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Metamath Proof Explorer |
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Ref | Description |
idi 1 | (_Note_: This inference r... |
a1ii 2 | (_Note_: This inference r... |
mp2 9 | A double modus ponens infe... |
mp2b 10 | A double modus ponens infe... |
a1i 11 | Inference introducing an a... |
2a1i 12 | Inference introducing two ... |
mp1i 13 | Inference detaching an ant... |
a2i 14 | Inference distributing an ... |
mpd 15 | A modus ponens deduction. ... |
imim2i 16 | Inference adding common an... |
syl 17 | An inference version of th... |
3syl 18 | Inference chaining two syl... |
4syl 19 | Inference chaining three s... |
mpi 20 | A nested modus ponens infe... |
mpisyl 21 | A syllogism combined with ... |
id 22 | Principle of identity. Th... |
idALT 23 | Alternate proof of ~ id . ... |
idd 24 | Principle of identity ~ id... |
a1d 25 | Deduction introducing an e... |
2a1d 26 | Deduction introducing two ... |
a1i13 27 | Add two antecedents to a w... |
2a1 28 | A double form of ~ ax-1 . ... |
a2d 29 | Deduction distributing an ... |
sylcom 30 | Syllogism inference with c... |
syl5com 31 | Syllogism inference with c... |
com12 32 | Inference that swaps (comm... |
syl11 33 | A syllogism inference. Co... |
syl5 34 | A syllogism rule of infere... |
syl6 35 | A syllogism rule of infere... |
syl56 36 | Combine ~ syl5 and ~ syl6 ... |
syl6com 37 | Syllogism inference with c... |
mpcom 38 | Modus ponens inference wit... |
syli 39 | Syllogism inference with c... |
syl2im 40 | Replace two antecedents. ... |
syl2imc 41 | A commuted version of ~ sy... |
pm2.27 42 | This theorem, sometimes ca... |
mpdd 43 | A nested modus ponens dedu... |
mpid 44 | A nested modus ponens dedu... |
mpdi 45 | A nested modus ponens dedu... |
mpii 46 | A doubly nested modus pone... |
syld 47 | Syllogism deduction. Dedu... |
syldc 48 | Syllogism deduction. Comm... |
mp2d 49 | A double modus ponens dedu... |
a1dd 50 | Double deduction introduci... |
2a1dd 51 | Double deduction introduci... |
pm2.43i 52 | Inference absorbing redund... |
pm2.43d 53 | Deduction absorbing redund... |
pm2.43a 54 | Inference absorbing redund... |
pm2.43b 55 | Inference absorbing redund... |
pm2.43 56 | Absorption of redundant an... |
imim2d 57 | Deduction adding nested an... |
imim2 58 | A closed form of syllogism... |
embantd 59 | Deduction embedding an ant... |
3syld 60 | Triple syllogism deduction... |
sylsyld 61 | A double syllogism inferen... |
imim12i 62 | Inference joining two impl... |
imim1i 63 | Inference adding common co... |
imim3i 64 | Inference adding three nes... |
sylc 65 | A syllogism inference comb... |
syl3c 66 | A syllogism inference comb... |
syl6mpi 67 | A syllogism inference. (C... |
mpsyl 68 | Modus ponens combined with... |
mpsylsyld 69 | Modus ponens combined with... |
syl6c 70 | Inference combining ~ syl6... |
syl6ci 71 | A syllogism inference comb... |
syldd 72 | Nested syllogism deduction... |
syl5d 73 | A nested syllogism deducti... |
syl7 74 | A syllogism rule of infere... |
syl6d 75 | A nested syllogism deducti... |
syl8 76 | A syllogism rule of infere... |
syl9 77 | A nested syllogism inferen... |
syl9r 78 | A nested syllogism inferen... |
syl10 79 | A nested syllogism inferen... |
a1ddd 80 | Triple deduction introduci... |
imim12d 81 | Deduction combining antece... |
imim1d 82 | Deduction adding nested co... |
imim1 83 | A closed form of syllogism... |
pm2.83 84 | Theorem *2.83 of [Whitehea... |
peirceroll 85 | Over minimal implicational... |
com23 86 | Commutation of antecedents... |
com3r 87 | Commutation of antecedents... |
com13 88 | Commutation of antecedents... |
com3l 89 | Commutation of antecedents... |
pm2.04 90 | Swap antecedents. Theorem... |
com34 91 | Commutation of antecedents... |
com4l 92 | Commutation of antecedents... |
com4t 93 | Commutation of antecedents... |
com4r 94 | Commutation of antecedents... |
com24 95 | Commutation of antecedents... |
com14 96 | Commutation of antecedents... |
com45 97 | Commutation of antecedents... |
com35 98 | Commutation of antecedents... |
com25 99 | Commutation of antecedents... |
com5l 100 | Commutation of antecedents... |
com15 101 | Commutation of antecedents... |
com52l 102 | Commutation of antecedents... |
com52r 103 | Commutation of antecedents... |
com5r 104 | Commutation of antecedents... |
imim12 105 | Closed form of ~ imim12i a... |
jarr 106 | Elimination of a nested an... |
jarri 107 | Inference associated with ... |
pm2.86d 108 | Deduction associated with ... |
pm2.86 109 | Converse of Axiom ~ ax-2 .... |
pm2.86i 110 | Inference associated with ... |
loolin 111 | The Linearity Axiom of the... |
loowoz 112 | An alternate for the Linea... |
con4 113 | Alias for ~ ax-3 to be use... |
con4i 114 | Inference associated with ... |
con4d 115 | Deduction associated with ... |
mt4 116 | The rule of modus tollens.... |
mt4d 117 | Modus tollens deduction. ... |
mt4i 118 | Modus tollens inference. ... |
pm2.21i 119 | A contradiction implies an... |
pm2.24ii 120 | A contradiction implies an... |
pm2.21d 121 | A contradiction implies an... |
pm2.21ddALT 122 | Alternate proof of ~ pm2.2... |
pm2.21 123 | From a wff and its negatio... |
pm2.24 124 | Theorem *2.24 of [Whitehea... |
jarl 125 | Elimination of a nested an... |
jarli 126 | Inference associated with ... |
pm2.18d 127 | Deduction form of the Clav... |
pm2.18 128 | Clavius law, or "consequen... |
pm2.18i 129 | Inference associated with ... |
notnotr 130 | Double negation eliminatio... |
notnotri 131 | Inference associated with ... |
notnotriALT 132 | Alternate proof of ~ notno... |
notnotrd 133 | Deduction associated with ... |
con2d 134 | A contraposition deduction... |
con2 135 | Contraposition. Theorem *... |
mt2d 136 | Modus tollens deduction. ... |
mt2i 137 | Modus tollens inference. ... |
nsyl3 138 | A negated syllogism infere... |
con2i 139 | A contraposition inference... |
nsyl 140 | A negated syllogism infere... |
nsyl2 141 | A negated syllogism infere... |
notnot 142 | Double negation introducti... |
notnoti 143 | Inference associated with ... |
notnotd 144 | Deduction associated with ... |
con1d 145 | A contraposition deduction... |
con1 146 | Contraposition. Theorem *... |
con1i 147 | A contraposition inference... |
mt3d 148 | Modus tollens deduction. ... |
mt3i 149 | Modus tollens inference. ... |
pm2.24i 150 | Inference associated with ... |
pm2.24d 151 | Deduction form of ~ pm2.24... |
con3d 152 | A contraposition deduction... |
con3 153 | Contraposition. Theorem *... |
con3i 154 | A contraposition inference... |
con3rr3 155 | Rotate through consequent ... |
nsyld 156 | A negated syllogism deduct... |
nsyli 157 | A negated syllogism infere... |
nsyl4 158 | A negated syllogism infere... |
nsyl5 159 | A negated syllogism infere... |
pm3.2im 160 | Theorem *3.2 of [Whitehead... |
jc 161 | Deduction joining the cons... |
jcn 162 | Theorem joining the conseq... |
jcnd 163 | Deduction joining the cons... |
impi 164 | An importation inference. ... |
expi 165 | An exportation inference. ... |
simprim 166 | Simplification. Similar t... |
simplim 167 | Simplification. Similar t... |
pm2.5g 168 | General instance of Theore... |
pm2.5 169 | Theorem *2.5 of [Whitehead... |
conax1 170 | Contrapositive of ~ ax-1 .... |
conax1k 171 | Weakening of ~ conax1 . G... |
pm2.51 172 | Theorem *2.51 of [Whitehea... |
pm2.52 173 | Theorem *2.52 of [Whitehea... |
pm2.521g 174 | A general instance of Theo... |
pm2.521g2 175 | A general instance of Theo... |
pm2.521 176 | Theorem *2.521 of [Whitehe... |
expt 177 | Exportation theorem ~ pm3.... |
impt 178 | Importation theorem ~ pm3.... |
pm2.61d 179 | Deduction eliminating an a... |
pm2.61d1 180 | Inference eliminating an a... |
pm2.61d2 181 | Inference eliminating an a... |
pm2.61i 182 | Inference eliminating an a... |
pm2.61ii 183 | Inference eliminating two ... |
pm2.61nii 184 | Inference eliminating two ... |
pm2.61iii 185 | Inference eliminating thre... |
ja 186 | Inference joining the ante... |
jad 187 | Deduction form of ~ ja . ... |
pm2.01 188 | Weak Clavius law. If a fo... |
pm2.01d 189 | Deduction based on reducti... |
pm2.6 190 | Theorem *2.6 of [Whitehead... |
pm2.61 191 | Theorem *2.61 of [Whitehea... |
pm2.65 192 | Theorem *2.65 of [Whitehea... |
pm2.65i 193 | Inference for proof by con... |
pm2.21dd 194 | A contradiction implies an... |
pm2.65d 195 | Deduction for proof by con... |
mto 196 | The rule of modus tollens.... |
mtod 197 | Modus tollens deduction. ... |
mtoi 198 | Modus tollens inference. ... |
mt2 199 | A rule similar to modus to... |
mt3 200 | A rule similar to modus to... |
peirce 201 | Peirce's axiom. A non-int... |
looinv 202 | The Inversion Axiom of the... |
bijust0 203 | A self-implication (see ~ ... |
bijust 204 | Theorem used to justify th... |
impbi 207 | Property of the biconditio... |
impbii 208 | Infer an equivalence from ... |
impbidd 209 | Deduce an equivalence from... |
impbid21d 210 | Deduce an equivalence from... |
impbid 211 | Deduce an equivalence from... |
dfbi1 212 | Relate the biconditional c... |
dfbi1ALT 213 | Alternate proof of ~ dfbi1... |
biimp 214 | Property of the biconditio... |
biimpi 215 | Infer an implication from ... |
sylbi 216 | A mixed syllogism inferenc... |
sylib 217 | A mixed syllogism inferenc... |
sylbb 218 | A mixed syllogism inferenc... |
biimpr 219 | Property of the biconditio... |
bicom1 220 | Commutative law for the bi... |
bicom 221 | Commutative law for the bi... |
bicomd 222 | Commute two sides of a bic... |
bicomi 223 | Inference from commutative... |
impbid1 224 | Infer an equivalence from ... |
impbid2 225 | Infer an equivalence from ... |
impcon4bid 226 | A variation on ~ impbid wi... |
biimpri 227 | Infer a converse implicati... |
biimpd 228 | Deduce an implication from... |
mpbi 229 | An inference from a bicond... |
mpbir 230 | An inference from a bicond... |
mpbid 231 | A deduction from a bicondi... |
mpbii 232 | An inference from a nested... |
sylibr 233 | A mixed syllogism inferenc... |
sylbir 234 | A mixed syllogism inferenc... |
sylbbr 235 | A mixed syllogism inferenc... |
sylbb1 236 | A mixed syllogism inferenc... |
sylbb2 237 | A mixed syllogism inferenc... |
sylibd 238 | A syllogism deduction. (C... |
sylbid 239 | A syllogism deduction. (C... |
mpbidi 240 | A deduction from a bicondi... |
biimtrid 241 | A mixed syllogism inferenc... |
biimtrrid 242 | A mixed syllogism inferenc... |
imbitrid 243 | A mixed syllogism inferenc... |
syl5ibcom 244 | A mixed syllogism inferenc... |
imbitrrid 245 | A mixed syllogism inferenc... |
syl5ibrcom 246 | A mixed syllogism inferenc... |
biimprd 247 | Deduce a converse implicat... |
biimpcd 248 | Deduce a commuted implicat... |
biimprcd 249 | Deduce a converse commuted... |
imbitrdi 250 | A mixed syllogism inferenc... |
imbitrrdi 251 | A mixed syllogism inferenc... |
biimtrdi 252 | A mixed syllogism inferenc... |
syl6bi 253 | A mixed syllogism inferenc... |
syl6bir 254 | A mixed syllogism inferenc... |
syl7bi 255 | A mixed syllogism inferenc... |
syl8ib 256 | A syllogism rule of infere... |
mpbird 257 | A deduction from a bicondi... |
mpbiri 258 | An inference from a nested... |
sylibrd 259 | A syllogism deduction. (C... |
sylbird 260 | A syllogism deduction. (C... |
biid 261 | Principle of identity for ... |
biidd 262 | Principle of identity with... |
pm5.1im 263 | Two propositions are equiv... |
2th 264 | Two truths are equivalent.... |
2thd 265 | Two truths are equivalent.... |
monothetic 266 | Two self-implications (see... |
ibi 267 | Inference that converts a ... |
ibir 268 | Inference that converts a ... |
ibd 269 | Deduction that converts a ... |
pm5.74 270 | Distribution of implicatio... |
pm5.74i 271 | Distribution of implicatio... |
pm5.74ri 272 | Distribution of implicatio... |
pm5.74d 273 | Distribution of implicatio... |
pm5.74rd 274 | Distribution of implicatio... |
bitri 275 | An inference from transiti... |
bitr2i 276 | An inference from transiti... |
bitr3i 277 | An inference from transiti... |
bitr4i 278 | An inference from transiti... |
bitrd 279 | Deduction form of ~ bitri ... |
bitr2d 280 | Deduction form of ~ bitr2i... |
bitr3d 281 | Deduction form of ~ bitr3i... |
bitr4d 282 | Deduction form of ~ bitr4i... |
bitrid 283 | A syllogism inference from... |
bitr2id 284 | A syllogism inference from... |
bitr3id 285 | A syllogism inference from... |
bitr3di 286 | A syllogism inference from... |
bitrdi 287 | A syllogism inference from... |
bitr2di 288 | A syllogism inference from... |
bitr4di 289 | A syllogism inference from... |
bitr4id 290 | A syllogism inference from... |
3imtr3i 291 | A mixed syllogism inferenc... |
3imtr4i 292 | A mixed syllogism inferenc... |
3imtr3d 293 | More general version of ~ ... |
3imtr4d 294 | More general version of ~ ... |
3imtr3g 295 | More general version of ~ ... |
3imtr4g 296 | More general version of ~ ... |
3bitri 297 | A chained inference from t... |
3bitrri 298 | A chained inference from t... |
3bitr2i 299 | A chained inference from t... |
3bitr2ri 300 | A chained inference from t... |
3bitr3i 301 | A chained inference from t... |
3bitr3ri 302 | A chained inference from t... |
3bitr4i 303 | A chained inference from t... |
3bitr4ri 304 | A chained inference from t... |
3bitrd 305 | Deduction from transitivit... |
3bitrrd 306 | Deduction from transitivit... |
3bitr2d 307 | Deduction from transitivit... |
3bitr2rd 308 | Deduction from transitivit... |
3bitr3d 309 | Deduction from transitivit... |
3bitr3rd 310 | Deduction from transitivit... |
3bitr4d 311 | Deduction from transitivit... |
3bitr4rd 312 | Deduction from transitivit... |
3bitr3g 313 | More general version of ~ ... |
3bitr4g 314 | More general version of ~ ... |
notnotb 315 | Double negation. Theorem ... |
con34b 316 | A biconditional form of co... |
con4bid 317 | A contraposition deduction... |
notbid 318 | Deduction negating both si... |
notbi 319 | Contraposition. Theorem *... |
notbii 320 | Negate both sides of a log... |
con4bii 321 | A contraposition inference... |
mtbi 322 | An inference from a bicond... |
mtbir 323 | An inference from a bicond... |
mtbid 324 | A deduction from a bicondi... |
mtbird 325 | A deduction from a bicondi... |
mtbii 326 | An inference from a bicond... |
mtbiri 327 | An inference from a bicond... |
sylnib 328 | A mixed syllogism inferenc... |
sylnibr 329 | A mixed syllogism inferenc... |
sylnbi 330 | A mixed syllogism inferenc... |
sylnbir 331 | A mixed syllogism inferenc... |
xchnxbi 332 | Replacement of a subexpres... |
xchnxbir 333 | Replacement of a subexpres... |
xchbinx 334 | Replacement of a subexpres... |
xchbinxr 335 | Replacement of a subexpres... |
imbi2i 336 | Introduce an antecedent to... |
bibi2i 337 | Inference adding a bicondi... |
bibi1i 338 | Inference adding a bicondi... |
bibi12i 339 | The equivalence of two equ... |
imbi2d 340 | Deduction adding an antece... |
imbi1d 341 | Deduction adding a consequ... |
bibi2d 342 | Deduction adding a bicondi... |
bibi1d 343 | Deduction adding a bicondi... |
imbi12d 344 | Deduction joining two equi... |
bibi12d 345 | Deduction joining two equi... |
imbi12 346 | Closed form of ~ imbi12i .... |
imbi1 347 | Theorem *4.84 of [Whitehea... |
imbi2 348 | Theorem *4.85 of [Whitehea... |
imbi1i 349 | Introduce a consequent to ... |
imbi12i 350 | Join two logical equivalen... |
bibi1 351 | Theorem *4.86 of [Whitehea... |
bitr3 352 | Closed nested implication ... |
con2bi 353 | Contraposition. Theorem *... |
con2bid 354 | A contraposition deduction... |
con1bid 355 | A contraposition deduction... |
con1bii 356 | A contraposition inference... |
con2bii 357 | A contraposition inference... |
con1b 358 | Contraposition. Bidirecti... |
con2b 359 | Contraposition. Bidirecti... |
biimt 360 | A wff is equivalent to its... |
pm5.5 361 | Theorem *5.5 of [Whitehead... |
a1bi 362 | Inference introducing a th... |
mt2bi 363 | A false consequent falsifi... |
mtt 364 | Modus-tollens-like theorem... |
imnot 365 | If a proposition is false,... |
pm5.501 366 | Theorem *5.501 of [Whitehe... |
ibib 367 | Implication in terms of im... |
ibibr 368 | Implication in terms of im... |
tbt 369 | A wff is equivalent to its... |
nbn2 370 | The negation of a wff is e... |
bibif 371 | Transfer negation via an e... |
nbn 372 | The negation of a wff is e... |
nbn3 373 | Transfer falsehood via equ... |
pm5.21im 374 | Two propositions are equiv... |
2false 375 | Two falsehoods are equival... |
2falsed 376 | Two falsehoods are equival... |
pm5.21ni 377 | Two propositions implying ... |
pm5.21nii 378 | Eliminate an antecedent im... |
pm5.21ndd 379 | Eliminate an antecedent im... |
bija 380 | Combine antecedents into a... |
pm5.18 381 | Theorem *5.18 of [Whitehea... |
xor3 382 | Two ways to express "exclu... |
nbbn 383 | Move negation outside of b... |
biass 384 | Associative law for the bi... |
biluk 385 | Lukasiewicz's shortest axi... |
pm5.19 386 | Theorem *5.19 of [Whitehea... |
bi2.04 387 | Logical equivalence of com... |
pm5.4 388 | Antecedent absorption impl... |
imdi 389 | Distributive law for impli... |
pm5.41 390 | Theorem *5.41 of [Whitehea... |
imbibi 391 | The antecedent of one side... |
pm4.8 392 | Theorem *4.8 of [Whitehead... |
pm4.81 393 | A formula is equivalent to... |
imim21b 394 | Simplify an implication be... |
pm4.63 397 | Theorem *4.63 of [Whitehea... |
pm4.67 398 | Theorem *4.67 of [Whitehea... |
imnan 399 | Express an implication in ... |
imnani 400 | Infer an implication from ... |
iman 401 | Implication in terms of co... |
pm3.24 402 | Law of noncontradiction. ... |
annim 403 | Express a conjunction in t... |
pm4.61 404 | Theorem *4.61 of [Whitehea... |
pm4.65 405 | Theorem *4.65 of [Whitehea... |
imp 406 | Importation inference. (C... |
impcom 407 | Importation inference with... |
con3dimp 408 | Variant of ~ con3d with im... |
mpnanrd 409 | Eliminate the right side o... |
impd 410 | Importation deduction. (C... |
impcomd 411 | Importation deduction with... |
ex 412 | Exportation inference. (T... |
expcom 413 | Exportation inference with... |
expdcom 414 | Commuted form of ~ expd . ... |
expd 415 | Exportation deduction. (C... |
expcomd 416 | Deduction form of ~ expcom... |
imp31 417 | An importation inference. ... |
imp32 418 | An importation inference. ... |
exp31 419 | An exportation inference. ... |
exp32 420 | An exportation inference. ... |
imp4b 421 | An importation inference. ... |
imp4a 422 | An importation inference. ... |
imp4c 423 | An importation inference. ... |
imp4d 424 | An importation inference. ... |
imp41 425 | An importation inference. ... |
imp42 426 | An importation inference. ... |
imp43 427 | An importation inference. ... |
imp44 428 | An importation inference. ... |
imp45 429 | An importation inference. ... |
exp4b 430 | An exportation inference. ... |
exp4a 431 | An exportation inference. ... |
exp4c 432 | An exportation inference. ... |
exp4d 433 | An exportation inference. ... |
exp41 434 | An exportation inference. ... |
exp42 435 | An exportation inference. ... |
exp43 436 | An exportation inference. ... |
exp44 437 | An exportation inference. ... |
exp45 438 | An exportation inference. ... |
imp5d 439 | An importation inference. ... |
imp5a 440 | An importation inference. ... |
imp5g 441 | An importation inference. ... |
imp55 442 | An importation inference. ... |
imp511 443 | An importation inference. ... |
exp5c 444 | An exportation inference. ... |
exp5j 445 | An exportation inference. ... |
exp5l 446 | An exportation inference. ... |
exp53 447 | An exportation inference. ... |
pm3.3 448 | Theorem *3.3 (Exp) of [Whi... |
pm3.31 449 | Theorem *3.31 (Imp) of [Wh... |
impexp 450 | Import-export theorem. Pa... |
impancom 451 | Mixed importation/commutat... |
expdimp 452 | A deduction version of exp... |
expimpd 453 | Exportation followed by a ... |
impr 454 | Import a wff into a right ... |
impl 455 | Export a wff from a left c... |
expr 456 | Export a wff from a right ... |
expl 457 | Export a wff from a left c... |
ancoms 458 | Inference commuting conjun... |
pm3.22 459 | Theorem *3.22 of [Whitehea... |
ancom 460 | Commutative law for conjun... |
ancomd 461 | Commutation of conjuncts i... |
biancomi 462 | Commuting conjunction in a... |
biancomd 463 | Commuting conjunction in a... |
ancomst 464 | Closed form of ~ ancoms . ... |
ancomsd 465 | Deduction commuting conjun... |
anasss 466 | Associative law for conjun... |
anassrs 467 | Associative law for conjun... |
anass 468 | Associative law for conjun... |
pm3.2 469 | Join antecedents with conj... |
pm3.2i 470 | Infer conjunction of premi... |
pm3.21 471 | Join antecedents with conj... |
pm3.43i 472 | Nested conjunction of ante... |
pm3.43 473 | Theorem *3.43 (Comp) of [W... |
dfbi2 474 | A theorem similar to the s... |
dfbi 475 | Definition ~ df-bi rewritt... |
biimpa 476 | Importation inference from... |
biimpar 477 | Importation inference from... |
biimpac 478 | Importation inference from... |
biimparc 479 | Importation inference from... |
adantr 480 | Inference adding a conjunc... |
adantl 481 | Inference adding a conjunc... |
simpl 482 | Elimination of a conjunct.... |
simpli 483 | Inference eliminating a co... |
simpr 484 | Elimination of a conjunct.... |
simpri 485 | Inference eliminating a co... |
intnan 486 | Introduction of conjunct i... |
intnanr 487 | Introduction of conjunct i... |
intnand 488 | Introduction of conjunct i... |
intnanrd 489 | Introduction of conjunct i... |
adantld 490 | Deduction adding a conjunc... |
adantrd 491 | Deduction adding a conjunc... |
pm3.41 492 | Theorem *3.41 of [Whitehea... |
pm3.42 493 | Theorem *3.42 of [Whitehea... |
simpld 494 | Deduction eliminating a co... |
simprd 495 | Deduction eliminating a co... |
simprbi 496 | Deduction eliminating a co... |
simplbi 497 | Deduction eliminating a co... |
simprbda 498 | Deduction eliminating a co... |
simplbda 499 | Deduction eliminating a co... |
simplbi2 500 | Deduction eliminating a co... |
simplbi2comt 501 | Closed form of ~ simplbi2c... |
simplbi2com 502 | A deduction eliminating a ... |
simpl2im 503 | Implication from an elimin... |
simplbiim 504 | Implication from an elimin... |
impel 505 | An inference for implicati... |
mpan9 506 | Modus ponens conjoining di... |
sylan9 507 | Nested syllogism inference... |
sylan9r 508 | Nested syllogism inference... |
sylan9bb 509 | Nested syllogism inference... |
sylan9bbr 510 | Nested syllogism inference... |
jca 511 | Deduce conjunction of the ... |
jcad 512 | Deduction conjoining the c... |
jca2 513 | Inference conjoining the c... |
jca31 514 | Join three consequents. (... |
jca32 515 | Join three consequents. (... |
jcai 516 | Deduction replacing implic... |
jcab 517 | Distributive law for impli... |
pm4.76 518 | Theorem *4.76 of [Whitehea... |
jctil 519 | Inference conjoining a the... |
jctir 520 | Inference conjoining a the... |
jccir 521 | Inference conjoining a con... |
jccil 522 | Inference conjoining a con... |
jctl 523 | Inference conjoining a the... |
jctr 524 | Inference conjoining a the... |
jctild 525 | Deduction conjoining a the... |
jctird 526 | Deduction conjoining a the... |
iba 527 | Introduction of antecedent... |
ibar 528 | Introduction of antecedent... |
biantru 529 | A wff is equivalent to its... |
biantrur 530 | A wff is equivalent to its... |
biantrud 531 | A wff is equivalent to its... |
biantrurd 532 | A wff is equivalent to its... |
bianfi 533 | A wff conjoined with false... |
bianfd 534 | A wff conjoined with false... |
baib 535 | Move conjunction outside o... |
baibr 536 | Move conjunction outside o... |
rbaibr 537 | Move conjunction outside o... |
rbaib 538 | Move conjunction outside o... |
baibd 539 | Move conjunction outside o... |
rbaibd 540 | Move conjunction outside o... |
bianabs 541 | Absorb a hypothesis into t... |
pm5.44 542 | Theorem *5.44 of [Whitehea... |
pm5.42 543 | Theorem *5.42 of [Whitehea... |
ancl 544 | Conjoin antecedent to left... |
anclb 545 | Conjoin antecedent to left... |
ancr 546 | Conjoin antecedent to righ... |
ancrb 547 | Conjoin antecedent to righ... |
ancli 548 | Deduction conjoining antec... |
ancri 549 | Deduction conjoining antec... |
ancld 550 | Deduction conjoining antec... |
ancrd 551 | Deduction conjoining antec... |
impac 552 | Importation with conjuncti... |
anc2l 553 | Conjoin antecedent to left... |
anc2r 554 | Conjoin antecedent to righ... |
anc2li 555 | Deduction conjoining antec... |
anc2ri 556 | Deduction conjoining antec... |
pm4.71 557 | Implication in terms of bi... |
pm4.71r 558 | Implication in terms of bi... |
pm4.71i 559 | Inference converting an im... |
pm4.71ri 560 | Inference converting an im... |
pm4.71d 561 | Deduction converting an im... |
pm4.71rd 562 | Deduction converting an im... |
pm4.24 563 | Theorem *4.24 of [Whitehea... |
anidm 564 | Idempotent law for conjunc... |
anidmdbi 565 | Conjunction idempotence wi... |
anidms 566 | Inference from idempotent ... |
imdistan 567 | Distribution of implicatio... |
imdistani 568 | Distribution of implicatio... |
imdistanri 569 | Distribution of implicatio... |
imdistand 570 | Distribution of implicatio... |
imdistanda 571 | Distribution of implicatio... |
pm5.3 572 | Theorem *5.3 of [Whitehead... |
pm5.32 573 | Distribution of implicatio... |
pm5.32i 574 | Distribution of implicatio... |
pm5.32ri 575 | Distribution of implicatio... |
pm5.32d 576 | Distribution of implicatio... |
pm5.32rd 577 | Distribution of implicatio... |
pm5.32da 578 | Distribution of implicatio... |
sylan 579 | A syllogism inference. (C... |
sylanb 580 | A syllogism inference. (C... |
sylanbr 581 | A syllogism inference. (C... |
sylanbrc 582 | Syllogism inference. (Con... |
syl2anc 583 | Syllogism inference combin... |
syl2anc2 584 | Double syllogism inference... |
sylancl 585 | Syllogism inference combin... |
sylancr 586 | Syllogism inference combin... |
sylancom 587 | Syllogism inference with c... |
sylanblc 588 | Syllogism inference combin... |
sylanblrc 589 | Syllogism inference combin... |
syldan 590 | A syllogism deduction with... |
sylbida 591 | A syllogism deduction. (C... |
sylan2 592 | A syllogism inference. (C... |
sylan2b 593 | A syllogism inference. (C... |
sylan2br 594 | A syllogism inference. (C... |
syl2an 595 | A double syllogism inferen... |
syl2anr 596 | A double syllogism inferen... |
syl2anb 597 | A double syllogism inferen... |
syl2anbr 598 | A double syllogism inferen... |
sylancb 599 | A syllogism inference comb... |
sylancbr 600 | A syllogism inference comb... |
syldanl 601 | A syllogism deduction with... |
syland 602 | A syllogism deduction. (C... |
sylani 603 | A syllogism inference. (C... |
sylan2d 604 | A syllogism deduction. (C... |
sylan2i 605 | A syllogism inference. (C... |
syl2ani 606 | A syllogism inference. (C... |
syl2and 607 | A syllogism deduction. (C... |
anim12d 608 | Conjoin antecedents and co... |
anim12d1 609 | Variant of ~ anim12d where... |
anim1d 610 | Add a conjunct to right of... |
anim2d 611 | Add a conjunct to left of ... |
anim12i 612 | Conjoin antecedents and co... |
anim12ci 613 | Variant of ~ anim12i with ... |
anim1i 614 | Introduce conjunct to both... |
anim1ci 615 | Introduce conjunct to both... |
anim2i 616 | Introduce conjunct to both... |
anim12ii 617 | Conjoin antecedents and co... |
anim12dan 618 | Conjoin antecedents and co... |
im2anan9 619 | Deduction joining nested i... |
im2anan9r 620 | Deduction joining nested i... |
pm3.45 621 | Theorem *3.45 (Fact) of [W... |
anbi2i 622 | Introduce a left conjunct ... |
anbi1i 623 | Introduce a right conjunct... |
anbi2ci 624 | Variant of ~ anbi2i with c... |
anbi1ci 625 | Variant of ~ anbi1i with c... |
anbi12i 626 | Conjoin both sides of two ... |
anbi12ci 627 | Variant of ~ anbi12i with ... |
anbi2d 628 | Deduction adding a left co... |
anbi1d 629 | Deduction adding a right c... |
anbi12d 630 | Deduction joining two equi... |
anbi1 631 | Introduce a right conjunct... |
anbi2 632 | Introduce a left conjunct ... |
anbi1cd 633 | Introduce a proposition as... |
an2anr 634 | Double commutation in conj... |
pm4.38 635 | Theorem *4.38 of [Whitehea... |
bi2anan9 636 | Deduction joining two equi... |
bi2anan9r 637 | Deduction joining two equi... |
bi2bian9 638 | Deduction joining two bico... |
bianass 639 | An inference to merge two ... |
bianassc 640 | An inference to merge two ... |
an21 641 | Swap two conjuncts. (Cont... |
an12 642 | Swap two conjuncts. Note ... |
an32 643 | A rearrangement of conjunc... |
an13 644 | A rearrangement of conjunc... |
an31 645 | A rearrangement of conjunc... |
an12s 646 | Swap two conjuncts in ante... |
ancom2s 647 | Inference commuting a nest... |
an13s 648 | Swap two conjuncts in ante... |
an32s 649 | Swap two conjuncts in ante... |
ancom1s 650 | Inference commuting a nest... |
an31s 651 | Swap two conjuncts in ante... |
anass1rs 652 | Commutative-associative la... |
an4 653 | Rearrangement of 4 conjunc... |
an42 654 | Rearrangement of 4 conjunc... |
an43 655 | Rearrangement of 4 conjunc... |
an3 656 | A rearrangement of conjunc... |
an4s 657 | Inference rearranging 4 co... |
an42s 658 | Inference rearranging 4 co... |
anabs1 659 | Absorption into embedded c... |
anabs5 660 | Absorption into embedded c... |
anabs7 661 | Absorption into embedded c... |
anabsan 662 | Absorption of antecedent w... |
anabss1 663 | Absorption of antecedent i... |
anabss4 664 | Absorption of antecedent i... |
anabss5 665 | Absorption of antecedent i... |
anabsi5 666 | Absorption of antecedent i... |
anabsi6 667 | Absorption of antecedent i... |
anabsi7 668 | Absorption of antecedent i... |
anabsi8 669 | Absorption of antecedent i... |
anabss7 670 | Absorption of antecedent i... |
anabsan2 671 | Absorption of antecedent w... |
anabss3 672 | Absorption of antecedent i... |
anandi 673 | Distribution of conjunctio... |
anandir 674 | Distribution of conjunctio... |
anandis 675 | Inference that undistribut... |
anandirs 676 | Inference that undistribut... |
sylanl1 677 | A syllogism inference. (C... |
sylanl2 678 | A syllogism inference. (C... |
sylanr1 679 | A syllogism inference. (C... |
sylanr2 680 | A syllogism inference. (C... |
syl6an 681 | A syllogism deduction comb... |
syl2an2r 682 | ~ syl2anr with antecedents... |
syl2an2 683 | ~ syl2an with antecedents ... |
mpdan 684 | An inference based on modu... |
mpancom 685 | An inference based on modu... |
mpidan 686 | A deduction which "stacks"... |
mpan 687 | An inference based on modu... |
mpan2 688 | An inference based on modu... |
mp2an 689 | An inference based on modu... |
mp4an 690 | An inference based on modu... |
mpan2d 691 | A deduction based on modus... |
mpand 692 | A deduction based on modus... |
mpani 693 | An inference based on modu... |
mpan2i 694 | An inference based on modu... |
mp2ani 695 | An inference based on modu... |
mp2and 696 | A deduction based on modus... |
mpanl1 697 | An inference based on modu... |
mpanl2 698 | An inference based on modu... |
mpanl12 699 | An inference based on modu... |
mpanr1 700 | An inference based on modu... |
mpanr2 701 | An inference based on modu... |
mpanr12 702 | An inference based on modu... |
mpanlr1 703 | An inference based on modu... |
mpbirand 704 | Detach truth from conjunct... |
mpbiran2d 705 | Detach truth from conjunct... |
mpbiran 706 | Detach truth from conjunct... |
mpbiran2 707 | Detach truth from conjunct... |
mpbir2an 708 | Detach a conjunction of tr... |
mpbi2and 709 | Detach a conjunction of tr... |
mpbir2and 710 | Detach a conjunction of tr... |
adantll 711 | Deduction adding a conjunc... |
adantlr 712 | Deduction adding a conjunc... |
adantrl 713 | Deduction adding a conjunc... |
adantrr 714 | Deduction adding a conjunc... |
adantlll 715 | Deduction adding a conjunc... |
adantllr 716 | Deduction adding a conjunc... |
adantlrl 717 | Deduction adding a conjunc... |
adantlrr 718 | Deduction adding a conjunc... |
adantrll 719 | Deduction adding a conjunc... |
adantrlr 720 | Deduction adding a conjunc... |
adantrrl 721 | Deduction adding a conjunc... |
adantrrr 722 | Deduction adding a conjunc... |
ad2antrr 723 | Deduction adding two conju... |
ad2antlr 724 | Deduction adding two conju... |
ad2antrl 725 | Deduction adding two conju... |
ad2antll 726 | Deduction adding conjuncts... |
ad3antrrr 727 | Deduction adding three con... |
ad3antlr 728 | Deduction adding three con... |
ad4antr 729 | Deduction adding 4 conjunc... |
ad4antlr 730 | Deduction adding 4 conjunc... |
ad5antr 731 | Deduction adding 5 conjunc... |
ad5antlr 732 | Deduction adding 5 conjunc... |
ad6antr 733 | Deduction adding 6 conjunc... |
ad6antlr 734 | Deduction adding 6 conjunc... |
ad7antr 735 | Deduction adding 7 conjunc... |
ad7antlr 736 | Deduction adding 7 conjunc... |
ad8antr 737 | Deduction adding 8 conjunc... |
ad8antlr 738 | Deduction adding 8 conjunc... |
ad9antr 739 | Deduction adding 9 conjunc... |
ad9antlr 740 | Deduction adding 9 conjunc... |
ad10antr 741 | Deduction adding 10 conjun... |
ad10antlr 742 | Deduction adding 10 conjun... |
ad2ant2l 743 | Deduction adding two conju... |
ad2ant2r 744 | Deduction adding two conju... |
ad2ant2lr 745 | Deduction adding two conju... |
ad2ant2rl 746 | Deduction adding two conju... |
adantl3r 747 | Deduction adding 1 conjunc... |
ad4ant13 748 | Deduction adding conjuncts... |
ad4ant14 749 | Deduction adding conjuncts... |
ad4ant23 750 | Deduction adding conjuncts... |
ad4ant24 751 | Deduction adding conjuncts... |
adantl4r 752 | Deduction adding 1 conjunc... |
ad5ant12 753 | Deduction adding conjuncts... |
ad5ant13 754 | Deduction adding conjuncts... |
ad5ant14 755 | Deduction adding conjuncts... |
ad5ant15 756 | Deduction adding conjuncts... |
ad5ant23 757 | Deduction adding conjuncts... |
ad5ant24 758 | Deduction adding conjuncts... |
ad5ant25 759 | Deduction adding conjuncts... |
adantl5r 760 | Deduction adding 1 conjunc... |
adantl6r 761 | Deduction adding 1 conjunc... |
pm3.33 762 | Theorem *3.33 (Syll) of [W... |
pm3.34 763 | Theorem *3.34 (Syll) of [W... |
simpll 764 | Simplification of a conjun... |
simplld 765 | Deduction form of ~ simpll... |
simplr 766 | Simplification of a conjun... |
simplrd 767 | Deduction eliminating a do... |
simprl 768 | Simplification of a conjun... |
simprld 769 | Deduction eliminating a do... |
simprr 770 | Simplification of a conjun... |
simprrd 771 | Deduction form of ~ simprr... |
simplll 772 | Simplification of a conjun... |
simpllr 773 | Simplification of a conjun... |
simplrl 774 | Simplification of a conjun... |
simplrr 775 | Simplification of a conjun... |
simprll 776 | Simplification of a conjun... |
simprlr 777 | Simplification of a conjun... |
simprrl 778 | Simplification of a conjun... |
simprrr 779 | Simplification of a conjun... |
simp-4l 780 | Simplification of a conjun... |
simp-4r 781 | Simplification of a conjun... |
simp-5l 782 | Simplification of a conjun... |
simp-5r 783 | Simplification of a conjun... |
simp-6l 784 | Simplification of a conjun... |
simp-6r 785 | Simplification of a conjun... |
simp-7l 786 | Simplification of a conjun... |
simp-7r 787 | Simplification of a conjun... |
simp-8l 788 | Simplification of a conjun... |
simp-8r 789 | Simplification of a conjun... |
simp-9l 790 | Simplification of a conjun... |
simp-9r 791 | Simplification of a conjun... |
simp-10l 792 | Simplification of a conjun... |
simp-10r 793 | Simplification of a conjun... |
simp-11l 794 | Simplification of a conjun... |
simp-11r 795 | Simplification of a conjun... |
pm2.01da 796 | Deduction based on reducti... |
pm2.18da 797 | Deduction based on reducti... |
impbida 798 | Deduce an equivalence from... |
pm5.21nd 799 | Eliminate an antecedent im... |
pm3.35 800 | Conjunctive detachment. T... |
pm5.74da 801 | Distribution of implicatio... |
bitr 802 | Theorem *4.22 of [Whitehea... |
biantr 803 | A transitive law of equiva... |
pm4.14 804 | Theorem *4.14 of [Whitehea... |
pm3.37 805 | Theorem *3.37 (Transp) of ... |
anim12 806 | Conjoin antecedents and co... |
pm3.4 807 | Conjunction implies implic... |
exbiri 808 | Inference form of ~ exbir ... |
pm2.61ian 809 | Elimination of an antecede... |
pm2.61dan 810 | Elimination of an antecede... |
pm2.61ddan 811 | Elimination of two anteced... |
pm2.61dda 812 | Elimination of two anteced... |
mtand 813 | A modus tollens deduction.... |
pm2.65da 814 | Deduction for proof by con... |
condan 815 | Proof by contradiction. (... |
biadan 816 | An implication is equivale... |
biadani 817 | Inference associated with ... |
biadaniALT 818 | Alternate proof of ~ biada... |
biadanii 819 | Inference associated with ... |
biadanid 820 | Deduction associated with ... |
pm5.1 821 | Two propositions are equiv... |
pm5.21 822 | Two propositions are equiv... |
pm5.35 823 | Theorem *5.35 of [Whitehea... |
abai 824 | Introduce one conjunct as ... |
pm4.45im 825 | Conjunction with implicati... |
impimprbi 826 | An implication and its rev... |
nan 827 | Theorem to move a conjunct... |
pm5.31 828 | Theorem *5.31 of [Whitehea... |
pm5.31r 829 | Variant of ~ pm5.31 . (Co... |
pm4.15 830 | Theorem *4.15 of [Whitehea... |
pm5.36 831 | Theorem *5.36 of [Whitehea... |
annotanannot 832 | A conjunction with a negat... |
pm5.33 833 | Theorem *5.33 of [Whitehea... |
syl12anc 834 | Syllogism combined with co... |
syl21anc 835 | Syllogism combined with co... |
syl22anc 836 | Syllogism combined with co... |
syl1111anc 837 | Four-hypothesis eliminatio... |
syldbl2 838 | Stacked hypotheseis implie... |
mpsyl4anc 839 | An elimination deduction. ... |
pm4.87 840 | Theorem *4.87 of [Whitehea... |
bimsc1 841 | Removal of conjunct from o... |
a2and 842 | Deduction distributing a c... |
animpimp2impd 843 | Deduction deriving nested ... |
pm4.64 846 | Theorem *4.64 of [Whitehea... |
pm4.66 847 | Theorem *4.66 of [Whitehea... |
pm2.53 848 | Theorem *2.53 of [Whitehea... |
pm2.54 849 | Theorem *2.54 of [Whitehea... |
imor 850 | Implication in terms of di... |
imori 851 | Infer disjunction from imp... |
imorri 852 | Infer implication from dis... |
pm4.62 853 | Theorem *4.62 of [Whitehea... |
jaoi 854 | Inference disjoining the a... |
jao1i 855 | Add a disjunct in the ante... |
jaod 856 | Deduction disjoining the a... |
mpjaod 857 | Eliminate a disjunction in... |
ori 858 | Infer implication from dis... |
orri 859 | Infer disjunction from imp... |
orrd 860 | Deduce disjunction from im... |
ord 861 | Deduce implication from di... |
orci 862 | Deduction introducing a di... |
olci 863 | Deduction introducing a di... |
orc 864 | Introduction of a disjunct... |
olc 865 | Introduction of a disjunct... |
pm1.4 866 | Axiom *1.4 of [WhiteheadRu... |
orcom 867 | Commutative law for disjun... |
orcomd 868 | Commutation of disjuncts i... |
orcoms 869 | Commutation of disjuncts i... |
orcd 870 | Deduction introducing a di... |
olcd 871 | Deduction introducing a di... |
orcs 872 | Deduction eliminating disj... |
olcs 873 | Deduction eliminating disj... |
olcnd 874 | A lemma for Conjunctive No... |
orcnd 875 | A lemma for Conjunctive No... |
mtord 876 | A modus tollens deduction ... |
pm3.2ni 877 | Infer negated disjunction ... |
pm2.45 878 | Theorem *2.45 of [Whitehea... |
pm2.46 879 | Theorem *2.46 of [Whitehea... |
pm2.47 880 | Theorem *2.47 of [Whitehea... |
pm2.48 881 | Theorem *2.48 of [Whitehea... |
pm2.49 882 | Theorem *2.49 of [Whitehea... |
norbi 883 | If neither of two proposit... |
nbior 884 | If two propositions are no... |
orel1 885 | Elimination of disjunction... |
pm2.25 886 | Theorem *2.25 of [Whitehea... |
orel2 887 | Elimination of disjunction... |
pm2.67-2 888 | Slight generalization of T... |
pm2.67 889 | Theorem *2.67 of [Whitehea... |
curryax 890 | A non-intuitionistic posit... |
exmid 891 | Law of excluded middle, al... |
exmidd 892 | Law of excluded middle in ... |
pm2.1 893 | Theorem *2.1 of [Whitehead... |
pm2.13 894 | Theorem *2.13 of [Whitehea... |
pm2.621 895 | Theorem *2.621 of [Whitehe... |
pm2.62 896 | Theorem *2.62 of [Whitehea... |
pm2.68 897 | Theorem *2.68 of [Whitehea... |
dfor2 898 | Logical 'or' expressed in ... |
pm2.07 899 | Theorem *2.07 of [Whitehea... |
pm1.2 900 | Axiom *1.2 of [WhiteheadRu... |
oridm 901 | Idempotent law for disjunc... |
