | The set of limits of a dia... |
| cmdfval2 49922 | The set of colimits of a d... |
| lmdpropd 49923 | If the categories have the... |
| cmdpropd 49924 | If the categories have the... |
| rellmd 49925 | The set of limits of a dia... |
| relcmd 49926 | The set of colimits of a d... |
| concl 49927 | A natural transformation f... |
| coccl 49928 | A natural transformation t... |
| concom 49929 | A cone to a diagram commut... |
| coccom 49930 | A co-cone to a diagram com... |
| islmd 49931 | The universal property of ... |
| iscmd 49932 | The universal property of ... |
| lmddu 49933 | The duality of limits and ... |
| cmddu 49934 | The duality of limits and ... |
| initocmd 49935 | Initial objects are the ob... |
| termolmd 49936 | Terminal objects are the o... |
| lmdran 49937 | To each limit of a diagram... |
| cmdlan 49938 | To each colimit of a diagr... |
| nfintd 49939 | Bound-variable hypothesis ... |
| nfiund 49940 | Bound-variable hypothesis ... |
| nfiundg 49941 | Bound-variable hypothesis ... |
| iunord 49942 | The indexed union of a col... |
| iunordi 49943 | The indexed union of a col... |
| spd 49944 | Specialization deduction, ... |
| spcdvw 49945 | A version of ~ spcdv where... |
| tfis2d 49946 | Transfinite Induction Sche... |
| bnd2d 49947 | Deduction form of ~ bnd2 .... |
| dffun3f 49948 | Alternate definition of fu... |
| setrecseq 49951 | Equality theorem for set r... |
| nfsetrecs 49952 | Bound-variable hypothesis ... |
| setrec1lem1 49953 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 49954 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 49955 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 49956 | Lemma for ~ setrec1 . If ... |
| setrec1 49957 | This is the first of two f... |
| setrec2fun 49958 | This is the second of two ... |
| setrec2lem1 49959 | Lemma for ~ setrec2 . The... |
| setrec2lem2 49960 | Lemma for ~ setrec2 . The... |
| setrec2 49961 | This is the second of two ... |
| setrec2v 49962 | Version of ~ setrec2 with ... |
| setrec2mpt 49963 | Version of ~ setrec2 where... |
| setis 49964 | Version of ~ setrec2 expre... |
| elsetrecslem 49965 | Lemma for ~ elsetrecs . A... |
| elsetrecs 49966 | A set ` A ` is an element ... |
| setrecsss 49967 | The ` setrecs ` operator r... |
| setrecsres 49968 | A recursively generated cl... |
| vsetrec 49969 | Construct ` _V ` using set... |
| 0setrec 49970 | If a function sends the em... |
| onsetreclem1 49971 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 49972 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 49973 | Lemma for ~ onsetrec . (C... |
| onsetrec 49974 | Construct ` On ` using set... |
| elpglem1 49977 | Lemma for ~ elpg . (Contr... |
| elpglem2 49978 | Lemma for ~ elpg . (Contr... |
| elpglem3 49979 | Lemma for ~ elpg . (Contr... |
| elpg 49980 | Membership in the class of... |
| pgindlem 49981 | Lemma for ~ pgind . (Cont... |
| pgindnf 49982 | Version of ~ pgind with ex... |
| pgind 49983 | Induction on partizan game... |
| sbidd 49984 | An identity theorem for su... |
| sbidd-misc 49985 | An identity theorem for su... |
| gte-lte 49990 | Simple relationship betwee... |
| gt-lt 49991 | Simple relationship betwee... |
| gte-lteh 49992 | Relationship between ` <_ ... |
| gt-lth 49993 | Relationship between ` < `... |
| ex-gt 49994 | Simple example of ` > ` , ... |
| ex-gte 49995 | Simple example of ` >_ ` ,... |
| sinhval-named 50002 | Value of the named sinh fu... |
| coshval-named 50003 | Value of the named cosh fu... |
| tanhval-named 50004 | Value of the named tanh fu... |
| sinh-conventional 50005 | Conventional definition of... |
| sinhpcosh 50006 | Prove that ` ( sinh `` A )... |
| secval 50013 | Value of the secant functi... |
| cscval 50014 | Value of the cosecant func... |
| cotval 50015 | Value of the cotangent fun... |
| seccl 50016 | The closure of the secant ... |
| csccl 50017 | The closure of the cosecan... |
| cotcl 50018 | The closure of the cotange... |
| reseccl 50019 | The closure of the secant ... |
| recsccl 50020 | The closure of the cosecan... |
| recotcl 50021 | The closure of the cotange... |
| recsec 50022 | The reciprocal of secant i... |
| reccsc 50023 | The reciprocal of cosecant... |
| reccot 50024 | The reciprocal of cotangen... |
| rectan 50025 | The reciprocal of tangent ... |
| sec0 50026 | The value of the secant fu... |
| onetansqsecsq 50027 | Prove the tangent squared ... |
| cotsqcscsq 50028 | Prove the tangent squared ... |
| ifnmfalse 50029 | If A is not a member of B,... |
| logb2aval 50030 | Define the value of the ` ... |
| mvlraddi 50037 | Move the right term in a s... |
| assraddsubi 50038 | Associate RHS addition-sub... |
| joinlmuladdmuli 50039 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 50040 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 50041 | Join AB-CB into (A-C) on L... |
| mvlrmuld 50042 | Move the right term in a p... |
| mvlrmuli 50043 | Move the right term in a p... |
| i2linesi 50044 | Solve for the intersection... |
| i2linesd 50045 | Solve for the intersection... |
| alimp-surprise 50046 | Demonstrate that when usin... |
| alimp-no-surprise 50047 | There is no "surprise" in ... |
| empty-surprise 50048 | Demonstrate that when usin... |
| empty-surprise2 50049 | "Prove" that false is true... |
| eximp-surprise 50050 | Show what implication insi... |
| eximp-surprise2 50051 | Show that "there exists" w... |
| alsconv 50056 | There is an equivalence be... |
| alsi1d 50057 | Deduction rule: Given "al... |
| alsi2d 50058 | Deduction rule: Given "al... |
| alsc1d 50059 | Deduction rule: Given "al... |
| alsc2d 50060 | Deduction rule: Given "al... |
| alscn0d 50061 | Deduction rule: Given "al... |
| alsi-no-surprise 50062 | Demonstrate that there is ... |
| 5m4e1 50063 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 50064 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 50065 | Resolution rule. This is ... |
| testable 50066 | In classical logic all wff... |
| aacllem 50067 | Lemma for other theorems a... |
| amgmwlem 50068 | Weighted version of ~ amgm... |
| amgmlemALT 50069 | Alternate proof of ~ amgml... |
| amgmw2d 50070 | Weighted arithmetic-geomet... |
| young2d 50071 | Young's inequality for ` n... |
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