pm4.25 902 | Theorem *4.25 of [Whitehea... |
pm2.4 903 | Theorem *2.4 of [Whitehead... |
pm2.41 904 | Theorem *2.41 of [Whitehea... |
orim12i 905 | Disjoin antecedents and co... |
orim1i 906 | Introduce disjunct to both... |
orim2i 907 | Introduce disjunct to both... |
orim12dALT 908 | Alternate proof of ~ orim1... |
orbi2i 909 | Inference adding a left di... |
orbi1i 910 | Inference adding a right d... |
orbi12i 911 | Infer the disjunction of t... |
orbi2d 912 | Deduction adding a left di... |
orbi1d 913 | Deduction adding a right d... |
orbi1 914 | Theorem *4.37 of [Whitehea... |
orbi12d 915 | Deduction joining two equi... |
pm1.5 916 | Axiom *1.5 (Assoc) of [Whi... |
or12 917 | Swap two disjuncts. (Cont... |
orass 918 | Associative law for disjun... |
pm2.31 919 | Theorem *2.31 of [Whitehea... |
pm2.32 920 | Theorem *2.32 of [Whitehea... |
pm2.3 921 | Theorem *2.3 of [Whitehead... |
or32 922 | A rearrangement of disjunc... |
or4 923 | Rearrangement of 4 disjunc... |
or42 924 | Rearrangement of 4 disjunc... |
orordi 925 | Distribution of disjunctio... |
orordir 926 | Distribution of disjunctio... |
orimdi 927 | Disjunction distributes ov... |
pm2.76 928 | Theorem *2.76 of [Whitehea... |
pm2.85 929 | Theorem *2.85 of [Whitehea... |
pm2.75 930 | Theorem *2.75 of [Whitehea... |
pm4.78 931 | Implication distributes ov... |
biort 932 | A disjunction with a true ... |
biorf 933 | A wff is equivalent to its... |
biortn 934 | A wff is equivalent to its... |
biorfi 935 | A wff is equivalent to its... |
pm2.26 936 | Theorem *2.26 of [Whitehea... |
pm2.63 937 | Theorem *2.63 of [Whitehea... |
pm2.64 938 | Theorem *2.64 of [Whitehea... |
pm2.42 939 | Theorem *2.42 of [Whitehea... |
pm5.11g 940 | A general instance of Theo... |
pm5.11 941 | Theorem *5.11 of [Whitehea... |
pm5.12 942 | Theorem *5.12 of [Whitehea... |
pm5.14 943 | Theorem *5.14 of [Whitehea... |
pm5.13 944 | Theorem *5.13 of [Whitehea... |
pm5.55 945 | Theorem *5.55 of [Whitehea... |
pm4.72 946 | Implication in terms of bi... |
imimorb 947 | Simplify an implication be... |
oibabs 948 | Absorption of disjunction ... |
orbidi 949 | Disjunction distributes ov... |
pm5.7 950 | Disjunction distributes ov... |
jaao 951 | Inference conjoining and d... |
jaoa 952 | Inference disjoining and c... |
jaoian 953 | Inference disjoining the a... |
jaodan 954 | Deduction disjoining the a... |
mpjaodan 955 | Eliminate a disjunction in... |
pm3.44 956 | Theorem *3.44 of [Whitehea... |
jao 957 | Disjunction of antecedents... |
jaob 958 | Disjunction of antecedents... |
pm4.77 959 | Theorem *4.77 of [Whitehea... |
pm3.48 960 | Theorem *3.48 of [Whitehea... |
orim12d 961 | Disjoin antecedents and co... |
orim1d 962 | Disjoin antecedents and co... |
orim2d 963 | Disjoin antecedents and co... |
orim2 964 | Axiom *1.6 (Sum) of [White... |
pm2.38 965 | Theorem *2.38 of [Whitehea... |
pm2.36 966 | Theorem *2.36 of [Whitehea... |
pm2.37 967 | Theorem *2.37 of [Whitehea... |
pm2.81 968 | Theorem *2.81 of [Whitehea... |
pm2.8 969 | Theorem *2.8 of [Whitehead... |
pm2.73 970 | Theorem *2.73 of [Whitehea... |
pm2.74 971 | Theorem *2.74 of [Whitehea... |
pm2.82 972 | Theorem *2.82 of [Whitehea... |
pm4.39 973 | Theorem *4.39 of [Whitehea... |
animorl 974 | Conjunction implies disjun... |
animorr 975 | Conjunction implies disjun... |
animorlr 976 | Conjunction implies disjun... |
animorrl 977 | Conjunction implies disjun... |
ianor 978 | Negated conjunction in ter... |
anor 979 | Conjunction in terms of di... |
ioran 980 | Negated disjunction in ter... |
pm4.52 981 | Theorem *4.52 of [Whitehea... |
pm4.53 982 | Theorem *4.53 of [Whitehea... |
pm4.54 983 | Theorem *4.54 of [Whitehea... |
pm4.55 984 | Theorem *4.55 of [Whitehea... |
pm4.56 985 | Theorem *4.56 of [Whitehea... |
oran 986 | Disjunction in terms of co... |
pm4.57 987 | Theorem *4.57 of [Whitehea... |
pm3.1 988 | Theorem *3.1 of [Whitehead... |
pm3.11 989 | Theorem *3.11 of [Whitehea... |
pm3.12 990 | Theorem *3.12 of [Whitehea... |
pm3.13 991 | Theorem *3.13 of [Whitehea... |
pm3.14 992 | Theorem *3.14 of [Whitehea... |
pm4.44 993 | Theorem *4.44 of [Whitehea... |
pm4.45 994 | Theorem *4.45 of [Whitehea... |
orabs 995 | Absorption of redundant in... |
oranabs 996 | Absorb a disjunct into a c... |
pm5.61 997 | Theorem *5.61 of [Whitehea... |
pm5.6 998 | Conjunction in antecedent ... |
orcanai 999 | Change disjunction in cons... |
pm4.79 1000 | Theorem *4.79 of [Whitehea... |
pm5.53 1001 | Theorem *5.53 of [Whitehea... |
ordi 1002 | Distributive law for disju... |
ordir 1003 | Distributive law for disju... |
andi 1004 | Distributive law for conju... |
andir 1005 | Distributive law for conju... |
orddi 1006 | Double distributive law fo... |
anddi 1007 | Double distributive law fo... |
pm5.17 1008 | Theorem *5.17 of [Whitehea... |
pm5.15 1009 | Theorem *5.15 of [Whitehea... |
pm5.16 1010 | Theorem *5.16 of [Whitehea... |
xor 1011 | Two ways to express exclus... |
nbi2 1012 | Two ways to express "exclu... |
xordi 1013 | Conjunction distributes ov... |
pm5.54 1014 | Theorem *5.54 of [Whitehea... |
pm5.62 1015 | Theorem *5.62 of [Whitehea... |
pm5.63 1016 | Theorem *5.63 of [Whitehea... |
niabn 1017 | Miscellaneous inference re... |
ninba 1018 | Miscellaneous inference re... |
pm4.43 1019 | Theorem *4.43 of [Whitehea... |
pm4.82 1020 | Theorem *4.82 of [Whitehea... |
pm4.83 1021 | Theorem *4.83 of [Whitehea... |
pclem6 1022 | Negation inferred from emb... |
bigolden 1023 | Dijkstra-Scholten's Golden... |
pm5.71 1024 | Theorem *5.71 of [Whitehea... |
pm5.75 1025 | Theorem *5.75 of [Whitehea... |
ecase2d 1026 | Deduction for elimination ... |
ecase2dOLD 1027 | Obsolete version of ~ ecas... |
ecase3 1028 | Inference for elimination ... |
ecase 1029 | Inference for elimination ... |
ecase3d 1030 | Deduction for elimination ... |
ecased 1031 | Deduction for elimination ... |
ecase3ad 1032 | Deduction for elimination ... |
ecase3adOLD 1033 | Obsolete version of ~ ecas... |
ccase 1034 | Inference for combining ca... |
ccased 1035 | Deduction for combining ca... |
ccase2 1036 | Inference for combining ca... |
4cases 1037 | Inference eliminating two ... |
4casesdan 1038 | Deduction eliminating two ... |
cases 1039 | Case disjunction according... |
dedlem0a 1040 | Lemma for an alternate ver... |
dedlem0b 1041 | Lemma for an alternate ver... |
dedlema 1042 | Lemma for weak deduction t... |
dedlemb 1043 | Lemma for weak deduction t... |
cases2 1044 | Case disjunction according... |
cases2ALT 1045 | Alternate proof of ~ cases... |
dfbi3 1046 | An alternate definition of... |
pm5.24 1047 | Theorem *5.24 of [Whitehea... |
4exmid 1048 | The disjunction of the fou... |
consensus 1049 | The consensus theorem. Th... |
pm4.42 1050 | Theorem *4.42 of [Whitehea... |
prlem1 1051 | A specialized lemma for se... |
prlem2 1052 | A specialized lemma for se... |
oplem1 1053 | A specialized lemma for se... |
dn1 1054 | A single axiom for Boolean... |
bianir 1055 | A closed form of ~ mpbir ,... |
jaoi2 1056 | Inference removing a negat... |
jaoi3 1057 | Inference separating a dis... |
ornld 1058 | Selecting one statement fr... |
dfifp2 1061 | Alternate definition of th... |
dfifp3 1062 | Alternate definition of th... |
dfifp4 1063 | Alternate definition of th... |
dfifp5 1064 | Alternate definition of th... |
dfifp6 1065 | Alternate definition of th... |
dfifp7 1066 | Alternate definition of th... |
ifpdfbi 1067 | Define the biconditional a... |
anifp 1068 | The conditional operator i... |
ifpor 1069 | The conditional operator i... |
ifpn 1070 | Conditional operator for t... |
ifpnOLD 1071 | Obsolete version of ~ ifpn... |
ifptru 1072 | Value of the conditional o... |
ifpfal 1073 | Value of the conditional o... |
ifpid 1074 | Value of the conditional o... |
casesifp 1075 | Version of ~ cases express... |
ifpbi123d 1076 | Equivalence deduction for ... |
ifpbi23d 1077 | Equivalence deduction for ... |
ifpimpda 1078 | Separation of the values o... |
1fpid3 1079 | The value of the condition... |
elimh 1080 | Hypothesis builder for the... |
dedt 1081 | The weak deduction theorem... |
con3ALT 1082 | Proof of ~ con3 from its a... |
3orass 1087 | Associative law for triple... |
3orel1 1088 | Partial elimination of a t... |
3orrot 1089 | Rotation law for triple di... |
3orcoma 1090 | Commutation law for triple... |
3orcomb 1091 | Commutation law for triple... |
3anass 1092 | Associative law for triple... |
3anan12 1093 | Convert triple conjunction... |
3anan32 1094 | Convert triple conjunction... |
3ancoma 1095 | Commutation law for triple... |
3ancomb 1096 | Commutation law for triple... |
3anrot 1097 | Rotation law for triple co... |
3anrev 1098 | Reversal law for triple co... |
anandi3 1099 | Distribution of triple con... |
anandi3r 1100 | Distribution of triple con... |
3anidm 1101 | Idempotent law for conjunc... |
3an4anass 1102 | Associative law for four c... |
3ioran 1103 | Negated triple disjunction... |
3ianor 1104 | Negated triple conjunction... |
3anor 1105 | Triple conjunction express... |
3oran 1106 | Triple disjunction in term... |
3impa 1107 | Importation from double to... |
3imp 1108 | Importation inference. (C... |
3imp31 1109 | The importation inference ... |
3imp231 1110 | Importation inference. (C... |
3imp21 1111 | The importation inference ... |
3impb 1112 | Importation from double to... |
3impib 1113 | Importation to triple conj... |
3impia 1114 | Importation to triple conj... |
3expa 1115 | Exportation from triple to... |
3exp 1116 | Exportation inference. (C... |
3expb 1117 | Exportation from triple to... |
3expia 1118 | Exportation from triple co... |
3expib 1119 | Exportation from triple co... |
3com12 1120 | Commutation in antecedent.... |
3com13 1121 | Commutation in antecedent.... |
3comr 1122 | Commutation in antecedent.... |
3com23 1123 | Commutation in antecedent.... |
3coml 1124 | Commutation in antecedent.... |
3jca 1125 | Join consequents with conj... |
3jcad 1126 | Deduction conjoining the c... |
3adant1 1127 | Deduction adding a conjunc... |
3adant2 1128 | Deduction adding a conjunc... |
3adant3 1129 | Deduction adding a conjunc... |
3ad2ant1 1130 | Deduction adding conjuncts... |
3ad2ant2 1131 | Deduction adding conjuncts... |
3ad2ant3 1132 | Deduction adding conjuncts... |
simp1 1133 | Simplification of triple c... |
simp2 1134 | Simplification of triple c... |
simp3 1135 | Simplification of triple c... |
simp1i 1136 | Infer a conjunct from a tr... |
simp2i 1137 | Infer a conjunct from a tr... |
simp3i 1138 | Infer a conjunct from a tr... |
simp1d 1139 | Deduce a conjunct from a t... |
simp2d 1140 | Deduce a conjunct from a t... |
simp3d 1141 | Deduce a conjunct from a t... |
simp1bi 1142 | Deduce a conjunct from a t... |
simp2bi 1143 | Deduce a conjunct from a t... |
simp3bi 1144 | Deduce a conjunct from a t... |
3simpa 1145 | Simplification of triple c... |
3simpb 1146 | Simplification of triple c... |
3simpc 1147 | Simplification of triple c... |
3anim123i 1148 | Join antecedents and conse... |
3anim1i 1149 | Add two conjuncts to antec... |
3anim2i 1150 | Add two conjuncts to antec... |
3anim3i 1151 | Add two conjuncts to antec... |
3anbi123i 1152 | Join 3 biconditionals with... |
3orbi123i 1153 | Join 3 biconditionals with... |
3anbi1i 1154 | Inference adding two conju... |
3anbi2i 1155 | Inference adding two conju... |
3anbi3i 1156 | Inference adding two conju... |
syl3an 1157 | A triple syllogism inferen... |
syl3anb 1158 | A triple syllogism inferen... |
syl3anbr 1159 | A triple syllogism inferen... |
syl3an1 1160 | A syllogism inference. (C... |
syl3an2 1161 | A syllogism inference. (C... |
syl3an3 1162 | A syllogism inference. (C... |
3adantl1 1163 | Deduction adding a conjunc... |
3adantl2 1164 | Deduction adding a conjunc... |
3adantl3 1165 | Deduction adding a conjunc... |
3adantr1 1166 | Deduction adding a conjunc... |
3adantr2 1167 | Deduction adding a conjunc... |
3adantr3 1168 | Deduction adding a conjunc... |
ad4ant123 1169 | Deduction adding conjuncts... |
ad4ant124 1170 | Deduction adding conjuncts... |
ad4ant134 1171 | Deduction adding conjuncts... |
ad4ant234 1172 | Deduction adding conjuncts... |
3adant1l 1173 | Deduction adding a conjunc... |
3adant1r 1174 | Deduction adding a conjunc... |
3adant2l 1175 | Deduction adding a conjunc... |
3adant2r 1176 | Deduction adding a conjunc... |
3adant3l 1177 | Deduction adding a conjunc... |
3adant3r 1178 | Deduction adding a conjunc... |
3adant3r1 1179 | Deduction adding a conjunc... |
3adant3r2 1180 | Deduction adding a conjunc... |
3adant3r3 1181 | Deduction adding a conjunc... |
3ad2antl1 1182 | Deduction adding conjuncts... |
3ad2antl2 1183 | Deduction adding conjuncts... |
3ad2antl3 1184 | Deduction adding conjuncts... |
3ad2antr1 1185 | Deduction adding conjuncts... |
3ad2antr2 1186 | Deduction adding conjuncts... |
3ad2antr3 1187 | Deduction adding conjuncts... |
simpl1 1188 | Simplification of conjunct... |
simpl2 1189 | Simplification of conjunct... |
simpl3 1190 | Simplification of conjunct... |
simpr1 1191 | Simplification of conjunct... |
simpr2 1192 | Simplification of conjunct... |
simpr3 1193 | Simplification of conjunct... |
simp1l 1194 | Simplification of triple c... |
simp1r 1195 | Simplification of triple c... |
simp2l 1196 | Simplification of triple c... |
simp2r 1197 | Simplification of triple c... |
simp3l 1198 | Simplification of triple c... |
simp3r 1199 | Simplification of triple c... |
simp11 1200 | Simplification of doubly t... |
simp12 1201 | Simplification of doubly t... |
simp13 1202 | Simplification of doubly t... |
simp21 1203 | Simplification of doubly t... |
simp22 1204 | Simplification of doubly t... |
simp23 1205 | Simplification of doubly t... |
simp31 1206 | Simplification of doubly t... |
simp32 1207 | Simplification of doubly t... |
simp33 1208 | Simplification of doubly t... |
simpll1 1209 | Simplification of conjunct... |
simpll2 1210 | Simplification of conjunct... |
simpll3 1211 | Simplification of conjunct... |
simplr1 1212 | Simplification of conjunct... |
simplr2 1213 | Simplification of conjunct... |
simplr3 1214 | Simplification of conjunct... |
simprl1 1215 | Simplification of conjunct... |
simprl2 1216 | Simplification of conjunct... |
simprl3 1217 | Simplification of conjunct... |
simprr1 1218 | Simplification of conjunct... |
simprr2 1219 | Simplification of conjunct... |
simprr3 1220 | Simplification of conjunct... |
simpl1l 1221 | Simplification of conjunct... |
simpl1r 1222 | Simplification of conjunct... |
simpl2l 1223 | Simplification of conjunct... |
simpl2r 1224 | Simplification of conjunct... |
simpl3l 1225 | Simplification of conjunct... |
simpl3r 1226 | Simplification of conjunct... |
simpr1l 1227 | Simplification of conjunct... |
simpr1r 1228 | Simplification of conjunct... |
simpr2l 1229 | Simplification of conjunct... |
simpr2r 1230 | Simplification of conjunct... |
simpr3l 1231 | Simplification of conjunct... |
simpr3r 1232 | Simplification of conjunct... |
simp1ll 1233 | Simplification of conjunct... |
simp1lr 1234 | Simplification of conjunct... |
simp1rl 1235 | Simplification of conjunct... |
simp1rr 1236 | Simplification of conjunct... |
simp2ll 1237 | Simplification of conjunct... |
simp2lr 1238 | Simplification of conjunct... |
simp2rl 1239 | Simplification of conjunct... |
simp2rr 1240 | Simplification of conjunct... |
simp3ll 1241 | Simplification of conjunct... |
simp3lr 1242 | Simplification of conjunct... |
simp3rl 1243 | Simplification of conjunct... |
simp3rr 1244 | Simplification of conjunct... |
simpl11 1245 | Simplification of conjunct... |
simpl12 1246 | Simplification of conjunct... |
simpl13 1247 | Simplification of conjunct... |
simpl21 1248 | Simplification of conjunct... |
simpl22 1249 | Simplification of conjunct... |
simpl23 1250 | Simplification of conjunct... |
simpl31 1251 | Simplification of conjunct... |
simpl32 1252 | Simplification of conjunct... |
simpl33 1253 | Simplification of conjunct... |
simpr11 1254 | Simplification of conjunct... |
simpr12 1255 | Simplification of conjunct... |
simpr13 1256 | Simplification of conjunct... |
simpr21 1257 | Simplification of conjunct... |
simpr22 1258 | Simplification of conjunct... |
simpr23 1259 | Simplification of conjunct... |
simpr31 1260 | Simplification of conjunct... |
simpr32 1261 | Simplification of conjunct... |
simpr33 1262 | Simplification of conjunct... |
simp1l1 1263 | Simplification of conjunct... |
simp1l2 1264 | Simplification of conjunct... |
simp1l3 1265 | Simplification of conjunct... |
simp1r1 1266 | Simplification of conjunct... |
simp1r2 1267 | Simplification of conjunct... |
simp1r3 1268 | Simplification of conjunct... |
simp2l1 1269 | Simplification of conjunct... |
simp2l2 1270 | Simplification of conjunct... |
simp2l3 1271 | Simplification of conjunct... |
simp2r1 1272 | Simplification of conjunct... |
simp2r2 1273 | Simplification of conjunct... |
simp2r3 1274 | Simplification of conjunct... |
simp3l1 1275 | Simplification of conjunct... |
simp3l2 1276 | Simplification of conjunct... |
simp3l3 1277 | Simplification of conjunct... |
simp3r1 1278 | Simplification of conjunct... |
simp3r2 1279 | Simplification of conjunct... |
simp3r3 1280 | Simplification of conjunct... |
simp11l 1281 | Simplification of conjunct... |
simp11r 1282 | Simplification of conjunct... |
simp12l 1283 | Simplification of conjunct... |
simp12r 1284 | Simplification of conjunct... |
simp13l 1285 | Simplification of conjunct... |
simp13r 1286 | Simplification of conjunct... |
simp21l 1287 | Simplification of conjunct... |
simp21r 1288 | Simplification of conjunct... |
simp22l 1289 | Simplification of conjunct... |
simp22r 1290 | Simplification of conjunct... |
simp23l 1291 | Simplification of conjunct... |
simp23r 1292 | Simplification of conjunct... |
simp31l 1293 | Simplification of conjunct... |
simp31r 1294 | Simplification of conjunct... |
simp32l 1295 | Simplification of conjunct... |
simp32r 1296 | Simplification of conjunct... |
simp33l 1297 | Simplification of conjunct... |
simp33r 1298 | Simplification of conjunct... |
simp111 1299 | Simplification of conjunct... |
simp112 1300 | Simplification of conjunct... |
simp113 1301 | Simplification of conjunct... |
simp121 1302 | Simplification of conjunct... |
simp122 1303 | Simplification of conjunct... |
simp123 1304 | Simplification of conjunct... |
simp131 1305 | Simplification of conjunct... |
simp132 1306 | Simplification of conjunct... |
simp133 1307 | Simplification of conjunct... |
simp211 1308 | Simplification of conjunct... |
simp212 1309 | Simplification of conjunct... |
simp213 1310 | Simplification of conjunct... |
simp221 1311 | Simplification of conjunct... |
simp222 1312 | Simplification of conjunct... |
simp223 1313 | Simplification of conjunct... |
simp231 1314 | Simplification of conjunct... |
simp232 1315 | Simplification of conjunct... |
simp233 1316 | Simplification of conjunct... |
simp311 1317 | Simplification of conjunct... |
simp312 1318 | Simplification of conjunct... |
simp313 1319 | Simplification of conjunct... |
simp321 1320 | Simplification of conjunct... |
simp322 1321 | Simplification of conjunct... |
simp323 1322 | Simplification of conjunct... |
simp331 1323 | Simplification of conjunct... |
simp332 1324 | Simplification of conjunct... |
simp333 1325 | Simplification of conjunct... |
3anibar 1326 | Remove a hypothesis from t... |
3mix1 1327 | Introduction in triple dis... |
3mix2 1328 | Introduction in triple dis... |
3mix3 1329 | Introduction in triple dis... |
3mix1i 1330 | Introduction in triple dis... |
3mix2i 1331 | Introduction in triple dis... |
3mix3i 1332 | Introduction in triple dis... |
3mix1d 1333 | Deduction introducing trip... |
3mix2d 1334 | Deduction introducing trip... |
3mix3d 1335 | Deduction introducing trip... |
3pm3.2i 1336 | Infer conjunction of premi... |
pm3.2an3 1337 | Version of ~ pm3.2 for a t... |
mpbir3an 1338 | Detach a conjunction of tr... |
mpbir3and 1339 | Detach a conjunction of tr... |
syl3anbrc 1340 | Syllogism inference. (Con... |
syl21anbrc 1341 | Syllogism inference. (Con... |
3imp3i2an 1342 | An elimination deduction. ... |
ex3 1343 | Apply ~ ex to a hypothesis... |
3imp1 1344 | Importation to left triple... |
3impd 1345 | Importation deduction for ... |
3imp2 1346 | Importation to right tripl... |
3impdi 1347 | Importation inference (und... |
3impdir 1348 | Importation inference (und... |
3exp1 1349 | Exportation from left trip... |
3expd 1350 | Exportation deduction for ... |
3exp2 1351 | Exportation from right tri... |
exp5o 1352 | A triple exportation infer... |
exp516 1353 | A triple exportation infer... |
exp520 1354 | A triple exportation infer... |
3impexp 1355 | Version of ~ impexp for a ... |
3an1rs 1356 | Swap conjuncts. (Contribu... |
3anassrs 1357 | Associative law for conjun... |
ad5ant245 1358 | Deduction adding conjuncts... |
ad5ant234 1359 | Deduction adding conjuncts... |
ad5ant235 1360 | Deduction adding conjuncts... |
ad5ant123 1361 | Deduction adding conjuncts... |
ad5ant124 1362 | Deduction adding conjuncts... |
ad5ant125 1363 | Deduction adding conjuncts... |
ad5ant134 1364 | Deduction adding conjuncts... |
ad5ant135 1365 | Deduction adding conjuncts... |
ad5ant145 1366 | Deduction adding conjuncts... |
ad5ant2345 1367 | Deduction adding conjuncts... |
syl3anc 1368 | Syllogism combined with co... |
syl13anc 1369 | Syllogism combined with co... |
syl31anc 1370 | Syllogism combined with co... |
syl112anc 1371 | Syllogism combined with co... |
syl121anc 1372 | Syllogism combined with co... |
syl211anc 1373 | Syllogism combined with co... |
syl23anc 1374 | Syllogism combined with co... |
syl32anc 1375 | Syllogism combined with co... |
syl122anc 1376 | Syllogism combined with co... |
syl212anc 1377 | Syllogism combined with co... |
syl221anc 1378 | Syllogism combined with co... |
syl113anc 1379 | Syllogism combined with co... |
syl131anc 1380 | Syllogism combined with co... |
syl311anc 1381 | Syllogism combined with co... |
syl33anc 1382 | Syllogism combined with co... |
syl222anc 1383 | Syllogism combined with co... |
syl123anc 1384 | Syllogism combined with co... |
syl132anc 1385 | Syllogism combined with co... |
syl213anc 1386 | Syllogism combined with co... |
syl231anc 1387 | Syllogism combined with co... |
syl312anc 1388 | Syllogism combined with co... |
syl321anc 1389 | Syllogism combined with co... |
syl133anc 1390 | Syllogism combined with co... |
syl313anc 1391 | Syllogism combined with co... |
syl331anc 1392 | Syllogism combined with co... |
syl223anc 1393 | Syllogism combined with co... |
syl232anc 1394 | Syllogism combined with co... |
syl322anc 1395 | Syllogism combined with co... |
syl233anc 1396 | Syllogism combined with co... |
syl323anc 1397 | Syllogism combined with co... |
syl332anc 1398 | Syllogism combined with co... |
syl333anc 1399 | A syllogism inference comb... |
syl3an1b 1400 | A syllogism inference. (C... |
syl3an2b 1401 | A syllogism inference. (C... |
syl3an3b 1402 | A syllogism inference. (C... |
syl3an1br 1403 | A syllogism inference. (C... |
syl3an2br 1404 | A syllogism inference. (C... |
syl3an3br 1405 | A syllogism inference. (C... |
syld3an3 1406 | A syllogism inference. (C... |
syld3an1 1407 | A syllogism inference. (C... |
syld3an2 1408 | A syllogism inference. (C... |
syl3anl1 1409 | A syllogism inference. (C... |
syl3anl2 1410 | A syllogism inference. (C... |
syl3anl3 1411 | A syllogism inference. (C... |
syl3anl 1412 | A triple syllogism inferen... |
syl3anr1 1413 | A syllogism inference. (C... |
syl3anr2 1414 | A syllogism inference. (C... |
syl3anr3 1415 | A syllogism inference. (C... |
3anidm12 1416 | Inference from idempotent ... |
3anidm13 1417 | Inference from idempotent ... |
3anidm23 1418 | Inference from idempotent ... |
syl2an3an 1419 | ~ syl3an with antecedents ... |
syl2an23an 1420 | Deduction related to ~ syl... |
3ori 1421 | Infer implication from tri... |
3jao 1422 | Disjunction of three antec... |
3jaob 1423 | Disjunction of three antec... |
3jaoi 1424 | Disjunction of three antec... |
3jaod 1425 | Disjunction of three antec... |
3jaoian 1426 | Disjunction of three antec... |
3jaodan 1427 | Disjunction of three antec... |
mpjao3dan 1428 | Eliminate a three-way disj... |
3jaao 1429 | Inference conjoining and d... |
syl3an9b 1430 | Nested syllogism inference... |
3orbi123d 1431 | Deduction joining 3 equiva... |
3anbi123d 1432 | Deduction joining 3 equiva... |
3anbi12d 1433 | Deduction conjoining and a... |
3anbi13d 1434 | Deduction conjoining and a... |
3anbi23d 1435 | Deduction conjoining and a... |
3anbi1d 1436 | Deduction adding conjuncts... |
3anbi2d 1437 | Deduction adding conjuncts... |
3anbi3d 1438 | Deduction adding conjuncts... |
3anim123d 1439 | Deduction joining 3 implic... |
3orim123d 1440 | Deduction joining 3 implic... |
an6 1441 | Rearrangement of 6 conjunc... |
3an6 1442 | Analogue of ~ an4 for trip... |
3or6 1443 | Analogue of ~ or4 for trip... |
mp3an1 1444 | An inference based on modu... |
mp3an2 1445 | An inference based on modu... |
mp3an3 1446 | An inference based on modu... |
mp3an12 1447 | An inference based on modu... |
mp3an13 1448 | An inference based on modu... |
mp3an23 1449 | An inference based on modu... |
mp3an1i 1450 | An inference based on modu... |
mp3anl1 1451 | An inference based on modu... |
mp3anl2 1452 | An inference based on modu... |
mp3anl3 1453 | An inference based on modu... |
mp3anr1 1454 | An inference based on modu... |
mp3anr2 1455 | An inference based on modu... |
mp3anr3 1456 | An inference based on modu... |
mp3an 1457 | An inference based on modu... |
mpd3an3 1458 | An inference based on modu... |
mpd3an23 1459 | An inference based on modu... |
mp3and 1460 | A deduction based on modus... |
mp3an12i 1461 | ~ mp3an with antecedents i... |
mp3an2i 1462 | ~ mp3an with antecedents i... |
mp3an3an 1463 | ~ mp3an with antecedents i... |
mp3an2ani 1464 | An elimination deduction. ... |
biimp3a 1465 | Infer implication from a l... |
biimp3ar 1466 | Infer implication from a l... |
3anandis 1467 | Inference that undistribut... |
3anandirs 1468 | Inference that undistribut... |
ecase23d 1469 | Deduction for elimination ... |
3ecase 1470 | Inference for elimination ... |
3bior1fd 1471 | A disjunction is equivalen... |
3bior1fand 1472 | A disjunction is equivalen... |
3bior2fd 1473 | A wff is equivalent to its... |
3biant1d 1474 | A conjunction is equivalen... |
intn3an1d 1475 | Introduction of a triple c... |
intn3an2d 1476 | Introduction of a triple c... |
intn3an3d 1477 | Introduction of a triple c... |
an3andi 1478 | Distribution of conjunctio... |
an33rean 1479 | Rearrange a 9-fold conjunc... |
3orel2 1480 | Partial elimination of a t... |
3orel3 1481 | Partial elimination of a t... |
3orel13 1482 | Elimination of two disjunc... |
3pm3.2ni 1483 | Triple negated disjunction... |
nanan 1486 | Conjunction in terms of al... |
dfnan2 1487 | Alternative denial in term... |
nanor 1488 | Alternative denial in term... |
nancom 1489 | Alternative denial is comm... |
nannan 1490 | Nested alternative denials... |
nanim 1491 | Implication in terms of al... |
nannot 1492 | Negation in terms of alter... |
nanbi 1493 | Biconditional in terms of ... |
nanbi1 1494 | Introduce a right anti-con... |
nanbi2 1495 | Introduce a left anti-conj... |
nanbi12 1496 | Join two logical equivalen... |
nanbi1i 1497 | Introduce a right anti-con... |
nanbi2i 1498 | Introduce a left anti-conj... |
nanbi12i 1499 | Join two logical equivalen... |
nanbi1d 1500 | Introduce a right anti-con... |
nanbi2d 1501 | Introduce a left anti-conj... |
nanbi12d 1502 | Join two logical equivalen... |
nanass 1503 | A characterization of when... |
xnor 1506 | Two ways to write XNOR (ex... |
xorcom 1507 | The connector ` \/_ ` is c... |
xorass 1508 | The connector ` \/_ ` is a... |
excxor 1509 | This tautology shows that ... |
xor2 1510 | Two ways to express "exclu... |
xoror 1511 | Exclusive disjunction impl... |
xornan 1512 | Exclusive disjunction impl... |
xornan2 1513 | XOR implies NAND (written ... |
xorneg2 1514 | The connector ` \/_ ` is n... |
xorneg1 1515 | The connector ` \/_ ` is n... |
xorneg 1516 | The connector ` \/_ ` is u... |
xorbi12i 1517 | Equality property for excl... |
xorbi12d 1518 | Equality property for excl... |
anxordi 1519 | Conjunction distributes ov... |
xorexmid 1520 | Exclusive-or variant of th... |
norcom 1523 | The connector ` -\/ ` is c... |
nornot 1524 | ` -. ` is expressible via ... |
noran 1525 | ` /\ ` is expressible via ... |
noror 1526 | ` \/ ` is expressible via ... |
norasslem1 1527 | This lemma shows the equiv... |
norasslem2 1528 | This lemma specializes ~ b... |
norasslem3 1529 | This lemma specializes ~ b... |
norass 1530 | A characterization of when... |
trujust 1535 | Soundness justification th... |
tru 1537 | The truth value ` T. ` is ... |
dftru2 1538 | An alternate definition of... |
trut 1539 | A proposition is equivalen... |
mptru 1540 | Eliminate ` T. ` as an ant... |
tbtru 1541 | A proposition is equivalen... |
bitru 1542 | A theorem is equivalent to... |
trud 1543 | Anything implies ` T. ` . ... |
truan 1544 | True can be removed from a... |
fal 1547 | The truth value ` F. ` is ... |
nbfal 1548 | The negation of a proposit... |
bifal 1549 | A contradiction is equival... |
falim 1550 | The truth value ` F. ` imp... |
falimd 1551 | The truth value ` F. ` imp... |
dfnot 1552 | Given falsum ` F. ` , we c... |
inegd 1553 | Negation introduction rule... |
efald 1554 | Deduction based on reducti... |
pm2.21fal 1555 | If a wff and its negation ... |
truimtru 1556 | A ` -> ` identity. (Contr... |
truimfal 1557 | A ` -> ` identity. (Contr... |
falimtru 1558 | A ` -> ` identity. (Contr... |
falimfal 1559 | A ` -> ` identity. (Contr... |
nottru 1560 | A ` -. ` identity. (Contr... |
notfal 1561 | A ` -. ` identity. (Contr... |
trubitru 1562 | A ` <-> ` identity. (Cont... |
falbitru 1563 | A ` <-> ` identity. (Cont... |
trubifal 1564 | A ` <-> ` identity. (Cont... |
falbifal 1565 | A ` <-> ` identity. (Cont... |
truantru 1566 | A ` /\ ` identity. (Contr... |
truanfal 1567 | A ` /\ ` identity. (Contr... |
falantru 1568 | A ` /\ ` identity. (Contr... |
falanfal 1569 | A ` /\ ` identity. (Contr... |
truortru 1570 | A ` \/ ` identity. (Contr... |
truorfal 1571 | A ` \/ ` identity. (Contr... |
falortru 1572 | A ` \/ ` identity. (Contr... |
falorfal 1573 | A ` \/ ` identity. (Contr... |
trunantru 1574 | A ` -/\ ` identity. (Cont... |
trunanfal 1575 | A ` -/\ ` identity. (Cont... |
falnantru 1576 | A ` -/\ ` identity. (Cont... |
falnanfal 1577 | A ` -/\ ` identity. (Cont... |
truxortru 1578 | A ` \/_ ` identity. (Cont... |
truxorfal 1579 | A ` \/_ ` identity. (Cont... |
falxortru 1580 | A ` \/_ ` identity. (Cont... |
falxorfal 1581 | A ` \/_ ` identity. (Cont... |
trunortru 1582 | A ` -\/ ` identity. (Cont... |
trunorfal 1583 | A ` -\/ ` identity. (Cont... |
falnortru 1584 | A ` -\/ ` identity. (Cont... |
falnorfal 1585 | A ` -\/ ` identity. (Cont... |
hadbi123d 1588 | Equality theorem for the a... |
hadbi123i 1589 | Equality theorem for the a... |
hadass 1590 | Associative law for the ad... |
hadbi 1591 | The adder sum is the same ... |
hadcoma 1592 | Commutative law for the ad... |
hadcomb 1593 | Commutative law for the ad... |
hadrot 1594 | Rotation law for the adder... |
hadnot 1595 | The adder sum distributes ... |
had1 1596 | If the first input is true... |
had0 1597 | If the first input is fals... |
hadifp 1598 | The value of the adder sum... |
cador 1601 | The adder carry in disjunc... |
cadan 1602 | The adder carry in conjunc... |
cadbi123d 1603 | Equality theorem for the a... |
cadbi123i 1604 | Equality theorem for the a... |
cadcoma 1605 | Commutative law for the ad... |
cadcomb 1606 | Commutative law for the ad... |
cadrot 1607 | Rotation law for the adder... |
cadnot 1608 | The adder carry distribute... |
cad11 1609 | If (at least) two inputs a... |
cad1 1610 | If one input is true, then... |
cad0 1611 | If one input is false, the... |
cad0OLD 1612 | Obsolete version of ~ cad0... |
cadifp 1613 | The value of the carry is,... |
cadtru 1614 | The adder carry is true as... |
minimp 1615 | A single axiom for minimal... |
minimp-syllsimp 1616 | Derivation of Syll-Simp ( ... |
minimp-ax1 1617 | Derivation of ~ ax-1 from ... |
minimp-ax2c 1618 | Derivation of a commuted f... |
minimp-ax2 1619 | Derivation of ~ ax-2 from ... |
minimp-pm2.43 1620 | Derivation of ~ pm2.43 (al... |
impsingle 1621 | The shortest single axiom ... |
impsingle-step4 1622 | Derivation of impsingle-st... |
impsingle-step8 1623 | Derivation of impsingle-st... |
impsingle-ax1 1624 | Derivation of impsingle-ax... |
impsingle-step15 1625 | Derivation of impsingle-st... |
impsingle-step18 1626 | Derivation of impsingle-st... |
impsingle-step19 1627 | Derivation of impsingle-st... |
impsingle-step20 1628 | Derivation of impsingle-st... |
impsingle-step21 1629 | Derivation of impsingle-st... |
impsingle-step22 1630 | Derivation of impsingle-st... |
impsingle-step25 1631 | Derivation of impsingle-st... |
impsingle-imim1 1632 | Derivation of impsingle-im... |
impsingle-peirce 1633 | Derivation of impsingle-pe... |
tarski-bernays-ax2 1634 | Derivation of ~ ax-2 from ... |
meredith 1635 | Carew Meredith's sole axio... |
merlem1 1636 | Step 3 of Meredith's proof... |
merlem2 1637 | Step 4 of Meredith's proof... |
merlem3 1638 | Step 7 of Meredith's proof... |
merlem4 1639 | Step 8 of Meredith's proof... |
merlem5 1640 | Step 11 of Meredith's proo... |
merlem6 1641 | Step 12 of Meredith's proo... |
merlem7 1642 | Between steps 14 and 15 of... |
merlem8 1643 | Step 15 of Meredith's proo... |
merlem9 1644 | Step 18 of Meredith's proo... |
merlem10 1645 | Step 19 of Meredith's proo... |
merlem11 1646 | Step 20 of Meredith's proo... |
merlem12 1647 | Step 28 of Meredith's proo... |
merlem13 1648 | Step 35 of Meredith's proo... |
luk-1 1649 | 1 of 3 axioms for proposit... |
luk-2 1650 | 2 of 3 axioms for proposit... |
luk-3 1651 | 3 of 3 axioms for proposit... |
luklem1 1652 | Used to rederive standard ... |
luklem2 1653 | Used to rederive standard ... |
luklem3 1654 | Used to rederive standard ... |
luklem4 1655 | Used to rederive standard ... |
luklem5 1656 | Used to rederive standard ... |
luklem6 1657 | Used to rederive standard ... |
luklem7 1658 | Used to rederive standard ... |
luklem8 1659 | Used to rederive standard ... |
ax1 1660 | Standard propositional axi... |
ax2 1661 | Standard propositional axi... |
ax3 1662 | Standard propositional axi... |
nic-dfim 1663 | This theorem "defines" imp... |
nic-dfneg 1664 | This theorem "defines" neg... |
nic-mp 1665 | Derive Nicod's rule of mod... |
nic-mpALT 1666 | A direct proof of ~ nic-mp... |
nic-ax 1667 | Nicod's axiom derived from... |
nic-axALT 1668 | A direct proof of ~ nic-ax... |
nic-imp 1669 | Inference for ~ nic-mp usi... |
nic-idlem1 1670 | Lemma for ~ nic-id . (Con... |
nic-idlem2 1671 | Lemma for ~ nic-id . Infe... |
nic-id 1672 | Theorem ~ id expressed wit... |
nic-swap 1673 | The connector ` -/\ ` is s... |
nic-isw1 1674 | Inference version of ~ nic... |
nic-isw2 1675 | Inference for swapping nes... |
nic-iimp1 1676 | Inference version of ~ nic... |
nic-iimp2 1677 | Inference version of ~ nic... |
nic-idel 1678 | Inference to remove the tr... |
nic-ich 1679 | Chained inference. (Contr... |
nic-idbl 1680 | Double the terms. Since d... |
nic-bijust 1681 | Biconditional justificatio... |
nic-bi1 1682 | Inference to extract one s... |
nic-bi2 1683 | Inference to extract the o... |
nic-stdmp 1684 | Derive the standard modus ... |
nic-luk1 1685 | Proof of ~ luk-1 from ~ ni... |
nic-luk2 1686 | Proof of ~ luk-2 from ~ ni... |
nic-luk3 1687 | Proof of ~ luk-3 from ~ ni... |
lukshef-ax1 1688 | This alternative axiom for... |
lukshefth1 1689 | Lemma for ~ renicax . (Co... |
lukshefth2 1690 | Lemma for ~ renicax . (Co... |
renicax 1691 | A rederivation of ~ nic-ax... |
tbw-bijust 1692 | Justification for ~ tbw-ne... |
tbw-negdf 1693 | The definition of negation... |
tbw-ax1 1694 | The first of four axioms i... |
tbw-ax2 1695 | The second of four axioms ... |
tbw-ax3 1696 | The third of four axioms i... |
tbw-ax4 1697 | The fourth of four axioms ... |
tbwsyl 1698 | Used to rederive the Lukas... |
tbwlem1 1699 | Used to rederive the Lukas... |
tbwlem2 1700 | Used to rederive the Lukas... |
tbwlem3 1701 | Used to rederive the Lukas... |
tbwlem4 1702 | Used to rederive the Lukas... |
tbwlem5 1703 | Used to rederive the Lukas... |
re1luk1 1704 | ~ luk-1 derived from the T... |
re1luk2 1705 | ~ luk-2 derived from the T... |
re1luk3 1706 | ~ luk-3 derived from the T... |
merco1 1707 | A single axiom for proposi... |
merco1lem1 1708 | Used to rederive the Tarsk... |
retbwax4 1709 | ~ tbw-ax4 rederived from ~... |
retbwax2 1710 | ~ tbw-ax2 rederived from ~... |
merco1lem2 1711 | Used to rederive the Tarsk... |
merco1lem3 1712 | Used to rederive the Tarsk... |
merco1lem4 1713 | Used to rederive the Tarsk... |
merco1lem5 1714 | Used to rederive the Tarsk... |
merco1lem6 1715 | Used to rederive the Tarsk... |
merco1lem7 1716 | Used to rederive the Tarsk... |
retbwax3 1717 | ~ tbw-ax3 rederived from ~... |
merco1lem8 1718 | Used to rederive the Tarsk... |
merco1lem9 1719 | Used to rederive the Tarsk... |
merco1lem10 1720 | Used to rederive the Tarsk... |
merco1lem11 1721 | Used to rederive the Tarsk... |
merco1lem12 1722 | Used to rederive the Tarsk... |
merco1lem13 1723 | Used to rederive the Tarsk... |
merco1lem14 1724 | Used to rederive the Tarsk... |
merco1lem15 1725 | Used to rederive the Tarsk... |
merco1lem16 1726 | Used to rederive the Tarsk... |
merco1lem17 1727 | Used to rederive the Tarsk... |
merco1lem18 1728 | Used to rederive the Tarsk... |
retbwax1 1729 | ~ tbw-ax1 rederived from ~... |
merco2 1730 | A single axiom for proposi... |
mercolem1 1731 | Used to rederive the Tarsk... |
mercolem2 1732 | Used to rederive the Tarsk... |
mercolem3 1733 | Used to rederive the Tarsk... |
mercolem4 1734 | Used to rederive the Tarsk... |
mercolem5 1735 | Used to rederive the Tarsk... |
mercolem6 1736 | Used to rederive the Tarsk... |
mercolem7 1737 | Used to rederive the Tarsk... |
mercolem8 1738 | Used to rederive the Tarsk... |
re1tbw1 1739 | ~ tbw-ax1 rederived from ~... |
re1tbw2 1740 | ~ tbw-ax2 rederived from ~... |
re1tbw3 1741 | ~ tbw-ax3 rederived from ~... |
re1tbw4 1742 | ~ tbw-ax4 rederived from ~... |
rb-bijust 1743 | Justification for ~ rb-imd... |
rb-imdf 1744 | The definition of implicat... |
anmp 1745 | Modus ponens for ` { \/ , ... |
rb-ax1 1746 | The first of four axioms i... |
rb-ax2 1747 | The second of four axioms ... |
rb-ax3 1748 | The third of four axioms i... |
rb-ax4 1749 | The fourth of four axioms ... |
rbsyl 1750 | Used to rederive the Lukas... |
rblem1 1751 | Used to rederive the Lukas... |
rblem2 1752 | Used to rederive the Lukas... |
rblem3 1753 | Used to rederive the Lukas... |
rblem4 1754 | Used to rederive the Lukas... |
rblem5 1755 | Used to rederive the Lukas... |
rblem6 1756 | Used to rederive the Lukas... |
rblem7 1757 | Used to rederive the Lukas... |
re1axmp 1758 | ~ ax-mp derived from Russe... |
re2luk1 1759 | ~ luk-1 derived from Russe... |
re2luk2 1760 | ~ luk-2 derived from Russe... |
re2luk3 1761 | ~ luk-3 derived from Russe... |
mptnan 1762 | Modus ponendo tollens 1, o... |
mptxor 1763 | Modus ponendo tollens 2, o... |
mtpor 1764 | Modus tollendo ponens (inc... |
mtpxor 1765 | Modus tollendo ponens (ori... |
stoic1a 1766 | Stoic logic Thema 1 (part ... |
stoic1b 1767 | Stoic logic Thema 1 (part ... |
stoic2a 1768 | Stoic logic Thema 2 versio... |
stoic2b 1769 | Stoic logic Thema 2 versio... |
stoic3 1770 | Stoic logic Thema 3. Stat... |
stoic4a 1771 | Stoic logic Thema 4 versio... |
stoic4b 1772 | Stoic logic Thema 4 versio... |
alnex 1775 | Universal quantification o... |
eximal 1776 | An equivalence between an ... |
nf2 1779 | Alternate definition of no... |
nf3 1780 | Alternate definition of no... |
nf4 1781 | Alternate definition of no... |
nfi 1782 | Deduce that ` x ` is not f... |
nfri 1783 | Consequence of the definit... |
nfd 1784 | Deduce that ` x ` is not f... |
nfrd 1785 | Consequence of the definit... |
nftht 1786 | Closed form of ~ nfth . (... |
nfntht 1787 | Closed form of ~ nfnth . ... |
nfntht2 1788 | Closed form of ~ nfnth . ... |
gen2 1790 | Generalization applied twi... |
mpg 1791 | Modus ponens combined with... |
mpgbi 1792 | Modus ponens on biconditio... |
mpgbir 1793 | Modus ponens on biconditio... |
nex 1794 | Generalization rule for ne... |
nfth 1795 | No variable is (effectivel... |
nfnth 1796 | No variable is (effectivel... |
hbth 1797 | No variable is (effectivel... |
nftru 1798 | The true constant has no f... |
nffal 1799 | The false constant has no ... |
sptruw 1800 | Version of ~ sp when ` ph ... |
altru 1801 | For all sets, ` T. ` is tr... |
alfal 1802 | For all sets, ` -. F. ` is... |
alim 1804 | Restatement of Axiom ~ ax-... |
alimi 1805 | Inference quantifying both... |
2alimi 1806 | Inference doubly quantifyi... |
ala1 1807 | Add an antecedent in a uni... |
al2im 1808 | Closed form of ~ al2imi . ... |
al2imi 1809 | Inference quantifying ante... |
alanimi 1810 | Variant of ~ al2imi with c... |
alimdh 1811 | Deduction form of Theorem ... |
albi 1812 | Theorem 19.15 of [Margaris... |
albii 1813 | Inference adding universal... |
2albii 1814 | Inference adding two unive... |
3albii 1815 | Inference adding three uni... |
sylgt 1816 | Closed form of ~ sylg . (... |
sylg 1817 | A syllogism combined with ... |
alrimih 1818 | Inference form of Theorem ... |
hbxfrbi 1819 | A utility lemma to transfe... |
alex 1820 | Universal quantifier in te... |
exnal 1821 | Existential quantification... |
2nalexn 1822 | Part of theorem *11.5 in [... |
2exnaln 1823 | Theorem *11.22 in [Whitehe... |
2nexaln 1824 | Theorem *11.25 in [Whitehe... |
alimex 1825 | An equivalence between an ... |
aleximi 1826 | A variant of ~ al2imi : in... |
alexbii 1827 | Biconditional form of ~ al... |
exim 1828 | Theorem 19.22 of [Margaris... |
eximi 1829 | Inference adding existenti... |
2eximi 1830 | Inference adding two exist... |
eximii 1831 | Inference associated with ... |
exa1 1832 | Add an antecedent in an ex... |
19.38 1833 | Theorem 19.38 of [Margaris... |
19.38a 1834 | Under a nonfreeness hypoth... |
19.38b 1835 | Under a nonfreeness hypoth... |
imnang 1836 | Quantified implication in ... |
alinexa 1837 | A transformation of quanti... |
exnalimn 1838 | Existential quantification... |
alexn 1839 | A relationship between two... |
2exnexn 1840 | Theorem *11.51 in [Whitehe... |
exbi 1841 | Theorem 19.18 of [Margaris... |
exbii 1842 | Inference adding existenti... |
2exbii 1843 | Inference adding two exist... |
3exbii 1844 | Inference adding three exi... |
nfbiit 1845 | Equivalence theorem for th... |
nfbii 1846 | Equality theorem for the n... |
nfxfr 1847 | A utility lemma to transfe... |
nfxfrd 1848 | A utility lemma to transfe... |
nfnbi 1849 | A variable is nonfree in a... |
nfnbiOLD 1850 | Obsolete version of ~ nfnb... |
nfnt 1851 | If a variable is nonfree i... |
nfn 1852 | Inference associated with ... |
nfnd 1853 | Deduction associated with ... |
exanali 1854 | A transformation of quanti... |
2exanali 1855 | Theorem *11.521 in [Whiteh... |
exancom 1856 | Commutation of conjunction... |
exan 1857 | Place a conjunct in the sc... |
alrimdh 1858 | Deduction form of Theorem ... |
eximdh 1859 | Deduction from Theorem 19.... |
nexdh 1860 | Deduction for generalizati... |
albidh 1861 | Formula-building rule for ... |
exbidh 1862 | Formula-building rule for ... |
exsimpl 1863 | Simplification of an exist... |
exsimpr 1864 | Simplification of an exist... |
19.26 1865 | Theorem 19.26 of [Margaris... |
19.26-2 1866 | Theorem ~ 19.26 with two q... |
19.26-3an 1867 | Theorem ~ 19.26 with tripl... |
19.29 1868 | Theorem 19.29 of [Margaris... |
19.29r 1869 | Variation of ~ 19.29 . (C... |
19.29r2 1870 | Variation of ~ 19.29r with... |
19.29x 1871 | Variation of ~ 19.29 with ... |
19.35 1872 | Theorem 19.35 of [Margaris... |
19.35i 1873 | Inference associated with ... |
19.35ri 1874 | Inference associated with ... |
19.25 1875 | Theorem 19.25 of [Margaris... |
19.30 1876 | Theorem 19.30 of [Margaris... |
19.43 1877 | Theorem 19.43 of [Margaris... |
19.43OLD 1878 | Obsolete proof of ~ 19.43 ... |
19.33 1879 | Theorem 19.33 of [Margaris... |
19.33b 1880 | The antecedent provides a ... |
19.40 1881 | Theorem 19.40 of [Margaris... |
19.40-2 1882 | Theorem *11.42 in [Whitehe... |
19.40b 1883 | The antecedent provides a ... |
albiim 1884 | Split a biconditional and ... |
2albiim 1885 | Split a biconditional and ... |
exintrbi 1886 | Add/remove a conjunct in t... |
exintr 1887 | Introduce a conjunct in th... |
alsyl 1888 | Universally quantified and... |
nfimd 1889 | If in a context ` x ` is n... |
nfimt 1890 | Closed form of ~ nfim and ... |
nfim 1891 | If ` x ` is not free in ` ... |
nfand 1892 | If in a context ` x ` is n... |
nf3and 1893 | Deduction form of bound-va... |
nfan 1894 | If ` x ` is not free in ` ... |
nfnan 1895 | If ` x ` is not free in ` ... |
nf3an 1896 | If ` x ` is not free in ` ... |
nfbid 1897 | If in a context ` x ` is n... |
nfbi 1898 | If ` x ` is not free in ` ... |
nfor 1899 | If ` x ` is not free in ` ... |
nf3or 1900 | If ` x ` is not free in ` ... |
empty 1901 | Two characterizations of t... |
emptyex 1902 | On the empty domain, any e... |
emptyal 1903 | On the empty domain, any u... |
emptynf 1904 | On the empty domain, any v... |
ax5d 1906 | Version of ~ ax-5 with ant... |
ax5e 1907 | A rephrasing of ~ ax-5 usi... |
ax5ea 1908 | If a formula holds for som... |
nfv 1909 | If ` x ` is not present in... |
nfvd 1910 | ~ nfv with antecedent. Us... |
alimdv 1911 | Deduction form of Theorem ... |
eximdv 1912 | Deduction form of Theorem ... |
2alimdv 1913 | Deduction form of Theorem ... |
2eximdv 1914 | Deduction form of Theorem ... |
albidv 1915 | Formula-building rule for ... |
exbidv 1916 | Formula-building rule for ... |
nfbidv 1917 | An equality theorem for no... |
2albidv 1918 | Formula-building rule for ... |
2exbidv 1919 | Formula-building rule for ... |
3exbidv 1920 | Formula-building rule for ... |
4exbidv 1921 | Formula-building rule for ... |
alrimiv 1922 | Inference form of Theorem ... |
alrimivv 1923 | Inference form of Theorem ... |
alrimdv 1924 | Deduction form of Theorem ... |
exlimiv 1925 | Inference form of Theorem ... |
exlimiiv 1926 | Inference (Rule C) associa... |
exlimivv 1927 | Inference form of Theorem ... |
exlimdv 1928 | Deduction form of Theorem ... |
exlimdvv 1929 | Deduction form of Theorem ... |
exlimddv 1930 | Existential elimination ru... |
nexdv 1931 | Deduction for generalizati... |
2ax5 1932 | Quantification of two vari... |
stdpc5v 1933 | Version of ~ stdpc5 with a... |
19.21v 1934 | Version of ~ 19.21 with a ... |
19.32v 1935 | Version of ~ 19.32 with a ... |
19.31v 1936 | Version of ~ 19.31 with a ... |
19.23v 1937 | Version of ~ 19.23 with a ... |
19.23vv 1938 | Theorem ~ 19.23v extended ... |
pm11.53v 1939 | Version of ~ pm11.53 with ... |
19.36imv 1940 | One direction of ~ 19.36v ... |
19.36imvOLD 1941 | Obsolete version of ~ 19.3... |
19.36iv 1942 | Inference associated with ... |
19.37imv 1943 | One direction of ~ 19.37v ... |
19.37iv 1944 | Inference associated with ... |
19.41v 1945 | Version of ~ 19.41 with a ... |
19.41vv 1946 | Version of ~ 19.41 with tw... |
19.41vvv 1947 | Version of ~ 19.41 with th... |
19.41vvvv 1948 | Version of ~ 19.41 with fo... |
19.42v 1949 | Version of ~ 19.42 with a ... |
exdistr 1950 | Distribution of existentia... |
exdistrv 1951 | Distribute a pair of exist... |
4exdistrv 1952 | Distribute two pairs of ex... |
19.42vv 1953 | Version of ~ 19.42 with tw... |
exdistr2 1954 | Distribution of existentia... |
19.42vvv 1955 | Version of ~ 19.42 with th... |
3exdistr 1956 | Distribution of existentia... |
4exdistr 1957 | Distribution of existentia... |
weq 1958 | Extend wff definition to i... |
speimfw 1959 | Specialization, with addit... |
speimfwALT 1960 | Alternate proof of ~ speim... |
spimfw 1961 | Specialization, with addit... |
ax12i 1962 | Inference that has ~ ax-12... |
ax6v 1964 | Axiom B7 of [Tarski] p. 75... |
ax6ev 1965 | At least one individual ex... |
spimw 1966 | Specialization. Lemma 8 o... |
spimew 1967 | Existential introduction, ... |
speiv 1968 | Inference from existential... |
speivw 1969 | Version of ~ spei with a d... |
exgen 1970 | Rule of existential genera... |
extru 1971 | There exists a variable su... |
19.2 1972 | Theorem 19.2 of [Margaris]... |
19.2d 1973 | Deduction associated with ... |
19.8w 1974 | Weak version of ~ 19.8a an... |
spnfw 1975 | Weak version of ~ sp . Us... |
spvw 1976 | Version of ~ sp when ` x `... |
19.3v 1977 | Version of ~ 19.3 with a d... |
19.8v 1978 | Version of ~ 19.8a with a ... |
19.9v 1979 | Version of ~ 19.9 with a d... |
19.39 1980 | Theorem 19.39 of [Margaris... |
19.24 1981 | Theorem 19.24 of [Margaris... |
19.34 1982 | Theorem 19.34 of [Margaris... |
19.36v 1983 | Version of ~ 19.36 with a ... |
19.12vvv 1984 | Version of ~ 19.12vv with ... |
19.27v 1985 | Version of ~ 19.27 with a ... |
19.28v 1986 | Version of ~ 19.28 with a ... |
19.37v 1987 | Version of ~ 19.37 with a ... |
19.44v 1988 | Version of ~ 19.44 with a ... |
19.45v 1989 | Version of ~ 19.45 with a ... |
spimevw 1990 | Existential introduction, ... |
spimvw 1991 | A weak form of specializat... |
spvv 1992 | Specialization, using impl... |
spfalw 1993 | Version of ~ sp when ` ph ... |
chvarvv 1994 | Implicit substitution of `... |
equs4v 1995 | Version of ~ equs4 with a ... |
alequexv 1996 | Version of ~ equs4v with i... |
exsbim 1997 | One direction of the equiv... |
equsv 1998 | If a formula does not cont... |
equsalvw 1999 | Version of ~ equsalv with ... |
equsexvw 2000 | Version of ~ equsexv with ... |
cbvaliw 2001 | Change bound variable. Us... |
cbvalivw 2002 | Change bound variable. Us... |
ax7v 2004 | Weakened version of ~ ax-7... |
ax7v1 2005 | First of two weakened vers... |
ax7v2 2006 | Second of two weakened ver... |
equid 2007 | Identity law for equality.... |
nfequid 2008 | Bound-variable hypothesis ... |
equcomiv 2009 | Weaker form of ~ equcomi w... |
ax6evr 2010 | A commuted form of ~ ax6ev... |
ax7 2011 | Proof of ~ ax-7 from ~ ax7... |
equcomi 2012 | Commutative law for equali... |
equcom 2013 | Commutative law for equali... |
equcomd 2014 | Deduction form of ~ equcom... |
equcoms 2015 | An inference commuting equ... |
equtr 2016 | A transitive law for equal... |
equtrr 2017 | A transitive law for equal... |
equeuclr 2018 | Commuted version of ~ eque... |
equeucl 2019 | Equality is a left-Euclide... |
equequ1 2020 | An equivalence law for equ... |
equequ2 2021 | An equivalence law for equ... |
equtr2 2022 | Equality is a left-Euclide... |
stdpc6 2023 | One of the two equality ax... |
equvinv 2024 | A variable introduction la... |
equvinva 2025 | A modified version of the ... |
equvelv 2026 | A biconditional form of ~ ... |
ax13b 2027 | An equivalence between two... |
spfw 2028 | Weak version of ~ sp . Us... |
spw 2029 | Weak version of the specia... |
cbvalw 2030 | Change bound variable. Us... |
cbvalvw 2031 | Change bound variable. Us... |
cbvexvw 2032 | Change bound variable. Us... |
cbvaldvaw 2033 | Rule used to change the bo... |
cbvexdvaw 2034 | Rule used to change the bo... |
cbval2vw 2035 | Rule used to change bound ... |
cbvex2vw 2036 | Rule used to change bound ... |
cbvex4vw 2037 | Rule used to change bound ... |
alcomiw 2038 | Weak version of ~ ax-11 . ... |
alcomw 2039 | Weak version of ~ alcom an... |
hbn1fw 2040 | Weak version of ~ ax-10 fr... |
hbn1w 2041 | Weak version of ~ hbn1 . ... |
hba1w 2042 | Weak version of ~ hba1 . ... |
hbe1w 2043 | Weak version of ~ hbe1 . ... |
hbalw 2044 | Weak version of ~ hbal . ... |
19.8aw 2045 | If a formula is true, then... |
exexw 2046 | Existential quantification... |
spaev 2047 | A special instance of ~ sp... |
cbvaev 2048 | Change bound variable in a... |
aevlem0 2049 | Lemma for ~ aevlem . Inst... |
aevlem 2050 | Lemma for ~ aev and ~ axc1... |
aeveq 2051 | The antecedent ` A. x x = ... |
aev 2052 | A "distinctor elimination"... |
aev2 2053 | A version of ~ aev with tw... |
hbaev 2054 | All variables are effectiv... |
naev 2055 | If some set variables can ... |
naev2 2056 | Generalization of ~ hbnaev... |
hbnaev 2057 | Any variable is free in ` ... |
sbjust 2058 | Justification theorem for ... |
sbt 2061 | A substitution into a theo... |
sbtru 2062 | The result of substituting... |
stdpc4 2063 | The specialization axiom o... |
sbtALT 2064 | Alternate proof of ~ sbt ,... |
2stdpc4 2065 | A double specialization us... |
sbi1 2066 | Distribute substitution ov... |
spsbim 2067 | Distribute substitution ov... |
spsbbi 2068 | Biconditional property for... |
sbimi 2069 | Distribute substitution ov... |
sb2imi 2070 | Distribute substitution ov... |
sbbii 2071 | Infer substitution into bo... |
2sbbii 2072 | Infer double substitution ... |
sbimdv 2073 | Deduction substituting bot... |
sbbidv 2074 | Deduction substituting bot... |
sban 2075 | Conjunction inside and out... |
sb3an 2076 | Threefold conjunction insi... |
spsbe 2077 | Existential generalization... |
sbequ 2078 | Equality property for subs... |
sbequi 2079 | An equality theorem for su... |
sb6 2080 | Alternate definition of su... |
2sb6 2081 | Equivalence for double sub... |
sb1v 2082 | One direction of ~ sb5 , p... |
sbv 2083 | Substitution for a variabl... |
sbcom4 2084 | Commutativity law for subs... |
pm11.07 2085 | Axiom *11.07 in [Whitehead... |
sbrimvw 2086 | Substitution in an implica... |
sbievw 2087 | Conversion of implicit sub... |
sbiedvw 2088 | Conversion of implicit sub... |
2sbievw 2089 | Conversion of double impli... |
sbcom3vv 2090 | Substituting ` y ` for ` x... |
sbievw2 2091 | ~ sbievw applied twice, av... |
sbco2vv 2092 | A composition law for subs... |
equsb3 2093 | Substitution in an equalit... |
equsb3r 2094 | Substitution applied to th... |
equsb1v 2095 | Substitution applied to an... |
nsb 2096 | Any substitution in an alw... |
sbn1 2097 | One direction of ~ sbn , u... |
wel 2099 | Extend wff definition to i... |
ax8v 2101 | Weakened version of ~ ax-8... |
ax8v1 2102 | First of two weakened vers... |
ax8v2 2103 | Second of two weakened ver... |
ax8 2104 | Proof of ~ ax-8 from ~ ax8... |
elequ1 2105 | An identity law for the no... |
elsb1 2106 | Substitution for the first... |
cleljust 2107 | When the class variables i... |
ax9v 2109 | Weakened version of ~ ax-9... |
ax9v1 2110 | First of two weakened vers... |
ax9v2 2111 | Second of two weakened ver... |
ax9 2112 | Proof of ~ ax-9 from ~ ax9... |
elequ2 2113 | An identity law for the no... |
elequ2g 2114 | A form of ~ elequ2 with a ... |
elsb2 2115 | Substitution for the secon... |
ax6dgen 2116 | Tarski's system uses the w... |
ax10w 2117 | Weak version of ~ ax-10 fr... |
ax11w 2118 | Weak version of ~ ax-11 fr... |
ax11dgen 2119 | Degenerate instance of ~ a... |
ax12wlem 2120 | Lemma for weak version of ... |
ax12w 2121 | Weak version of ~ ax-12 fr... |
ax12dgen 2122 | Degenerate instance of ~ a... |
ax12wdemo 2123 | Example of an application ... |
ax13w 2124 | Weak version (principal in... |
ax13dgen1 2125 | Degenerate instance of ~ a... |
ax13dgen2 2126 | Degenerate instance of ~ a... |
ax13dgen3 2127 | Degenerate instance of ~ a... |
ax13dgen4 2128 | Degenerate instance of ~ a... |
hbn1 2130 | Alias for ~ ax-10 to be us... |
hbe1 2131 | The setvar ` x ` is not fr... |
hbe1a 2132 | Dual statement of ~ hbe1 .... |
nf5-1 2133 | One direction of ~ nf5 can... |
nf5i 2134 | Deduce that ` x ` is not f... |
nf5dh 2135 | Deduce that ` x ` is not f... |
nf5dv 2136 | Apply the definition of no... |
nfnaew 2137 | All variables are effectiv... |
nfnaewOLD 2138 | Obsolete version of ~ nfna... |
nfe1 2139 | The setvar ` x ` is not fr... |
nfa1 2140 | The setvar ` x ` is not fr... |
nfna1 2141 | A convenience theorem part... |
nfia1 2142 | Lemma 23 of [Monk2] p. 114... |
nfnf1 2143 | The setvar ` x ` is not fr... |
modal5 2144 | The analogue in our predic... |
nfs1v 2145 | The setvar ` x ` is not fr... |
alcoms 2147 | Swap quantifiers in an ant... |
alcom 2148 | Theorem 19.5 of [Margaris]... |
alrot3 2149 | Theorem *11.21 in [Whitehe... |
alrot4 2150 | Rotate four universal quan... |
sbal 2151 | Move universal quantifier ... |
sbalv 2152 | Quantify with new variable... |
sbcom2 2153 | Commutativity law for subs... |
excom 2154 | Theorem 19.11 of [Margaris... |
excomim 2155 | One direction of Theorem 1... |
excom13 2156 | Swap 1st and 3rd existenti... |
exrot3 2157 | Rotate existential quantif... |
exrot4 2158 | Rotate existential quantif... |
hbal 2159 | If ` x ` is not free in ` ... |
hbald 2160 | Deduction form of bound-va... |
hbsbw 2161 | If ` z ` is not free in ` ... |
nfa2 2162 | Lemma 24 of [Monk2] p. 114... |
ax12v 2164 | This is essentially Axiom ... |
ax12v2 2165 | It is possible to remove a... |
19.8a 2166 | If a wff is true, it is tr... |
19.8ad 2167 | If a wff is true, it is tr... |
sp 2168 | Specialization. A univers... |
spi 2169 | Inference rule of universa... |
sps 2170 | Generalization of antecede... |
2sp 2171 | A double specialization (s... |
spsd 2172 | Deduction generalizing ant... |
19.2g 2173 | Theorem 19.2 of [Margaris]... |
19.21bi 2174 | Inference form of ~ 19.21 ... |
19.21bbi 2175 | Inference removing two uni... |
19.23bi 2176 | Inference form of Theorem ... |
nexr 2177 | Inference associated with ... |
qexmid 2178 | Quantified excluded middle... |
nf5r 2179 | Consequence of the definit... |
nf5ri 2180 | Consequence of the definit... |
nf5rd 2181 | Consequence of the definit... |
spimedv 2182 | Deduction version of ~ spi... |
spimefv 2183 | Version of ~ spime with a ... |
nfim1 2184 | A closed form of ~ nfim . ... |
nfan1 2185 | A closed form of ~ nfan . ... |
19.3t 2186 | Closed form of ~ 19.3 and ... |
19.3 2187 | A wff may be quantified wi... |
19.9d 2188 | A deduction version of one... |
19.9t 2189 | Closed form of ~ 19.9 and ... |
19.9 2190 | A wff may be existentially... |
19.21t 2191 | Closed form of Theorem 19.... |
19.21 2192 | Theorem 19.21 of [Margaris... |
stdpc5 2193 | An axiom scheme of standar... |
19.21-2 2194 | Version of ~ 19.21 with tw... |
19.23t 2195 | Closed form of Theorem 19.... |
19.23 2196 | Theorem 19.23 of [Margaris... |
alimd 2197 | Deduction form of Theorem ... |
alrimi 2198 | Inference form of Theorem ... |
alrimdd 2199 | Deduction form of Theorem ... |
alrimd 2200 | Deduction form of Theorem ... |
eximd 2201 | Deduction form of Theorem ... |
exlimi 2202 | Inference associated with ... |
exlimd 2203 | Deduction form of Theorem ... |
exlimimdd 2204 | Existential elimination ru... |
exlimdd 2205 | Existential elimination ru... |
nexd 2206 | Deduction for generalizati... |
albid 2207 | Formula-building rule for ... |
exbid 2208 | Formula-building rule for ... |
nfbidf 2209 | An equality theorem for ef... |
19.16 2210 | Theorem 19.16 of [Margaris... |
19.17 2211 | Theorem 19.17 of [Margaris... |
19.27 2212 | Theorem 19.27 of [Margaris... |
19.28 2213 | Theorem 19.28 of [Margaris... |
19.19 2214 | Theorem 19.19 of [Margaris... |
19.36 2215 | Theorem 19.36 of [Margaris... |
19.36i 2216 | Inference associated with ... |
19.37 2217 | Theorem 19.37 of [Margaris... |
19.32 2218 | Theorem 19.32 of [Margaris... |
19.31 2219 | Theorem 19.31 of [Margaris... |
19.41 2220 | Theorem 19.41 of [Margaris... |
19.42 2221 | Theorem 19.42 of [Margaris... |
19.44 2222 | Theorem 19.44 of [Margaris... |
19.45 2223 | Theorem 19.45 of [Margaris... |
spimfv 2224 | Specialization, using impl... |
chvarfv 2225 | Implicit substitution of `... |
cbv3v2 2226 | Version of ~ cbv3 with two... |
sbalex 2227 | Equivalence of two ways to... |
sb4av 2228 | Version of ~ sb4a with a d... |
sbimd 2229 | Deduction substituting bot... |
sbbid 2230 | Deduction substituting bot... |
2sbbid 2231 | Deduction doubly substitut... |
sbequ1 2232 | An equality theorem for su... |
sbequ2 2233 | An equality theorem for su... |
stdpc7 2234 | One of the two equality ax... |
sbequ12 2235 | An equality theorem for su... |
sbequ12r 2236 | An equality theorem for su... |
sbelx 2237 | Elimination of substitutio... |
sbequ12a 2238 | An equality theorem for su... |
sbid 2239 | An identity theorem for su... |
sbcov 2240 | A composition law for subs... |
sb6a 2241 | Equivalence for substituti... |
sbid2vw 2242 | Reverting substitution yie... |
axc16g 2243 | Generalization of ~ axc16 ... |
axc16 2244 | Proof of older axiom ~ ax-... |
axc16gb 2245 | Biconditional strengthenin... |
axc16nf 2246 | If ~ dtru is false, then t... |
axc11v 2247 | Version of ~ axc11 with a ... |
axc11rv 2248 | Version of ~ axc11r with a... |
drsb2 2249 | Formula-building lemma for... |
equsalv 2250 | An equivalence related to ... |
equsexv 2251 | An equivalence related to ... |
equsexvOLD 2252 | Obsolete version of ~ equs... |
sbft 2253 | Substitution has no effect... |
sbf 2254 | Substitution for a variabl... |
sbf2 2255 | Substitution has no effect... |
sbh 2256 | Substitution for a variabl... |
hbs1 2257 | The setvar ` x ` is not fr... |
nfs1f 2258 | If ` x ` is not free in ` ... |
sb5 2259 | Alternate definition of su... |
sb5OLD 2260 | Obsolete version of ~ sb5 ... |
sb56OLD 2261 | Obsolete version of ~ sbal... |
equs5av 2262 | A property related to subs... |
2sb5 2263 | Equivalence for double sub... |
sbco4lem 2264 | Lemma for ~ sbco4 . It re... |
sbco4lemOLD 2265 | Obsolete version of ~ sbco... |
sbco4 2266 | Two ways of exchanging two... |
dfsb7 2267 | An alternate definition of... |
sbn 2268 | Negation inside and outsid... |
sbex 2269 | Move existential quantifie... |
nf5 2270 | Alternate definition of ~ ... |
nf6 2271 | An alternate definition of... |
nf5d 2272 | Deduce that ` x ` is not f... |
nf5di 2273 | Since the converse holds b... |
19.9h 2274 | A wff may be existentially... |
19.21h 2275 | Theorem 19.21 of [Margaris... |
19.23h 2276 | Theorem 19.23 of [Margaris... |
exlimih 2277 | Inference associated with ... |
exlimdh 2278 | Deduction form of Theorem ... |
equsalhw 2279 | Version of ~ equsalh with ... |
equsexhv 2280 | An equivalence related to ... |
hba1 2281 | The setvar ` x ` is not fr... |
hbnt 2282 | Closed theorem version of ... |
hbn 2283 | If ` x ` is not free in ` ... |
hbnd 2284 | Deduction form of bound-va... |
hbim1 2285 | A closed form of ~ hbim . ... |
hbimd 2286 | Deduction form of bound-va... |
hbim 2287 | If ` x ` is not free in ` ... |
hban 2288 | If ` x ` is not free in ` ... |
hb3an 2289 | If ` x ` is not free in ` ... |
sbi2 2290 | Introduction of implicatio... |
sbim 2291 | Implication inside and out... |
sbrim 2292 | Substitution in an implica... |
sbrimOLD 2293 | Obsolete version of ~ sbri... |
sblim 2294 | Substitution in an implica... |
sbor 2295 | Disjunction inside and out... |
sbbi 2296 | Equivalence inside and out... |
sblbis 2297 | Introduce left bicondition... |
sbrbis 2298 | Introduce right biconditio... |
sbrbif 2299 | Introduce right biconditio... |
sbiev 2300 | Conversion of implicit sub... |
sbiedw 2301 | Conversion of implicit sub... |
axc7 2302 | Show that the original axi... |
axc7e 2303 | Abbreviated version of ~ a... |
modal-b 2304 | The analogue in our predic... |
19.9ht 2305 | A closed version of ~ 19.9... |
axc4 2306 | Show that the original axi... |
axc4i 2307 | Inference version of ~ axc... |
nfal 2308 | If ` x ` is not free in ` ... |
nfex 2309 | If ` x ` is not free in ` ... |
hbex 2310 | If ` x ` is not free in ` ... |
nfnf 2311 | If ` x ` is not free in ` ... |
19.12 2312 | Theorem 19.12 of [Margaris... |
nfald 2313 | Deduction form of ~ nfal .... |
nfexd 2314 | If ` x ` is not free in ` ... |
nfsbv 2315 | If ` z ` is not free in ` ... |
nfsbvOLD 2316 | Obsolete version of ~ nfsb... |
hbsbwOLD 2317 | Obsolete version of ~ hbsb... |
sbco2v 2318 | A composition law for subs... |
aaan 2319 | Distribute universal quant... |
aaanOLD 2320 | Obsolete version of ~ aaan... |
eeor 2321 | Distribute existential qua... |
eeorOLD 2322 | Obsolete version of ~ eeor... |
cbv3v 2323 | Rule used to change bound ... |
cbv1v 2324 | Rule used to change bound ... |
cbv2w 2325 | Rule used to change bound ... |
cbvaldw 2326 | Deduction used to change b... |
cbvexdw 2327 | Deduction used to change b... |
cbv3hv 2328 | Rule used to change bound ... |
cbvalv1 2329 | Rule used to change bound ... |
cbvexv1 2330 | Rule used to change bound ... |
cbval2v 2331 | Rule used to change bound ... |
cbvex2v 2332 | Rule used to change bound ... |
dvelimhw 2333 | Proof of ~ dvelimh without... |
pm11.53 2334 | Theorem *11.53 in [Whitehe... |
19.12vv 2335 | Special case of ~ 19.12 wh... |
eean 2336 | Distribute existential qua... |
eeanv 2337 | Distribute a pair of exist... |
eeeanv 2338 | Distribute three existenti... |
ee4anv 2339 | Distribute two pairs of ex... |
sb8v 2340 | Substitution of variable i... |
sb8f 2341 | Substitution of variable i... |
sb8fOLD 2342 | Obsolete version of ~ sb8f... |
sb8ef 2343 | Substitution of variable i... |
2sb8ef 2344 | An equivalent expression f... |
sb6rfv 2345 | Reversed substitution. Ve... |
sbnf2 2346 | Two ways of expressing " `... |
exsb 2347 | An equivalent expression f... |
2exsb 2348 | An equivalent expression f... |
sbbib 2349 | Reversal of substitution. ... |
sbbibvv 2350 | Reversal of substitution. ... |
cbvsbv 2351 | Change the bound variable ... |
cbvsbvf 2352 | Change the bound variable ... |
cleljustALT 2353 | Alternate proof of ~ clelj... |
cleljustALT2 2354 | Alternate proof of ~ clelj... |
equs5aALT 2355 | Alternate proof of ~ equs5... |
equs5eALT 2356 | Alternate proof of ~ equs5... |
axc11r 2357 | Same as ~ axc11 but with r... |
dral1v 2358 | Formula-building lemma for... |
dral1vOLD 2359 | Obsolete version of ~ dral... |
drex1v 2360 | Formula-building lemma for... |
drnf1v 2361 | Formula-building lemma for... |
drnf1vOLD 2362 | Obsolete version of ~ drnf... |
ax13v 2364 | A weaker version of ~ ax-1... |
ax13lem1 2365 | A version of ~ ax13v with ... |
ax13 2366 | Derive ~ ax-13 from ~ ax13... |
ax13lem2 2367 | Lemma for ~ nfeqf2 . This... |
nfeqf2 2368 | An equation between setvar... |
dveeq2 2369 | Quantifier introduction wh... |
nfeqf1 2370 | An equation between setvar... |
dveeq1 2371 | Quantifier introduction wh... |
nfeqf 2372 | A variable is effectively ... |
axc9 2373 | Derive set.mm's original ~... |
ax6e 2374 | At least one individual ex... |
ax6 2375 | Theorem showing that ~ ax-... |
axc10 2376 | Show that the original axi... |
spimt 2377 | Closed theorem form of ~ s... |
spim 2378 | Specialization, using impl... |
spimed 2379 | Deduction version of ~ spi... |
spime 2380 | Existential introduction, ... |
spimv 2381 | A version of ~ spim with a... |
spimvALT 2382 | Alternate proof of ~ spimv... |
spimev 2383 | Distinct-variable version ... |
spv 2384 | Specialization, using impl... |
spei 2385 | Inference from existential... |
chvar 2386 | Implicit substitution of `... |
chvarv 2387 | Implicit substitution of `... |
cbv3 2388 | Rule used to change bound ... |
cbval 2389 | Rule used to change bound ... |
cbvex 2390 | Rule used to change bound ... |
cbvalv 2391 | Rule used to change bound ... |
cbvexv 2392 | Rule used to change bound ... |
cbv1 2393 | Rule used to change bound ... |
cbv2 2394 | Rule used to change bound ... |
cbv3h 2395 | Rule used to change bound ... |
cbv1h 2396 | Rule used to change bound ... |
cbv2h 2397 | Rule used to change bound ... |
cbvald 2398 | Deduction used to change b... |
cbvexd 2399 | Deduction used to change b... |
cbvaldva 2400 | Rule used to change the bo... |
cbvexdva 2401 | Rule used to change the bo... |
cbval2 2402 | Rule used to change bound ... |
cbvex2 2403 | Rule used to change bound ... |
cbval2vv 2404 | Rule used to change bound ... |
cbvex2vv 2405 | Rule used to change bound ... |
cbvex4v 2406 | Rule used to change bound ... |
equs4 2407 | Lemma used in proofs of im... |
equsal 2408 | An equivalence related to ... |
equsex 2409 | An equivalence related to ... |
equsexALT 2410 | Alternate proof of ~ equse... |
equsalh 2411 | An equivalence related to ... |
equsexh 2412 | An equivalence related to ... |
axc15 2413 | Derivation of set.mm's ori... |
ax12 2414 | Rederivation of Axiom ~ ax... |
ax12b 2415 | A bidirectional version of... |
ax13ALT 2416 | Alternate proof of ~ ax13 ... |
axc11n 2417 | Derive set.mm's original ~... |
aecom 2418 | Commutation law for identi... |
aecoms 2419 | A commutation rule for ide... |
naecoms 2420 | A commutation rule for dis... |
axc11 2421 | Show that ~ ax-c11 can be ... |
hbae 2422 | All variables are effectiv... |
hbnae 2423 | All variables are effectiv... |
nfae 2424 | All variables are effectiv... |
nfnae 2425 | All variables are effectiv... |
hbnaes 2426 | Rule that applies ~ hbnae ... |
axc16i 2427 | Inference with ~ axc16 as ... |
axc16nfALT 2428 | Alternate proof of ~ axc16... |
dral2 2429 | Formula-building lemma for... |
dral1 2430 | Formula-building lemma for... |
dral1ALT 2431 | Alternate proof of ~ dral1... |
drex1 2432 | Formula-building lemma for... |
drex2 2433 | Formula-building lemma for... |
drnf1 2434 | Formula-building lemma for... |
drnf2 2435 | Formula-building lemma for... |
nfald2 2436 | Variation on ~ nfald which... |
nfexd2 2437 | Variation on ~ nfexd which... |
exdistrf 2438 | Distribution of existentia... |
dvelimf 2439 | Version of ~ dvelimv witho... |
dvelimdf 2440 | Deduction form of ~ dvelim... |
dvelimh 2441 | Version of ~ dvelim withou... |
dvelim 2442 | This theorem can be used t... |
dvelimv 2443 | Similar to ~ dvelim with f... |
dvelimnf 2444 | Version of ~ dvelim using ... |
dveeq2ALT 2445 | Alternate proof of ~ dveeq... |
equvini 2446 | A variable introduction la... |
equvel 2447 | A variable elimination law... |
equs5a 2448 | A property related to subs... |
equs5e 2449 | A property related to subs... |
equs45f 2450 | Two ways of expressing sub... |
equs5 2451 | Lemma used in proofs of su... |
dveel1 2452 | Quantifier introduction wh... |
dveel2 2453 | Quantifier introduction wh... |
axc14 2454 | Axiom ~ ax-c14 is redundan... |
sb6x 2455 | Equivalence involving subs... |
sbequ5 2456 | Substitution does not chan... |
sbequ6 2457 | Substitution does not chan... |
sb5rf 2458 | Reversed substitution. Us... |
sb6rf 2459 | Reversed substitution. Fo... |
ax12vALT 2460 | Alternate proof of ~ ax12v... |
2ax6elem 2461 | We can always find values ... |
2ax6e 2462 | We can always find values ... |
2sb5rf 2463 | Reversed double substituti... |
2sb6rf 2464 | Reversed double substituti... |
sbel2x 2465 | Elimination of double subs... |
sb4b 2466 | Simplified definition of s... |
sb3b 2467 | Simplified definition of s... |
sb3 2468 | One direction of a simplif... |
sb1 2469 | One direction of a simplif... |
sb2 2470 | One direction of a simplif... |
sb4a 2471 | A version of one implicati... |
dfsb1 2472 | Alternate definition of su... |
hbsb2 2473 | Bound-variable hypothesis ... |
nfsb2 2474 | Bound-variable hypothesis ... |
hbsb2a 2475 | Special case of a bound-va... |
sb4e 2476 | One direction of a simplif... |
hbsb2e 2477 | Special case of a bound-va... |
hbsb3 2478 | If ` y ` is not free in ` ... |
nfs1 2479 | If ` y ` is not free in ` ... |
axc16ALT 2480 | Alternate proof of ~ axc16... |
axc16gALT 2481 | Alternate proof of ~ axc16... |
equsb1 2482 | Substitution applied to an... |
equsb2 2483 | Substitution applied to an... |
dfsb2 2484 | An alternate definition of... |
dfsb3 2485 | An alternate definition of... |
drsb1 2486 | Formula-building lemma for... |
sb2ae 2487 | In the case of two success... |
sb6f 2488 | Equivalence for substituti... |
sb5f 2489 | Equivalence for substituti... |
nfsb4t 2490 | A variable not free in a p... |
nfsb4 2491 | A variable not free in a p... |
sbequ8 2492 | Elimination of equality fr... |
sbie 2493 | Conversion of implicit sub... |
sbied 2494 | Conversion of implicit sub... |
sbiedv 2495 | Conversion of implicit sub... |
2sbiev 2496 | Conversion of double impli... |
sbcom3 2497 | Substituting ` y ` for ` x... |
sbco 2498 | A composition law for subs... |
sbid2 2499 | An identity law for substi... |
sbid2v 2500 | An identity law for substi... |
sbidm 2501 | An idempotent law for subs... |
sbco2 2502 | A composition law for subs... |
sbco2d 2503 | A composition law for subs... |
sbco3 2504 | A composition law for subs... |
sbcom 2505 | A commutativity law for su... |
sbtrt 2506 | Partially closed form of ~... |
sbtr 2507 | A partial converse to ~ sb... |
sb8 2508 | Substitution of variable i... |
sb8e 2509 | Substitution of variable i... |
sb9 2510 | Commutation of quantificat... |
sb9i 2511 | Commutation of quantificat... |
sbhb 2512 | Two ways of expressing " `... |
nfsbd 2513 | Deduction version of ~ nfs... |
nfsb 2514 | If ` z ` is not free in ` ... |
hbsb 2515 | If ` z ` is not free in ` ... |
sb7f 2516 | This version of ~ dfsb7 do... |
sb7h 2517 | This version of ~ dfsb7 do... |
sb10f 2518 | Hao Wang's identity axiom ... |
sbal1 2519 | Check out ~ sbal for a ver... |
sbal2 2520 | Move quantifier in and out... |
2sb8e 2521 | An equivalent expression f... |
dfmoeu 2522 | An elementary proof of ~ m... |
dfeumo 2523 | An elementary proof showin... |
mojust 2525 | Soundness justification th... |
nexmo 2527 | Nonexistence implies uniqu... |
exmo 2528 | Any proposition holds for ... |
moabs 2529 | Absorption of existence co... |
moim 2530 | The at-most-one quantifier... |
moimi 2531 | The at-most-one quantifier... |
moimdv 2532 | The at-most-one quantifier... |
mobi 2533 | Equivalence theorem for th... |
mobii 2534 | Formula-building rule for ... |
mobidv 2535 | Formula-building rule for ... |
mobid 2536 | Formula-building rule for ... |
moa1 2537 | If an implication holds fo... |
moan 2538 | "At most one" is still the... |
moani 2539 | "At most one" is still tru... |
moor 2540 | "At most one" is still the... |
mooran1 2541 | "At most one" imports disj... |
mooran2 2542 | "At most one" exports disj... |
nfmo1 2543 | Bound-variable hypothesis ... |
nfmod2 2544 | Bound-variable hypothesis ... |
nfmodv 2545 | Bound-variable hypothesis ... |
nfmov 2546 | Bound-variable hypothesis ... |
nfmod 2547 | Bound-variable hypothesis ... |
nfmo 2548 | Bound-variable hypothesis ... |
mof 2549 | Version of ~ df-mo with di... |
mo3 2550 | Alternate definition of th... |
mo 2551 | Equivalent definitions of ... |
mo4 2552 | At-most-one quantifier exp... |
mo4f 2553 | At-most-one quantifier exp... |
eu3v 2556 | An alternate way to expres... |
eujust 2557 | Soundness justification th... |
eujustALT 2558 | Alternate proof of ~ eujus... |
eu6lem 2559 | Lemma of ~ eu6im . A diss... |
eu6 2560 | Alternate definition of th... |
eu6im 2561 | One direction of ~ eu6 nee... |
euf 2562 | Version of ~ eu6 with disj... |
euex 2563 | Existential uniqueness imp... |
eumo 2564 | Existential uniqueness imp... |
eumoi 2565 | Uniqueness inferred from e... |
exmoeub 2566 | Existence implies that uni... |
exmoeu 2567 | Existence is equivalent to... |
moeuex 2568 | Uniqueness implies that ex... |
moeu 2569 | Uniqueness is equivalent t... |
eubi 2570 | Equivalence theorem for th... |
eubii 2571 | Introduce unique existenti... |
eubidv 2572 | Formula-building rule for ... |
eubid 2573 | Formula-building rule for ... |
nfeu1 2574 | Bound-variable hypothesis ... |
nfeu1ALT 2575 | Alternate proof of ~ nfeu1... |
nfeud2 2576 | Bound-variable hypothesis ... |
nfeudw 2577 | Bound-variable hypothesis ... |
nfeud 2578 | Bound-variable hypothesis ... |
nfeuw 2579 | Bound-variable hypothesis ... |
nfeu 2580 | Bound-variable hypothesis ... |
dfeu 2581 | Rederive ~ df-eu from the ... |
dfmo 2582 | Rederive ~ df-mo from the ... |
euequ 2583 | There exists a unique set ... |
sb8eulem 2584 | Lemma. Factor out the com... |
sb8euv 2585 | Variable substitution in u... |
sb8eu 2586 | Variable substitution in u... |
sb8mo 2587 | Variable substitution for ... |
cbvmovw 2588 | Change bound variable. Us... |
cbvmow 2589 | Rule used to change bound ... |
cbvmowOLD 2590 | Obsolete version of ~ cbvm... |
cbvmo 2591 | Rule used to change bound ... |
cbveuvw 2592 | Change bound variable. Us... |
cbveuw 2593 | Version of ~ cbveu with a ... |
cbveuwOLD 2594 | Obsolete version of ~ cbve... |
cbveu 2595 | Rule used to change bound ... |
cbveuALT 2596 | Alternative proof of ~ cbv... |
eu2 2597 | An alternate way of defini... |
eu1 2598 | An alternate way to expres... |
euor 2599 | Introduce a disjunct into ... |
euorv 2600 | Introduce a disjunct into ... |
euor2 2601 | Introduce or eliminate a d... |
sbmo 2602 | Substitution into an at-mo... |
eu4 2603 | Uniqueness using implicit ... |
euimmo 2604 | Existential uniqueness imp... |
euim 2605 | Add unique existential qua... |
moanimlem 2606 | Factor out the common proo... |
moanimv 2607 | Introduction of a conjunct... |
moanim 2608 | Introduction of a conjunct... |
euan 2609 | Introduction of a conjunct... |
moanmo 2610 | Nested at-most-one quantif... |
moaneu 2611 | Nested at-most-one and uni... |
euanv 2612 | Introduction of a conjunct... |
mopick 2613 | "At most one" picks a vari... |
moexexlem 2614 | Factor out the proof skele... |
2moexv 2615 | Double quantification with... |
moexexvw 2616 | "At most one" double quant... |
2moswapv 2617 | A condition allowing to sw... |
2euswapv 2618 | A condition allowing to sw... |
2euexv 2619 | Double quantification with... |
2exeuv 2620 | Double existential uniquen... |
eupick 2621 | Existential uniqueness "pi... |
eupicka 2622 | Version of ~ eupick with c... |
eupickb 2623 | Existential uniqueness "pi... |
eupickbi 2624 | Theorem *14.26 in [Whitehe... |
mopick2 2625 | "At most one" can show the... |
moexex 2626 | "At most one" double quant... |
moexexv 2627 | "At most one" double quant... |
2moex 2628 | Double quantification with... |
2euex 2629 | Double quantification with... |
2eumo 2630 | Nested unique existential ... |
2eu2ex 2631 | Double existential uniquen... |
2moswap 2632 | A condition allowing to sw... |
2euswap 2633 | A condition allowing to sw... |
2exeu 2634 | Double existential uniquen... |
2mo2 2635 | Two ways of expressing "th... |
2mo 2636 | Two ways of expressing "th... |
2mos 2637 | Double "there exists at mo... |
2eu1 2638 | Double existential uniquen... |
2eu1v 2639 | Double existential uniquen... |
2eu2 2640 | Double existential uniquen... |
2eu3 2641 | Double existential uniquen... |
2eu4 2642 | This theorem provides us w... |
2eu5 2643 | An alternate definition of... |
2eu6 2644 | Two equivalent expressions... |
2eu7 2645 | Two equivalent expressions... |
2eu8 2646 | Two equivalent expressions... |
euae 2647 | Two ways to express "exact... |
exists1 2648 | Two ways to express "exact... |
exists2 2649 | A condition implying that ... |
barbara 2650 | "Barbara", one of the fund... |
celarent 2651 | "Celarent", one of the syl... |
darii 2652 | "Darii", one of the syllog... |
dariiALT 2653 | Alternate proof of ~ darii... |
ferio 2654 | "Ferio" ("Ferioque"), one ... |
barbarilem 2655 | Lemma for ~ barbari and th... |
barbari 2656 | "Barbari", one of the syll... |
barbariALT 2657 | Alternate proof of ~ barba... |
celaront 2658 | "Celaront", one of the syl... |
cesare 2659 | "Cesare", one of the syllo... |
camestres 2660 | "Camestres", one of the sy... |
festino 2661 | "Festino", one of the syll... |
festinoALT 2662 | Alternate proof of ~ festi... |
baroco 2663 | "Baroco", one of the syllo... |
barocoALT 2664 | Alternate proof of ~ festi... |
cesaro 2665 | "Cesaro", one of the syllo... |
camestros 2666 | "Camestros", one of the sy... |
datisi 2667 | "Datisi", one of the syllo... |
disamis 2668 | "Disamis", one of the syll... |
ferison 2669 | "Ferison", one of the syll... |
bocardo 2670 | "Bocardo", one of the syll... |
darapti 2671 | "Darapti", one of the syll... |
daraptiALT 2672 | Alternate proof of ~ darap... |
felapton 2673 | "Felapton", one of the syl... |
calemes 2674 | "Calemes", one of the syll... |
dimatis 2675 | "Dimatis", one of the syll... |
fresison 2676 | "Fresison", one of the syl... |
calemos 2677 | "Calemos", one of the syll... |
fesapo 2678 | "Fesapo", one of the syllo... |
bamalip 2679 | "Bamalip", one of the syll... |
axia1 2680 | Left 'and' elimination (in... |
axia2 2681 | Right 'and' elimination (i... |
axia3 2682 | 'And' introduction (intuit... |
axin1 2683 | 'Not' introduction (intuit... |
axin2 2684 | 'Not' elimination (intuiti... |
axio 2685 | Definition of 'or' (intuit... |
axi4 2686 | Specialization (intuitioni... |
axi5r 2687 | Converse of ~ axc4 (intuit... |
axial 2688 | The setvar ` x ` is not fr... |
axie1 2689 | The setvar ` x ` is not fr... |
axie2 2690 | A key property of existent... |
axi9 2691 | Axiom of existence (intuit... |
axi10 2692 | Axiom of Quantifier Substi... |
axi12 2693 | Axiom of Quantifier Introd... |
axbnd 2694 | Axiom of Bundling (intuiti... |
axexte 2696 | The axiom of extensionalit... |
axextg 2697 | A generalization of the ax... |
axextb 2698 | A bidirectional version of... |
axextmo 2699 | There exists at most one s... |
nulmo 2700 | There exists at most one e... |
eleq1ab 2703 | Extension (in the sense of... |
cleljustab 2704 | Extension of ~ cleljust fr... |
abid 2705 | Simplification of class ab... |
vexwt 2706 | A standard theorem of pred... |
vexw 2707 | If ` ph ` is a theorem, th... |
vextru 2708 | Every setvar is a member o... |
nfsab1 2709 | Bound-variable hypothesis ... |
hbab1 2710 | Bound-variable hypothesis ... |
hbab1OLD 2711 | Obsolete version of ~ hbab... |
hbab 2712 | Bound-variable hypothesis ... |
hbabg 2713 | Bound-variable hypothesis ... |
nfsab 2714 | Bound-variable hypothesis ... |
nfsabg 2715 | Bound-variable hypothesis ... |
dfcleq 2717 | The defining characterizat... |
cvjust 2718 | Every set is a class. Pro... |
ax9ALT 2719 | Proof of ~ ax-9 from Tarsk... |
eleq2w2 2720 | A weaker version of ~ eleq... |
eqriv 2721 | Infer equality of classes ... |
eqrdv 2722 | Deduce equality of classes... |
eqrdav 2723 | Deduce equality of classes... |
eqid 2724 | Law of identity (reflexivi... |
eqidd 2725 | Class identity law with an... |
eqeq1d 2726 | Deduction from equality to... |
eqeq1dALT 2727 | Alternate proof of ~ eqeq1... |
eqeq1 2728 | Equality implies equivalen... |
eqeq1i 2729 | Inference from equality to... |
eqcomd 2730 | Deduction from commutative... |
eqcom 2731 | Commutative law for class ... |
eqcoms 2732 | Inference applying commuta... |
eqcomi 2733 | Inference from commutative... |
neqcomd 2734 | Commute an inequality. (C... |
eqeq2d 2735 | Deduction from equality to... |
eqeq2 2736 | Equality implies equivalen... |
eqeq2i 2737 | Inference from equality to... |
eqeqan12d 2738 | A useful inference for sub... |
eqeqan12rd 2739 | A useful inference for sub... |
eqeq12d 2740 | A useful inference for sub... |
eqeq12 2741 | Equality relationship amon... |
eqeq12i 2742 | A useful inference for sub... |
eqeq12OLD 2743 | Obsolete version of ~ eqeq... |
eqeq12dOLD 2744 | Obsolete version of ~ eqeq... |
eqeqan12dOLD 2745 | Obsolete version of ~ eqeq... |
eqeqan12dALT 2746 | Alternate proof of ~ eqeqa... |
eqtr 2747 | Transitive law for class e... |
eqtr2 2748 | A transitive law for class... |
eqtr2OLD 2749 | Obsolete version of eqtr2 ... |
eqtr3 2750 | A transitive law for class... |
eqtr3OLD 2751 | Obsolete version of ~ eqtr... |
eqtri 2752 | An equality transitivity i... |
eqtr2i 2753 | An equality transitivity i... |
eqtr3i 2754 | An equality transitivity i... |
eqtr4i 2755 | An equality transitivity i... |
3eqtri 2756 | An inference from three ch... |
3eqtrri 2757 | An inference from three ch... |
3eqtr2i 2758 | An inference from three ch... |
3eqtr2ri 2759 | An inference from three ch... |
3eqtr3i 2760 | An inference from three ch... |
3eqtr3ri 2761 | An inference from three ch... |
3eqtr4i 2762 | An inference from three ch... |
3eqtr4ri 2763 | An inference from three ch... |
eqtrd 2764 | An equality transitivity d... |
eqtr2d 2765 | An equality transitivity d... |
eqtr3d 2766 | An equality transitivity e... |
eqtr4d 2767 | An equality transitivity e... |
3eqtrd 2768 | A deduction from three cha... |
3eqtrrd 2769 | A deduction from three cha... |
3eqtr2d 2770 | A deduction from three cha... |
3eqtr2rd 2771 | A deduction from three cha... |
3eqtr3d 2772 | A deduction from three cha... |
3eqtr3rd 2773 | A deduction from three cha... |
3eqtr4d 2774 | A deduction from three cha... |
3eqtr4rd 2775 | A deduction from three cha... |
eqtrid 2776 | An equality transitivity d... |
eqtr2id 2777 | An equality transitivity d... |
eqtr3id 2778 | An equality transitivity d... |
eqtr3di 2779 | An equality transitivity d... |
eqtrdi 2780 | An equality transitivity d... |
eqtr2di 2781 | An equality transitivity d... |
eqtr4di 2782 | An equality transitivity d... |
eqtr4id 2783 | An equality transitivity d... |
sylan9eq 2784 | An equality transitivity d... |
sylan9req 2785 | An equality transitivity d... |
sylan9eqr 2786 | An equality transitivity d... |
3eqtr3g 2787 | A chained equality inferen... |
3eqtr3a 2788 | A chained equality inferen... |
3eqtr4g 2789 | A chained equality inferen... |
3eqtr4a 2790 | A chained equality inferen... |
eq2tri 2791 | A compound transitive infe... |
abbi 2792 | Equivalent formulas yield ... |
abbidv 2793 | Equivalent wff's yield equ... |
abbii 2794 | Equivalent wff's yield equ... |
abbid 2795 | Equivalent wff's yield equ... |
abbib 2796 | Equal class abstractions r... |
cbvabv 2797 | Rule used to change bound ... |
cbvabw 2798 | Rule used to change bound ... |
cbvabwOLD 2799 | Obsolete version of ~ cbva... |
cbvab 2800 | Rule used to change bound ... |
eqabbw 2801 | Version of ~ eqabb using i... |
dfclel 2803 | Characterization of the el... |
elex2 2804 | If a class contains anothe... |
issetlem 2805 | Lemma for ~ elisset and ~ ... |
elissetv 2806 | An element of a class exis... |
elisset 2807 | An element of a class exis... |
eleq1w 2808 | Weaker version of ~ eleq1 ... |
eleq2w 2809 | Weaker version of ~ eleq2 ... |
eleq1d 2810 | Deduction from equality to... |
eleq2d 2811 | Deduction from equality to... |
eleq2dALT 2812 | Alternate proof of ~ eleq2... |
eleq1 2813 | Equality implies equivalen... |
eleq2 2814 | Equality implies equivalen... |
eleq12 2815 | Equality implies equivalen... |
eleq1i 2816 | Inference from equality to... |
eleq2i 2817 | Inference from equality to... |
eleq12i 2818 | Inference from equality to... |
eleq12d 2819 | Deduction from equality to... |
eleq1a 2820 | A transitive-type law rela... |
eqeltri 2821 | Substitution of equal clas... |
eqeltrri 2822 | Substitution of equal clas... |
eleqtri 2823 | Substitution of equal clas... |
eleqtrri 2824 | Substitution of equal clas... |
eqeltrd 2825 | Substitution of equal clas... |
eqeltrrd 2826 | Deduction that substitutes... |
eleqtrd 2827 | Deduction that substitutes... |
eleqtrrd 2828 | Deduction that substitutes... |
eqeltrid 2829 | A membership and equality ... |
eqeltrrid 2830 | A membership and equality ... |
eleqtrid 2831 | A membership and equality ... |
eleqtrrid 2832 | A membership and equality ... |
eqeltrdi 2833 | A membership and equality ... |
eqeltrrdi 2834 | A membership and equality ... |
eleqtrdi 2835 | A membership and equality ... |
eleqtrrdi 2836 | A membership and equality ... |
3eltr3i 2837 | Substitution of equal clas... |
3eltr4i 2838 | Substitution of equal clas... |
3eltr3d 2839 | Substitution of equal clas... |
3eltr4d 2840 | Substitution of equal clas... |
3eltr3g 2841 | Substitution of equal clas... |
3eltr4g 2842 | Substitution of equal clas... |
eleq2s 2843 | Substitution of equal clas... |
eqneltri 2844 | If a class is not an eleme... |
eqneltrd 2845 | If a class is not an eleme... |
eqneltrrd 2846 | If a class is not an eleme... |
neleqtrd 2847 | If a class is not an eleme... |
neleqtrrd 2848 | If a class is not an eleme... |
nelneq 2849 | A way of showing two class... |
nelneq2 2850 | A way of showing two class... |
eqsb1 2851 | Substitution for the left-... |
clelsb1 2852 | Substitution for the first... |
clelsb2 2853 | Substitution for the secon... |
clelsb2OLD 2854 | Obsolete version of ~ clel... |
cleqh 2855 | Establish equality between... |
hbxfreq 2856 | A utility lemma to transfe... |
hblem 2857 | Change the free variable o... |
hblemg 2858 | Change the free variable o... |
eqabdv 2859 | Deduction from a wff to a ... |
eqabcdv 2860 | Deduction from a wff to a ... |
eqabi 2861 | Equality of a class variab... |
abid1 2862 | Every class is equal to a ... |
abid2 2863 | A simplification of class ... |
eqab 2864 | One direction of ~ eqabb i... |
eqabb 2865 | Equality of a class variab... |
eqabbOLD 2866 | Obsolete version of ~ eqab... |
eqabcb 2867 | Equality of a class variab... |
eqabrd 2868 | Equality of a class variab... |
eqabri 2869 | Equality of a class variab... |
eqabcri 2870 | Equality of a class variab... |
clelab 2871 | Membership of a class vari... |
clelabOLD 2872 | Obsolete version of ~ clel... |
clabel 2873 | Membership of a class abst... |
sbab 2874 | The right-hand side of the... |
nfcjust 2876 | Justification theorem for ... |
nfci 2878 | Deduce that a class ` A ` ... |
nfcii 2879 | Deduce that a class ` A ` ... |
nfcr 2880 | Consequence of the not-fre... |
nfcrALT 2881 | Alternate version of ~ nfc... |
nfcri 2882 | Consequence of the not-fre... |
nfcd 2883 | Deduce that a class ` A ` ... |
nfcrd 2884 | Consequence of the not-fre... |
nfcriOLD 2885 | Obsolete version of ~ nfcr... |
nfcriOLDOLD 2886 | Obsolete version of ~ nfcr... |
nfcrii 2887 | Consequence of the not-fre... |
nfcriiOLD 2888 | Obsolete version of ~ nfcr... |
nfcriOLDOLDOLD 2889 | Obsolete version of ~ nfcr... |
nfceqdf 2890 | An equality theorem for ef... |
nfceqdfOLD 2891 | Obsolete version of ~ nfce... |
nfceqi 2892 | Equality theorem for class... |
nfcxfr 2893 | A utility lemma to transfe... |
nfcxfrd 2894 | A utility lemma to transfe... |
nfcv 2895 | If ` x ` is disjoint from ... |
nfcvd 2896 | If ` x ` is disjoint from ... |
nfab1 2897 | Bound-variable hypothesis ... |
nfnfc1 2898 | The setvar ` x ` is bound ... |
clelsb1fw 2899 | Substitution for the first... |
clelsb1f 2900 | Substitution for the first... |
nfab 2901 | Bound-variable hypothesis ... |
nfabg 2902 | Bound-variable hypothesis ... |
nfaba1 2903 | Bound-variable hypothesis ... |
nfaba1g 2904 | Bound-variable hypothesis ... |
nfeqd 2905 | Hypothesis builder for equ... |
nfeld 2906 | Hypothesis builder for ele... |
nfnfc 2907 | Hypothesis builder for ` F... |
nfeq 2908 | Hypothesis builder for equ... |
nfel 2909 | Hypothesis builder for ele... |
nfeq1 2910 | Hypothesis builder for equ... |
nfel1 2911 | Hypothesis builder for ele... |
nfeq2 2912 | Hypothesis builder for equ... |
nfel2 2913 | Hypothesis builder for ele... |
drnfc1 2914 | Formula-building lemma for... |
drnfc1OLD 2915 | Obsolete version of ~ drnf... |
drnfc2 2916 | Formula-building lemma for... |
drnfc2OLD 2917 | Obsolete version of ~ drnf... |
nfabdw 2918 | Bound-variable hypothesis ... |
nfabdwOLD 2919 | Obsolete version of ~ nfab... |
nfabd 2920 | Bound-variable hypothesis ... |
nfabd2 2921 | Bound-variable hypothesis ... |
dvelimdc 2922 | Deduction form of ~ dvelim... |
dvelimc 2923 | Version of ~ dvelim for cl... |
nfcvf 2924 | If ` x ` and ` y ` are dis... |
nfcvf2 2925 | If ` x ` and ` y ` are dis... |
cleqf 2926 | Establish equality between... |
eqabf 2927 | Equality of a class variab... |
abid2f 2928 | A simplification of class ... |
abid2fOLD 2929 | Obsolete version of ~ abid... |
sbabel 2930 | Theorem to move a substitu... |
sbabelOLD 2931 | Obsolete version of ~ sbab... |
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... |
necon3aiOLD 2958 | Obsolete version of ~ neco... |
necon3bi 2959 | Contrapositive inference f... |
necon1ai 2960 | Contrapositive inference f... |
necon1bi 2961 | Contrapositive inference f... |
necon2ai 2962 | Contrapositive inference f... |
necon2bi 2963 | Contrapositive inference f... |
necon4ai 2964 | Contrapositive inference f... |
necon3i 2965 | Contrapositive inference f... |
necon1i 2966 | Contrapositive inference f... |
necon2i 2967 | Contrapositive inference f... |
necon4i 2968 | Contrapositive inference f... |
necon3abid 2969 | Deduction from equality to... |
necon3bbid 2970 | Deduction from equality to... |
necon1abid 2971 | Contrapositive deduction f... |
necon1bbid 2972 | Contrapositive inference f... |
necon4abid 2973 | Contrapositive law deducti... |
necon4bbid 2974 | Contrapositive law deducti... |
necon2abid 2975 | Contrapositive deduction f... |
necon2bbid 2976 | Contrapositive deduction f... |
necon3bid 2977 | Deduction from equality to... |
necon4bid 2978 | Contrapositive law deducti... |
necon3abii 2979 | Deduction from equality to... |
necon3bbii 2980 | Deduction from equality to... |
necon1abii 2981 | Contrapositive inference f... |
necon1bbii 2982 | Contrapositive inference f... |
necon2abii 2983 | Contrapositive inference f... |
necon2bbii 2984 | Contrapositive inference f... |
necon3bii 2985 | Inference from equality to... |
necom 2986 | Commutation of inequality.... |
necomi 2987 | Inference from commutative... |
necomd 2988 | Deduction from commutative... |
nesym 2989 | Characterization of inequa... |
nesymi 2990 | Inference associated with ... |
nesymir 2991 | Inference associated with ... |
neeq1d 2992 | Deduction for inequality. ... |
neeq2d 2993 | Deduction for inequality. ... |
neeq12d 2994 | Deduction for inequality. ... |
neeq1 2995 | Equality theorem for inequ... |
neeq2 2996 | Equality theorem for inequ... |
neeq1i 2997 | Inference for inequality. ... |
neeq2i 2998 | Inference for inequality. ... |
neeq12i 2999 | Inference for inequality. ... |
eqnetrd 3000 | Substitution of equal clas... |
eqnetrrd 3001 | Substitution of equal clas... |
neeqtrd 3002 | Substitution of equal clas... |
eqnetri 3003 | Substitution of equal clas... |
eqnetrri 3004 | Substitution of equal clas... |
neeqtri 3005 | Substitution of equal clas... |
neeqtrri 3006 | Substitution of equal clas... |
neeqtrrd 3007 | Substitution of equal clas... |
eqnetrrid 3008 | A chained equality inferen... |
3netr3d 3009 | Substitution of equality i... |
3netr4d 3010 | Substitution of equality i... |
3netr3g 3011 | Substitution of equality i... |
3netr4g 3012 | Substitution of equality i... |
nebi 3013 | Contraposition law for ine... |
pm13.18 3014 | Theorem *13.18 in [Whitehe... |
pm13.181 3015 | Theorem *13.181 in [Whiteh... |
pm13.181OLD 3016 | Obsolete version of ~ pm13... |
pm2.61ine 3017 | Inference eliminating an i... |
pm2.21ddne 3018 | A contradiction implies an... |
pm2.61ne 3019 | Deduction eliminating an i... |
pm2.61dne 3020 | Deduction eliminating an i... |
pm2.61dane 3021 | Deduction eliminating an i... |
pm2.61da2ne 3022 | Deduction eliminating two ... |
pm2.61da3ne 3023 | Deduction eliminating thre... |
pm2.61iine 3024 | Equality version of ~ pm2.... |
mteqand 3025 | A modus tollens deduction ... |
neor 3026 | Logical OR with an equalit... |
neanior 3027 | A De Morgan's law for ineq... |
ne3anior 3028 | A De Morgan's law for ineq... |
neorian 3029 | A De Morgan's law for ineq... |
nemtbir 3030 | An inference from an inequ... |
nelne1 3031 | Two classes are different ... |
nelne2 3032 | Two classes are different ... |
nelelne 3033 | Two classes are different ... |
neneor 3034 | If two classes are differe... |
nfne 3035 | Bound-variable hypothesis ... |
nfned 3036 | Bound-variable hypothesis ... |
nabbib 3037 | Not equivalent wff's corre... |
neli 3040 | Inference associated with ... |
nelir 3041 | Inference associated with ... |
nelcon3d 3042 | Contrapositive law deducti... |
neleq12d 3043 | Equality theorem for negat... |
neleq1 3044 | Equality theorem for negat... |
neleq2 3045 | Equality theorem for negat... |
nfnel 3046 | Bound-variable hypothesis ... |
nfneld 3047 | Bound-variable hypothesis ... |
nnel 3048 | Negation of negated member... |
elnelne1 3049 | Two classes are different ... |
elnelne2 3050 | Two classes are different ... |
pm2.24nel 3051 | A contradiction concerning... |
pm2.61danel 3052 | Deduction eliminating an e... |
rgen 3055 | Generalization rule for re... |
ralel 3056 | All elements of a class ar... |
rgenw 3057 | Generalization rule for re... |
rgen2w 3058 | Generalization rule for re... |
mprg 3059 | Modus ponens combined with... |
mprgbir 3060 | Modus ponens on biconditio... |
raln 3061 | Restricted universally qua... |
ralnex 3064 | Relationship between restr... |
dfrex2 3065 | Relationship between restr... |
nrex 3066 | Inference adding restricte... |
alral 3067 | Universal quantification i... |
rexex 3068 | Restricted existence impli... |
rextru 3069 | Two ways of expressing tha... |
ralimi2 3070 | Inference quantifying both... |
reximi2 3071 | Inference quantifying both... |
ralimia 3072 | Inference quantifying both... |
reximia 3073 | Inference quantifying both... |
ralimiaa 3074 | Inference quantifying both... |
ralimi 3075 | Inference quantifying both... |
reximi 3076 | Inference quantifying both... |
ral2imi 3077 | Inference quantifying ante... |
ralim 3078 | Distribution of restricted... |
rexim 3079 | Theorem 19.22 of [Margaris... |
reximiaOLD 3080 | Obsolete version of ~ rexi... |
ralbii2 3081 | Inference adding different... |
rexbii2 3082 | Inference adding different... |
ralbiia 3083 | Inference adding restricte... |
rexbiia 3084 | Inference adding restricte... |
ralbii 3085 | Inference adding restricte... |
rexbii 3086 | Inference adding restricte... |
ralanid 3087 | Cancellation law for restr... |
rexanid 3088 | Cancellation law for restr... |
ralcom3 3089 | A commutation law for rest... |
ralcom3OLD 3090 | Obsolete version of ~ ralc... |
dfral2 3091 | Relationship between restr... |
rexnal 3092 | Relationship between restr... |
ralinexa 3093 | A transformation of restri... |
rexanali 3094 | A transformation of restri... |
ralbi 3095 | Distribute a restricted un... |
rexbi 3096 | Distribute restricted quan... |
rexbiOLD 3097 | Obsolete version of ~ rexb... |
ralrexbid 3098 | Formula-building rule for ... |
ralrexbidOLD 3099 | Obsolete version of ~ ralr... |
r19.35 3100 | Restricted quantifier vers... |
r19.35OLD 3101 | Obsolete version of ~ 19.3... |
r19.26m 3102 | Version of ~ 19.26 and ~ r... |
r19.26 3103 | Restricted quantifier vers... |
r19.26-3 3104 | Version of ~ r19.26 with t... |
ralbiim 3105 | Split a biconditional and ... |
r19.29 3106 | Restricted quantifier vers... |
r19.29OLD 3107 | Obsolete version of ~ r19.... |
r19.29r 3108 | Restricted quantifier vers... |
r19.29rOLD 3109 | Obsolete version of ~ r19.... |
r19.29imd 3110 | Theorem 19.29 of [Margaris... |
r19.40 3111 | Restricted quantifier vers... |
r19.30 3112 | Restricted quantifier vers... |
r19.30OLD 3113 | Obsolete version of ~ 19.3... |
r19.43 3114 | Restricted quantifier vers... |
2ralimi 3115 | Inference quantifying both... |
3ralimi 3116 | Inference quantifying both... |
4ralimi 3117 | Inference quantifying both... |
5ralimi 3118 | Inference quantifying both... |
6ralimi 3119 | Inference quantifying both... |
2ralbii 3120 | Inference adding two restr... |
2rexbii 3121 | Inference adding two restr... |
3ralbii 3122 | Inference adding three res... |
4ralbii 3123 | Inference adding four rest... |
2ralbiim 3124 | Split a biconditional and ... |
ralnex2 3125 | Relationship between two r... |
ralnex3 3126 | Relationship between three... |
rexnal2 3127 | Relationship between two r... |
rexnal3 3128 | Relationship between three... |
nrexralim 3129 | Negation of a complex pred... |
r19.26-2 3130 | Restricted quantifier vers... |
2r19.29 3131 | Theorem ~ r19.29 with two ... |
r19.29d2r 3132 | Theorem 19.29 of [Margaris... |
r19.29d2rOLD 3133 | Obsolete version of ~ r19.... |
r2allem 3134 | Lemma factoring out common... |
r2exlem 3135 | Lemma factoring out common... |
hbralrimi 3136 | Inference from Theorem 19.... |
ralrimiv 3137 | Inference from Theorem 19.... |
ralrimiva 3138 | Inference from Theorem 19.... |
rexlimiva 3139 | Inference from Theorem 19.... |
rexlimiv 3140 | Inference from Theorem 19.... |
nrexdv 3141 | Deduction adding restricte... |
ralrimivw 3142 | Inference from Theorem 19.... |
rexlimivw 3143 | Weaker version of ~ rexlim... |
ralrimdv 3144 | Inference from Theorem 19.... |
rexlimdv 3145 | Inference from Theorem 19.... |
ralrimdva 3146 | Inference from Theorem 19.... |
rexlimdva 3147 | Inference from Theorem 19.... |
rexlimdvaa 3148 | Inference from Theorem 19.... |
rexlimdva2 3149 | Inference from Theorem 19.... |
r19.29an 3150 | A commonly used pattern in... |
rexlimdv3a 3151 | Inference from Theorem 19.... |
rexlimdvw 3152 | Inference from Theorem 19.... |
rexlimddv 3153 | Restricted existential eli... |
r19.29a 3154 | A commonly used pattern in... |
ralimdv2 3155 | Inference quantifying both... |
reximdv2 3156 | Deduction quantifying both... |
reximdvai 3157 | Deduction quantifying both... |
reximdvaiOLD 3158 | Obsolete version of ~ rexi... |
ralimdva 3159 | Deduction quantifying both... |
reximdva 3160 | Deduction quantifying both... |
ralimdv 3161 | Deduction quantifying both... |
reximdv 3162 | Deduction from Theorem 19.... |
reximddv 3163 | Deduction from Theorem 19.... |
reximssdv 3164 | Derivation of a restricted... |
ralbidv2 3165 | Formula-building rule for ... |
rexbidv2 3166 | Formula-building rule for ... |
ralbidva 3167 | Formula-building rule for ... |
rexbidva 3168 | Formula-building rule for ... |
ralbidv 3169 | Formula-building rule for ... |
rexbidv 3170 | Formula-building rule for ... |
r19.21v 3171 | Restricted quantifier vers... |
r19.21vOLD 3172 | Obsolete version of ~ r19.... |
r19.37v 3173 | Restricted quantifier vers... |
r19.23v 3174 | Restricted quantifier vers... |
r19.36v 3175 | Restricted quantifier vers... |
rexlimivOLD 3176 | Obsolete version of ~ rexl... |
rexlimivaOLD 3177 | Obsolete version of ~ rexl... |
rexlimivwOLD 3178 | Obsolete version of ~ rexl... |
r19.27v 3179 | Restricted quantitifer ver... |
r19.41v 3180 | Restricted quantifier vers... |
r19.28v 3181 | Restricted quantifier vers... |
r19.42v 3182 | Restricted quantifier vers... |
r19.32v 3183 | Restricted quantifier vers... |
r19.45v 3184 | Restricted quantifier vers... |
r19.44v 3185 | One direction of a restric... |
r2al 3186 | Double restricted universa... |
r2ex 3187 | Double restricted existent... |
r3al 3188 | Triple restricted universa... |
rgen2 3189 | Generalization rule for re... |
ralrimivv 3190 | Inference from Theorem 19.... |
rexlimivv 3191 | Inference from Theorem 19.... |
ralrimivva 3192 | Inference from Theorem 19.... |
ralrimdvv 3193 | Inference from Theorem 19.... |
rgen3 3194 | Generalization rule for re... |
ralrimivvva 3195 | Inference from Theorem 19.... |
ralimdvva 3196 | Deduction doubly quantifyi... |
reximdvva 3197 | Deduction doubly quantifyi... |
ralimdvv 3198 | Deduction doubly quantifyi... |
ralimd4v 3199 | Deduction quadrupally quan... |
ralimd6v 3200 | Deduction sextupally quant... |
ralrimdvva 3201 | Inference from Theorem 19.... |
rexlimdvv 3202 | Inference from Theorem 19.... |
rexlimdvva 3203 | Inference from Theorem 19.... |
reximddv2 3204 | Double deduction from Theo... |
r19.29vva 3205 | A commonly used pattern ba... |
r19.29vvaOLD 3206 | Obsolete version of ~ r19.... |
2rexbiia 3207 | Inference adding two restr... |
2ralbidva 3208 | Formula-building rule for ... |
2rexbidva 3209 | Formula-building rule for ... |
2ralbidv 3210 | Formula-building rule for ... |
2rexbidv 3211 | Formula-building rule for ... |
rexralbidv 3212 | Formula-building rule for ... |
3ralbidv 3213 | Formula-building rule for ... |
4ralbidv 3214 | Formula-building rule for ... |
6ralbidv 3215 | Formula-building rule for ... |
r19.41vv 3216 | Version of ~ r19.41v with ... |
reeanlem 3217 | Lemma factoring out common... |
reeanv 3218 | Rearrange restricted exist... |
3reeanv 3219 | Rearrange three restricted... |
2ralor 3220 | Distribute restricted univ... |
2ralorOLD 3221 | Obsolete version of ~ 2ral... |
risset 3222 | Two ways to say " ` A ` be... |
nelb 3223 | A definition of ` -. A e. ... |
nelbOLD 3224 | Obsolete version of ~ nelb... |
rspw 3225 | Restricted specialization.... |
cbvralvw 3226 | Change the bound variable ... |
cbvrexvw 3227 | Change the bound variable ... |
cbvraldva 3228 | Rule used to change the bo... |
cbvrexdva 3229 | Rule used to change the bo... |
cbvral2vw 3230 | Change bound variables of ... |
cbvrex2vw 3231 | Change bound variables of ... |
cbvral3vw 3232 | Change bound variables of ... |
cbvral4vw 3233 | Change bound variables of ... |
cbvral6vw 3234 | Change bound variables of ... |
cbvral8vw 3235 | Change bound variables of ... |
rsp 3236 | Restricted specialization.... |
rspa 3237 | Restricted specialization.... |
rspe 3238 | Restricted specialization.... |
rspec 3239 | Specialization rule for re... |
r19.21bi 3240 | Inference from Theorem 19.... |
r19.21be 3241 | Inference from Theorem 19.... |
r19.21t 3242 | Restricted quantifier vers... |
r19.21 3243 | Restricted quantifier vers... |
r19.23t 3244 | Closed theorem form of ~ r... |
r19.23 3245 | Restricted quantifier vers... |
ralrimi 3246 | Inference from Theorem 19.... |
ralrimia 3247 | Inference from Theorem 19.... |
rexlimi 3248 | Restricted quantifier vers... |
ralimdaa 3249 | Deduction quantifying both... |
reximdai 3250 | Deduction from Theorem 19.... |
r19.37 3251 | Restricted quantifier vers... |
r19.41 3252 | Restricted quantifier vers... |
ralrimd 3253 | Inference from Theorem 19.... |
rexlimd2 3254 | Version of ~ rexlimd with ... |
rexlimd 3255 | Deduction form of ~ rexlim... |
r19.29af2 3256 | A commonly used pattern ba... |
r19.29af 3257 | A commonly used pattern ba... |
reximd2a 3258 | Deduction quantifying both... |
ralbida 3259 | Formula-building rule for ... |
ralbidaOLD 3260 | Obsolete version of ~ ralb... |
rexbida 3261 | Formula-building rule for ... |
ralbid 3262 | Formula-building rule for ... |
rexbid 3263 | Formula-building rule for ... |
rexbidvALT 3264 | Alternate proof of ~ rexbi... |
rexbidvaALT 3265 | Alternate proof of ~ rexbi... |
rsp2 3266 | Restricted specialization,... |
rsp2e 3267 | Restricted specialization.... |
rspec2 3268 | Specialization rule for re... |
rspec3 3269 | Specialization rule for re... |
r2alf 3270 | Double restricted universa... |
r2exf 3271 | Double restricted existent... |
2ralbida 3272 | Formula-building rule for ... |
nfra1 3273 | The setvar ` x ` is not fr... |
nfre1 3274 | The setvar ` x ` is not fr... |
ralcom4 3275 | Commutation of restricted ... |
ralcom4OLD 3276 | Obsolete version of ~ ralc... |
rexcom4 3277 | Commutation of restricted ... |
ralcom 3278 | Commutation of restricted ... |
rexcom 3279 | Commutation of restricted ... |
rexcomOLD 3280 | Obsolete version of ~ rexc... |
rexcom4a 3281 | Specialized existential co... |
ralrot3 3282 | Rotate three restricted un... |
ralcom13 3283 | Swap first and third restr... |
ralcom13OLD 3284 | Obsolete version of ~ ralc... |
rexcom13 3285 | Swap first and third restr... |
rexrot4 3286 | Rotate four restricted exi... |
2ex2rexrot 3287 | Rotate two existential qua... |
nfra2w 3288 | Similar to Lemma 24 of [Mo... |
nfra2wOLD 3289 | Obsolete version of ~ nfra... |
hbra1 3290 | The setvar ` x ` is not fr... |
ralcomf 3291 | Commutation of restricted ... |
rexcomf 3292 | Commutation of restricted ... |
cbvralfw 3293 | Rule used to change bound ... |
cbvrexfw 3294 | Rule used to change bound ... |
cbvralw 3295 | Rule used to change bound ... |
cbvrexw 3296 | Rule used to change bound ... |
hbral 3297 | Bound-variable hypothesis ... |
nfraldw 3298 | Deduction version of ~ nfr... |
nfrexdw 3299 | Deduction version of ~ nfr... |
nfralw 3300 | Bound-variable hypothesis ... |
nfralwOLD 3301 | Obsolete version of ~ nfra... |
nfrexw 3302 | Bound-variable hypothesis ... |
r19.12 3303 | Restricted quantifier vers... |
r19.12OLD 3304 | Obsolete version of ~ 19.1... |
reean 3305 | Rearrange restricted exist... |
cbvralsvw 3306 | Change bound variable by u... |
cbvrexsvw 3307 | Change bound variable by u... |
cbvralsvwOLD 3308 | Obsolete version of ~ cbvr... |
cbvrexsvwOLD 3309 | Obsolete version of ~ cbvr... |
nfraldwOLD 3310 | Obsolete version of ~ nfra... |
nfra2wOLDOLD 3311 | Obsolete version of ~ nfra... |
cbvralfwOLD 3312 | Obsolete version of ~ cbvr... |
rexeq 3313 | Equality theorem for restr... |
raleq 3314 | Equality theorem for restr... |
raleqi 3315 | Equality inference for res... |
rexeqi 3316 | Equality inference for res... |
raleqdv 3317 | Equality deduction for res... |
rexeqdv 3318 | Equality deduction for res... |
raleqbidva 3319 | Equality deduction for res... |
rexeqbidva 3320 | Equality deduction for res... |
raleqbidvv 3321 | Version of ~ raleqbidv wit... |
raleqbidvvOLD 3322 | Obsolete version of ~ rale... |
rexeqbidvv 3323 | Version of ~ rexeqbidv wit... |
rexeqbidvvOLD 3324 | Obsolete version of ~ rexe... |
raleqbi1dv 3325 | Equality deduction for res... |
rexeqbi1dv 3326 | Equality deduction for res... |
raleqOLD 3327 | Obsolete version of ~ rale... |
rexeqOLD 3328 | Obsolete version of ~ rale... |
raleleq 3329 | All elements of a class ar... |
raleqbii 3330 | Equality deduction for res... |
rexeqbii 3331 | Equality deduction for res... |
raleleqOLD 3332 | Obsolete version of ~ rale... |
raleleqALT 3333 | Alternate proof of ~ ralel... |
raleqbidv 3334 | Equality deduction for res... |
rexeqbidv 3335 | Equality deduction for res... |
cbvraldva2 3336 | Rule used to change the bo... |
cbvrexdva2 3337 | Rule used to change the bo... |
cbvrexdva2OLD 3338 | Obsolete version of ~ cbvr... |
cbvraldvaOLD 3339 | Obsolete version of ~ cbvr... |
cbvrexdvaOLD 3340 | Obsolete version of ~ cbvr... |
raleqf 3341 | Equality theorem for restr... |
rexeqf 3342 | Equality theorem for restr... |
rexeqfOLD 3343 | Obsolete version of ~ rexe... |
raleqbid 3344 | Equality deduction for res... |
rexeqbid 3345 | Equality deduction for res... |
sbralie 3346 | Implicit to explicit subst... |
sbralieALT 3347 | Alternative shorter proof ... |
cbvralf 3348 | Rule used to change bound ... |
cbvrexf 3349 | Rule used to change bound ... |
cbvral 3350 | Rule used to change bound ... |
cbvrex 3351 | Rule used to change bound ... |
cbvralv 3352 | Change the bound variable ... |
cbvrexv 3353 | Change the bound variable ... |
cbvralsv 3354 | Change bound variable by u... |
cbvrexsv 3355 | Change bound variable by u... |
cbvral2v 3356 | Change bound variables of ... |
cbvrex2v 3357 | Change bound variables of ... |
cbvral3v 3358 | Change bound variables of ... |
rgen2a 3359 | Generalization rule for re... |
nfrald 3360 | Deduction version of ~ nfr... |
nfrexd 3361 | Deduction version of ~ nfr... |
nfral 3362 | Bound-variable hypothesis ... |
nfrex 3363 | Bound-variable hypothesis ... |
nfra2 3364 | Similar to Lemma 24 of [Mo... |
ralcom2 3365 | Commutation of restricted ... |
reu5 3370 | Restricted uniqueness in t... |
reurmo 3371 | Restricted existential uni... |
reurex 3372 | Restricted unique existenc... |
mormo 3373 | Unrestricted "at most one"... |
rmobiia 3374 | Formula-building rule for ... |
reubiia 3375 | Formula-building rule for ... |
rmobii 3376 | Formula-building rule for ... |
reubii 3377 | Formula-building rule for ... |
rmoanid 3378 | Cancellation law for restr... |
reuanid 3379 | Cancellation law for restr... |
rmoanidOLD 3380 | Obsolete version of ~ rmoa... |
reuanidOLD 3381 | Obsolete version of ~ reua... |
2reu2rex 3382 | Double restricted existent... |
rmobidva 3383 | Formula-building rule for ... |
reubidva 3384 | Formula-building rule for ... |
rmobidv 3385 | Formula-building rule for ... |
reubidv 3386 | Formula-building rule for ... |
reueubd 3387 | Restricted existential uni... |
rmo5 3388 | Restricted "at most one" i... |
nrexrmo 3389 | Nonexistence implies restr... |
moel 3390 | "At most one" element in a... |
cbvrmovw 3391 | Change the bound variable ... |
cbvreuvw 3392 | Change the bound variable ... |
moelOLD 3393 | Obsolete version of ~ moel... |
rmobida 3394 | Formula-building rule for ... |
reubida 3395 | Formula-building rule for ... |
rmobidvaOLD 3396 | Obsolete version of ~ rmob... |
cbvrmow 3397 | Change the bound variable ... |
cbvreuw 3398 | Change the bound variable ... |
nfrmo1 3399 | The setvar ` x ` is not fr... |
nfreu1 3400 | The setvar ` x ` is not fr... |
nfrmow 3401 | Bound-variable hypothesis ... |
nfreuw 3402 | Bound-variable hypothesis ... |
cbvrmowOLD 3403 | Obsolete version of ~ cbvr... |
cbvreuwOLD 3404 | Obsolete version of ~ cbvr... |
cbvreuvwOLD 3405 | Obsolete version of ~ cbvr... |
rmoeq1 3406 | Equality theorem for restr... |
reueq1 3407 | Equality theorem for restr... |
rmoeq1OLD 3408 | Obsolete version of ~ rmoe... |
reueq1OLD 3409 | Obsolete version of ~ reue... |
rmoeqd 3410 | Equality deduction for res... |
reueqd 3411 | Equality deduction for res... |
rmoeq1f 3412 | Equality theorem for restr... |
reueq1f 3413 | Equality theorem for restr... |
nfreuwOLD 3414 | Obsolete version of ~ nfre... |
nfrmowOLD 3415 | Obsolete version of ~ nfrm... |
cbvreu 3416 | Change the bound variable ... |
cbvrmo 3417 | Change the bound variable ... |
cbvrmov 3418 | Change the bound variable ... |
cbvreuv 3419 | Change the bound variable ... |
nfrmod 3420 | Deduction version of ~ nfr... |
nfreud 3421 | Deduction version of ~ nfr... |
nfrmo 3422 | Bound-variable hypothesis ... |
nfreu 3423 | Bound-variable hypothesis ... |
rabbidva2 3426 | Equivalent wff's yield equ... |
rabbia2 3427 | Equivalent wff's yield equ... |
rabbiia 3428 | Equivalent formulas yield ... |
rabbiiaOLD 3429 | Obsolete version of ~ rabb... |
rabbii 3430 | Equivalent wff's correspon... |
rabbidva 3431 | Equivalent wff's yield equ... |
rabbidv 3432 | Equivalent wff's yield equ... |
rabswap 3433 | Swap with a membership rel... |
cbvrabv 3434 | Rule to change the bound v... |
rabeqcda 3435 | When ` ps ` is always true... |
rabeqc 3436 | A restricted class abstrac... |
rabeqi 3437 | Equality theorem for restr... |
rabeq 3438 | Equality theorem for restr... |
rabeqdv 3439 | Equality of restricted cla... |
rabeqbidva 3440 | Equality of restricted cla... |
rabeqbidv 3441 | Equality of restricted cla... |
rabrabi 3442 | Abstract builder restricte... |
nfrab1 3443 | The abstraction variable i... |
rabid 3444 | An "identity" law of concr... |
rabidim1 3445 | Membership in a restricted... |
reqabi 3446 | Inference from equality of... |
rabrab 3447 | Abstract builder restricte... |
rabrabiOLD 3448 | Obsolete version of ~ rabr... |
rabbida4 3449 | Version of ~ rabbidva2 wit... |
rabbida 3450 | Equivalent wff's yield equ... |
rabbid 3451 | Version of ~ rabbidv with ... |
rabeqd 3452 | Deduction form of ~ rabeq ... |
rabeqbida 3453 | Version of ~ rabeqbidva wi... |
rabbi 3454 | Equivalent wff's correspon... |
rabid2f 3455 | An "identity" law for rest... |
rabid2 3456 | An "identity" law for rest... |
rabid2OLD 3457 | Obsolete version of ~ rabi... |
rabeqf 3458 | Equality theorem for restr... |
cbvrabw 3459 | Rule to change the bound v... |
nfrabw 3460 | A variable not free in a w... |
nfrabwOLD 3461 | Obsolete version of ~ nfra... |
rabbidaOLD 3462 | Obsolete version of ~ rabb... |
rabeqiOLD 3463 | Obsolete version of ~ rabe... |
nfrab 3464 | A variable not free in a w... |
cbvrab 3465 | Rule to change the bound v... |
vjust 3467 | Justification theorem for ... |
dfv2 3469 | Alternate definition of th... |
vex 3470 | All setvar variables are s... |
vexOLD 3471 | Obsolete version of ~ vex ... |
elv 3472 | If a proposition is implie... |
elvd 3473 | If a proposition is implie... |
el2v 3474 | If a proposition is implie... |
eqv 3475 | The universe contains ever... |
eqvf 3476 | The universe contains ever... |
abv 3477 | The class of sets verifyin... |
abvALT 3478 | Alternate proof of ~ abv ,... |
isset 3479 | Two ways to express that "... |
issetft 3480 | Closed theorem form of ~ i... |
issetf 3481 | A version of ~ isset that ... |
isseti 3482 | A way to say " ` A ` is a ... |
issetri 3483 | A way to say " ` A ` is a ... |
eqvisset 3484 | A class equal to a variabl... |
elex 3485 | If a class is a member of ... |
elexi 3486 | If a class is a member of ... |
elexd 3487 | If a class is a member of ... |
elex2OLD 3488 | Obsolete version of ~ elex... |
elex22 3489 | If two classes each contai... |
prcnel 3490 | A proper class doesn't bel... |
ralv 3491 | A universal quantifier res... |
rexv 3492 | An existential quantifier ... |
reuv 3493 | A unique existential quant... |
rmov 3494 | An at-most-one quantifier ... |
rabab 3495 | A class abstraction restri... |
rexcom4b 3496 | Specialized existential co... |
ceqsal1t 3497 | One direction of ~ ceqsalt... |
ceqsalt 3498 | Closed theorem version of ... |
ceqsralt 3499 | Restricted quantifier vers... |
ceqsalg 3500 | A representation of explic... |
ceqsalgALT 3501 | Alternate proof of ~ ceqsa... |
ceqsal 3502 | A representation of explic... |
ceqsalALT 3503 | A representation of explic... |
ceqsalv 3504 | A representation of explic... |
ceqsalvOLD 3505 | Obsolete version of ~ ceqs... |
ceqsralv 3506 | Restricted quantifier vers... |
ceqsralvOLD 3507 | Obsolete version of ~ ceqs... |
gencl 3508 | Implicit substitution for ... |
2gencl 3509 | Implicit substitution for ... |
3gencl 3510 | Implicit substitution for ... |
cgsexg 3511 | Implicit substitution infe... |
cgsex2g 3512 | Implicit substitution infe... |
cgsex4g 3513 | An implicit substitution i... |
cgsex4gOLD 3514 | Obsolete version of ~ cgse... |
cgsex4gOLDOLD 3515 | Obsolete version of ~ cgse... |
ceqsex 3516 | Elimination of an existent... |
ceqsexOLD 3517 | Obsolete version of ~ ceqs... |
ceqsexv 3518 | Elimination of an existent... |
ceqsexvOLD 3519 | Obsolete version of ~ ceqs... |
ceqsexvOLDOLD 3520 | Obsolete version of ~ ceqs... |
ceqsexv2d 3521 | Elimination of an existent... |
ceqsex2 3522 | Elimination of two existen... |
ceqsex2v 3523 | Elimination of two existen... |
ceqsex3v 3524 | Elimination of three exist... |
ceqsex4v 3525 | Elimination of four existe... |
ceqsex6v 3526 | Elimination of six existen... |
ceqsex8v 3527 | Elimination of eight exist... |
gencbvex 3528 | Change of bound variable u... |
gencbvex2 3529 | Restatement of ~ gencbvex ... |
gencbval 3530 | Change of bound variable u... |
sbhypf 3531 | Introduce an explicit subs... |
sbhypfOLD 3532 | Obsolete version of ~ sbhy... |
vtoclgft 3533 | Closed theorem form of ~ v... |
vtocleg 3534 | Implicit substitution of a... |
vtoclg 3535 | Implicit substitution of a... |
vtocle 3536 | Implicit substitution of a... |
vtoclbg 3537 | Implicit substitution of a... |
vtocl 3538 | Implicit substitution of a... |
vtocldf 3539 | Implicit substitution of a... |
vtocld 3540 | Implicit substitution of a... |
vtocldOLD 3541 | Obsolete version of ~ vtoc... |
vtocl2d 3542 | Implicit substitution of t... |
vtoclef 3543 | Implicit substitution of a... |
vtoclf 3544 | Implicit substitution of a... |
vtoclfOLD 3545 | Obsolete version of ~ vtoc... |
vtoclALT 3546 | Alternate proof of ~ vtocl... |
vtocl2 3547 | Implicit substitution of c... |
vtocl3 3548 | Implicit substitution of c... |
vtoclb 3549 | Implicit substitution of a... |
vtoclgf 3550 | Implicit substitution of a... |
vtoclg1f 3551 | Version of ~ vtoclgf with ... |
vtoclgOLD 3552 | Obsolete version of ~ vtoc... |
vtocl2gf 3553 | Implicit substitution of a... |
vtocl3gf 3554 | Implicit substitution of a... |
vtocl2g 3555 | Implicit substitution of 2... |
vtocl3g 3556 | Implicit substitution of a... |
vtoclgaf 3557 | Implicit substitution of a... |
vtoclga 3558 | Implicit substitution of a... |
vtocl2ga 3559 | Implicit substitution of 2... |
vtocl2gaf 3560 | Implicit substitution of 2... |
vtocl3gaf 3561 | Implicit substitution of 3... |
vtocl3ga 3562 | Implicit substitution of 3... |
vtocl3gaOLD 3563 | Obsolete version of ~ vtoc... |
vtocl4g 3564 | Implicit substitution of 4... |
vtocl4ga 3565 | Implicit substitution of 4... |
vtoclegft 3566 | Implicit substitution of a... |
vtoclegftOLD 3567 | Obsolete version of ~ vtoc... |
vtoclri 3568 | Implicit substitution of a... |
spcimgft 3569 | A closed version of ~ spci... |
spcgft 3570 | A closed version of ~ spcg... |
spcimgf 3571 | Rule of specialization, us... |
spcimegf 3572 | Existential specialization... |
spcgf 3573 | Rule of specialization, us... |
spcegf 3574 | Existential specialization... |
spcimdv 3575 | Restricted specialization,... |
spcdv 3576 | Rule of specialization, us... |
spcimedv 3577 | Restricted existential spe... |
spcgv 3578 | Rule of specialization, us... |
spcegv 3579 | Existential specialization... |
spcedv 3580 | Existential specialization... |
spc2egv 3581 | Existential specialization... |
spc2gv 3582 | Specialization with two qu... |
spc2ed 3583 | Existential specialization... |
spc2d 3584 | Specialization with 2 quan... |
spc3egv 3585 | Existential specialization... |
spc3gv 3586 | Specialization with three ... |
spcv 3587 | Rule of specialization, us... |
spcev 3588 | Existential specialization... |
spc2ev 3589 | Existential specialization... |
rspct 3590 | A closed version of ~ rspc... |
rspcdf 3591 | Restricted specialization,... |
rspc 3592 | Restricted specialization,... |
rspce 3593 | Restricted existential spe... |
rspcimdv 3594 | Restricted specialization,... |
rspcimedv 3595 | Restricted existential spe... |
rspcdv 3596 | Restricted specialization,... |
rspcedv 3597 | Restricted existential spe... |
rspcebdv 3598 | Restricted existential spe... |
rspcdv2 3599 | Restricted specialization,... |
rspcv 3600 | Restricted specialization,... |
rspccv 3601 | Restricted specialization,... |
rspcva 3602 | Restricted specialization,... |
rspccva 3603 | Restricted specialization,... |
rspcev 3604 | Restricted existential spe... |
rspcdva 3605 | Restricted specialization,... |
rspcedvd 3606 | Restricted existential spe... |
rspcedvdw 3607 | Version of ~ rspcedvd wher... |
rspcime 3608 | Prove a restricted existen... |
rspceaimv 3609 | Restricted existential spe... |
rspcedeq1vd 3610 | Restricted existential spe... |
rspcedeq2vd 3611 | Restricted existential spe... |
rspc2 3612 | Restricted specialization ... |
rspc2gv 3613 | Restricted specialization ... |
rspc2v 3614 | 2-variable restricted spec... |
rspc2va 3615 | 2-variable restricted spec... |
rspc2ev 3616 | 2-variable restricted exis... |
2rspcedvdw 3617 | Double application of ~ rs... |
rspc2dv 3618 | 2-variable restricted spec... |
rspc3v 3619 | 3-variable restricted spec... |
rspc3ev 3620 | 3-variable restricted exis... |
rspc3dv 3621 | 3-variable restricted spec... |
rspc4v 3622 | 4-variable restricted spec... |
rspc6v 3623 | 6-variable restricted spec... |
rspc8v 3624 | 8-variable restricted spec... |
rspceeqv 3625 | Restricted existential spe... |
ralxpxfr2d 3626 | Transfer a universal quant... |
rexraleqim 3627 | Statement following from e... |
eqvincg 3628 | A variable introduction la... |
eqvinc 3629 | A variable introduction la... |
eqvincf 3630 | A variable introduction la... |
alexeqg 3631 | Two ways to express substi... |
ceqex 3632 | Equality implies equivalen... |
ceqsexg 3633 | A representation of explic... |
ceqsexgv 3634 | Elimination of an existent... |
ceqsrexv 3635 | Elimination of a restricte... |
ceqsrexbv 3636 | Elimination of a restricte... |
ceqsralbv 3637 | Elimination of a restricte... |
ceqsrex2v 3638 | Elimination of a restricte... |
clel2g 3639 | Alternate definition of me... |
clel2gOLD 3640 | Obsolete version of ~ clel... |
clel2 3641 | Alternate definition of me... |
clel3g 3642 | Alternate definition of me... |
clel3 3643 | Alternate definition of me... |
clel4g 3644 | Alternate definition of me... |
clel4 3645 | Alternate definition of me... |
clel4OLD 3646 | Obsolete version of ~ clel... |
clel5 3647 | Alternate definition of cl... |
pm13.183 3648 | Compare theorem *13.183 in... |
rr19.3v 3649 | Restricted quantifier vers... |
rr19.28v 3650 | Restricted quantifier vers... |
elab6g 3651 | Membership in a class abst... |
elabd2 3652 | Membership in a class abst... |
elabd3 3653 | Membership in a class abst... |
elabgt 3654 | Membership in a class abst... |
elabgtOLD 3655 | Obsolete version of ~ elab... |
elabgf 3656 | Membership in a class abst... |
elabf 3657 | Membership in a class abst... |
elabg 3658 | Membership in a class abst... |
elabgOLD 3659 | Obsolete version of ~ elab... |
elab 3660 | Membership in a class abst... |
elabOLD 3661 | Obsolete version of ~ elab... |
elab2g 3662 | Membership in a class abst... |
elabd 3663 | Explicit demonstration the... |
elab2 3664 | Membership in a class abst... |
elab4g 3665 | Membership in a class abst... |
elab3gf 3666 | Membership in a class abst... |
elab3g 3667 | Membership in a class abst... |
elab3 3668 | Membership in a class abst... |
elrabi 3669 | Implication for the member... |
elrabiOLD 3670 | Obsolete version of ~ elra... |
elrabf 3671 | Membership in a restricted... |
rabtru 3672 | Abstract builder using the... |
rabeqcOLD 3673 | Obsolete version of ~ rabe... |
elrab3t 3674 | Membership in a restricted... |
elrab 3675 | Membership in a restricted... |
elrab3 3676 | Membership in a restricted... |
elrabd 3677 | Membership in a restricted... |
elrab2 3678 | Membership in a restricted... |
ralab 3679 | Universal quantification o... |
ralabOLD 3680 | Obsolete version of ~ rala... |
ralrab 3681 | Universal quantification o... |
rexab 3682 | Existential quantification... |
rexabOLD 3683 | Obsolete version of ~ rexa... |
rexrab 3684 | Existential quantification... |
ralab2 3685 | Universal quantification o... |
ralrab2 3686 | Universal quantification o... |
rexab2 3687 | Existential quantification... |
rexrab2 3688 | Existential quantification... |
reurab 3689 | Restricted existential uni... |
abidnf 3690 | Identity used to create cl... |
dedhb 3691 | A deduction theorem for co... |
class2seteq 3692 | Writing a set as a class a... |
nelrdva 3693 | Deduce negative membership... |
eqeu 3694 | A condition which implies ... |
moeq 3695 | There exists at most one s... |
eueq 3696 | A class is a set if and on... |
eueqi 3697 | There exists a unique set ... |
eueq2 3698 | Equality has existential u... |
eueq3 3699 | Equality has existential u... |
moeq3 3700 | "At most one" property of ... |
mosub 3701 | "At most one" remains true... |
mo2icl 3702 | Theorem for inferring "at ... |
mob2 3703 | Consequence of "at most on... |
moi2 3704 | Consequence of "at most on... |
mob 3705 | Equality implied by "at mo... |
moi 3706 | Equality implied by "at mo... |
morex 3707 | Derive membership from uni... |
euxfr2w 3708 | Transfer existential uniqu... |
euxfrw 3709 | Transfer existential uniqu... |
euxfr2 3710 | Transfer existential uniqu... |
euxfr 3711 | Transfer existential uniqu... |
euind 3712 | Existential uniqueness via... |
reu2 3713 | A way to express restricte... |
reu6 3714 | A way to express restricte... |
reu3 3715 | A way to express restricte... |
reu6i 3716 | A condition which implies ... |
eqreu 3717 | A condition which implies ... |
rmo4 3718 | Restricted "at most one" u... |
reu4 3719 | Restricted uniqueness usin... |
reu7 3720 | Restricted uniqueness usin... |
reu8 3721 | Restricted uniqueness usin... |
rmo3f 3722 | Restricted "at most one" u... |
rmo4f 3723 | Restricted "at most one" u... |
reu2eqd 3724 | Deduce equality from restr... |
reueq 3725 | Equality has existential u... |
rmoeq 3726 | Equality's restricted exis... |
rmoan 3727 | Restricted "at most one" s... |
rmoim 3728 | Restricted "at most one" i... |
rmoimia 3729 | Restricted "at most one" i... |
rmoimi 3730 | Restricted "at most one" i... |
rmoimi2 3731 | Restricted "at most one" i... |
2reu5a 3732 | Double restricted existent... |
reuimrmo 3733 | Restricted uniqueness impl... |
2reuswap 3734 | A condition allowing swap ... |
2reuswap2 3735 | A condition allowing swap ... |
reuxfrd 3736 | Transfer existential uniqu... |
reuxfr 3737 | Transfer existential uniqu... |
reuxfr1d 3738 | Transfer existential uniqu... |
reuxfr1ds 3739 | Transfer existential uniqu... |
reuxfr1 3740 | Transfer existential uniqu... |
reuind 3741 | Existential uniqueness via... |
2rmorex 3742 | Double restricted quantifi... |
2reu5lem1 3743 | Lemma for ~ 2reu5 . Note ... |
2reu5lem2 3744 | Lemma for ~ 2reu5 . (Cont... |
2reu5lem3 3745 | Lemma for ~ 2reu5 . This ... |
2reu5 3746 | Double restricted existent... |
2reurmo 3747 | Double restricted quantifi... |
2reurex 3748 | Double restricted quantifi... |
2rmoswap 3749 | A condition allowing to sw... |
2rexreu 3750 | Double restricted existent... |
cdeqi 3753 | Deduce conditional equalit... |
cdeqri 3754 | Property of conditional eq... |
cdeqth 3755 | Deduce conditional equalit... |
cdeqnot 3756 | Distribute conditional equ... |
cdeqal 3757 | Distribute conditional equ... |
cdeqab 3758 | Distribute conditional equ... |
cdeqal1 3759 | Distribute conditional equ... |
cdeqab1 3760 | Distribute conditional equ... |
cdeqim 3761 | Distribute conditional equ... |
cdeqcv 3762 | Conditional equality for s... |
cdeqeq 3763 | Distribute conditional equ... |
cdeqel 3764 | Distribute conditional equ... |
nfcdeq 3765 | If we have a conditional e... |
nfccdeq 3766 | Variation of ~ nfcdeq for ... |
rru 3767 | Relative version of Russel... |
ru 3768 | Russell's Paradox. Propos... |
dfsbcq 3771 | Proper substitution of a c... |
dfsbcq2 3772 | This theorem, which is sim... |
sbsbc 3773 | Show that ~ df-sb and ~ df... |
sbceq1d 3774 | Equality theorem for class... |
sbceq1dd 3775 | Equality theorem for class... |
sbceqbid 3776 | Equality theorem for class... |
sbc8g 3777 | This is the closest we can... |
sbc2or 3778 | The disjunction of two equ... |
sbcex 3779 | By our definition of prope... |
sbceq1a 3780 | Equality theorem for class... |
sbceq2a 3781 | Equality theorem for class... |
spsbc 3782 | Specialization: if a formu... |
spsbcd 3783 | Specialization: if a formu... |
sbcth 3784 | A substitution into a theo... |
sbcthdv 3785 | Deduction version of ~ sbc... |
sbcid 3786 | An identity theorem for su... |
nfsbc1d 3787 | Deduction version of ~ nfs... |
nfsbc1 3788 | Bound-variable hypothesis ... |
nfsbc1v 3789 | Bound-variable hypothesis ... |
nfsbcdw 3790 | Deduction version of ~ nfs... |
nfsbcw 3791 | Bound-variable hypothesis ... |
sbccow 3792 | A composition law for clas... |
nfsbcd 3793 | Deduction version of ~ nfs... |
nfsbc 3794 | Bound-variable hypothesis ... |
sbcco 3795 | A composition law for clas... |
sbcco2 3796 | A composition law for clas... |
sbc5 3797 | An equivalence for class s... |
sbc5ALT 3798 | Alternate proof of ~ sbc5 ... |
sbc6g 3799 | An equivalence for class s... |
sbc6gOLD 3800 | Obsolete version of ~ sbc6... |
sbc6 3801 | An equivalence for class s... |
sbc7 3802 | An equivalence for class s... |
cbvsbcw 3803 | Change bound variables in ... |
cbvsbcvw 3804 | Change the bound variable ... |
cbvsbc 3805 | Change bound variables in ... |
cbvsbcv 3806 | Change the bound variable ... |
sbciegft 3807 | Conversion of implicit sub... |
sbciegf 3808 | Conversion of implicit sub... |
sbcieg 3809 | Conversion of implicit sub... |
sbciegOLD 3810 | Obsolete version of ~ sbci... |
sbcie2g 3811 | Conversion of implicit sub... |
sbcie 3812 | Conversion of implicit sub... |
sbciedf 3813 | Conversion of implicit sub... |
sbcied 3814 | Conversion of implicit sub... |
sbciedOLD 3815 | Obsolete version of ~ sbci... |
sbcied2 3816 | Conversion of implicit sub... |
elrabsf 3817 | Membership in a restricted... |
eqsbc1 3818 | Substitution for the left-... |
sbcng 3819 | Move negation in and out o... |
sbcimg 3820 | Distribution of class subs... |
sbcan 3821 | Distribution of class subs... |
sbcor 3822 | Distribution of class subs... |
sbcbig 3823 | Distribution of class subs... |
sbcn1 3824 | Move negation in and out o... |
sbcim1 3825 | Distribution of class subs... |
sbcim1OLD 3826 | Obsolete version of ~ sbci... |
sbcbid 3827 | Formula-building deduction... |
sbcbidv 3828 | Formula-building deduction... |
sbcbii 3829 | Formula-building inference... |
sbcbi1 3830 | Distribution of class subs... |
sbcbi2 3831 | Substituting into equivale... |
sbcbi2OLD 3832 | Obsolete proof of ~ sbcbi2... |
sbcal 3833 | Move universal quantifier ... |
sbcex2 3834 | Move existential quantifie... |
sbceqal 3835 | Class version of one impli... |
sbceqalOLD 3836 | Obsolete version of ~ sbce... |
sbeqalb 3837 | Theorem *14.121 in [Whiteh... |
eqsbc2 3838 | Substitution for the right... |
sbc3an 3839 | Distribution of class subs... |
sbcel1v 3840 | Class substitution into a ... |
sbcel2gv 3841 | Class substitution into a ... |
sbcel21v 3842 | Class substitution into a ... |
sbcimdv 3843 | Substitution analogue of T... |
sbcimdvOLD 3844 | Obsolete version of ~ sbci... |
sbctt 3845 | Substitution for a variabl... |
sbcgf 3846 | Substitution for a variabl... |
sbc19.21g 3847 | Substitution for a variabl... |
sbcg 3848 | Substitution for a variabl... |
sbcgOLD 3849 | Obsolete version of ~ sbcg... |
sbcgfi 3850 | Substitution for a variabl... |
sbc2iegf 3851 | Conversion of implicit sub... |
sbc2ie 3852 | Conversion of implicit sub... |
sbc2ieOLD 3853 | Obsolete version of ~ sbc2... |
sbc2iedv 3854 | Conversion of implicit sub... |
sbc3ie 3855 | Conversion of implicit sub... |
sbccomlem 3856 | Lemma for ~ sbccom . (Con... |
sbccom 3857 | Commutative law for double... |
sbcralt 3858 | Interchange class substitu... |
sbcrext 3859 | Interchange class substitu... |
sbcralg 3860 | Interchange class substitu... |
sbcrex 3861 | Interchange class substitu... |
sbcreu 3862 | Interchange class substitu... |
reu8nf 3863 | Restricted uniqueness usin... |
sbcabel 3864 | Interchange class substitu... |
rspsbc 3865 | Restricted quantifier vers... |
rspsbca 3866 | Restricted quantifier vers... |
rspesbca 3867 | Existence form of ~ rspsbc... |
spesbc 3868 | Existence form of ~ spsbc ... |
spesbcd 3869 | form of ~ spsbc . (Contri... |
sbcth2 3870 | A substitution into a theo... |
ra4v 3871 | Version of ~ ra4 with a di... |
ra4 3872 | Restricted quantifier vers... |
rmo2 3873 | Alternate definition of re... |
rmo2i 3874 | Condition implying restric... |
rmo3 3875 | Restricted "at most one" u... |
rmob 3876 | Consequence of "at most on... |
rmoi 3877 | Consequence of "at most on... |
rmob2 3878 | Consequence of "restricted... |
rmoi2 3879 | Consequence of "restricted... |
rmoanim 3880 | Introduction of a conjunct... |
rmoanimALT 3881 | Alternate proof of ~ rmoan... |
reuan 3882 | Introduction of a conjunct... |
2reu1 3883 | Double restricted existent... |
2reu2 3884 | Double restricted existent... |
csb2 3887 | Alternate expression for t... |
csbeq1 3888 | Analogue of ~ dfsbcq for p... |
csbeq1d 3889 | Equality deduction for pro... |
csbeq2 3890 | Substituting into equivale... |
csbeq2d 3891 | Formula-building deduction... |
csbeq2dv 3892 | Formula-building deduction... |
csbeq2i 3893 | Formula-building inference... |
csbeq12dv 3894 | Formula-building inference... |
cbvcsbw 3895 | Change bound variables in ... |
cbvcsb 3896 | Change bound variables in ... |
cbvcsbv 3897 | Change the bound variable ... |
csbid 3898 | Analogue of ~ sbid for pro... |
csbeq1a 3899 | Equality theorem for prope... |
csbcow 3900 | Composition law for chaine... |
csbco 3901 | Composition law for chaine... |
csbtt 3902 | Substitution doesn't affec... |
csbconstgf 3903 | Substitution doesn't affec... |
csbconstg 3904 | Substitution doesn't affec... |
csbconstgOLD 3905 | Obsolete version of ~ csbc... |
csbgfi 3906 | Substitution for a variabl... |
csbconstgi 3907 | The proper substitution of... |
nfcsb1d 3908 | Bound-variable hypothesis ... |
nfcsb1 3909 | Bound-variable hypothesis ... |
nfcsb1v 3910 | Bound-variable hypothesis ... |
nfcsbd 3911 | Deduction version of ~ nfc... |
nfcsbw 3912 | Bound-variable hypothesis ... |
nfcsb 3913 | Bound-variable hypothesis ... |
csbhypf 3914 | Introduce an explicit subs... |
csbiebt 3915 | Conversion of implicit sub... |
csbiedf 3916 | Conversion of implicit sub... |
csbieb 3917 | Bidirectional conversion b... |
csbiebg 3918 | Bidirectional conversion b... |
csbiegf 3919 | Conversion of implicit sub... |
csbief 3920 | Conversion of implicit sub... |
csbie 3921 | Conversion of implicit sub... |
csbieOLD 3922 | Obsolete version of ~ csbi... |
csbied 3923 | Conversion of implicit sub... |
csbiedOLD 3924 | Obsolete version of ~ csbi... |
csbied2 3925 | Conversion of implicit sub... |
csbie2t 3926 | Conversion of implicit sub... |
csbie2 3927 | Conversion of implicit sub... |
csbie2g 3928 | Conversion of implicit sub... |
cbvrabcsfw 3929 | Version of ~ cbvrabcsf wit... |
cbvralcsf 3930 | A more general version of ... |
cbvrexcsf 3931 | A more general version of ... |
cbvreucsf 3932 | A more general version of ... |
cbvrabcsf 3933 | A more general version of ... |
cbvralv2 3934 | Rule used to change the bo... |
cbvrexv2 3935 | Rule used to change the bo... |
rspc2vd 3936 | Deduction version of 2-var... |
difjust 3942 | Soundness justification th... |
unjust 3944 | Soundness justification th... |
injust 3946 | Soundness justification th... |
dfin5 3948 | Alternate definition for t... |
dfdif2 3949 | Alternate definition of cl... |
eldif 3950 | Expansion of membership in... |
eldifd 3951 | If a class is in one class... |
eldifad 3952 | If a class is in the diffe... |
eldifbd 3953 | If a class is in the diffe... |
elneeldif 3954 | The elements of a set diff... |
velcomp 3955 | Characterization of setvar... |
elin 3956 | Expansion of membership in... |
dfss 3958 | Variant of subclass defini... |
dfss2 3960 | Alternate definition of th... |
dfss2OLD 3961 | Obsolete version of ~ dfss... |
dfss3 3962 | Alternate definition of su... |
dfss6 3963 | Alternate definition of su... |
dfss2f 3964 | Equivalence for subclass r... |
dfss3f 3965 | Equivalence for subclass r... |
nfss 3966 | If ` x ` is not free in ` ... |
ssel 3967 | Membership relationships f... |
sselOLD 3968 | Obsolete version of ~ ssel... |
ssel2 3969 | Membership relationships f... |
sseli 3970 | Membership implication fro... |
sselii 3971 | Membership inference from ... |
sselid 3972 | Membership inference from ... |
sseld 3973 | Membership deduction from ... |
sselda 3974 | Membership deduction from ... |
sseldd 3975 | Membership inference from ... |
ssneld 3976 | If a class is not in anoth... |
ssneldd 3977 | If an element is not in a ... |
ssriv 3978 | Inference based on subclas... |
ssrd 3979 | Deduction based on subclas... |
ssrdv 3980 | Deduction based on subclas... |
sstr2 3981 | Transitivity of subclass r... |
sstr 3982 | Transitivity of subclass r... |
sstri 3983 | Subclass transitivity infe... |
sstrd 3984 | Subclass transitivity dedu... |
sstrid 3985 | Subclass transitivity dedu... |
sstrdi 3986 | Subclass transitivity dedu... |
sylan9ss 3987 | A subclass transitivity de... |
sylan9ssr 3988 | A subclass transitivity de... |
eqss 3989 | The subclass relationship ... |
eqssi 3990 | Infer equality from two su... |
eqssd 3991 | Equality deduction from tw... |
sssseq 3992 | If a class is a subclass o... |
eqrd 3993 | Deduce equality of classes... |
eqri 3994 | Infer equality of classes ... |
eqelssd 3995 | Equality deduction from su... |
ssid 3996 | Any class is a subclass of... |
ssidd 3997 | Weakening of ~ ssid . (Co... |
ssv 3998 | Any class is a subclass of... |
sseq1 3999 | Equality theorem for subcl... |
sseq2 4000 | Equality theorem for the s... |
sseq12 4001 | Equality theorem for the s... |
sseq1i 4002 | An equality inference for ... |
sseq2i 4003 | An equality inference for ... |
sseq12i 4004 | An equality inference for ... |
sseq1d 4005 | An equality deduction for ... |
sseq2d 4006 | An equality deduction for ... |
sseq12d 4007 | An equality deduction for ... |
eqsstri 4008 | Substitution of equality i... |
eqsstrri 4009 | Substitution of equality i... |
sseqtri 4010 | Substitution of equality i... |
sseqtrri 4011 | Substitution of equality i... |
eqsstrd 4012 | Substitution of equality i... |
eqsstrrd 4013 | Substitution of equality i... |
sseqtrd 4014 | Substitution of equality i... |
sseqtrrd 4015 | Substitution of equality i... |
3sstr3i 4016 | Substitution of equality i... |
3sstr4i 4017 | Substitution of equality i... |
3sstr3g 4018 | Substitution of equality i... |
3sstr4g 4019 | Substitution of equality i... |
3sstr3d 4020 | Substitution of equality i... |
3sstr4d 4021 | Substitution of equality i... |
eqsstrid 4022 | A chained subclass and equ... |
eqsstrrid 4023 | A chained subclass and equ... |
sseqtrdi 4024 | A chained subclass and equ... |
sseqtrrdi 4025 | A chained subclass and equ... |
sseqtrid 4026 | Subclass transitivity dedu... |
sseqtrrid 4027 | Subclass transitivity dedu... |
eqsstrdi 4028 | A chained subclass and equ... |
eqsstrrdi 4029 | A chained subclass and equ... |
eqimssd 4030 | Equality implies inclusion... |
eqimsscd 4031 | Equality implies inclusion... |
eqimss 4032 | Equality implies inclusion... |
eqimss2 4033 | Equality implies inclusion... |
eqimssi 4034 | Infer subclass relationshi... |
eqimss2i 4035 | Infer subclass relationshi... |
nssne1 4036 | Two classes are different ... |
nssne2 4037 | Two classes are different ... |
nss 4038 | Negation of subclass relat... |
nelss 4039 | Demonstrate by witnesses t... |
ssrexf 4040 | Restricted existential qua... |
ssrmof 4041 | "At most one" existential ... |
ssralv 4042 | Quantification restricted ... |
ssrexv 4043 | Existential quantification... |
ss2ralv 4044 | Two quantifications restri... |
ss2rexv 4045 | Two existential quantifica... |
ralss 4046 | Restricted universal quant... |
rexss 4047 | Restricted existential qua... |
ss2ab 4048 | Class abstractions in a su... |
abss 4049 | Class abstraction in a sub... |
ssab 4050 | Subclass of a class abstra... |
ssabral 4051 | The relation for a subclas... |
ss2abdv 4052 | Deduction of abstraction s... |
ss2abdvALT 4053 | Alternate proof of ~ ss2ab... |
ss2abdvOLD 4054 | Obsolete version of ~ ss2a... |
ss2abi 4055 | Inference of abstraction s... |
ss2abiOLD 4056 | Obsolete version of ~ ss2a... |
abssdv 4057 | Deduction of abstraction s... |
abssdvOLD 4058 | Obsolete version of ~ abss... |
abssi 4059 | Inference of abstraction s... |
ss2rab 4060 | Restricted abstraction cla... |
rabss 4061 | Restricted class abstracti... |
ssrab 4062 | Subclass of a restricted c... |
ssrabdv 4063 | Subclass of a restricted c... |
rabssdv 4064 | Subclass of a restricted c... |
ss2rabdv 4065 | Deduction of restricted ab... |
ss2rabi 4066 | Inference of restricted ab... |
rabss2 4067 | Subclass law for restricte... |
ssab2 4068 | Subclass relation for the ... |
ssrab2 4069 | Subclass relation for a re... |
ssrab2OLD 4070 | Obsolete version of ~ ssra... |
rabss3d 4071 | Subclass law for restricte... |
ssrab3 4072 | Subclass relation for a re... |
rabssrabd 4073 | Subclass of a restricted c... |
ssrabeq 4074 | If the restricting class o... |
rabssab 4075 | A restricted class is a su... |
uniiunlem 4076 | A subset relationship usef... |
dfpss2 4077 | Alternate definition of pr... |
dfpss3 4078 | Alternate definition of pr... |
psseq1 4079 | Equality theorem for prope... |
psseq2 4080 | Equality theorem for prope... |
psseq1i 4081 | An equality inference for ... |
psseq2i 4082 | An equality inference for ... |
psseq12i 4083 | An equality inference for ... |
psseq1d 4084 | An equality deduction for ... |
psseq2d 4085 | An equality deduction for ... |
psseq12d 4086 | An equality deduction for ... |
pssss 4087 | A proper subclass is a sub... |
pssne 4088 | Two classes in a proper su... |
pssssd 4089 | Deduce subclass from prope... |
pssned 4090 | Proper subclasses are uneq... |
sspss 4091 | Subclass in terms of prope... |
pssirr 4092 | Proper subclass is irrefle... |
pssn2lp 4093 | Proper subclass has no 2-c... |
sspsstri 4094 | Two ways of stating tricho... |
ssnpss 4095 | Partial trichotomy law for... |
psstr 4096 | Transitive law for proper ... |
sspsstr 4097 | Transitive law for subclas... |
psssstr 4098 | Transitive law for subclas... |
psstrd 4099 | Proper subclass inclusion ... |
sspsstrd 4100 | Transitivity involving sub... |
psssstrd 4101 | Transitivity involving sub... |
npss 4102 | A class is not a proper su... |
ssnelpss 4103 | A subclass missing a membe... |
ssnelpssd 4104 | Subclass inclusion with on... |
ssexnelpss 4105 | If there is an element of ... |
dfdif3 4106 | Alternate definition of cl... |
difeq1 4107 | Equality theorem for class... |
difeq2 4108 | Equality theorem for class... |
difeq12 4109 | Equality theorem for class... |
difeq1i 4110 | Inference adding differenc... |
difeq2i 4111 | Inference adding differenc... |
difeq12i 4112 | Equality inference for cla... |
difeq1d 4113 | Deduction adding differenc... |
difeq2d 4114 | Deduction adding differenc... |
difeq12d 4115 | Equality deduction for cla... |
difeqri 4116 | Inference from membership ... |
nfdif 4117 | Bound-variable hypothesis ... |
eldifi 4118 | Implication of membership ... |
eldifn 4119 | Implication of membership ... |
elndif 4120 | A set does not belong to a... |
neldif 4121 | Implication of membership ... |
difdif 4122 | Double class difference. ... |
difss 4123 | Subclass relationship for ... |
difssd 4124 | A difference of two classe... |
difss2 4125 | If a class is contained in... |
difss2d 4126 | If a class is contained in... |
ssdifss 4127 | Preservation of a subclass... |
ddif 4128 | Double complement under un... |
ssconb 4129 | Contraposition law for sub... |
sscon 4130 | Contraposition law for sub... |
ssdif 4131 | Difference law for subsets... |
ssdifd 4132 | If ` A ` is contained in `... |
sscond 4133 | If ` A ` is contained in `... |
ssdifssd 4134 | If ` A ` is contained in `... |
ssdif2d 4135 | If ` A ` is contained in `... |
raldifb 4136 | Restricted universal quant... |
rexdifi 4137 | Restricted existential qua... |
complss 4138 | Complementation reverses i... |
compleq 4139 | Two classes are equal if a... |
elun 4140 | Expansion of membership in... |
elunnel1 4141 | A member of a union that i... |
elunnel2 4142 | A member of a union that i... |
uneqri 4143 | Inference from membership ... |
unidm 4144 | Idempotent law for union o... |
uncom 4145 | Commutative law for union ... |
equncom 4146 | If a class equals the unio... |
equncomi 4147 | Inference form of ~ equnco... |
uneq1 4148 | Equality theorem for the u... |
uneq2 4149 | Equality theorem for the u... |
uneq12 4150 | Equality theorem for the u... |
uneq1i 4151 | Inference adding union to ... |
uneq2i 4152 | Inference adding union to ... |
uneq12i 4153 | Equality inference for the... |
uneq1d 4154 | Deduction adding union to ... |
uneq2d 4155 | Deduction adding union to ... |
uneq12d 4156 | Equality deduction for the... |
nfun 4157 | Bound-variable hypothesis ... |
unass 4158 | Associative law for union ... |
un12 4159 | A rearrangement of union. ... |
un23 4160 | A rearrangement of union. ... |
un4 4161 | A rearrangement of the uni... |
unundi 4162 | Union distributes over its... |
unundir 4163 | Union distributes over its... |
ssun1 4164 | Subclass relationship for ... |
ssun2 4165 | Subclass relationship for ... |
ssun3 4166 | Subclass law for union of ... |
ssun4 4167 | Subclass law for union of ... |
elun1 4168 | Membership law for union o... |
elun2 4169 | Membership law for union o... |
elunant 4170 | A statement is true for ev... |
unss1 4171 | Subclass law for union of ... |
ssequn1 4172 | A relationship between sub... |
unss2 4173 | Subclass law for union of ... |
unss12 4174 | Subclass law for union of ... |
ssequn2 4175 | A relationship between sub... |
unss 4176 | The union of two subclasse... |
unssi 4177 | An inference showing the u... |
unssd 4178 | A deduction showing the un... |
unssad 4179 | If ` ( A u. B ) ` is conta... |
unssbd 4180 | If ` ( A u. B ) ` is conta... |
ssun 4181 | A condition that implies i... |
rexun 4182 | Restricted existential qua... |
ralunb 4183 | Restricted quantification ... |
ralun 4184 | Restricted quantification ... |
elini 4185 | Membership in an intersect... |
elind 4186 | Deduce membership in an in... |
elinel1 4187 | Membership in an intersect... |
elinel2 4188 | Membership in an intersect... |
elin2 4189 | Membership in a class defi... |
elin1d 4190 | Elementhood in the first s... |
elin2d 4191 | Elementhood in the first s... |
elin3 4192 | Membership in a class defi... |
incom 4193 | Commutative law for inters... |
ineqcom 4194 | Two ways of expressing tha... |
ineqcomi 4195 | Two ways of expressing tha... |
ineqri 4196 | Inference from membership ... |
ineq1 4197 | Equality theorem for inter... |
ineq2 4198 | Equality theorem for inter... |
ineq12 4199 | Equality theorem for inter... |
ineq1i 4200 | Equality inference for int... |
ineq2i 4201 | Equality inference for int... |
ineq12i 4202 | Equality inference for int... |
ineq1d 4203 | Equality deduction for int... |
ineq2d 4204 | Equality deduction for int... |
ineq12d 4205 | Equality deduction for int... |
ineqan12d 4206 | Equality deduction for int... |
sseqin2 4207 | A relationship between sub... |
nfin 4208 | Bound-variable hypothesis ... |
rabbi2dva 4209 | Deduction from a wff to a ... |
inidm 4210 | Idempotent law for interse... |
inass 4211 | Associative law for inters... |
in12 4212 | A rearrangement of interse... |
in32 4213 | A rearrangement of interse... |
in13 4214 | A rearrangement of interse... |
in31 4215 | A rearrangement of interse... |
inrot 4216 | Rotate the intersection of... |
in4 4217 | Rearrangement of intersect... |
inindi 4218 | Intersection distributes o... |
inindir 4219 | Intersection distributes o... |
inss1 4220 | The intersection of two cl... |
inss2 4221 | The intersection of two cl... |
ssin 4222 | Subclass of intersection. ... |
ssini 4223 | An inference showing that ... |
ssind 4224 | A deduction showing that a... |
ssrin 4225 | Add right intersection to ... |
sslin 4226 | Add left intersection to s... |
ssrind 4227 | Add right intersection to ... |
ss2in 4228 | Intersection of subclasses... |
ssinss1 4229 | Intersection preserves sub... |
inss 4230 | Inclusion of an intersecti... |
rexin 4231 | Restricted existential qua... |
dfss7 4232 | Alternate definition of su... |
symdifcom 4235 | Symmetric difference commu... |
symdifeq1 4236 | Equality theorem for symme... |
symdifeq2 4237 | Equality theorem for symme... |
nfsymdif 4238 | Hypothesis builder for sym... |
elsymdif 4239 | Membership in a symmetric ... |
dfsymdif4 4240 | Alternate definition of th... |
elsymdifxor 4241 | Membership in a symmetric ... |
dfsymdif2 4242 | Alternate definition of th... |
symdifass 4243 | Symmetric difference is as... |
difsssymdif 4244 | The symmetric difference c... |
difsymssdifssd 4245 | If the symmetric differenc... |
unabs 4246 | Absorption law for union. ... |
inabs 4247 | Absorption law for interse... |
nssinpss 4248 | Negation of subclass expre... |
nsspssun 4249 | Negation of subclass expre... |
dfss4 4250 | Subclass defined in terms ... |
dfun2 4251 | An alternate definition of... |
dfin2 4252 | An alternate definition of... |
difin 4253 | Difference with intersecti... |
ssdifim 4254 | Implication of a class dif... |
ssdifsym 4255 | Symmetric class difference... |
dfss5 4256 | Alternate definition of su... |
dfun3 4257 | Union defined in terms of ... |
dfin3 4258 | Intersection defined in te... |
dfin4 4259 | Alternate definition of th... |
invdif 4260 | Intersection with universa... |
indif 4261 | Intersection with class di... |
indif2 4262 | Bring an intersection in a... |
indif1 4263 | Bring an intersection in a... |
indifcom 4264 | Commutation law for inters... |
indi 4265 | Distributive law for inter... |
undi 4266 | Distributive law for union... |
indir 4267 | Distributive law for inter... |
undir 4268 | Distributive law for union... |
unineq 4269 | Infer equality from equali... |
uneqin 4270 | Equality of union and inte... |
difundi 4271 | Distributive law for class... |
difundir 4272 | Distributive law for class... |
difindi 4273 | Distributive law for class... |
difindir 4274 | Distributive law for class... |
indifdi 4275 | Distribute intersection ov... |
indifdir 4276 | Distribute intersection ov... |
indifdirOLD 4277 | Obsolete version of ~ indi... |
difdif2 4278 | Class difference by a clas... |
undm 4279 | De Morgan's law for union.... |
indm 4280 | De Morgan's law for inters... |
difun1 4281 | A relationship involving d... |
undif3 4282 | An equality involving clas... |
difin2 4283 | Represent a class differen... |
dif32 4284 | Swap second and third argu... |
difabs 4285 | Absorption-like law for cl... |
sscon34b 4286 | Relative complementation r... |
rcompleq 4287 | Two subclasses are equal i... |
dfsymdif3 4288 | Alternate definition of th... |
unabw 4289 | Union of two class abstrac... |
unab 4290 | Union of two class abstrac... |
inab 4291 | Intersection of two class ... |
difab 4292 | Difference of two class ab... |
abanssl 4293 | A class abstraction with a... |
abanssr 4294 | A class abstraction with a... |
notabw 4295 | A class abstraction define... |
notab 4296 | A class abstraction define... |
unrab 4297 | Union of two restricted cl... |
inrab 4298 | Intersection of two restri... |
inrab2 4299 | Intersection with a restri... |
difrab 4300 | Difference of two restrict... |
dfrab3 4301 | Alternate definition of re... |
dfrab2 4302 | Alternate definition of re... |
notrab 4303 | Complementation of restric... |
dfrab3ss 4304 | Restricted class abstracti... |
rabun2 4305 | Abstraction restricted to ... |
reuun2 4306 | Transfer uniqueness to a s... |
reuss2 4307 | Transfer uniqueness to a s... |
reuss 4308 | Transfer uniqueness to a s... |
reuun1 4309 | Transfer uniqueness to a s... |
reupick 4310 | Restricted uniqueness "pic... |
reupick3 4311 | Restricted uniqueness "pic... |
reupick2 4312 | Restricted uniqueness "pic... |
euelss 4313 | Transfer uniqueness of an ... |
dfnul4 4316 | Alternate definition of th... |
dfnul2 4317 | Alternate definition of th... |
dfnul3 4318 | Alternate definition of th... |
dfnul2OLD 4319 | Obsolete version of ~ dfnu... |
dfnul3OLD 4320 | Obsolete version of ~ dfnu... |
dfnul4OLD 4321 | Obsolete version of ~ dfnu... |
noel 4322 | The empty set has no eleme... |
noelOLD 4323 | Obsolete version of ~ noel... |
nel02 4324 | The empty set has no eleme... |
n0i 4325 | If a class has elements, t... |
ne0i 4326 | If a class has elements, t... |
ne0d 4327 | Deduction form of ~ ne0i .... |
n0ii 4328 | If a class has elements, t... |
ne0ii 4329 | If a class has elements, t... |
vn0 4330 | The universal class is not... |
vn0ALT 4331 | Alternate proof of ~ vn0 .... |
eq0f 4332 | A class is equal to the em... |
neq0f 4333 | A class is not empty if an... |
n0f 4334 | A class is nonempty if and... |
eq0 4335 | A class is equal to the em... |
eq0ALT 4336 | Alternate proof of ~ eq0 .... |
neq0 4337 | A class is not empty if an... |
n0 4338 | A class is nonempty if and... |
eq0OLDOLD 4339 | Obsolete version of ~ eq0 ... |
neq0OLD 4340 | Obsolete version of ~ neq0... |
n0OLD 4341 | Obsolete version of ~ n0 a... |
nel0 4342 | From the general negation ... |
reximdva0 4343 | Restricted existence deduc... |
rspn0 4344 | Specialization for restric... |
rspn0OLD 4345 | Obsolete version of ~ rspn... |
n0rex 4346 | There is an element in a n... |
ssn0rex 4347 | There is an element in a c... |
n0moeu 4348 | A case of equivalence of "... |
rex0 4349 | Vacuous restricted existen... |
reu0 4350 | Vacuous restricted uniquen... |
rmo0 4351 | Vacuous restricted at-most... |
0el 4352 | Membership of the empty se... |
n0el 4353 | Negated membership of the ... |
eqeuel 4354 | A condition which implies ... |
ssdif0 4355 | Subclass expressed in term... |
difn0 4356 | If the difference of two s... |
pssdifn0 4357 | A proper subclass has a no... |
pssdif 4358 | A proper subclass has a no... |
ndisj 4359 | Express that an intersecti... |
difin0ss 4360 | Difference, intersection, ... |
inssdif0 4361 | Intersection, subclass, an... |
difid 4362 | The difference between a c... |
difidALT 4363 | Alternate proof of ~ difid... |
dif0 4364 | The difference between a c... |
ab0w 4365 | The class of sets verifyin... |
ab0 4366 | The class of sets verifyin... |
ab0OLD 4367 | Obsolete version of ~ ab0 ... |
ab0ALT 4368 | Alternate proof of ~ ab0 ,... |
dfnf5 4369 | Characterization of nonfre... |
ab0orv 4370 | The class abstraction defi... |
ab0orvALT 4371 | Alternate proof of ~ ab0or... |
abn0 4372 | Nonempty class abstraction... |
abn0OLD 4373 | Obsolete version of ~ abn0... |
rab0 4374 | Any restricted class abstr... |
rabeq0w 4375 | Condition for a restricted... |
rabeq0 4376 | Condition for a restricted... |
rabn0 4377 | Nonempty restricted class ... |
rabxm 4378 | Law of excluded middle, in... |
rabnc 4379 | Law of noncontradiction, i... |
elneldisj 4380 | The set of elements ` s ` ... |
elnelun 4381 | The union of the set of el... |
un0 4382 | The union of a class with ... |
in0 4383 | The intersection of a clas... |
0un 4384 | The union of the empty set... |
0in 4385 | The intersection of the em... |
inv1 4386 | The intersection of a clas... |
unv 4387 | The union of a class with ... |
0ss 4388 | The null set is a subset o... |
ss0b 4389 | Any subset of the empty se... |
ss0 4390 | Any subset of the empty se... |
sseq0 4391 | A subclass of an empty cla... |
ssn0 4392 | A class with a nonempty su... |
0dif 4393 | The difference between the... |
abf 4394 | A class abstraction determ... |
abfOLD 4395 | Obsolete version of ~ abf ... |
eq0rdv 4396 | Deduction for equality to ... |
eq0rdvALT 4397 | Alternate proof of ~ eq0rd... |
csbprc 4398 | The proper substitution of... |
csb0 4399 | The proper substitution of... |
sbcel12 4400 | Distribute proper substitu... |
sbceqg 4401 | Distribute proper substitu... |
sbceqi 4402 | Distribution of class subs... |
sbcnel12g 4403 | Distribute proper substitu... |
sbcne12 4404 | Distribute proper substitu... |
sbcel1g 4405 | Move proper substitution i... |
sbceq1g 4406 | Move proper substitution t... |
sbcel2 4407 | Move proper substitution i... |
sbceq2g 4408 | Move proper substitution t... |
csbcom 4409 | Commutative law for double... |
sbcnestgfw 4410 | Nest the composition of tw... |
csbnestgfw 4411 | Nest the composition of tw... |
sbcnestgw 4412 | Nest the composition of tw... |
csbnestgw 4413 | Nest the composition of tw... |
sbcco3gw 4414 | Composition of two substit... |
sbcnestgf 4415 | Nest the composition of tw... |
csbnestgf 4416 | Nest the composition of tw... |
sbcnestg 4417 | Nest the composition of tw... |
csbnestg 4418 | Nest the composition of tw... |
sbcco3g 4419 | Composition of two substit... |
csbco3g 4420 | Composition of two class s... |
csbnest1g 4421 | Nest the composition of tw... |
csbidm 4422 | Idempotent law for class s... |
csbvarg 4423 | The proper substitution of... |
csbvargi 4424 | The proper substitution of... |
sbccsb 4425 | Substitution into a wff ex... |
sbccsb2 4426 | Substitution into a wff ex... |
rspcsbela 4427 | Special case related to ~ ... |
sbnfc2 4428 | Two ways of expressing " `... |
csbab 4429 | Move substitution into a c... |
csbun 4430 | Distribution of class subs... |
csbin 4431 | Distribute proper substitu... |
csbie2df 4432 | Conversion of implicit sub... |
2nreu 4433 | If there are two different... |
un00 4434 | Two classes are empty iff ... |
vss 4435 | Only the universal class h... |
0pss 4436 | The null set is a proper s... |
npss0 4437 | No set is a proper subset ... |
pssv 4438 | Any non-universal class is... |
disj 4439 | Two ways of saying that tw... |
disjOLD 4440 | Obsolete version of ~ disj... |
disjr 4441 | Two ways of saying that tw... |
disj1 4442 | Two ways of saying that tw... |
reldisj 4443 | Two ways of saying that tw... |
reldisjOLD 4444 | Obsolete version of ~ reld... |
disj3 4445 | Two ways of saying that tw... |
disjne 4446 | Members of disjoint sets a... |
disjeq0 4447 | Two disjoint sets are equa... |
disjel 4448 | A set can't belong to both... |
disj2 4449 | Two ways of saying that tw... |
disj4 4450 | Two ways of saying that tw... |
ssdisj 4451 | Intersection with a subcla... |
disjpss 4452 | A class is a proper subset... |
undisj1 4453 | The union of disjoint clas... |
undisj2 4454 | The union of disjoint clas... |
ssindif0 4455 | Subclass expressed in term... |
inelcm 4456 | The intersection of classe... |
minel 4457 | A minimum element of a cla... |
undif4 4458 | Distribute union over diff... |
disjssun 4459 | Subset relation for disjoi... |
vdif0 4460 | Universal class equality i... |
difrab0eq 4461 | If the difference between ... |
pssnel 4462 | A proper subclass has a me... |
disjdif 4463 | A class and its relative c... |
disjdifr 4464 | A class and its relative c... |
difin0 4465 | The difference of a class ... |
unvdif 4466 | The union of a class and i... |
undif1 4467 | Absorption of difference b... |
undif2 4468 | Absorption of difference b... |
undifabs 4469 | Absorption of difference b... |
inundif 4470 | The intersection and class... |
disjdif2 4471 | The difference of a class ... |
difun2 4472 | Absorption of union by dif... |
undif 4473 | Union of complementary par... |
undifr 4474 | Union of complementary par... |
undifrOLD 4475 | Obsolete version of ~ undi... |
undif5 4476 | An equality involving clas... |
ssdifin0 4477 | A subset of a difference d... |
ssdifeq0 4478 | A class is a subclass of i... |
ssundif 4479 | A condition equivalent to ... |
difcom 4480 | Swap the arguments of a cl... |
pssdifcom1 4481 | Two ways to express overla... |
pssdifcom2 4482 | Two ways to express non-co... |
difdifdir 4483 | Distributive law for class... |
uneqdifeq 4484 | Two ways to say that ` A `... |
raldifeq 4485 | Equality theorem for restr... |
r19.2z 4486 | Theorem 19.2 of [Margaris]... |
r19.2zb 4487 | A response to the notion t... |
r19.3rz 4488 | Restricted quantification ... |
r19.28z 4489 | Restricted quantifier vers... |
r19.3rzv 4490 | Restricted quantification ... |
r19.9rzv 4491 | Restricted quantification ... |
r19.28zv 4492 | Restricted quantifier vers... |
r19.37zv 4493 | Restricted quantifier vers... |
r19.45zv 4494 | Restricted version of Theo... |
r19.44zv 4495 | Restricted version of Theo... |
r19.27z 4496 | Restricted quantifier vers... |
r19.27zv 4497 | Restricted quantifier vers... |
r19.36zv 4498 | Restricted quantifier vers... |
ralidmw 4499 | Idempotent law for restric... |
rzal 4500 | Vacuous quantification is ... |
rzalALT 4501 | Alternate proof of ~ rzal ... |
rexn0 4502 | Restricted existential qua... |
ralidm 4503 | Idempotent law for restric... |
ral0 4504 | Vacuous universal quantifi... |
ralf0 4505 | The quantification of a fa... |
rexn0OLD 4506 | Obsolete version of ~ rexn... |
ralidmOLD 4507 | Obsolete version of ~ rali... |
ral0OLD 4508 | Obsolete version of ~ ral0... |
ralf0OLD 4509 | Obsolete version of ~ ralf... |
ralnralall 4510 | A contradiction concerning... |
falseral0 4511 | A false statement can only... |
raaan 4512 | Rearrange restricted quant... |
raaanv 4513 | Rearrange restricted quant... |
sbss 4514 | Set substitution into the ... |
sbcssg 4515 | Distribute proper substitu... |
raaan2 4516 | Rearrange restricted quant... |
2reu4lem 4517 | Lemma for ~ 2reu4 . (Cont... |
2reu4 4518 | Definition of double restr... |
csbdif 4519 | Distribution of class subs... |
dfif2 4522 | An alternate definition of... |
dfif6 4523 | An alternate definition of... |
ifeq1 4524 | Equality theorem for condi... |
ifeq2 4525 | Equality theorem for condi... |
iftrue 4526 | Value of the conditional o... |
iftruei 4527 | Inference associated with ... |
iftrued 4528 | Value of the conditional o... |
iffalse 4529 | Value of the conditional o... |
iffalsei 4530 | Inference associated with ... |
iffalsed 4531 | Value of the conditional o... |
ifnefalse 4532 | When values are unequal, b... |
ifsb 4533 | Distribute a function over... |
dfif3 4534 | Alternate definition of th... |
dfif4 4535 | Alternate definition of th... |
dfif5 4536 | Alternate definition of th... |
ifssun 4537 | A conditional class is inc... |
ifeq12 4538 | Equality theorem for condi... |
ifeq1d 4539 | Equality deduction for con... |
ifeq2d 4540 | Equality deduction for con... |
ifeq12d 4541 | Equality deduction for con... |
ifbi 4542 | Equivalence theorem for co... |
ifbid 4543 | Equivalence deduction for ... |
ifbieq1d 4544 | Equivalence/equality deduc... |
ifbieq2i 4545 | Equivalence/equality infer... |
ifbieq2d 4546 | Equivalence/equality deduc... |
ifbieq12i 4547 | Equivalence deduction for ... |
ifbieq12d 4548 | Equivalence deduction for ... |
nfifd 4549 | Deduction form of ~ nfif .... |
nfif 4550 | Bound-variable hypothesis ... |
ifeq1da 4551 | Conditional equality. (Co... |
ifeq2da 4552 | Conditional equality. (Co... |
ifeq12da 4553 | Equivalence deduction for ... |
ifbieq12d2 4554 | Equivalence deduction for ... |
ifclda 4555 | Conditional closure. (Con... |
ifeqda 4556 | Separation of the values o... |
elimif 4557 | Elimination of a condition... |
ifbothda 4558 | A wff ` th ` containing a ... |
ifboth 4559 | A wff ` th ` containing a ... |
ifid 4560 | Identical true and false a... |
eqif 4561 | Expansion of an equality w... |
ifval 4562 | Another expression of the ... |
elif 4563 | Membership in a conditiona... |
ifel 4564 | Membership of a conditiona... |
ifcl 4565 | Membership (closure) of a ... |
ifcld 4566 | Membership (closure) of a ... |
ifcli 4567 | Inference associated with ... |
ifexd 4568 | Existence of the condition... |
ifexg 4569 | Existence of the condition... |
ifex 4570 | Existence of the condition... |
ifeqor 4571 | The possible values of a c... |
ifnot 4572 | Negating the first argumen... |
ifan 4573 | Rewrite a conjunction in a... |
ifor 4574 | Rewrite a disjunction in a... |
2if2 4575 | Resolve two nested conditi... |
ifcomnan 4576 | Commute the conditions in ... |
csbif 4577 | Distribute proper substitu... |
dedth 4578 | Weak deduction theorem tha... |
dedth2h 4579 | Weak deduction theorem eli... |
dedth3h 4580 | Weak deduction theorem eli... |
dedth4h 4581 | Weak deduction theorem eli... |
dedth2v 4582 | Weak deduction theorem for... |
dedth3v 4583 | Weak deduction theorem for... |
dedth4v 4584 | Weak deduction theorem for... |
elimhyp 4585 | Eliminate a hypothesis con... |
elimhyp2v 4586 | Eliminate a hypothesis con... |
elimhyp3v 4587 | Eliminate a hypothesis con... |
elimhyp4v 4588 | Eliminate a hypothesis con... |
elimel 4589 | Eliminate a membership hyp... |
elimdhyp 4590 | Version of ~ elimhyp where... |
keephyp 4591 | Transform a hypothesis ` p... |
keephyp2v 4592 | Keep a hypothesis containi... |
keephyp3v 4593 | Keep a hypothesis containi... |
pwjust 4595 | Soundness justification th... |
elpwg 4597 | Membership in a power clas... |
elpw 4598 | Membership in a power clas... |
velpw 4599 | Setvar variable membership... |
elpwd 4600 | Membership in a power clas... |
elpwi 4601 | Subset relation implied by... |
elpwb 4602 | Characterization of the el... |
elpwid 4603 | An element of a power clas... |
elelpwi 4604 | If ` A ` belongs to a part... |
sspw 4605 | The powerclass preserves i... |
sspwi 4606 | The powerclass preserves i... |
sspwd 4607 | The powerclass preserves i... |
pweq 4608 | Equality theorem for power... |
pweqALT 4609 | Alternate proof of ~ pweq ... |
pweqi 4610 | Equality inference for pow... |
pweqd 4611 | Equality deduction for pow... |
pwunss 4612 | The power class of the uni... |
nfpw 4613 | Bound-variable hypothesis ... |
pwidg 4614 | A set is an element of its... |
pwidb 4615 | A class is an element of i... |
pwid 4616 | A set is a member of its p... |
pwss 4617 | Subclass relationship for ... |
pwundif 4618 | Break up the power class o... |
snjust 4619 | Soundness justification th... |
sneq 4630 | Equality theorem for singl... |
sneqi 4631 | Equality inference for sin... |
sneqd 4632 | Equality deduction for sin... |
dfsn2 4633 | Alternate definition of si... |
elsng 4634 | There is exactly one eleme... |
elsn 4635 | There is exactly one eleme... |
velsn 4636 | There is only one element ... |
elsni 4637 | There is at most one eleme... |
absn 4638 | Condition for a class abst... |
dfpr2 4639 | Alternate definition of a ... |
dfsn2ALT 4640 | Alternate definition of si... |
elprg 4641 | A member of a pair of clas... |
elpri 4642 | If a class is an element o... |
elpr 4643 | A member of a pair of clas... |
elpr2g 4644 | A member of a pair of sets... |
elpr2 4645 | A member of a pair of sets... |
elpr2OLD 4646 | Obsolete version of ~ elpr... |
nelpr2 4647 | If a class is not an eleme... |
nelpr1 4648 | If a class is not an eleme... |
nelpri 4649 | If an element doesn't matc... |
prneli 4650 | If an element doesn't matc... |
nelprd 4651 | If an element doesn't matc... |
eldifpr 4652 | Membership in a set with t... |
rexdifpr 4653 | Restricted existential qua... |
snidg 4654 | A set is a member of its s... |
snidb 4655 | A class is a set iff it is... |
snid 4656 | A set is a member of its s... |
vsnid 4657 | A setvar variable is a mem... |
elsn2g 4658 | There is exactly one eleme... |
elsn2 4659 | There is exactly one eleme... |
nelsn 4660 | If a class is not equal to... |
rabeqsn 4661 | Conditions for a restricte... |
rabsssn 4662 | Conditions for a restricte... |
rabeqsnd 4663 | Conditions for a restricte... |
ralsnsg 4664 | Substitution expressed in ... |
rexsns 4665 | Restricted existential qua... |
rexsngf 4666 | Restricted existential qua... |
ralsngf 4667 | Restricted universal quant... |
reusngf 4668 | Restricted existential uni... |
ralsng 4669 | Substitution expressed in ... |
rexsng 4670 | Restricted existential qua... |
reusng 4671 | Restricted existential uni... |
2ralsng 4672 | Substitution expressed in ... |
ralsngOLD 4673 | Obsolete version of ~ rals... |
rexsngOLD 4674 | Obsolete version of ~ rexs... |
rexreusng 4675 | Restricted existential uni... |
exsnrex 4676 | There is a set being the e... |
ralsn 4677 | Convert a universal quanti... |
rexsn 4678 | Convert an existential qua... |
elpwunsn 4679 | Membership in an extension... |
eqoreldif 4680 | An element of a set is eit... |
eltpg 4681 | Members of an unordered tr... |
eldiftp 4682 | Membership in a set with t... |
eltpi 4683 | A member of an unordered t... |
eltp 4684 | A member of an unordered t... |
dftp2 4685 | Alternate definition of un... |
nfpr 4686 | Bound-variable hypothesis ... |
ifpr 4687 | Membership of a conditiona... |
ralprgf 4688 | Convert a restricted unive... |
rexprgf 4689 | Convert a restricted exist... |
ralprg 4690 | Convert a restricted unive... |
ralprgOLD 4691 | Obsolete version of ~ ralp... |
rexprg 4692 | Convert a restricted exist... |
rexprgOLD 4693 | Obsolete version of ~ rexp... |
raltpg 4694 | Convert a restricted unive... |
rextpg 4695 | Convert a restricted exist... |
ralpr 4696 | Convert a restricted unive... |
rexpr 4697 | Convert a restricted exist... |
reuprg0 4698 | Convert a restricted exist... |
reuprg 4699 | Convert a restricted exist... |
reurexprg 4700 | Convert a restricted exist... |
raltp 4701 | Convert a universal quanti... |
rextp 4702 | Convert an existential qua... |
nfsn 4703 | Bound-variable hypothesis ... |
csbsng 4704 | Distribute proper substitu... |
csbprg 4705 | Distribute proper substitu... |
elinsn 4706 | If the intersection of two... |
disjsn 4707 | Intersection with the sing... |
disjsn2 4708 | Two distinct singletons ar... |
disjpr2 4709 | Two completely distinct un... |
disjprsn 4710 | The disjoint intersection ... |
disjtpsn 4711 | The disjoint intersection ... |
disjtp2 4712 | Two completely distinct un... |
snprc 4713 | The singleton of a proper ... |
snnzb 4714 | A singleton is nonempty if... |
rmosn 4715 | A restricted at-most-one q... |
r19.12sn 4716 | Special case of ~ r19.12 w... |
rabsn 4717 | Condition where a restrict... |
rabsnifsb 4718 | A restricted class abstrac... |
rabsnif 4719 | A restricted class abstrac... |
rabrsn 4720 | A restricted class abstrac... |
euabsn2 4721 | Another way to express exi... |
euabsn 4722 | Another way to express exi... |
reusn 4723 | A way to express restricte... |
absneu 4724 | Restricted existential uni... |
rabsneu 4725 | Restricted existential uni... |
eusn 4726 | Two ways to express " ` A ... |
rabsnt 4727 | Truth implied by equality ... |
prcom 4728 | Commutative law for unorde... |
preq1 4729 | Equality theorem for unord... |
preq2 4730 | Equality theorem for unord... |
preq12 4731 | Equality theorem for unord... |
preq1i 4732 | Equality inference for uno... |
preq2i 4733 | Equality inference for uno... |
preq12i 4734 | Equality inference for uno... |
preq1d 4735 | Equality deduction for uno... |
preq2d 4736 | Equality deduction for uno... |
preq12d 4737 | Equality deduction for uno... |
tpeq1 4738 | Equality theorem for unord... |
tpeq2 4739 | Equality theorem for unord... |
tpeq3 4740 | Equality theorem for unord... |
tpeq1d 4741 | Equality theorem for unord... |
tpeq2d 4742 | Equality theorem for unord... |
tpeq3d 4743 | Equality theorem for unord... |
tpeq123d 4744 | Equality theorem for unord... |
tprot 4745 | Rotation of the elements o... |
tpcoma 4746 | Swap 1st and 2nd members o... |
tpcomb 4747 | Swap 2nd and 3rd members o... |
tpass 4748 | Split off the first elemen... |
qdass 4749 | Two ways to write an unord... |
qdassr 4750 | Two ways to write an unord... |
tpidm12 4751 | Unordered triple ` { A , A... |
tpidm13 4752 | Unordered triple ` { A , B... |
tpidm23 4753 | Unordered triple ` { A , B... |
tpidm 4754 | Unordered triple ` { A , A... |
tppreq3 4755 | An unordered triple is an ... |
prid1g 4756 | An unordered pair contains... |
prid2g 4757 | An unordered pair contains... |
prid1 4758 | An unordered pair contains... |
prid2 4759 | An unordered pair contains... |
ifpprsnss 4760 | An unordered pair is a sin... |
prprc1 4761 | A proper class vanishes in... |
prprc2 4762 | A proper class vanishes in... |
prprc 4763 | An unordered pair containi... |
tpid1 4764 | One of the three elements ... |
tpid1g 4765 | Closed theorem form of ~ t... |
tpid2 4766 | One of the three elements ... |
tpid2g 4767 | Closed theorem form of ~ t... |
tpid3g 4768 | Closed theorem form of ~ t... |
tpid3 4769 | One of the three elements ... |
snnzg 4770 | The singleton of a set is ... |
snn0d 4771 | The singleton of a set is ... |
snnz 4772 | The singleton of a set is ... |
prnz 4773 | A pair containing a set is... |
prnzg 4774 | A pair containing a set is... |
tpnz 4775 | An unordered triple contai... |
tpnzd 4776 | An unordered triple contai... |
raltpd 4777 | Convert a universal quanti... |
snssb 4778 | Characterization of the in... |
snssg 4779 | The singleton formed on a ... |
snssgOLD 4780 | Obsolete version of ~ snss... |
snss 4781 | The singleton of an elemen... |
eldifsn 4782 | Membership in a set with a... |
ssdifsn 4783 | Subset of a set with an el... |
elpwdifsn 4784 | A subset of a set is an el... |
eldifsni 4785 | Membership in a set with a... |
eldifsnneq 4786 | An element of a difference... |
neldifsn 4787 | The class ` A ` is not in ... |
neldifsnd 4788 | The class ` A ` is not in ... |
rexdifsn 4789 | Restricted existential qua... |
raldifsni 4790 | Rearrangement of a propert... |
raldifsnb 4791 | Restricted universal quant... |
eldifvsn 4792 | A set is an element of the... |
difsn 4793 | An element not in a set ca... |
difprsnss 4794 | Removal of a singleton fro... |
difprsn1 4795 | Removal of a singleton fro... |
difprsn2 4796 | Removal of a singleton fro... |
diftpsn3 4797 | Removal of a singleton fro... |
difpr 4798 | Removing two elements as p... |
tpprceq3 4799 | An unordered triple is an ... |
tppreqb 4800 | An unordered triple is an ... |
difsnb 4801 | ` ( B \ { A } ) ` equals `... |
difsnpss 4802 | ` ( B \ { A } ) ` is a pro... |
snssi 4803 | The singleton of an elemen... |
snssd 4804 | The singleton of an elemen... |
difsnid 4805 | If we remove a single elem... |
eldifeldifsn 4806 | An element of a difference... |
pw0 4807 | Compute the power set of t... |
pwpw0 4808 | Compute the power set of t... |
snsspr1 4809 | A singleton is a subset of... |
snsspr2 4810 | A singleton is a subset of... |
snsstp1 4811 | A singleton is a subset of... |
snsstp2 4812 | A singleton is a subset of... |
snsstp3 4813 | A singleton is a subset of... |
prssg 4814 | A pair of elements of a cl... |
prss 4815 | A pair of elements of a cl... |
prssi 4816 | A pair of elements of a cl... |
prssd 4817 | Deduction version of ~ prs... |
prsspwg 4818 | An unordered pair belongs ... |
ssprss 4819 | A pair as subset of a pair... |
ssprsseq 4820 | A proper pair is a subset ... |
sssn 4821 | The subsets of a singleton... |
ssunsn2 4822 | The property of being sand... |
ssunsn 4823 | Possible values for a set ... |
eqsn 4824 | Two ways to express that a... |
issn 4825 | A sufficient condition for... |
n0snor2el 4826 | A nonempty set is either a... |
ssunpr 4827 | Possible values for a set ... |
sspr 4828 | The subsets of a pair. (C... |
sstp 4829 | The subsets of an unordere... |
tpss 4830 | An unordered triple of ele... |
tpssi 4831 | An unordered triple of ele... |
sneqrg 4832 | Closed form of ~ sneqr . ... |
sneqr 4833 | If the singletons of two s... |
snsssn 4834 | If a singleton is a subset... |
mosneq 4835 | There exists at most one s... |
sneqbg 4836 | Two singletons of sets are... |
snsspw 4837 | The singleton of a class i... |
prsspw 4838 | An unordered pair belongs ... |
preq1b 4839 | Biconditional equality lem... |
preq2b 4840 | Biconditional equality lem... |
preqr1 4841 | Reverse equality lemma for... |
preqr2 4842 | Reverse equality lemma for... |
preq12b 4843 | Equality relationship for ... |
opthpr 4844 | An unordered pair has the ... |
preqr1g 4845 | Reverse equality lemma for... |
preq12bg 4846 | Closed form of ~ preq12b .... |
prneimg 4847 | Two pairs are not equal if... |
prnebg 4848 | A (proper) pair is not equ... |
pr1eqbg 4849 | A (proper) pair is equal t... |
pr1nebg 4850 | A (proper) pair is not equ... |
preqsnd 4851 | Equivalence for a pair equ... |
prnesn 4852 | A proper unordered pair is... |
prneprprc 4853 | A proper unordered pair is... |
preqsn 4854 | Equivalence for a pair equ... |
preq12nebg 4855 | Equality relationship for ... |
prel12g 4856 | Equality of two unordered ... |
opthprneg 4857 | An unordered pair has the ... |
elpreqprlem 4858 | Lemma for ~ elpreqpr . (C... |
elpreqpr 4859 | Equality and membership ru... |
elpreqprb 4860 | A set is an element of an ... |
elpr2elpr 4861 | For an element ` A ` of an... |
dfopif 4862 | Rewrite ~ df-op using ` if... |
dfopg 4863 | Value of the ordered pair ... |
dfop 4864 | Value of an ordered pair w... |
opeq1 4865 | Equality theorem for order... |
opeq2 4866 | Equality theorem for order... |
opeq12 4867 | Equality theorem for order... |
opeq1i 4868 | Equality inference for ord... |
opeq2i 4869 | Equality inference for ord... |
opeq12i 4870 | Equality inference for ord... |
opeq1d 4871 | Equality deduction for ord... |
opeq2d 4872 | Equality deduction for ord... |
opeq12d 4873 | Equality deduction for ord... |
oteq1 4874 | Equality theorem for order... |
oteq2 4875 | Equality theorem for order... |
oteq3 4876 | Equality theorem for order... |
oteq1d 4877 | Equality deduction for ord... |
oteq2d 4878 | Equality deduction for ord... |
oteq3d 4879 | Equality deduction for ord... |
oteq123d 4880 | Equality deduction for ord... |
nfop 4881 | Bound-variable hypothesis ... |
nfopd 4882 | Deduction version of bound... |
csbopg 4883 | Distribution of class subs... |
opidg 4884 | The ordered pair ` <. A , ... |
opid 4885 | The ordered pair ` <. A , ... |
ralunsn 4886 | Restricted quantification ... |
2ralunsn 4887 | Double restricted quantifi... |
opprc 4888 | Expansion of an ordered pa... |
opprc1 4889 | Expansion of an ordered pa... |
opprc2 4890 | Expansion of an ordered pa... |
oprcl 4891 | If an ordered pair has an ... |
pwsn 4892 | The power set of a singlet... |
pwpr 4893 | The power set of an unorde... |
pwtp 4894 | The power set of an unorde... |
pwpwpw0 4895 | Compute the power set of t... |
pwv 4896 | The power class of the uni... |
prproe 4897 | For an element of a proper... |
3elpr2eq 4898 | If there are three element... |
dfuni2 4901 | Alternate definition of cl... |
eluni 4902 | Membership in class union.... |
eluni2 4903 | Membership in class union.... |
elunii 4904 | Membership in class union.... |
nfunid 4905 | Deduction version of ~ nfu... |
nfuni 4906 | Bound-variable hypothesis ... |
uniss 4907 | Subclass relationship for ... |
unissi 4908 | Subclass relationship for ... |
unissd 4909 | Subclass relationship for ... |
unieq 4910 | Equality theorem for class... |
unieqi 4911 | Inference of equality of t... |
unieqd 4912 | Deduction of equality of t... |
eluniab 4913 | Membership in union of a c... |
elunirab 4914 | Membership in union of a c... |
uniprg 4915 | The union of a pair is the... |
unipr 4916 | The union of a pair is the... |
uniprOLD 4917 | Obsolete version of ~ unip... |
uniprgOLD 4918 | Obsolete version of ~ unip... |
unisng 4919 | A set equals the union of ... |
unisn 4920 | A set equals the union of ... |
unisnv 4921 | A set equals the union of ... |
unisn3 4922 | Union of a singleton in th... |
dfnfc2 4923 | An alternative statement o... |
uniun 4924 | The class union of the uni... |
uniin 4925 | The class union of the int... |
ssuni 4926 | Subclass relationship for ... |
uni0b 4927 | The union of a set is empt... |
uni0c 4928 | The union of a set is empt... |
uni0 4929 | The union of the empty set... |
csbuni 4930 | Distribute proper substitu... |
elssuni 4931 | An element of a class is a... |
unissel 4932 | Condition turning a subcla... |
unissb 4933 | Relationship involving mem... |
unissbOLD 4934 | Obsolete version of ~ unis... |
uniss2 4935 | A subclass condition on th... |
unidif 4936 | If the difference ` A \ B ... |
ssunieq 4937 | Relationship implying unio... |
unimax 4938 | Any member of a class is t... |
pwuni 4939 | A class is a subclass of t... |
dfint2 4942 | Alternate definition of cl... |
inteq 4943 | Equality law for intersect... |
inteqi 4944 | Equality inference for cla... |
inteqd 4945 | Equality deduction for cla... |
elint 4946 | Membership in class inters... |
elint2 4947 | Membership in class inters... |
elintg 4948 | Membership in class inters... |
elinti 4949 | Membership in class inters... |
nfint 4950 | Bound-variable hypothesis ... |
elintabg 4951 | Two ways of saying a set i... |
elintab 4952 | Membership in the intersec... |
elintabOLD 4953 | Obsolete version of ~ elin... |
elintrab 4954 | Membership in the intersec... |
elintrabg 4955 | Membership in the intersec... |
int0 4956 | The intersection of the em... |
intss1 4957 | An element of a class incl... |
ssint 4958 | Subclass of a class inters... |
ssintab 4959 | Subclass of the intersecti... |
ssintub 4960 | Subclass of the least uppe... |
ssmin 4961 | Subclass of the minimum va... |
intmin 4962 | Any member of a class is t... |
intss 4963 | Intersection of subclasses... |
intssuni 4964 | The intersection of a none... |
ssintrab 4965 | Subclass of the intersecti... |
unissint 4966 | If the union of a class is... |
intssuni2 4967 | Subclass relationship for ... |
intminss 4968 | Under subset ordering, the... |
intmin2 4969 | Any set is the smallest of... |
intmin3 4970 | Under subset ordering, the... |
intmin4 4971 | Elimination of a conjunct ... |
intab 4972 | The intersection of a spec... |
int0el 4973 | The intersection of a clas... |
intun 4974 | The class intersection of ... |
intprg 4975 | The intersection of a pair... |
intpr 4976 | The intersection of a pair... |
intprOLD 4977 | Obsolete version of ~ intp... |
intprgOLD 4978 | Obsolete version of ~ intp... |
intsng 4979 | Intersection of a singleto... |
intsn 4980 | The intersection of a sing... |
uniintsn 4981 | Two ways to express " ` A ... |
uniintab 4982 | The union and the intersec... |
intunsn 4983 | Theorem joining a singleto... |
rint0 4984 | Relative intersection of a... |
elrint 4985 | Membership in a restricted... |
elrint2 4986 | Membership in a restricted... |
eliun 4991 | Membership in indexed unio... |
eliin 4992 | Membership in indexed inte... |
eliuni 4993 | Membership in an indexed u... |
iuncom 4994 | Commutation of indexed uni... |
iuncom4 4995 | Commutation of union with ... |
iunconst 4996 | Indexed union of a constan... |
iinconst 4997 | Indexed intersection of a ... |
iuneqconst 4998 | Indexed union of identical... |
iuniin 4999 | Law combining indexed unio... |
iinssiun 5000 | An indexed intersection is... |
iunss1 5001 | Subclass theorem for index... |
iinss1 5002 | Subclass theorem for index... |
iuneq1 5003 | Equality theorem for index... |
iineq1 5004 | Equality theorem for index... |
ss2iun 5005 | Subclass theorem for index... |
iuneq2 5006 | Equality theorem for index... |
iineq2 5007 | Equality theorem for index... |
iuneq2i 5008 | Equality inference for ind... |
iineq2i 5009 | Equality inference for ind... |
iineq2d 5010 | Equality deduction for ind... |
iuneq2dv 5011 | Equality deduction for ind... |
iineq2dv 5012 | Equality deduction for ind... |
iuneq12df 5013 | Equality deduction for ind... |
iuneq1d 5014 | Equality theorem for index... |
iuneq12d 5015 | Equality deduction for ind... |
iuneq2d 5016 | Equality deduction for ind... |
nfiun 5017 | Bound-variable hypothesis ... |
nfiin 5018 | Bound-variable hypothesis ... |
nfiung 5019 | Bound-variable hypothesis ... |
nfiing 5020 | Bound-variable hypothesis ... |
nfiu1 5021 | Bound-variable hypothesis ... |
nfii1 5022 | Bound-variable hypothesis ... |
dfiun2g 5023 | Alternate definition of in... |
dfiun2gOLD 5024 | Obsolete version of ~ dfiu... |
dfiin2g 5025 | Alternate definition of in... |
dfiun2 5026 | Alternate definition of in... |
dfiin2 5027 | Alternate definition of in... |
dfiunv2 5028 | Define double indexed unio... |
cbviun 5029 | Rule used to change the bo... |
cbviin 5030 | Change bound variables in ... |
cbviung 5031 | Rule used to change the bo... |
cbviing 5032 | Change bound variables in ... |
cbviunv 5033 | Rule used to change the bo... |
cbviinv 5034 | Change bound variables in ... |
cbviunvg 5035 | Rule used to change the bo... |
cbviinvg 5036 | Change bound variables in ... |
iunssf 5037 | Subset theorem for an inde... |
iunss 5038 | Subset theorem for an inde... |
ssiun 5039 | Subset implication for an ... |
ssiun2 5040 | Identity law for subset of... |
ssiun2s 5041 | Subset relationship for an... |
iunss2 5042 | A subclass condition on th... |
iunssd 5043 | Subset theorem for an inde... |
iunab 5044 | The indexed union of a cla... |
iunrab 5045 | The indexed union of a res... |
iunxdif2 5046 | Indexed union with a class... |
ssiinf 5047 | Subset theorem for an inde... |
ssiin 5048 | Subset theorem for an inde... |
iinss 5049 | Subset implication for an ... |
iinss2 5050 | An indexed intersection is... |
uniiun 5051 | Class union in terms of in... |
intiin 5052 | Class intersection in term... |
iunid 5053 | An indexed union of single... |
iunidOLD 5054 | Obsolete version of ~ iuni... |
iun0 5055 | An indexed union of the em... |
0iun 5056 | An empty indexed union is ... |
0iin 5057 | An empty indexed intersect... |
viin 5058 | Indexed intersection with ... |
iunsn 5059 | Indexed union of a singlet... |
iunn0 5060 | There is a nonempty class ... |
iinab 5061 | Indexed intersection of a ... |
iinrab 5062 | Indexed intersection of a ... |
iinrab2 5063 | Indexed intersection of a ... |
iunin2 5064 | Indexed union of intersect... |
iunin1 5065 | Indexed union of intersect... |
iinun2 5066 | Indexed intersection of un... |
iundif2 5067 | Indexed union of class dif... |
iindif1 5068 | Indexed intersection of cl... |
2iunin 5069 | Rearrange indexed unions o... |
iindif2 5070 | Indexed intersection of cl... |
iinin2 5071 | Indexed intersection of in... |
iinin1 5072 | Indexed intersection of in... |
iinvdif 5073 | The indexed intersection o... |
elriin 5074 | Elementhood in a relative ... |
riin0 5075 | Relative intersection of a... |
riinn0 5076 | Relative intersection of a... |
riinrab 5077 | Relative intersection of a... |
symdif0 5078 | Symmetric difference with ... |
symdifv 5079 | The symmetric difference w... |
symdifid 5080 | The symmetric difference o... |
iinxsng 5081 | A singleton index picks ou... |
iinxprg 5082 | Indexed intersection with ... |
iunxsng 5083 | A singleton index picks ou... |
iunxsn 5084 | A singleton index picks ou... |
iunxsngf 5085 | A singleton index picks ou... |
iunun 5086 | Separate a union in an ind... |
iunxun 5087 | Separate a union in the in... |
iunxdif3 5088 | An indexed union where som... |
iunxprg 5089 | A pair index picks out two... |
iunxiun 5090 | Separate an indexed union ... |
iinuni 5091 | A relationship involving u... |
iununi 5092 | A relationship involving u... |
sspwuni 5093 | Subclass relationship for ... |
pwssb 5094 | Two ways to express a coll... |
elpwpw 5095 | Characterization of the el... |
pwpwab 5096 | The double power class wri... |
pwpwssunieq 5097 | The class of sets whose un... |
elpwuni 5098 | Relationship for power cla... |
iinpw 5099 | The power class of an inte... |
iunpwss 5100 | Inclusion of an indexed un... |
intss2 5101 | A nonempty intersection of... |
rintn0 5102 | Relative intersection of a... |
dfdisj2 5105 | Alternate definition for d... |
disjss2 5106 | If each element of a colle... |
disjeq2 5107 | Equality theorem for disjo... |
disjeq2dv 5108 | Equality deduction for dis... |
disjss1 5109 | A subset of a disjoint col... |
disjeq1 5110 | Equality theorem for disjo... |
disjeq1d 5111 | Equality theorem for disjo... |
disjeq12d 5112 | Equality theorem for disjo... |
cbvdisj 5113 | Change bound variables in ... |
cbvdisjv 5114 | Change bound variables in ... |
nfdisjw 5115 | Bound-variable hypothesis ... |
nfdisj 5116 | Bound-variable hypothesis ... |
nfdisj1 5117 | Bound-variable hypothesis ... |
disjor 5118 | Two ways to say that a col... |
disjors 5119 | Two ways to say that a col... |
disji2 5120 | Property of a disjoint col... |
disji 5121 | Property of a disjoint col... |
invdisj 5122 | If there is a function ` C... |
invdisjrabw 5123 | Version of ~ invdisjrab wi... |
invdisjrab 5124 | The restricted class abstr... |
disjiun 5125 | A disjoint collection yiel... |
disjord 5126 | Conditions for a collectio... |
disjiunb 5127 | Two ways to say that a col... |
disjiund 5128 | Conditions for a collectio... |
sndisj 5129 | Any collection of singleto... |
0disj 5130 | Any collection of empty se... |
disjxsn 5131 | A singleton collection is ... |
disjx0 5132 | An empty collection is dis... |
disjprgw 5133 | Version of ~ disjprg with ... |
disjprg 5134 | A pair collection is disjo... |
disjxiun 5135 | An indexed union of a disj... |
disjxun 5136 | The union of two disjoint ... |
disjss3 5137 | Expand a disjoint collecti... |
breq 5140 | Equality theorem for binar... |
breq1 5141 | Equality theorem for a bin... |
breq2 5142 | Equality theorem for a bin... |
breq12 5143 | Equality theorem for a bin... |
breqi 5144 | Equality inference for bin... |
breq1i 5145 | Equality inference for a b... |
breq2i 5146 | Equality inference for a b... |
breq12i 5147 | Equality inference for a b... |
breq1d 5148 | Equality deduction for a b... |
breqd 5149 | Equality deduction for a b... |
breq2d 5150 | Equality deduction for a b... |
breq12d 5151 | Equality deduction for a b... |
breq123d 5152 | Equality deduction for a b... |
breqdi 5153 | Equality deduction for a b... |
breqan12d 5154 | Equality deduction for a b... |
breqan12rd 5155 | Equality deduction for a b... |
eqnbrtrd 5156 | Substitution of equal clas... |
nbrne1 5157 | Two classes are different ... |
nbrne2 5158 | Two classes are different ... |
eqbrtri 5159 | Substitution of equal clas... |
eqbrtrd 5160 | Substitution of equal clas... |
eqbrtrri 5161 | Substitution of equal clas... |
eqbrtrrd 5162 | Substitution of equal clas... |
breqtri 5163 | Substitution of equal clas... |
breqtrd 5164 | Substitution of equal clas... |
breqtrri 5165 | Substitution of equal clas... |
breqtrrd 5166 | Substitution of equal clas... |
3brtr3i 5167 | Substitution of equality i... |
3brtr4i 5168 | Substitution of equality i... |
3brtr3d 5169 | Substitution of equality i... |
3brtr4d 5170 | Substitution of equality i... |
3brtr3g 5171 | Substitution of equality i... |
3brtr4g 5172 | Substitution of equality i... |
eqbrtrid 5173 | A chained equality inferen... |
eqbrtrrid 5174 | A chained equality inferen... |
breqtrid 5175 | A chained equality inferen... |
breqtrrid 5176 | A chained equality inferen... |
eqbrtrdi 5177 | A chained equality inferen... |
eqbrtrrdi 5178 | A chained equality inferen... |
breqtrdi 5179 | A chained equality inferen... |
breqtrrdi 5180 | A chained equality inferen... |
ssbrd 5181 | Deduction from a subclass ... |
ssbr 5182 | Implication from a subclas... |
ssbri 5183 | Inference from a subclass ... |
nfbrd 5184 | Deduction version of bound... |
nfbr 5185 | Bound-variable hypothesis ... |
brab1 5186 | Relationship between a bin... |
br0 5187 | The empty binary relation ... |
brne0 5188 | If two sets are in a binar... |
brun 5189 | The union of two binary re... |
brin 5190 | The intersection of two re... |
brdif 5191 | The difference of two bina... |
sbcbr123 5192 | Move substitution in and o... |
sbcbr 5193 | Move substitution in and o... |
sbcbr12g 5194 | Move substitution in and o... |
sbcbr1g 5195 | Move substitution in and o... |
sbcbr2g 5196 | Move substitution in and o... |
brsymdif 5197 | Characterization of the sy... |
brralrspcev 5198 | Restricted existential spe... |
brimralrspcev 5199 | Restricted existential spe... |
opabss 5202 | The collection of ordered ... |
opabbid 5203 | Equivalent wff's yield equ... |
opabbidv 5204 | Equivalent wff's yield equ... |
opabbii 5205 | Equivalent wff's yield equ... |
nfopabd 5206 | Bound-variable hypothesis ... |
nfopab 5207 | Bound-variable hypothesis ... |
nfopab1 5208 | The first abstraction vari... |
nfopab2 5209 | The second abstraction var... |
cbvopab 5210 | Rule used to change bound ... |
cbvopabv 5211 | Rule used to change bound ... |
cbvopabvOLD 5212 | Obsolete version of ~ cbvo... |
cbvopab1 5213 | Change first bound variabl... |
cbvopab1g 5214 | Change first bound variabl... |
cbvopab2 5215 | Change second bound variab... |
cbvopab1s 5216 | Change first bound variabl... |
cbvopab1v 5217 | Rule used to change the fi... |
cbvopab1vOLD 5218 | Obsolete version of ~ cbvo... |
cbvopab2v 5219 | Rule used to change the se... |
unopab 5220 | Union of two ordered pair ... |
mpteq12da 5223 | An equality inference for ... |
mpteq12df 5224 | An equality inference for ... |
mpteq12dfOLD 5225 | Obsolete version of ~ mpte... |
mpteq12f 5226 | An equality theorem for th... |
mpteq12dva 5227 | An equality inference for ... |
mpteq12dvaOLD 5228 | Obsolete version of ~ mpte... |
mpteq12dv 5229 | An equality inference for ... |
mpteq12 5230 | An equality theorem for th... |
mpteq1 5231 | An equality theorem for th... |
mpteq1OLD 5232 | Obsolete version of ~ mpte... |
mpteq1d 5233 | An equality theorem for th... |
mpteq1i 5234 | An equality theorem for th... |
mpteq1iOLD 5235 | Obsolete version of ~ mpte... |
mpteq2da 5236 | Slightly more general equa... |
mpteq2daOLD 5237 | Obsolete version of ~ mpte... |
mpteq2dva 5238 | Slightly more general equa... |
mpteq2dvaOLD 5239 | Obsolete version of ~ mpte... |
mpteq2dv 5240 | An equality inference for ... |
mpteq2ia 5241 | An equality inference for ... |
mpteq2iaOLD 5242 | Obsolete version of ~ mpte... |
mpteq2i 5243 | An equality inference for ... |
mpteq12i 5244 | An equality inference for ... |
nfmpt 5245 | Bound-variable hypothesis ... |
nfmpt1 5246 | Bound-variable hypothesis ... |
cbvmptf 5247 | Rule to change the bound v... |
cbvmptfg 5248 | Rule to change the bound v... |
cbvmpt 5249 | Rule to change the bound v... |
cbvmptg 5250 | Rule to change the bound v... |
cbvmptv 5251 | Rule to change the bound v... |
cbvmptvOLD 5252 | Obsolete version of ~ cbvm... |
cbvmptvg 5253 | Rule to change the bound v... |
mptv 5254 | Function with universal do... |
dftr2 5257 | An alternate way of defini... |
dftr2c 5258 | Variant of ~ dftr2 with co... |
dftr5 5259 | An alternate way of defini... |
dftr5OLD 5260 | Obsolete version of ~ dftr... |
dftr3 5261 | An alternate way of defini... |
dftr4 5262 | An alternate way of defini... |
treq 5263 | Equality theorem for the t... |
trel 5264 | In a transitive class, the... |
trel3 5265 | In a transitive class, the... |
trss 5266 | An element of a transitive... |
trin 5267 | The intersection of transi... |
tr0 5268 | The empty set is transitiv... |
trv 5269 | The universe is transitive... |
triun 5270 | An indexed union of a clas... |
truni 5271 | The union of a class of tr... |
triin 5272 | An indexed intersection of... |
trint 5273 | The intersection of a clas... |
trintss 5274 | Any nonempty transitive cl... |
axrep1 5276 | The version of the Axiom o... |
axreplem 5277 | Lemma for ~ axrep2 and ~ a... |
axrep2 5278 | Axiom of Replacement expre... |
axrep3 5279 | Axiom of Replacement sligh... |
axrep4 5280 | A more traditional version... |
axrep5 5281 | Axiom of Replacement (simi... |
axrep6 5282 | A condensed form of ~ ax-r... |
axrep6g 5283 | ~ axrep6 in class notation... |
zfrepclf 5284 | An inference based on the ... |
zfrep3cl 5285 | An inference based on the ... |
zfrep4 5286 | A version of Replacement u... |
axsepgfromrep 5287 | A more general version ~ a... |
axsep 5288 | Axiom scheme of separation... |
axsepg 5290 | A more general version of ... |
zfauscl 5291 | Separation Scheme (Aussond... |
bm1.3ii 5292 | Convert implication to equ... |
ax6vsep 5293 | Derive ~ ax6v (a weakened ... |
axnulALT 5294 | Alternate proof of ~ axnul... |
axnul 5295 | The Null Set Axiom of ZF s... |
0ex 5297 | The Null Set Axiom of ZF s... |
al0ssb 5298 | The empty set is the uniqu... |
sseliALT 5299 | Alternate proof of ~ sseli... |
csbexg 5300 | The existence of proper su... |
csbex 5301 | The existence of proper su... |
unisn2 5302 | A version of ~ unisn witho... |
nalset 5303 | No set contains all sets. ... |
vnex 5304 | The universal class does n... |
vprc 5305 | The universal class is not... |
nvel 5306 | The universal class does n... |
inex1 5307 | Separation Scheme (Aussond... |
inex2 5308 | Separation Scheme (Aussond... |
inex1g 5309 | Closed-form, generalized S... |
inex2g 5310 | Sufficient condition for a... |
ssex 5311 | The subset of a set is als... |
ssexi 5312 | The subset of a set is als... |
ssexg 5313 | The subset of a set is als... |
ssexd 5314 | A subclass of a set is a s... |
prcssprc 5315 | The superclass of a proper... |
sselpwd 5316 | Elementhood to a power set... |
difexg 5317 | Existence of a difference.... |
difexi 5318 | Existence of a difference,... |
difexd 5319 | Existence of a difference.... |
zfausab 5320 | Separation Scheme (Aussond... |
rabexg 5321 | Separation Scheme in terms... |
rabex 5322 | Separation Scheme in terms... |
rabexd 5323 | Separation Scheme in terms... |
rabex2 5324 | Separation Scheme in terms... |
rab2ex 5325 | A class abstraction based ... |
elssabg 5326 | Membership in a class abst... |
intex 5327 | The intersection of a none... |
intnex 5328 | If a class intersection is... |
intexab 5329 | The intersection of a none... |
intexrab 5330 | The intersection of a none... |
iinexg 5331 | The existence of a class i... |
intabs 5332 | Absorption of a redundant ... |
inuni 5333 | The intersection of a unio... |
elpw2g 5334 | Membership in a power clas... |
elpw2 5335 | Membership in a power clas... |
elpwi2 5336 | Membership in a power clas... |
elpwi2OLD 5337 | Obsolete version of ~ elpw... |
axpweq 5338 | Two equivalent ways to exp... |
pwnss 5339 | The power set of a set is ... |
pwne 5340 | No set equals its power se... |
difelpw 5341 | A difference is an element... |
rabelpw 5342 | A restricted class abstrac... |
class2set 5343 | The class of elements of `... |
0elpw 5344 | Every power class contains... |
pwne0 5345 | A power class is never emp... |
0nep0 5346 | The empty set and its powe... |
0inp0 5347 | Something cannot be equal ... |
unidif0 5348 | The removal of the empty s... |
eqsnuniex 5349 | If a class is equal to the... |
iin0 5350 | An indexed intersection of... |
notzfaus 5351 | In the Separation Scheme ~... |
intv 5352 | The intersection of the un... |
zfpow 5354 | Axiom of Power Sets expres... |
axpow2 5355 | A variant of the Axiom of ... |
axpow3 5356 | A variant of the Axiom of ... |
elALT2 5357 | Alternate proof of ~ el us... |
dtruALT2 5358 | Alternate proof of ~ dtru ... |
dtrucor 5359 | Corollary of ~ dtru . Thi... |
dtrucor2 5360 | The theorem form of the de... |
dvdemo1 5361 | Demonstration of a theorem... |
dvdemo2 5362 | Demonstration of a theorem... |
nfnid 5363 | A setvar variable is not f... |
nfcvb 5364 | The "distinctor" expressio... |
vpwex 5365 | Power set axiom: the power... |
pwexg 5366 | Power set axiom expressed ... |
pwexd 5367 | Deduction version of the p... |
pwex 5368 | Power set axiom expressed ... |
pwel 5369 | Quantitative version of ~ ... |
abssexg 5370 | Existence of a class of su... |
snexALT 5371 | Alternate proof of ~ snex ... |
p0ex 5372 | The power set of the empty... |
p0exALT 5373 | Alternate proof of ~ p0ex ... |
pp0ex 5374 | The power set of the power... |
ord3ex 5375 | The ordinal number 3 is a ... |
dtruALT 5376 | Alternate proof of ~ dtru ... |
axc16b 5377 | This theorem shows that Ax... |
eunex 5378 | Existential uniqueness imp... |
eusv1 5379 | Two ways to express single... |
eusvnf 5380 | Even if ` x ` is free in `... |
eusvnfb 5381 | Two ways to say that ` A (... |
eusv2i 5382 | Two ways to express single... |
eusv2nf 5383 | Two ways to express single... |
eusv2 5384 | Two ways to express single... |
reusv1 5385 | Two ways to express single... |
reusv2lem1 5386 | Lemma for ~ reusv2 . (Con... |
reusv2lem2 5387 | Lemma for ~ reusv2 . (Con... |
reusv2lem3 5388 | Lemma for ~ reusv2 . (Con... |
reusv2lem4 5389 | Lemma for ~ reusv2 . (Con... |
reusv2lem5 5390 | Lemma for ~ reusv2 . (Con... |
reusv2 5391 | Two ways to express single... |
reusv3i 5392 | Two ways of expressing exi... |
reusv3 5393 | Two ways to express single... |
eusv4 5394 | Two ways to express single... |
alxfr 5395 | Transfer universal quantif... |
ralxfrd 5396 | Transfer universal quantif... |
rexxfrd 5397 | Transfer universal quantif... |
ralxfr2d 5398 | Transfer universal quantif... |
rexxfr2d 5399 | Transfer universal quantif... |
ralxfrd2 5400 | Transfer universal quantif... |
rexxfrd2 5401 | Transfer existence from a ... |
ralxfr 5402 | Transfer universal quantif... |
ralxfrALT 5403 | Alternate proof of ~ ralxf... |
rexxfr 5404 | Transfer existence from a ... |
rabxfrd 5405 | Membership in a restricted... |
rabxfr 5406 | Membership in a restricted... |
reuhypd 5407 | A theorem useful for elimi... |
reuhyp 5408 | A theorem useful for elimi... |
zfpair 5409 | The Axiom of Pairing of Ze... |
axprALT 5410 | Alternate proof of ~ axpr ... |
axprlem1 5411 | Lemma for ~ axpr . There ... |
axprlem2 5412 | Lemma for ~ axpr . There ... |
axprlem3 5413 | Lemma for ~ axpr . Elimin... |
axprlem4 5414 | Lemma for ~ axpr . The fi... |
axprlem5 5415 | Lemma for ~ axpr . The se... |
axpr 5416 | Unabbreviated version of t... |
zfpair2 5418 | Derive the abbreviated ver... |
vsnex 5419 | A singleton built on a set... |
snexg 5420 | A singleton built on a set... |
snex 5421 | A singleton is a set. The... |
prex 5422 | The Axiom of Pairing using... |
exel 5423 | There exist two sets, one ... |
exexneq 5424 | There exist two different ... |
exneq 5425 | Given any set (the " ` y `... |
dtru 5426 | Given any set (the " ` y `... |
el 5427 | Any set is an element of s... |
sels 5428 | If a class is a set, then ... |
selsALT 5429 | Alternate proof of ~ sels ... |
elALT 5430 | Alternate proof of ~ el , ... |
dtruOLD 5431 | Obsolete proof of ~ dtru a... |
snelpwg 5432 | A singleton of a set is a ... |
snelpwi 5433 | If a set is a member of a ... |
snelpwiOLD 5434 | Obsolete version of ~ snel... |
snelpw 5435 | A singleton of a set is a ... |
prelpw 5436 | An unordered pair of two s... |
prelpwi 5437 | If two sets are members of... |
rext 5438 | A theorem similar to exten... |
sspwb 5439 | The powerclass constructio... |
unipw 5440 | A class equals the union o... |
univ 5441 | The union of the universe ... |
pwtr 5442 | A class is transitive iff ... |
ssextss 5443 | An extensionality-like pri... |
ssext 5444 | An extensionality-like pri... |
nssss 5445 | Negation of subclass relat... |
pweqb 5446 | Classes are equal if and o... |
intidg 5447 | The intersection of all se... |
intidOLD 5448 | Obsolete version of ~ inti... |
moabex 5449 | "At most one" existence im... |
rmorabex 5450 | Restricted "at most one" e... |
euabex 5451 | The abstraction of a wff w... |
nnullss 5452 | A nonempty class (even if ... |
exss 5453 | Restricted existence in a ... |
opex 5454 | An ordered pair of classes... |
otex 5455 | An ordered triple of class... |
elopg 5456 | Characterization of the el... |
elop 5457 | Characterization of the el... |
opi1 5458 | One of the two elements in... |
opi2 5459 | One of the two elements of... |
opeluu 5460 | Each member of an ordered ... |
op1stb 5461 | Extract the first member o... |
brv 5462 | Two classes are always in ... |
opnz 5463 | An ordered pair is nonempt... |
opnzi 5464 | An ordered pair is nonempt... |
opth1 5465 | Equality of the first memb... |
opth 5466 | The ordered pair theorem. ... |
opthg 5467 | Ordered pair theorem. ` C ... |
opth1g 5468 | Equality of the first memb... |
opthg2 5469 | Ordered pair theorem. (Co... |
opth2 5470 | Ordered pair theorem. (Co... |
opthneg 5471 | Two ordered pairs are not ... |
opthne 5472 | Two ordered pairs are not ... |
otth2 5473 | Ordered triple theorem, wi... |
otth 5474 | Ordered triple theorem. (... |
otthg 5475 | Ordered triple theorem, cl... |
otthne 5476 | Contrapositive of the orde... |
eqvinop 5477 | A variable introduction la... |
sbcop1 5478 | The proper substitution of... |
sbcop 5479 | The proper substitution of... |
copsexgw 5480 | Version of ~ copsexg with ... |
copsexg 5481 | Substitution of class ` A ... |
copsex2t 5482 | Closed theorem form of ~ c... |
copsex2g 5483 | Implicit substitution infe... |
copsex2gOLD 5484 | Obsolete version of ~ cops... |
copsex4g 5485 | An implicit substitution i... |
0nelop 5486 | A property of ordered pair... |
opwo0id 5487 | An ordered pair is equal t... |
opeqex 5488 | Equivalence of existence i... |
oteqex2 5489 | Equivalence of existence i... |
oteqex 5490 | Equivalence of existence i... |
opcom 5491 | An ordered pair commutes i... |
moop2 5492 | "At most one" property of ... |
opeqsng 5493 | Equivalence for an ordered... |
opeqsn 5494 | Equivalence for an ordered... |
opeqpr 5495 | Equivalence for an ordered... |
snopeqop 5496 | Equivalence for an ordered... |
propeqop 5497 | Equivalence for an ordered... |
propssopi 5498 | If a pair of ordered pairs... |
snopeqopsnid 5499 | Equivalence for an ordered... |
mosubopt 5500 | "At most one" remains true... |
mosubop 5501 | "At most one" remains true... |
euop2 5502 | Transfer existential uniqu... |
euotd 5503 | Prove existential uniquene... |
opthwiener 5504 | Justification theorem for ... |
uniop 5505 | The union of an ordered pa... |
uniopel 5506 | Ordered pair membership is... |
opthhausdorff 5507 | Justification theorem for ... |
opthhausdorff0 5508 | Justification theorem for ... |
otsndisj 5509 | The singletons consisting ... |
otiunsndisj 5510 | The union of singletons co... |
iunopeqop 5511 | Implication of an ordered ... |
brsnop 5512 | Binary relation for an ord... |
brtp 5513 | A necessary and sufficient... |
opabidw 5514 | The law of concretion. Sp... |
opabid 5515 | The law of concretion. Sp... |
elopabw 5516 | Membership in a class abst... |
elopab 5517 | Membership in a class abst... |
rexopabb 5518 | Restricted existential qua... |
vopelopabsb 5519 | The law of concretion in t... |
opelopabsb 5520 | The law of concretion in t... |
brabsb 5521 | The law of concretion in t... |
opelopabt 5522 | Closed theorem form of ~ o... |
opelopabga 5523 | The law of concretion. Th... |
brabga 5524 | The law of concretion for ... |
opelopab2a 5525 | Ordered pair membership in... |
opelopaba 5526 | The law of concretion. Th... |
braba 5527 | The law of concretion for ... |
opelopabg 5528 | The law of concretion. Th... |
brabg 5529 | The law of concretion for ... |
opelopabgf 5530 | The law of concretion. Th... |
opelopab2 5531 | Ordered pair membership in... |
opelopab 5532 | The law of concretion. Th... |
brab 5533 | The law of concretion for ... |
opelopabaf 5534 | The law of concretion. Th... |
opelopabf 5535 | The law of concretion. Th... |
ssopab2 5536 | Equivalence of ordered pai... |
ssopab2bw 5537 | Equivalence of ordered pai... |
eqopab2bw 5538 | Equivalence of ordered pai... |
ssopab2b 5539 | Equivalence of ordered pai... |
ssopab2i 5540 | Inference of ordered pair ... |
ssopab2dv 5541 | Inference of ordered pair ... |
eqopab2b 5542 | Equivalence of ordered pai... |
opabn0 5543 | Nonempty ordered pair clas... |
opab0 5544 | Empty ordered pair class a... |
csbopab 5545 | Move substitution into a c... |
csbopabgALT 5546 | Move substitution into a c... |
csbmpt12 5547 | Move substitution into a m... |
csbmpt2 5548 | Move substitution into the... |
iunopab 5549 | Move indexed union inside ... |
iunopabOLD 5550 | Obsolete version of ~ iuno... |
elopabr 5551 | Membership in an ordered-p... |
elopabran 5552 | Membership in an ordered-p... |
elopabrOLD 5553 | Obsolete version of ~ elop... |
rbropapd 5554 | Properties of a pair in an... |
rbropap 5555 | Properties of a pair in a ... |
2rbropap 5556 | Properties of a pair in a ... |
0nelopab 5557 | The empty set is never an ... |
0nelopabOLD 5558 | Obsolete version of ~ 0nel... |
brabv 5559 | If two classes are in a re... |
pwin 5560 | The power class of the int... |
pwssun 5561 | The power class of the uni... |
pwun 5562 | The power class of the uni... |
dfid4 5565 | The identity function expr... |
dfid2 5566 | Alternate definition of th... |
dfid3 5567 | A stronger version of ~ df... |
dfid2OLD 5568 | Obsolete version of ~ dfid... |
epelg 5571 | The membership relation an... |
epeli 5572 | The membership relation an... |
epel 5573 | The membership relation an... |
0sn0ep 5574 | An example for the members... |
epn0 5575 | The membership relation is... |
poss 5580 | Subset theorem for the par... |
poeq1 5581 | Equality theorem for parti... |
poeq2 5582 | Equality theorem for parti... |
nfpo 5583 | Bound-variable hypothesis ... |
nfso 5584 | Bound-variable hypothesis ... |
pocl 5585 | Characteristic properties ... |
poclOLD 5586 | Obsolete version of ~ pocl... |
ispod 5587 | Sufficient conditions for ... |
swopolem 5588 | Perform the substitutions ... |
swopo 5589 | A strict weak order is a p... |
poirr 5590 | A partial order is irrefle... |
potr 5591 | A partial order is a trans... |
po2nr 5592 | A partial order has no 2-c... |
po3nr 5593 | A partial order has no 3-c... |
po2ne 5594 | Two sets related by a part... |
po0 5595 | Any relation is a partial ... |
pofun 5596 | The inverse image of a par... |
sopo 5597 | A strict linear order is a... |
soss 5598 | Subset theorem for the str... |
soeq1 5599 | Equality theorem for the s... |
soeq2 5600 | Equality theorem for the s... |
sonr 5601 | A strict order relation is... |
sotr 5602 | A strict order relation is... |
solin 5603 | A strict order relation is... |
so2nr 5604 | A strict order relation ha... |
so3nr 5605 | A strict order relation ha... |
sotric 5606 | A strict order relation sa... |
sotrieq 5607 | Trichotomy law for strict ... |
sotrieq2 5608 | Trichotomy law for strict ... |
soasym 5609 | Asymmetry law for strict o... |
sotr2 5610 | A transitivity relation. ... |
issod 5611 | An irreflexive, transitive... |
issoi 5612 | An irreflexive, transitive... |
isso2i 5613 | Deduce strict ordering fro... |
so0 5614 | Any relation is a strict o... |
somo 5615 | A totally ordered set has ... |
sotrine 5616 | Trichotomy law for strict ... |
sotr3 5617 | Transitivity law for stric... |
dffr6 5624 | Alternate definition of ~ ... |
frd 5625 | A nonempty subset of an ` ... |
fri 5626 | A nonempty subset of an ` ... |
friOLD 5627 | Obsolete version of ~ fri ... |
seex 5628 | The ` R ` -preimage of an ... |
exse 5629 | Any relation on a set is s... |
dffr2 5630 | Alternate definition of we... |
dffr2ALT 5631 | Alternate proof of ~ dffr2... |
frc 5632 | Property of well-founded r... |
frss 5633 | Subset theorem for the wel... |
sess1 5634 | Subset theorem for the set... |
sess2 5635 | Subset theorem for the set... |
freq1 5636 | Equality theorem for the w... |
freq2 5637 | Equality theorem for the w... |
seeq1 5638 | Equality theorem for the s... |
seeq2 5639 | Equality theorem for the s... |
nffr 5640 | Bound-variable hypothesis ... |
nfse 5641 | Bound-variable hypothesis ... |
nfwe 5642 | Bound-variable hypothesis ... |
frirr 5643 | A well-founded relation is... |
fr2nr 5644 | A well-founded relation ha... |
fr0 5645 | Any relation is well-found... |
frminex 5646 | If an element of a well-fo... |
efrirr 5647 | A well-founded class does ... |
efrn2lp 5648 | A well-founded class conta... |
epse 5649 | The membership relation is... |
tz7.2 5650 | Similar to Theorem 7.2 of ... |
dfepfr 5651 | An alternate way of saying... |
epfrc 5652 | A subset of a well-founded... |
wess 5653 | Subset theorem for the wel... |
weeq1 5654 | Equality theorem for the w... |
weeq2 5655 | Equality theorem for the w... |
wefr 5656 | A well-ordering is well-fo... |
weso 5657 | A well-ordering is a stric... |
wecmpep 5658 | The elements of a class we... |
wetrep 5659 | On a class well-ordered by... |
wefrc 5660 | A nonempty subclass of a c... |
we0 5661 | Any relation is a well-ord... |
wereu 5662 | A nonempty subset of an ` ... |
wereu2 5663 | A nonempty subclass of an ... |
xpeq1 5680 | Equality theorem for Carte... |
xpss12 5681 | Subset theorem for Cartesi... |
xpss 5682 | A Cartesian product is inc... |
inxpssres 5683 | Intersection with a Cartes... |
relxp 5684 | A Cartesian product is a r... |
xpss1 5685 | Subset relation for Cartes... |
xpss2 5686 | Subset relation for Cartes... |
xpeq2 5687 | Equality theorem for Carte... |
elxpi 5688 | Membership in a Cartesian ... |
elxp 5689 | Membership in a Cartesian ... |
elxp2 5690 | Membership in a Cartesian ... |
xpeq12 5691 | Equality theorem for Carte... |
xpeq1i 5692 | Equality inference for Car... |
xpeq2i 5693 | Equality inference for Car... |
xpeq12i 5694 | Equality inference for Car... |
xpeq1d 5695 | Equality deduction for Car... |
xpeq2d 5696 | Equality deduction for Car... |
xpeq12d 5697 | Equality deduction for Car... |
sqxpeqd 5698 | Equality deduction for a C... |
nfxp 5699 | Bound-variable hypothesis ... |
0nelxp 5700 | The empty set is not a mem... |
0nelelxp 5701 | A member of a Cartesian pr... |
opelxp 5702 | Ordered pair membership in... |
opelxpi 5703 | Ordered pair membership in... |
opelxpii 5704 | Ordered pair membership in... |
opelxpd 5705 | Ordered pair membership in... |
opelvv 5706 | Ordered pair membership in... |
opelvvg 5707 | Ordered pair membership in... |
opelxp1 5708 | The first member of an ord... |
opelxp2 5709 | The second member of an or... |
otelxp 5710 | Ordered triple membership ... |
otelxp1 5711 | The first member of an ord... |
otel3xp 5712 | An ordered triple is an el... |
opabssxpd 5713 | An ordered-pair class abst... |
rabxp 5714 | Class abstraction restrict... |
brxp 5715 | Binary relation on a Carte... |
pwvrel 5716 | A set is a binary relation... |
pwvabrel 5717 | The powerclass of the cart... |
brrelex12 5718 | Two classes related by a b... |
brrelex1 5719 | If two classes are related... |
brrelex2 5720 | If two classes are related... |
brrelex12i 5721 | Two classes that are relat... |
brrelex1i 5722 | The first argument of a bi... |
brrelex2i 5723 | The second argument of a b... |
nprrel12 5724 | Proper classes are not rel... |
nprrel 5725 | No proper class is related... |
0nelrel0 5726 | A binary relation does not... |
0nelrel 5727 | A binary relation does not... |
fconstmpt 5728 | Representation of a consta... |
vtoclr 5729 | Variable to class conversi... |
opthprc 5730 | Justification theorem for ... |
brel 5731 | Two things in a binary rel... |
elxp3 5732 | Membership in a Cartesian ... |
opeliunxp 5733 | Membership in a union of C... |
xpundi 5734 | Distributive law for Carte... |
xpundir 5735 | Distributive law for Carte... |
xpiundi 5736 | Distributive law for Carte... |
xpiundir 5737 | Distributive law for Carte... |
iunxpconst 5738 | Membership in a union of C... |
xpun 5739 | The Cartesian product of t... |
elvv 5740 | Membership in universal cl... |
elvvv 5741 | Membership in universal cl... |
elvvuni 5742 | An ordered pair contains i... |
brinxp2 5743 | Intersection of binary rel... |
brinxp 5744 | Intersection of binary rel... |
opelinxp 5745 | Ordered pair element in an... |
poinxp 5746 | Intersection of partial or... |
soinxp 5747 | Intersection of total orde... |
frinxp 5748 | Intersection of well-found... |
seinxp 5749 | Intersection of set-like r... |
weinxp 5750 | Intersection of well-order... |
posn 5751 | Partial ordering of a sing... |
sosn 5752 | Strict ordering on a singl... |
frsn 5753 | Founded relation on a sing... |
wesn 5754 | Well-ordering of a singlet... |
elopaelxp 5755 | Membership in an ordered-p... |
elopaelxpOLD 5756 | Obsolete version of ~ elop... |
bropaex12 5757 | Two classes related by an ... |
opabssxp 5758 | An abstraction relation is... |
brab2a 5759 | The law of concretion for ... |
optocl 5760 | Implicit substitution of c... |
2optocl 5761 | Implicit substitution of c... |
3optocl 5762 | Implicit substitution of c... |
opbrop 5763 | Ordered pair membership in... |
0xp 5764 | The Cartesian product with... |
csbxp 5765 | Distribute proper substitu... |
releq 5766 | Equality theorem for the r... |
releqi 5767 | Equality inference for the... |
releqd 5768 | Equality deduction for the... |
nfrel 5769 | Bound-variable hypothesis ... |
sbcrel 5770 | Distribute proper substitu... |
relss 5771 | Subclass theorem for relat... |
ssrel 5772 | A subclass relationship de... |
ssrelOLD 5773 | Obsolete version of ~ ssre... |
eqrel 5774 | Extensionality principle f... |
ssrel2 5775 | A subclass relationship de... |
ssrel3 5776 | Subclass relation in anoth... |
relssi 5777 | Inference from subclass pr... |
relssdv 5778 | Deduction from subclass pr... |
eqrelriv 5779 | Inference from extensional... |
eqrelriiv 5780 | Inference from extensional... |
eqbrriv 5781 | Inference from extensional... |
eqrelrdv 5782 | Deduce equality of relatio... |
eqbrrdv 5783 | Deduction from extensional... |
eqbrrdiv 5784 | Deduction from extensional... |
eqrelrdv2 5785 | A version of ~ eqrelrdv . ... |
ssrelrel 5786 | A subclass relationship de... |
eqrelrel 5787 | Extensionality principle f... |
elrel 5788 | A member of a relation is ... |
rel0 5789 | The empty set is a relatio... |
nrelv 5790 | The universal class is not... |
relsng 5791 | A singleton is a relation ... |
relsnb 5792 | An at-most-singleton is a ... |
relsnopg 5793 | A singleton of an ordered ... |
relsn 5794 | A singleton is a relation ... |
relsnop 5795 | A singleton of an ordered ... |
copsex2gb 5796 | Implicit substitution infe... |
copsex2ga 5797 | Implicit substitution infe... |
elopaba 5798 | Membership in an ordered-p... |
xpsspw 5799 | A Cartesian product is inc... |
unixpss 5800 | The double class union of ... |
relun 5801 | The union of two relations... |
relin1 5802 | The intersection with a re... |
relin2 5803 | The intersection with a re... |
relinxp 5804 | Intersection with a Cartes... |
reldif 5805 | A difference cutting down ... |
reliun 5806 | An indexed union is a rela... |
reliin 5807 | An indexed intersection is... |
reluni 5808 | The union of a class is a ... |
relint 5809 | The intersection of a clas... |
relopabiv 5810 | A class of ordered pairs i... |
relopabv 5811 | A class of ordered pairs i... |
relopabi 5812 | A class of ordered pairs i... |
relopabiALT 5813 | Alternate proof of ~ relop... |
relopab 5814 | A class of ordered pairs i... |
mptrel 5815 | The maps-to notation alway... |
reli 5816 | The identity relation is a... |
rele 5817 | The membership relation is... |
opabid2 5818 | A relation expressed as an... |
inopab 5819 | Intersection of two ordere... |
difopab 5820 | Difference of two ordered-... |
difopabOLD 5821 | Obsolete version of ~ difo... |
inxp 5822 | Intersection of two Cartes... |
xpindi 5823 | Distributive law for Carte... |
xpindir 5824 | Distributive law for Carte... |
xpiindi 5825 | Distributive law for Carte... |
xpriindi 5826 | Distributive law for Carte... |
eliunxp 5827 | Membership in a union of C... |
opeliunxp2 5828 | Membership in a union of C... |
raliunxp 5829 | Write a double restricted ... |
rexiunxp 5830 | Write a double restricted ... |
ralxp 5831 | Universal quantification r... |
rexxp 5832 | Existential quantification... |
exopxfr 5833 | Transfer ordered-pair exis... |
exopxfr2 5834 | Transfer ordered-pair exis... |
djussxp 5835 | Disjoint union is a subset... |
ralxpf 5836 | Version of ~ ralxp with bo... |
rexxpf 5837 | Version of ~ rexxp with bo... |
iunxpf 5838 | Indexed union on a Cartesi... |
opabbi2dv 5839 | Deduce equality of a relat... |
relop 5840 | A necessary and sufficient... |
ideqg 5841 | For sets, the identity rel... |
ideq 5842 | For sets, the identity rel... |
ididg 5843 | A set is identical to itse... |
issetid 5844 | Two ways of expressing set... |
coss1 5845 | Subclass theorem for compo... |
coss2 5846 | Subclass theorem for compo... |
coeq1 5847 | Equality theorem for compo... |
coeq2 5848 | Equality theorem for compo... |
coeq1i 5849 | Equality inference for com... |
coeq2i 5850 | Equality inference for com... |
coeq1d 5851 | Equality deduction for com... |
coeq2d 5852 | Equality deduction for com... |
coeq12i 5853 | Equality inference for com... |
coeq12d 5854 | Equality deduction for com... |
nfco 5855 | Bound-variable hypothesis ... |
brcog 5856 | Ordered pair membership in... |
opelco2g 5857 | Ordered pair membership in... |
brcogw 5858 | Ordered pair membership in... |
eqbrrdva 5859 | Deduction from extensional... |
brco 5860 | Binary relation on a compo... |
opelco 5861 | Ordered pair membership in... |
cnvss 5862 | Subset theorem for convers... |
cnveq 5863 | Equality theorem for conve... |
cnveqi 5864 | Equality inference for con... |
cnveqd 5865 | Equality deduction for con... |
elcnv 5866 | Membership in a converse r... |
elcnv2 5867 | Membership in a converse r... |
nfcnv 5868 | Bound-variable hypothesis ... |
brcnvg 5869 | The converse of a binary r... |
opelcnvg 5870 | Ordered-pair membership in... |
opelcnv 5871 | Ordered-pair membership in... |
brcnv 5872 | The converse of a binary r... |
csbcnv 5873 | Move class substitution in... |
csbcnvgALT 5874 | Move class substitution in... |
cnvco 5875 | Distributive law of conver... |
cnvuni 5876 | The converse of a class un... |
dfdm3 5877 | Alternate definition of do... |
dfrn2 5878 | Alternate definition of ra... |
dfrn3 5879 | Alternate definition of ra... |
elrn2g 5880 | Membership in a range. (C... |
elrng 5881 | Membership in a range. (C... |
elrn2 5882 | Membership in a range. (C... |
elrn 5883 | Membership in a range. (C... |
ssrelrn 5884 | If a relation is a subset ... |
dfdm4 5885 | Alternate definition of do... |
dfdmf 5886 | Definition of domain, usin... |
csbdm 5887 | Distribute proper substitu... |
eldmg 5888 | Domain membership. Theore... |
eldm2g 5889 | Domain membership. Theore... |
eldm 5890 | Membership in a domain. T... |
eldm2 5891 | Membership in a domain. T... |
dmss 5892 | Subset theorem for domain.... |
dmeq 5893 | Equality theorem for domai... |
dmeqi 5894 | Equality inference for dom... |
dmeqd 5895 | Equality deduction for dom... |
opeldmd 5896 | Membership of first of an ... |
opeldm 5897 | Membership of first of an ... |
breldm 5898 | Membership of first of a b... |
breldmg 5899 | Membership of first of a b... |
dmun 5900 | The domain of a union is t... |
dmin 5901 | The domain of an intersect... |
breldmd 5902 | Membership of first of a b... |
dmiun 5903 | The domain of an indexed u... |
dmuni 5904 | The domain of a union. Pa... |
dmopab 5905 | The domain of a class of o... |
dmopabelb 5906 | A set is an element of the... |
dmopab2rex 5907 | The domain of an ordered p... |
dmopabss 5908 | Upper bound for the domain... |
dmopab3 5909 | The domain of a restricted... |
dm0 5910 | The domain of the empty se... |
dmi 5911 | The domain of the identity... |
dmv 5912 | The domain of the universe... |
dmep 5913 | The domain of the membersh... |
dm0rn0 5914 | An empty domain is equival... |
rn0 5915 | The range of the empty set... |
rnep 5916 | The range of the membershi... |
reldm0 5917 | A relation is empty iff it... |
dmxp 5918 | The domain of a Cartesian ... |
dmxpid 5919 | The domain of a Cartesian ... |
dmxpin 5920 | The domain of the intersec... |
xpid11 5921 | The Cartesian square is a ... |
dmcnvcnv 5922 | The domain of the double c... |
rncnvcnv 5923 | The range of the double co... |
elreldm 5924 | The first member of an ord... |
rneq 5925 | Equality theorem for range... |
rneqi 5926 | Equality inference for ran... |
rneqd 5927 | Equality deduction for ran... |
rnss 5928 | Subset theorem for range. ... |
rnssi 5929 | Subclass inference for ran... |
brelrng 5930 | The second argument of a b... |
brelrn 5931 | The second argument of a b... |
opelrn 5932 | Membership of second membe... |
releldm 5933 | The first argument of a bi... |
relelrn 5934 | The second argument of a b... |
releldmb 5935 | Membership in a domain. (... |
relelrnb 5936 | Membership in a range. (C... |
releldmi 5937 | The first argument of a bi... |
relelrni 5938 | The second argument of a b... |
dfrnf 5939 | Definition of range, using... |
nfdm 5940 | Bound-variable hypothesis ... |
nfrn 5941 | Bound-variable hypothesis ... |
dmiin 5942 | Domain of an intersection.... |
rnopab 5943 | The range of a class of or... |
rnmpt 5944 | The range of a function in... |
elrnmpt 5945 | The range of a function in... |
elrnmpt1s 5946 | Elementhood in an image se... |
elrnmpt1 5947 | Elementhood in an image se... |
elrnmptg 5948 | Membership in the range of... |
elrnmpti 5949 | Membership in the range of... |
elrnmptd 5950 | The range of a function in... |
elrnmptdv 5951 | Elementhood in the range o... |
elrnmpt2d 5952 | Elementhood in the range o... |
dfiun3g 5953 | Alternate definition of in... |
dfiin3g 5954 | Alternate definition of in... |
dfiun3 5955 | Alternate definition of in... |
dfiin3 5956 | Alternate definition of in... |
riinint 5957 | Express a relative indexed... |
relrn0 5958 | A relation is empty iff it... |
dmrnssfld 5959 | The domain and range of a ... |
dmcoss 5960 | Domain of a composition. ... |
rncoss 5961 | Range of a composition. (... |
dmcosseq 5962 | Domain of a composition. ... |
dmcoeq 5963 | Domain of a composition. ... |
rncoeq 5964 | Range of a composition. (... |
reseq1 5965 | Equality theorem for restr... |
reseq2 5966 | Equality theorem for restr... |
reseq1i 5967 | Equality inference for res... |
reseq2i 5968 | Equality inference for res... |
reseq12i 5969 | Equality inference for res... |
reseq1d 5970 | Equality deduction for res... |
reseq2d 5971 | Equality deduction for res... |
reseq12d 5972 | Equality deduction for res... |
nfres 5973 | Bound-variable hypothesis ... |
csbres 5974 | Distribute proper substitu... |
res0 5975 | A restriction to the empty... |
dfres3 5976 | Alternate definition of re... |
opelres 5977 | Ordered pair elementhood i... |
brres 5978 | Binary relation on a restr... |
opelresi 5979 | Ordered pair membership in... |
brresi 5980 | Binary relation on a restr... |
opres 5981 | Ordered pair membership in... |
resieq 5982 | A restricted identity rela... |
opelidres 5983 | ` <. A , A >. ` belongs to... |
resres 5984 | The restriction of a restr... |
resundi 5985 | Distributive law for restr... |
resundir 5986 | Distributive law for restr... |
resindi 5987 | Class restriction distribu... |
resindir 5988 | Class restriction distribu... |
inres 5989 | Move intersection into cla... |
resdifcom 5990 | Commutative law for restri... |
resiun1 5991 | Distribution of restrictio... |
resiun2 5992 | Distribution of restrictio... |
dmres 5993 | The domain of a restrictio... |
ssdmres 5994 | A domain restricted to a s... |
dmresexg 5995 | The domain of a restrictio... |
resss 5996 | A class includes its restr... |
rescom 5997 | Commutative law for restri... |
ssres 5998 | Subclass theorem for restr... |
ssres2 5999 | Subclass theorem for restr... |
relres 6000 | A restriction is a relatio... |
resabs1 6001 | Absorption law for restric... |
resabs1d 6002 | Absorption law for restric... |
resabs2 6003 | Absorption law for restric... |
residm 6004 | Idempotent law for restric... |
resima 6005 | A restriction to an image.... |
resima2 6006 | Image under a restricted c... |
rnresss 6007 | The range of a restriction... |
xpssres 6008 | Restriction of a constant ... |
elinxp 6009 | Membership in an intersect... |
elres 6010 | Membership in a restrictio... |
elsnres 6011 | Membership in restriction ... |
relssres 6012 | Simplification law for res... |
dmressnsn 6013 | The domain of a restrictio... |
eldmressnsn 6014 | The element of the domain ... |
eldmeldmressn 6015 | An element of the domain (... |
resdm 6016 | A relation restricted to i... |
resexg 6017 | The restriction of a set i... |
resexd 6018 | The restriction of a set i... |
resex 6019 | The restriction of a set i... |
resindm 6020 | When restricting a relatio... |
resdmdfsn 6021 | Restricting a relation to ... |
reldisjun 6022 | Split a relation into two ... |
relresdm1 6023 | Restriction of a disjoint ... |
resopab 6024 | Restriction of a class abs... |
iss 6025 | A subclass of the identity... |
resopab2 6026 | Restriction of a class abs... |
resmpt 6027 | Restriction of the mapping... |
resmpt3 6028 | Unconditional restriction ... |
resmptf 6029 | Restriction of the mapping... |
resmptd 6030 | Restriction of the mapping... |
dfres2 6031 | Alternate definition of th... |
mptss 6032 | Sufficient condition for i... |
elidinxp 6033 | Characterization of the el... |
elidinxpid 6034 | Characterization of the el... |
elrid 6035 | Characterization of the el... |
idinxpres 6036 | The intersection of the id... |
idinxpresid 6037 | The intersection of the id... |
idssxp 6038 | A diagonal set as a subset... |
opabresid 6039 | The restricted identity re... |
mptresid 6040 | The restricted identity re... |
dmresi 6041 | The domain of a restricted... |
restidsing 6042 | Restriction of the identit... |
iresn0n0 6043 | The identity function rest... |
imaeq1 6044 | Equality theorem for image... |
imaeq2 6045 | Equality theorem for image... |
imaeq1i 6046 | Equality theorem for image... |
imaeq2i 6047 | Equality theorem for image... |
imaeq1d 6048 | Equality theorem for image... |
imaeq2d 6049 | Equality theorem for image... |
imaeq12d 6050 | Equality theorem for image... |
dfima2 6051 | Alternate definition of im... |
dfima3 6052 | Alternate definition of im... |
elimag 6053 | Membership in an image. T... |
elima 6054 | Membership in an image. T... |
elima2 6055 | Membership in an image. T... |
elima3 6056 | Membership in an image. T... |
nfima 6057 | Bound-variable hypothesis ... |
nfimad 6058 | Deduction version of bound... |
imadmrn 6059 | The image of the domain of... |
imassrn 6060 | The image of a class is a ... |
mptima 6061 | Image of a function in map... |
mptimass 6062 | Image of a function in map... |
imai 6063 | Image under the identity r... |
rnresi 6064 | The range of the restricte... |
resiima 6065 | The image of a restriction... |
ima0 6066 | Image of the empty set. T... |
0ima 6067 | Image under the empty rela... |
csbima12 6068 | Move class substitution in... |
imadisj 6069 | A class whose image under ... |
cnvimass 6070 | A preimage under any class... |
cnvimarndm 6071 | The preimage of the range ... |
imasng 6072 | The image of a singleton. ... |
relimasn 6073 | The image of a singleton. ... |
elrelimasn 6074 | Elementhood in the image o... |
elimasng1 6075 | Membership in an image of ... |
elimasn1 6076 | Membership in an image of ... |
elimasng 6077 | Membership in an image of ... |
elimasn 6078 | Membership in an image of ... |
elimasngOLD 6079 | Obsolete version of ~ elim... |
elimasni 6080 | Membership in an image of ... |
args 6081 | Two ways to express the cl... |
elinisegg 6082 | Membership in the inverse ... |
eliniseg 6083 | Membership in the inverse ... |
epin 6084 | Any set is equal to its pr... |
epini 6085 | Any set is equal to its pr... |
iniseg 6086 | An idiom that signifies an... |
inisegn0 6087 | Nonemptiness of an initial... |
dffr3 6088 | Alternate definition of we... |
dfse2 6089 | Alternate definition of se... |
imass1 6090 | Subset theorem for image. ... |
imass2 6091 | Subset theorem for image. ... |
ndmima 6092 | The image of a singleton o... |
relcnv 6093 | A converse is a relation. ... |
relbrcnvg 6094 | When ` R ` is a relation, ... |
eliniseg2 6095 | Eliminate the class existe... |
relbrcnv 6096 | When ` R ` is a relation, ... |
relco 6097 | A composition is a relatio... |
cotrg 6098 | Two ways of saying that th... |
cotrgOLD 6099 | Obsolete version of ~ cotr... |
cotrgOLDOLD 6100 | Obsolete version of ~ cotr... |
cotr 6101 | Two ways of saying a relat... |
idrefALT 6102 | Alternate proof of ~ idref... |
cnvsym 6103 | Two ways of saying a relat... |
cnvsymOLD 6104 | Obsolete proof of ~ cnvsym... |
cnvsymOLDOLD 6105 | Obsolete proof of ~ cnvsym... |
intasym 6106 | Two ways of saying a relat... |
asymref 6107 | Two ways of saying a relat... |
asymref2 6108 | Two ways of saying a relat... |
intirr 6109 | Two ways of saying a relat... |
brcodir 6110 | Two ways of saying that tw... |
codir 6111 | Two ways of saying a relat... |
qfto 6112 | A quantifier-free way of e... |
xpidtr 6113 | A Cartesian square is a tr... |
trin2 6114 | The intersection of two tr... |
poirr2 6115 | A partial order is irrefle... |
trinxp 6116 | The relation induced by a ... |
soirri 6117 | A strict order relation is... |
sotri 6118 | A strict order relation is... |
son2lpi 6119 | A strict order relation ha... |
sotri2 6120 | A transitivity relation. ... |
sotri3 6121 | A transitivity relation. ... |
poleloe 6122 | Express "less than or equa... |
poltletr 6123 | Transitive law for general... |
somin1 6124 | Property of a minimum in a... |
somincom 6125 | Commutativity of minimum i... |
somin2 6126 | Property of a minimum in a... |
soltmin 6127 | Being less than a minimum,... |
cnvopab 6128 | The converse of a class ab... |
mptcnv 6129 | The converse of a mapping ... |
cnv0 6130 | The converse of the empty ... |
cnvi 6131 | The converse of the identi... |
cnvun 6132 | The converse of a union is... |
cnvdif 6133 | Distributive law for conve... |
cnvin 6134 | Distributive law for conve... |
rnun 6135 | Distributive law for range... |
rnin 6136 | The range of an intersecti... |
rniun 6137 | The range of an indexed un... |
rnuni 6138 | The range of a union. Par... |
imaundi 6139 | Distributive law for image... |
imaundir 6140 | The image of a union. (Co... |
cnvimassrndm 6141 | The preimage of a superset... |
dminss 6142 | An upper bound for interse... |
imainss 6143 | An upper bound for interse... |
inimass 6144 | The image of an intersecti... |
inimasn 6145 | The intersection of the im... |
cnvxp 6146 | The converse of a Cartesia... |
xp0 6147 | The Cartesian product with... |
xpnz 6148 | The Cartesian product of n... |
xpeq0 6149 | At least one member of an ... |
xpdisj1 6150 | Cartesian products with di... |
xpdisj2 6151 | Cartesian products with di... |
xpsndisj 6152 | Cartesian products with tw... |
difxp 6153 | Difference of Cartesian pr... |
difxp1 6154 | Difference law for Cartesi... |
difxp2 6155 | Difference law for Cartesi... |
djudisj 6156 | Disjoint unions with disjo... |
xpdifid 6157 | The set of distinct couple... |
resdisj 6158 | A double restriction to di... |
rnxp 6159 | The range of a Cartesian p... |
dmxpss 6160 | The domain of a Cartesian ... |
rnxpss 6161 | The range of a Cartesian p... |
rnxpid 6162 | The range of a Cartesian s... |
ssxpb 6163 | A Cartesian product subcla... |
xp11 6164 | The Cartesian product of n... |
xpcan 6165 | Cancellation law for Carte... |
xpcan2 6166 | Cancellation law for Carte... |
ssrnres 6167 | Two ways to express surjec... |
rninxp 6168 | Two ways to express surjec... |
dminxp 6169 | Two ways to express totali... |
imainrect 6170 | Image by a restricted and ... |
xpima 6171 | Direct image by a Cartesia... |
xpima1 6172 | Direct image by a Cartesia... |
xpima2 6173 | Direct image by a Cartesia... |
xpimasn 6174 | Direct image of a singleto... |
sossfld 6175 | The base set of a strict o... |
sofld 6176 | The base set of a nonempty... |
cnvcnv3 6177 | The set of all ordered pai... |
dfrel2 6178 | Alternate definition of re... |
dfrel4v 6179 | A relation can be expresse... |
dfrel4 6180 | A relation can be expresse... |
cnvcnv 6181 | The double converse of a c... |
cnvcnv2 6182 | The double converse of a c... |
cnvcnvss 6183 | The double converse of a c... |
cnvrescnv 6184 | Two ways to express the co... |
cnveqb 6185 | Equality theorem for conve... |
cnveq0 6186 | A relation empty iff its c... |
dfrel3 6187 | Alternate definition of re... |
elid 6188 | Characterization of the el... |
dmresv 6189 | The domain of a universal ... |
rnresv 6190 | The range of a universal r... |
dfrn4 6191 | Range defined in terms of ... |
csbrn 6192 | Distribute proper substitu... |
rescnvcnv 6193 | The restriction of the dou... |
cnvcnvres 6194 | The double converse of the... |
imacnvcnv 6195 | The image of the double co... |
dmsnn0 6196 | The domain of a singleton ... |
rnsnn0 6197 | The range of a singleton i... |
dmsn0 6198 | The domain of the singleto... |
cnvsn0 6199 | The converse of the single... |
dmsn0el 6200 | The domain of a singleton ... |
relsn2 6201 | A singleton is a relation ... |
dmsnopg 6202 | The domain of a singleton ... |
dmsnopss 6203 | The domain of a singleton ... |
dmpropg 6204 | The domain of an unordered... |
dmsnop 6205 | The domain of a singleton ... |
dmprop 6206 | The domain of an unordered... |
dmtpop 6207 | The domain of an unordered... |
cnvcnvsn 6208 | Double converse of a singl... |
dmsnsnsn 6209 | The domain of the singleto... |
rnsnopg 6210 | The range of a singleton o... |
rnpropg 6211 | The range of a pair of ord... |
cnvsng 6212 | Converse of a singleton of... |
rnsnop 6213 | The range of a singleton o... |
op1sta 6214 | Extract the first member o... |
cnvsn 6215 | Converse of a singleton of... |
op2ndb 6216 | Extract the second member ... |
op2nda 6217 | Extract the second member ... |
opswap 6218 | Swap the members of an ord... |
cnvresima 6219 | An image under the convers... |
resdm2 6220 | A class restricted to its ... |
resdmres 6221 | Restriction to the domain ... |
resresdm 6222 | A restriction by an arbitr... |
imadmres 6223 | The image of the domain of... |
resdmss 6224 | Subset relationship for th... |
resdifdi 6225 | Distributive law for restr... |
resdifdir 6226 | Distributive law for restr... |
mptpreima 6227 | The preimage of a function... |
mptiniseg 6228 | Converse singleton image o... |
dmmpt 6229 | The domain of the mapping ... |
dmmptss 6230 | The domain of a mapping is... |
dmmptg 6231 | The domain of the mapping ... |
rnmpt0f 6232 | The range of a function in... |
rnmptn0 6233 | The range of a function in... |
dfco2 6234 | Alternate definition of a ... |
dfco2a 6235 | Generalization of ~ dfco2 ... |
coundi 6236 | Class composition distribu... |
coundir 6237 | Class composition distribu... |
cores 6238 | Restricted first member of... |
resco 6239 | Associative law for the re... |
imaco 6240 | Image of the composition o... |
rnco 6241 | The range of the compositi... |
rnco2 6242 | The range of the compositi... |
dmco 6243 | The domain of a compositio... |
coeq0 6244 | A composition of two relat... |
coiun 6245 | Composition with an indexe... |
cocnvcnv1 6246 | A composition is not affec... |
cocnvcnv2 6247 | A composition is not affec... |
cores2 6248 | Absorption of a reverse (p... |
co02 6249 | Composition with the empty... |
co01 6250 | Composition with the empty... |
coi1 6251 | Composition with the ident... |
coi2 6252 | Composition with the ident... |
coires1 6253 | Composition with a restric... |
coass 6254 | Associative law for class ... |
relcnvtrg 6255 | General form of ~ relcnvtr... |
relcnvtr 6256 | A relation is transitive i... |
relssdmrn 6257 | A relation is included in ... |
relssdmrnOLD 6258 | Obsolete version of ~ rels... |
resssxp 6259 | If the ` R ` -image of a c... |
cnvssrndm 6260 | The converse is a subset o... |
cossxp 6261 | Composition as a subset of... |
relrelss 6262 | Two ways to describe the s... |
unielrel 6263 | The membership relation fo... |
relfld 6264 | The double union of a rela... |
relresfld 6265 | Restriction of a relation ... |
relcoi2 6266 | Composition with the ident... |
relcoi1 6267 | Composition with the ident... |
unidmrn 6268 | The double union of the co... |
relcnvfld 6269 | if ` R ` is a relation, it... |
dfdm2 6270 | Alternate definition of do... |
unixp 6271 | The double class union of ... |
unixp0 6272 | A Cartesian product is emp... |
unixpid 6273 | Field of a Cartesian squar... |
ressn 6274 | Restriction of a class to ... |
cnviin 6275 | The converse of an interse... |
cnvpo 6276 | The converse of a partial ... |
cnvso 6277 | The converse of a strict o... |
xpco 6278 | Composition of two Cartesi... |
xpcoid 6279 | Composition of two Cartesi... |
elsnxp 6280 | Membership in a Cartesian ... |
reu3op 6281 | There is a unique ordered ... |
reuop 6282 | There is a unique ordered ... |
opreu2reurex 6283 | There is a unique ordered ... |
opreu2reu 6284 | If there is a unique order... |
dfpo2 6285 | Quantifier-free definition... |
csbcog 6286 | Distribute proper substitu... |
snres0 6287 | Condition for restriction ... |
imaindm 6288 | The image is unaffected by... |
predeq123 6291 | Equality theorem for the p... |
predeq1 6292 | Equality theorem for the p... |
predeq2 6293 | Equality theorem for the p... |
predeq3 6294 | Equality theorem for the p... |
nfpred 6295 | Bound-variable hypothesis ... |
csbpredg 6296 | Move class substitution in... |
predpredss 6297 | If ` A ` is a subset of ` ... |
predss 6298 | The predecessor class of `... |
sspred 6299 | Another subset/predecessor... |
dfpred2 6300 | An alternate definition of... |
dfpred3 6301 | An alternate definition of... |
dfpred3g 6302 | An alternate definition of... |
elpredgg 6303 | Membership in a predecesso... |
elpredg 6304 | Membership in a predecesso... |
elpredimg 6305 | Membership in a predecesso... |
elpredim 6306 | Membership in a predecesso... |
elpred 6307 | Membership in a predecesso... |
predexg 6308 | The predecessor class exis... |
predasetexOLD 6309 | Obsolete form of ~ predexg... |
dffr4 6310 | Alternate definition of we... |
predel 6311 | Membership in the predeces... |
predbrg 6312 | Closed form of ~ elpredim ... |
predtrss 6313 | If ` R ` is transitive ove... |
predpo 6314 | Property of the predecesso... |
predso 6315 | Property of the predecesso... |
setlikespec 6316 | If ` R ` is set-like in ` ... |
predidm 6317 | Idempotent law for the pre... |
predin 6318 | Intersection law for prede... |
predun 6319 | Union law for predecessor ... |
preddif 6320 | Difference law for predece... |
predep 6321 | The predecessor under the ... |
trpred 6322 | The class of predecessors ... |
preddowncl 6323 | A property of classes that... |
predpoirr 6324 | Given a partial ordering, ... |
predfrirr 6325 | Given a well-founded relat... |
pred0 6326 | The predecessor class over... |
dfse3 6327 | Alternate definition of se... |
predrelss 6328 | Subset carries from relati... |
predprc 6329 | The predecessor of a prope... |
predres 6330 | Predecessor class is unaff... |
frpomin 6331 | Every nonempty (possibly p... |
frpomin2 6332 | Every nonempty (possibly p... |
frpoind 6333 | The principle of well-foun... |
frpoinsg 6334 | Well-Founded Induction Sch... |
frpoins2fg 6335 | Well-Founded Induction sch... |
frpoins2g 6336 | Well-Founded Induction sch... |
frpoins3g 6337 | Well-Founded Induction sch... |
tz6.26 6338 | All nonempty subclasses of... |
tz6.26OLD 6339 | Obsolete proof of ~ tz6.26... |
tz6.26i 6340 | All nonempty subclasses of... |
wfi 6341 | The Principle of Well-Orde... |
wfiOLD 6342 | Obsolete proof of ~ wfi as... |
wfii 6343 | The Principle of Well-Orde... |
wfisg 6344 | Well-Ordered Induction Sch... |
wfisgOLD 6345 | Obsolete version of ~ wfis... |
wfis 6346 | Well-Ordered Induction Sch... |
wfis2fg 6347 | Well-Ordered Induction Sch... |
wfis2fgOLD 6348 | Obsolete version of ~ wfis... |
wfis2f 6349 | Well-Ordered Induction sch... |
wfis2g 6350 | Well-Ordered Induction Sch... |
wfis2 6351 | Well-Ordered Induction sch... |
wfis3 6352 | Well-Ordered Induction sch... |
ordeq 6361 | Equality theorem for the o... |
elong 6362 | An ordinal number is an or... |
elon 6363 | An ordinal number is an or... |
eloni 6364 | An ordinal number has the ... |
elon2 6365 | An ordinal number is an or... |
limeq 6366 | Equality theorem for the l... |
ordwe 6367 | Membership well-orders eve... |
ordtr 6368 | An ordinal class is transi... |
ordfr 6369 | Membership is well-founded... |
ordelss 6370 | An element of an ordinal c... |
trssord 6371 | A transitive subclass of a... |
ordirr 6372 | No ordinal class is a memb... |
nordeq 6373 | A member of an ordinal cla... |
ordn2lp 6374 | An ordinal class cannot be... |
tz7.5 6375 | A nonempty subclass of an ... |
ordelord 6376 | An element of an ordinal c... |
tron 6377 | The class of all ordinal n... |
ordelon 6378 | An element of an ordinal c... |
onelon 6379 | An element of an ordinal n... |
tz7.7 6380 | A transitive class belongs... |
ordelssne 6381 | For ordinal classes, membe... |
ordelpss 6382 | For ordinal classes, membe... |
ordsseleq 6383 | For ordinal classes, inclu... |
ordin 6384 | The intersection of two or... |
onin 6385 | The intersection of two or... |
ordtri3or 6386 | A trichotomy law for ordin... |
ordtri1 6387 | A trichotomy law for ordin... |
ontri1 6388 | A trichotomy law for ordin... |
ordtri2 6389 | A trichotomy law for ordin... |
ordtri3 6390 | A trichotomy law for ordin... |
ordtri4 6391 | A trichotomy law for ordin... |
orddisj 6392 | An ordinal class and its s... |
onfr 6393 | The ordinal class is well-... |
onelpss 6394 | Relationship between membe... |
onsseleq 6395 | Relationship between subse... |
onelss 6396 | An element of an ordinal n... |
ordtr1 6397 | Transitive law for ordinal... |
ordtr2 6398 | Transitive law for ordinal... |
ordtr3 6399 | Transitive law for ordinal... |
ontr1 6400 | Transitive law for ordinal... |
ontr2 6401 | Transitive law for ordinal... |
onelssex 6402 | Ordinal less than is equiv... |
ordunidif 6403 | The union of an ordinal st... |
ordintdif 6404 | If ` B ` is smaller than `... |
onintss 6405 | If a property is true for ... |
oneqmini 6406 | A way to show that an ordi... |
ord0 6407 | The empty set is an ordina... |
0elon 6408 | The empty set is an ordina... |
ord0eln0 6409 | A nonempty ordinal contain... |
on0eln0 6410 | An ordinal number contains... |
dflim2 6411 | An alternate definition of... |
inton 6412 | The intersection of the cl... |
nlim0 6413 | The empty set is not a lim... |
limord 6414 | A limit ordinal is ordinal... |
limuni 6415 | A limit ordinal is its own... |
limuni2 6416 | The union of a limit ordin... |
0ellim 6417 | A limit ordinal contains t... |
limelon 6418 | A limit ordinal class that... |
onn0 6419 | The class of all ordinal n... |
suceq 6420 | Equality of successors. (... |
elsuci 6421 | Membership in a successor.... |
elsucg 6422 | Membership in a successor.... |
elsuc2g 6423 | Variant of membership in a... |
elsuc 6424 | Membership in a successor.... |
elsuc2 6425 | Membership in a successor.... |
nfsuc 6426 | Bound-variable hypothesis ... |
elelsuc 6427 | Membership in a successor.... |
sucel 6428 | Membership of a successor ... |
suc0 6429 | The successor of the empty... |
sucprc 6430 | A proper class is its own ... |
unisucs 6431 | The union of the successor... |
unisucg 6432 | A transitive class is equa... |
unisuc 6433 | A transitive class is equa... |
sssucid 6434 | A class is included in its... |
sucidg 6435 | Part of Proposition 7.23 o... |
sucid 6436 | A set belongs to its succe... |
nsuceq0 6437 | No successor is empty. (C... |
eqelsuc 6438 | A set belongs to the succe... |
iunsuc 6439 | Inductive definition for t... |
suctr 6440 | The successor of a transit... |
trsuc 6441 | A set whose successor belo... |
trsucss 6442 | A member of the successor ... |
ordsssuc 6443 | An ordinal is a subset of ... |
onsssuc 6444 | A subset of an ordinal num... |
ordsssuc2 6445 | An ordinal subset of an or... |
onmindif 6446 | When its successor is subt... |
ordnbtwn 6447 | There is no set between an... |
onnbtwn 6448 | There is no set between an... |
sucssel 6449 | A set whose successor is a... |
orddif 6450 | Ordinal derived from its s... |
orduniss 6451 | An ordinal class includes ... |
ordtri2or 6452 | A trichotomy law for ordin... |
ordtri2or2 6453 | A trichotomy law for ordin... |
ordtri2or3 6454 | A consequence of total ord... |
ordelinel 6455 | The intersection of two or... |
ordssun 6456 | Property of a subclass of ... |
ordequn 6457 | The maximum (i.e. union) o... |
ordun 6458 | The maximum (i.e., union) ... |
onunel 6459 | The union of two ordinals ... |
ordunisssuc 6460 | A subclass relationship fo... |
suc11 6461 | The successor operation be... |
onun2 6462 | The union of two ordinals ... |
ontr 6463 | An ordinal number is a tra... |
onunisuc 6464 | An ordinal number is equal... |
onordi 6465 | An ordinal number is an or... |
ontrciOLD 6466 | Obsolete version of ~ ontr... |
onirri 6467 | An ordinal number is not a... |
oneli 6468 | A member of an ordinal num... |
onelssi 6469 | A member of an ordinal num... |
onssneli 6470 | An ordering law for ordina... |
onssnel2i 6471 | An ordering law for ordina... |
onelini 6472 | An element of an ordinal n... |
oneluni 6473 | An ordinal number equals i... |
onunisuci 6474 | An ordinal number is equal... |
onsseli 6475 | Subset is equivalent to me... |
onun2i 6476 | The union of two ordinal n... |
unizlim 6477 | An ordinal equal to its ow... |
on0eqel 6478 | An ordinal number either e... |
snsn0non 6479 | The singleton of the singl... |
onxpdisj 6480 | Ordinal numbers and ordere... |
onnev 6481 | The class of ordinal numbe... |
onnevOLD 6482 | Obsolete version of ~ onne... |
iotajust 6484 | Soundness justification th... |
dfiota2 6486 | Alternate definition for d... |
nfiota1 6487 | Bound-variable hypothesis ... |
nfiotadw 6488 | Deduction version of ~ nfi... |
nfiotaw 6489 | Bound-variable hypothesis ... |
nfiotad 6490 | Deduction version of ~ nfi... |
nfiota 6491 | Bound-variable hypothesis ... |
cbviotaw 6492 | Change bound variables in ... |
cbviotavw 6493 | Change bound variables in ... |
cbviotavwOLD 6494 | Obsolete version of ~ cbvi... |
cbviota 6495 | Change bound variables in ... |
cbviotav 6496 | Change bound variables in ... |
sb8iota 6497 | Variable substitution in d... |
iotaeq 6498 | Equality theorem for descr... |
iotabi 6499 | Equivalence theorem for de... |
uniabio 6500 | Part of Theorem 8.17 in [Q... |
iotaval2 6501 | Version of ~ iotaval using... |
iotauni2 6502 | Version of ~ iotauni using... |
iotanul2 6503 | Version of ~ iotanul using... |
iotaval 6504 | Theorem 8.19 in [Quine] p.... |
iotassuni 6505 | The ` iota ` class is a su... |
iotaex 6506 | Theorem 8.23 in [Quine] p.... |
iotavalOLD 6507 | Obsolete version of ~ iota... |
iotauni 6508 | Equivalence between two di... |
iotaint 6509 | Equivalence between two di... |
iota1 6510 | Property of iota. (Contri... |
iotanul 6511 | Theorem 8.22 in [Quine] p.... |
iotassuniOLD 6512 | Obsolete version of ~ iota... |
iotaexOLD 6513 | Obsolete version of ~ iota... |
iota4 6514 | Theorem *14.22 in [Whitehe... |
iota4an 6515 | Theorem *14.23 in [Whitehe... |
iota5 6516 | A method for computing iot... |
iotabidv 6517 | Formula-building deduction... |
iotabii 6518 | Formula-building deduction... |
iotacl 6519 | Membership law for descrip... |
iota2df 6520 | A condition that allows to... |
iota2d 6521 | A condition that allows to... |
iota2 6522 | The unique element such th... |
iotan0 6523 | Representation of "the uni... |
sniota 6524 | A class abstraction with a... |
dfiota4 6525 | The ` iota ` operation usi... |
csbiota 6526 | Class substitution within ... |
dffun2 6543 | Alternate definition of a ... |
dffun2OLD 6544 | Obsolete version of ~ dffu... |
dffun2OLDOLD 6545 | Obsolete version of ~ dffu... |
dffun6 6546 | Alternate definition of a ... |
dffun3 6547 | Alternate definition of fu... |
dffun3OLD 6548 | Obsolete version of ~ dffu... |
dffun4 6549 | Alternate definition of a ... |
dffun5 6550 | Alternate definition of fu... |
dffun6f 6551 | Definition of function, us... |
dffun6OLD 6552 | Obsolete version of ~ dffu... |
funmo 6553 | A function has at most one... |
funmoOLD 6554 | Obsolete version of ~ funm... |
funrel 6555 | A function is a relation. ... |
0nelfun 6556 | A function does not contai... |
funss 6557 | Subclass theorem for funct... |
funeq 6558 | Equality theorem for funct... |
funeqi 6559 | Equality inference for the... |
funeqd 6560 | Equality deduction for the... |
nffun 6561 | Bound-variable hypothesis ... |
sbcfung 6562 | Distribute proper substitu... |
funeu 6563 | There is exactly one value... |
funeu2 6564 | There is exactly one value... |
dffun7 6565 | Alternate definition of a ... |
dffun8 6566 | Alternate definition of a ... |
dffun9 6567 | Alternate definition of a ... |
funfn 6568 | A class is a function if a... |
funfnd 6569 | A function is a function o... |
funi 6570 | The identity relation is a... |
nfunv 6571 | The universal class is not... |
funopg 6572 | A Kuratowski ordered pair ... |
funopab 6573 | A class of ordered pairs i... |
funopabeq 6574 | A class of ordered pairs o... |
funopab4 6575 | A class of ordered pairs o... |
funmpt 6576 | A function in maps-to nota... |
funmpt2 6577 | Functionality of a class g... |
funco 6578 | The composition of two fun... |
funresfunco 6579 | Composition of two functio... |
funres 6580 | A restriction of a functio... |
funresd 6581 | A restriction of a functio... |
funssres 6582 | The restriction of a funct... |
fun2ssres 6583 | Equality of restrictions o... |
funun 6584 | The union of functions wit... |
fununmo 6585 | If the union of classes is... |
fununfun 6586 | If the union of classes is... |
fundif 6587 | A function with removed el... |
funcnvsn 6588 | The converse singleton of ... |
funsng 6589 | A singleton of an ordered ... |
fnsng 6590 | Functionality and domain o... |
funsn 6591 | A singleton of an ordered ... |
funprg 6592 | A set of two pairs is a fu... |
funtpg 6593 | A set of three pairs is a ... |
funpr 6594 | A function with a domain o... |
funtp 6595 | A function with a domain o... |
fnsn 6596 | Functionality and domain o... |
fnprg 6597 | Function with a domain of ... |
fntpg 6598 | Function with a domain of ... |
fntp 6599 | A function with a domain o... |
funcnvpr 6600 | The converse pair of order... |
funcnvtp 6601 | The converse triple of ord... |
funcnvqp 6602 | The converse quadruple of ... |
fun0 6603 | The empty set is a functio... |
funcnv0 6604 | The converse of the empty ... |
funcnvcnv 6605 | The double converse of a f... |
funcnv2 6606 | A simpler equivalence for ... |
funcnv 6607 | The converse of a class is... |
funcnv3 6608 | A condition showing a clas... |
fun2cnv 6609 | The double converse of a c... |
svrelfun 6610 | A single-valued relation i... |
fncnv 6611 | Single-rootedness (see ~ f... |
fun11 6612 | Two ways of stating that `... |
fununi 6613 | The union of a chain (with... |
funin 6614 | The intersection with a fu... |
funres11 6615 | The restriction of a one-t... |
funcnvres 6616 | The converse of a restrict... |
cnvresid 6617 | Converse of a restricted i... |
funcnvres2 6618 | The converse of a restrict... |
funimacnv 6619 | The image of the preimage ... |
funimass1 6620 | A kind of contraposition l... |
funimass2 6621 | A kind of contraposition l... |
imadif 6622 | The image of a difference ... |
imain 6623 | The image of an intersecti... |
funimaexg 6624 | Axiom of Replacement using... |
funimaexgOLD 6625 | Obsolete version of ~ funi... |
funimaex 6626 | The image of a set under a... |
isarep1 6627 | Part of a study of the Axi... |
isarep1OLD 6628 | Obsolete version of ~ isar... |
isarep2 6629 | Part of a study of the Axi... |
fneq1 6630 | Equality theorem for funct... |
fneq2 6631 | Equality theorem for funct... |
fneq1d 6632 | Equality deduction for fun... |
fneq2d 6633 | Equality deduction for fun... |
fneq12d 6634 | Equality deduction for fun... |
fneq12 6635 | Equality theorem for funct... |
fneq1i 6636 | Equality inference for fun... |
fneq2i 6637 | Equality inference for fun... |
nffn 6638 | Bound-variable hypothesis ... |
fnfun 6639 | A function with domain is ... |
fnfund 6640 | A function with domain is ... |
fnrel 6641 | A function with domain is ... |
fndm 6642 | The domain of a function. ... |
fndmi 6643 | The domain of a function. ... |
fndmd 6644 | The domain of a function. ... |
funfni 6645 | Inference to convert a fun... |
fndmu 6646 | A function has a unique do... |
fnbr 6647 | The first argument of bina... |
fnop 6648 | The first argument of an o... |
fneu 6649 | There is exactly one value... |
fneu2 6650 | There is exactly one value... |
fnunres1 6651 | Restriction of a disjoint ... |
fnunres2 6652 | Restriction of a disjoint ... |
fnun 6653 | The union of two functions... |
fnund 6654 | The union of two functions... |
fnunop 6655 | Extension of a function wi... |
fncofn 6656 | Composition of a function ... |
fnco 6657 | Composition of two functio... |
fncoOLD 6658 | Obsolete version of ~ fnco... |
fnresdm 6659 | A function does not change... |
fnresdisj 6660 | A function restricted to a... |
2elresin 6661 | Membership in two function... |
fnssresb 6662 | Restriction of a function ... |
fnssres 6663 | Restriction of a function ... |
fnssresd 6664 | Restriction of a function ... |
fnresin1 6665 | Restriction of a function'... |
fnresin2 6666 | Restriction of a function'... |
fnres 6667 | An equivalence for functio... |
idfn 6668 | The identity relation is a... |
fnresi 6669 | The restricted identity re... |
fnima 6670 | The image of a function's ... |
fn0 6671 | A function with empty doma... |
fnimadisj 6672 | A class that is disjoint w... |
fnimaeq0 6673 | Images under a function ne... |
dfmpt3 6674 | Alternate definition for t... |
mptfnf 6675 | The maps-to notation defin... |
fnmptf 6676 | The maps-to notation defin... |
fnopabg 6677 | Functionality and domain o... |
fnopab 6678 | Functionality and domain o... |
mptfng 6679 | The maps-to notation defin... |
fnmpt 6680 | The maps-to notation defin... |
fnmptd 6681 | The maps-to notation defin... |
mpt0 6682 | A mapping operation with e... |
fnmpti 6683 | Functionality and domain o... |
dmmpti 6684 | Domain of the mapping oper... |
dmmptd 6685 | The domain of the mapping ... |
mptun 6686 | Union of mappings which ar... |
partfun 6687 | Rewrite a function defined... |
feq1 6688 | Equality theorem for funct... |
feq2 6689 | Equality theorem for funct... |
feq3 6690 | Equality theorem for funct... |
feq23 6691 | Equality theorem for funct... |
feq1d 6692 | Equality deduction for fun... |
feq2d 6693 | Equality deduction for fun... |
feq3d 6694 | Equality deduction for fun... |
feq12d 6695 | Equality deduction for fun... |
feq123d 6696 | Equality deduction for fun... |
feq123 6697 | Equality theorem for funct... |
feq1i 6698 | Equality inference for fun... |
feq2i 6699 | Equality inference for fun... |
feq12i 6700 | Equality inference for fun... |
feq23i 6701 | Equality inference for fun... |
feq23d 6702 | Equality deduction for fun... |
nff 6703 | Bound-variable hypothesis ... |
sbcfng 6704 | Distribute proper substitu... |
sbcfg 6705 | Distribute proper substitu... |
elimf 6706 | Eliminate a mapping hypoth... |
ffn 6707 | A mapping is a function wi... |
ffnd 6708 | A mapping is a function wi... |
dffn2 6709 | Any function is a mapping ... |
ffun 6710 | A mapping is a function. ... |
ffund 6711 | A mapping is a function, d... |
frel 6712 | A mapping is a relation. ... |
freld 6713 | A mapping is a relation. ... |
frn 6714 | The range of a mapping. (... |
frnd 6715 | Deduction form of ~ frn . ... |
fdm 6716 | The domain of a mapping. ... |
fdmOLD 6717 | Obsolete version of ~ fdm ... |
fdmd 6718 | Deduction form of ~ fdm . ... |
fdmi 6719 | Inference associated with ... |
dffn3 6720 | A function maps to its ran... |
ffrn 6721 | A function maps to its ran... |
ffrnb 6722 | Characterization of a func... |
ffrnbd 6723 | A function maps to its ran... |
fss 6724 | Expanding the codomain of ... |
fssd 6725 | Expanding the codomain of ... |
fssdmd 6726 | Expressing that a class is... |
fssdm 6727 | Expressing that a class is... |
fimass 6728 | The image of a class under... |
fimacnv 6729 | The preimage of the codoma... |
fcof 6730 | Composition of a function ... |
fco 6731 | Composition of two functio... |
fcoOLD 6732 | Obsolete version of ~ fco ... |
fcod 6733 | Composition of two mapping... |
fco2 6734 | Functionality of a composi... |
fssxp 6735 | A mapping is a class of or... |
funssxp 6736 | Two ways of specifying a p... |
ffdm 6737 | A mapping is a partial fun... |
ffdmd 6738 | The domain of a function. ... |
fdmrn 6739 | A different way to write `... |
funcofd 6740 | Composition of two functio... |
fco3OLD 6741 | Obsolete version of ~ func... |
opelf 6742 | The members of an ordered ... |
fun 6743 | The union of two functions... |
fun2 6744 | The union of two functions... |
fun2d 6745 | The union of functions wit... |
fnfco 6746 | Composition of two functio... |
fssres 6747 | Restriction of a function ... |
fssresd 6748 | Restriction of a function ... |
fssres2 6749 | Restriction of a restricte... |
fresin 6750 | An identity for the mappin... |
resasplit 6751 | If two functions agree on ... |
fresaun 6752 | The union of two functions... |
fresaunres2 6753 | From the union of two func... |
fresaunres1 6754 | From the union of two func... |
fcoi1 6755 | Composition of a mapping a... |
fcoi2 6756 | Composition of restricted ... |
feu 6757 | There is exactly one value... |
fcnvres 6758 | The converse of a restrict... |
fimacnvdisj 6759 | The preimage of a class di... |
fint 6760 | Function into an intersect... |
fin 6761 | Mapping into an intersecti... |
f0 6762 | The empty function. (Cont... |
f00 6763 | A class is a function with... |
f0bi 6764 | A function with empty doma... |
f0dom0 6765 | A function is empty iff it... |
f0rn0 6766 | If there is no element in ... |
fconst 6767 | A Cartesian product with a... |
fconstg 6768 | A Cartesian product with a... |
fnconstg 6769 | A Cartesian product with a... |
fconst6g 6770 | Constant function with loo... |
fconst6 6771 | A constant function as a m... |
f1eq1 6772 | Equality theorem for one-t... |
f1eq2 6773 | Equality theorem for one-t... |
f1eq3 6774 | Equality theorem for one-t... |
nff1 6775 | Bound-variable hypothesis ... |
dff12 6776 | Alternate definition of a ... |
f1f 6777 | A one-to-one mapping is a ... |
f1fn 6778 | A one-to-one mapping is a ... |
f1fun 6779 | A one-to-one mapping is a ... |
f1rel 6780 | A one-to-one onto mapping ... |
f1dm 6781 | The domain of a one-to-one... |
f1dmOLD 6782 | Obsolete version of ~ f1dm... |
f1ss 6783 | A function that is one-to-... |
f1ssr 6784 | A function that is one-to-... |
f1ssres 6785 | A function that is one-to-... |
f1resf1 6786 | The restriction of an inje... |
f1cnvcnv 6787 | Two ways to express that a... |
f1cof1 6788 | Composition of two one-to-... |
f1co 6789 | Composition of one-to-one ... |
f1coOLD 6790 | Obsolete version of ~ f1co... |
foeq1 6791 | Equality theorem for onto ... |
foeq2 6792 | Equality theorem for onto ... |
foeq3 6793 | Equality theorem for onto ... |
nffo 6794 | Bound-variable hypothesis ... |
fof 6795 | An onto mapping is a mappi... |
fofun 6796 | An onto mapping is a funct... |
fofn 6797 | An onto mapping is a funct... |
forn 6798 | The codomain of an onto fu... |
dffo2 6799 | Alternate definition of an... |
foima 6800 | The image of the domain of... |
dffn4 6801 | A function maps onto its r... |
funforn 6802 | A function maps its domain... |
fodmrnu 6803 | An onto function has uniqu... |
fimadmfo 6804 | A function is a function o... |
fores 6805 | Restriction of an onto fun... |
fimadmfoALT 6806 | Alternate proof of ~ fimad... |
focnvimacdmdm 6807 | The preimage of the codoma... |
focofo 6808 | Composition of onto functi... |
foco 6809 | Composition of onto functi... |
foconst 6810 | A nonzero constant functio... |
f1oeq1 6811 | Equality theorem for one-t... |
f1oeq2 6812 | Equality theorem for one-t... |
f1oeq3 6813 | Equality theorem for one-t... |
f1oeq23 6814 | Equality theorem for one-t... |
f1eq123d 6815 | Equality deduction for one... |
foeq123d 6816 | Equality deduction for ont... |
f1oeq123d 6817 | Equality deduction for one... |
f1oeq1d 6818 | Equality deduction for one... |
f1oeq2d 6819 | Equality deduction for one... |
f1oeq3d 6820 | Equality deduction for one... |
nff1o 6821 | Bound-variable hypothesis ... |
f1of1 6822 | A one-to-one onto mapping ... |
f1of 6823 | A one-to-one onto mapping ... |
f1ofn 6824 | A one-to-one onto mapping ... |
f1ofun 6825 | A one-to-one onto mapping ... |
f1orel 6826 | A one-to-one onto mapping ... |
f1odm 6827 | The domain of a one-to-one... |
dff1o2 6828 | Alternate definition of on... |
dff1o3 6829 | Alternate definition of on... |
f1ofo 6830 | A one-to-one onto function... |
dff1o4 6831 | Alternate definition of on... |
dff1o5 6832 | Alternate definition of on... |
f1orn 6833 | A one-to-one function maps... |
f1f1orn 6834 | A one-to-one function maps... |
f1ocnv 6835 | The converse of a one-to-o... |
f1ocnvb 6836 | A relation is a one-to-one... |
f1ores 6837 | The restriction of a one-t... |
f1orescnv 6838 | The converse of a one-to-o... |
f1imacnv 6839 | Preimage of an image. (Co... |
foimacnv 6840 | A reverse version of ~ f1i... |
foun 6841 | The union of two onto func... |
f1oun 6842 | The union of two one-to-on... |
f1un 6843 | The union of two one-to-on... |
resdif 6844 | The restriction of a one-t... |
resin 6845 | The restriction of a one-t... |
f1oco 6846 | Composition of one-to-one ... |
f1cnv 6847 | The converse of an injecti... |
funcocnv2 6848 | Composition with the conve... |
fococnv2 6849 | The composition of an onto... |
f1ococnv2 6850 | The composition of a one-t... |
f1cocnv2 6851 | Composition of an injectiv... |
f1ococnv1 6852 | The composition of a one-t... |
f1cocnv1 6853 | Composition of an injectiv... |
funcoeqres 6854 | Express a constraint on a ... |
f1ssf1 6855 | A subset of an injective f... |
f10 6856 | The empty set maps one-to-... |
f10d 6857 | The empty set maps one-to-... |
f1o00 6858 | One-to-one onto mapping of... |
fo00 6859 | Onto mapping of the empty ... |
f1o0 6860 | One-to-one onto mapping of... |
f1oi 6861 | A restriction of the ident... |
f1ovi 6862 | The identity relation is a... |
f1osn 6863 | A singleton of an ordered ... |
f1osng 6864 | A singleton of an ordered ... |
f1sng 6865 | A singleton of an ordered ... |
fsnd 6866 | A singleton of an ordered ... |
f1oprswap 6867 | A two-element swap is a bi... |
f1oprg 6868 | An unordered pair of order... |
tz6.12-2 6869 | Function value when ` F ` ... |
fveu 6870 | The value of a function at... |
brprcneu 6871 | If ` A ` is a proper class... |
brprcneuALT 6872 | Alternate proof of ~ brprc... |
fvprc 6873 | A function's value at a pr... |
fvprcALT 6874 | Alternate proof of ~ fvprc... |
rnfvprc 6875 | The range of a function va... |
fv2 6876 | Alternate definition of fu... |
dffv3 6877 | A definition of function v... |
dffv4 6878 | The previous definition of... |
elfv 6879 | Membership in a function v... |
fveq1 6880 | Equality theorem for funct... |
fveq2 6881 | Equality theorem for funct... |
fveq1i 6882 | Equality inference for fun... |
fveq1d 6883 | Equality deduction for fun... |
fveq2i 6884 | Equality inference for fun... |
fveq2d 6885 | Equality deduction for fun... |
2fveq3 6886 | Equality theorem for neste... |
fveq12i 6887 | Equality deduction for fun... |
fveq12d 6888 | Equality deduction for fun... |
fveqeq2d 6889 | Equality deduction for fun... |
fveqeq2 6890 | Equality deduction for fun... |
nffv 6891 | Bound-variable hypothesis ... |
nffvmpt1 6892 | Bound-variable hypothesis ... |
nffvd 6893 | Deduction version of bound... |
fvex 6894 | The value of a class exist... |
fvexi 6895 | The value of a class exist... |
fvexd 6896 | The value of a class exist... |
fvif 6897 | Move a conditional outside... |
iffv 6898 | Move a conditional outside... |
fv3 6899 | Alternate definition of th... |
fvres 6900 | The value of a restricted ... |
fvresd 6901 | The value of a restricted ... |
funssfv 6902 | The value of a member of t... |
tz6.12c 6903 | Corollary of Theorem 6.12(... |
tz6.12-1 6904 | Function value. Theorem 6... |
tz6.12-1OLD 6905 | Obsolete version of ~ tz6.... |
tz6.12 6906 | Function value. Theorem 6... |
tz6.12f 6907 | Function value, using boun... |
tz6.12cOLD 6908 | Obsolete version of ~ tz6.... |
tz6.12i 6909 | Corollary of Theorem 6.12(... |
fvbr0 6910 | Two possibilities for the ... |
fvrn0 6911 | A function value is a memb... |
fvn0fvelrn 6912 | If the value of a function... |
elfvunirn 6913 | A function value is a subs... |
fvssunirn 6914 | The result of a function v... |
fvssunirnOLD 6915 | Obsolete version of ~ fvss... |
ndmfv 6916 | The value of a class outsi... |
ndmfvrcl 6917 | Reverse closure law for fu... |
elfvdm 6918 | If a function value has a ... |
elfvex 6919 | If a function value has a ... |
elfvexd 6920 | If a function value has a ... |
eliman0 6921 | A nonempty function value ... |
nfvres 6922 | The value of a non-member ... |
nfunsn 6923 | If the restriction of a cl... |
fvfundmfvn0 6924 | If the "value of a class" ... |
0fv 6925 | Function value of the empt... |
fv2prc 6926 | A function value of a func... |
elfv2ex 6927 | If a function value of a f... |
fveqres 6928 | Equal values imply equal v... |
csbfv12 6929 | Move class substitution in... |
csbfv2g 6930 | Move class substitution in... |
csbfv 6931 | Substitution for a functio... |
funbrfv 6932 | The second argument of a b... |
funopfv 6933 | The second element in an o... |
fnbrfvb 6934 | Equivalence of function va... |
fnopfvb 6935 | Equivalence of function va... |
funbrfvb 6936 | Equivalence of function va... |
funopfvb 6937 | Equivalence of function va... |
fnbrfvb2 6938 | Version of ~ fnbrfvb for f... |
funbrfv2b 6939 | Function value in terms of... |
dffn5 6940 | Representation of a functi... |
fnrnfv 6941 | The range of a function ex... |
fvelrnb 6942 | A member of a function's r... |
foelcdmi 6943 | A member of a surjective f... |
dfimafn 6944 | Alternate definition of th... |
dfimafn2 6945 | Alternate definition of th... |
funimass4 6946 | Membership relation for th... |
fvelima 6947 | Function value in an image... |
funimassd 6948 | Sufficient condition for t... |
fvelimad 6949 | Function value in an image... |
feqmptd 6950 | Deduction form of ~ dffn5 ... |
feqresmpt 6951 | Express a restricted funct... |
feqmptdf 6952 | Deduction form of ~ dffn5f... |
dffn5f 6953 | Representation of a functi... |
fvelimab 6954 | Function value in an image... |
fvelimabd 6955 | Deduction form of ~ fvelim... |
unima 6956 | Image of a union. (Contri... |
fvi 6957 | The value of the identity ... |
fviss 6958 | The value of the identity ... |
fniinfv 6959 | The indexed intersection o... |
fnsnfv 6960 | Singleton of function valu... |
fnsnfvOLD 6961 | Obsolete version of ~ fnsn... |
opabiotafun 6962 | Define a function whose va... |
opabiotadm 6963 | Define a function whose va... |
opabiota 6964 | Define a function whose va... |
fnimapr 6965 | The image of a pair under ... |
ssimaex 6966 | The existence of a subimag... |
ssimaexg 6967 | The existence of a subimag... |
funfv 6968 | A simplified expression fo... |
funfv2 6969 | The value of a function. ... |
funfv2f 6970 | The value of a function. ... |
fvun 6971 | Value of the union of two ... |
fvun1 6972 | The value of a union when ... |
fvun2 6973 | The value of a union when ... |
fvun1d 6974 | The value of a union when ... |
fvun2d 6975 | The value of a union when ... |
dffv2 6976 | Alternate definition of fu... |
dmfco 6977 | Domains of a function comp... |
fvco2 6978 | Value of a function compos... |
fvco 6979 | Value of a function compos... |
fvco3 6980 | Value of a function compos... |
fvco3d 6981 | Value of a function compos... |
fvco4i 6982 | Conditions for a compositi... |
fvopab3g 6983 | Value of a function given ... |
fvopab3ig 6984 | Value of a function given ... |
brfvopabrbr 6985 | The binary relation of a f... |
fvmptg 6986 | Value of a function given ... |
fvmpti 6987 | Value of a function given ... |
fvmpt 6988 | Value of a function given ... |
fvmpt2f 6989 | Value of a function given ... |
fvtresfn 6990 | Functionality of a tuple-r... |
fvmpts 6991 | Value of a function given ... |
fvmpt3 6992 | Value of a function given ... |
fvmpt3i 6993 | Value of a function given ... |
fvmptdf 6994 | Deduction version of ~ fvm... |
fvmptd 6995 | Deduction version of ~ fvm... |
fvmptd2 6996 | Deduction version of ~ fvm... |
mptrcl 6997 | Reverse closure for a mapp... |
fvmpt2i 6998 | Value of a function given ... |
fvmpt2 6999 | Value of a function given ... |
fvmptss 7000 | If all the values of the m... |
fvmpt2d 7001 | Deduction version of ~ fvm... |
fvmptex 7002 | Express a function ` F ` w... |
fvmptd3f 7003 | Alternate deduction versio... |
fvmptd2f 7004 | Alternate deduction versio... |
fvmptdv 7005 | Alternate deduction versio... |
fvmptdv2 7006 | Alternate deduction versio... |
mpteqb 7007 | Bidirectional equality the... |
fvmptt 7008 | Closed theorem form of ~ f... |
fvmptf 7009 | Value of a function given ... |
fvmptnf 7010 | The value of a function gi... |
fvmptd3 7011 | Deduction version of ~ fvm... |
fvmptn 7012 | This somewhat non-intuitiv... |
fvmptss2 7013 | A mapping always evaluates... |
elfvmptrab1w 7014 | Implications for the value... |
elfvmptrab1 7015 | Implications for the value... |
elfvmptrab 7016 | Implications for the value... |
fvopab4ndm 7017 | Value of a function given ... |
fvmptndm 7018 | Value of a function given ... |
fvmptrabfv 7019 | Value of a function mappin... |
fvopab5 7020 | The value of a function th... |
fvopab6 7021 | Value of a function given ... |
eqfnfv 7022 | Equality of functions is d... |
eqfnfv2 7023 | Equality of functions is d... |
eqfnfv3 7024 | Derive equality of functio... |
eqfnfvd 7025 | Deduction for equality of ... |
eqfnfv2f 7026 | Equality of functions is d... |
eqfunfv 7027 | Equality of functions is d... |
eqfnun 7028 | Two functions on ` A u. B ... |
fvreseq0 7029 | Equality of restricted fun... |
fvreseq1 7030 | Equality of a function res... |
fvreseq 7031 | Equality of restricted fun... |
fnmptfvd 7032 | A function with a given do... |
fndmdif 7033 | Two ways to express the lo... |
fndmdifcom 7034 | The difference set between... |
fndmdifeq0 7035 | The difference set of two ... |
fndmin 7036 | Two ways to express the lo... |
fneqeql 7037 | Two functions are equal if... |
fneqeql2 7038 | Two functions are equal if... |
fnreseql 7039 | Two functions are equal on... |
chfnrn 7040 | The range of a choice func... |
funfvop 7041 | Ordered pair with function... |
funfvbrb 7042 | Two ways to say that ` A `... |
fvimacnvi 7043 | A member of a preimage is ... |
fvimacnv 7044 | The argument of a function... |
funimass3 7045 | A kind of contraposition l... |
funimass5 7046 | A subclass of a preimage i... |
funconstss 7047 | Two ways of specifying tha... |
fvimacnvALT 7048 | Alternate proof of ~ fvima... |
elpreima 7049 | Membership in the preimage... |
elpreimad 7050 | Membership in the preimage... |
fniniseg 7051 | Membership in the preimage... |
fncnvima2 7052 | Inverse images under funct... |
fniniseg2 7053 | Inverse point images under... |
unpreima 7054 | Preimage of a union. (Con... |
inpreima 7055 | Preimage of an intersectio... |
difpreima 7056 | Preimage of a difference. ... |
respreima 7057 | The preimage of a restrict... |
cnvimainrn 7058 | The preimage of the inters... |
sspreima 7059 | The preimage of a subset i... |
iinpreima 7060 | Preimage of an intersectio... |
intpreima 7061 | Preimage of an intersectio... |
fimacnvOLD 7062 | Obsolete version of ~ fima... |
fimacnvinrn 7063 | Taking the converse image ... |
fimacnvinrn2 7064 | Taking the converse image ... |
rescnvimafod 7065 | The restriction of a funct... |
fvn0ssdmfun 7066 | If a class' function value... |
fnopfv 7067 | Ordered pair with function... |
fvelrn 7068 | A function's value belongs... |
nelrnfvne 7069 | A function value cannot be... |
fveqdmss 7070 | If the empty set is not co... |
fveqressseq 7071 | If the empty set is not co... |
fnfvelrn 7072 | A function's value belongs... |
ffvelcdm 7073 | A function's value belongs... |
fnfvelrnd 7074 | A function's value belongs... |
ffvelcdmi 7075 | A function's value belongs... |
ffvelcdmda 7076 | A function's value belongs... |
ffvelcdmd 7077 | A function's value belongs... |
rexrn 7078 | Restricted existential qua... |
ralrn 7079 | Restricted universal quant... |
elrnrexdm 7080 | For any element in the ran... |
elrnrexdmb 7081 | For any element in the ran... |
eldmrexrn 7082 | For any element in the dom... |
eldmrexrnb 7083 | For any element in the dom... |
fvcofneq 7084 | The values of two function... |
ralrnmptw 7085 | A restricted quantifier ov... |
rexrnmptw 7086 | A restricted quantifier ov... |
ralrnmpt 7087 | A restricted quantifier ov... |
rexrnmpt 7088 | A restricted quantifier ov... |
f0cli 7089 | Unconditional closure of a... |
dff2 7090 | Alternate definition of a ... |
dff3 7091 | Alternate definition of a ... |
dff4 7092 | Alternate definition of a ... |
dffo3 7093 | An onto mapping expressed ... |
dffo4 7094 | Alternate definition of an... |
dffo5 7095 | Alternate definition of an... |
exfo 7096 | A relation equivalent to t... |
dffo3f 7097 | An onto mapping expressed ... |
foelrn 7098 | Property of a surjective f... |
foelrnf 7099 | Property of a surjective f... |
foco2 7100 | If a composition of two fu... |
fmpt 7101 | Functionality of the mappi... |
f1ompt 7102 | Express bijection for a ma... |
fmpti 7103 | Functionality of the mappi... |
fvmptelcdm 7104 | The value of a function at... |
fmptd 7105 | Domain and codomain of the... |
fmpttd 7106 | Version of ~ fmptd with in... |
fmpt3d 7107 | Domain and codomain of the... |
fmptdf 7108 | A version of ~ fmptd using... |
fompt 7109 | Express being onto for a m... |
ffnfv 7110 | A function maps to a class... |
ffnfvf 7111 | A function maps to a class... |
fnfvrnss 7112 | An upper bound for range d... |
fcdmssb 7113 | A function is a function i... |
rnmptss 7114 | The range of an operation ... |
fmpt2d 7115 | Domain and codomain of the... |
ffvresb 7116 | A necessary and sufficient... |
f1oresrab 7117 | Build a bijection between ... |
f1ossf1o 7118 | Restricting a bijection, w... |
fmptco 7119 | Composition of two functio... |
fmptcof 7120 | Version of ~ fmptco where ... |
fmptcos 7121 | Composition of two functio... |
cofmpt 7122 | Express composition of a m... |
fcompt 7123 | Express composition of two... |
fcoconst 7124 | Composition with a constan... |
fsn 7125 | A function maps a singleto... |
fsn2 7126 | A function that maps a sin... |
fsng 7127 | A function maps a singleto... |
fsn2g 7128 | A function that maps a sin... |
xpsng 7129 | The Cartesian product of t... |
xpprsng 7130 | The Cartesian product of a... |
xpsn 7131 | The Cartesian product of t... |
f1o2sn 7132 | A singleton consisting in ... |
residpr 7133 | Restriction of the identit... |
dfmpt 7134 | Alternate definition for t... |
fnasrn 7135 | A function expressed as th... |
idref 7136 | Two ways to state that a r... |
funiun 7137 | A function is a union of s... |
funopsn 7138 | If a function is an ordere... |
funop 7139 | An ordered pair is a funct... |
funopdmsn 7140 | The domain of a function w... |
funsndifnop 7141 | A singleton of an ordered ... |
funsneqopb 7142 | A singleton of an ordered ... |
ressnop0 7143 | If ` A ` is not in ` C ` ,... |
fpr 7144 | A function with a domain o... |
fprg 7145 | A function with a domain o... |
ftpg 7146 | A function with a domain o... |
ftp 7147 | A function with a domain o... |
fnressn 7148 | A function restricted to a... |
funressn 7149 | A function restricted to a... |
fressnfv 7150 | The value of a function re... |
fvrnressn 7151 | If the value of a function... |
fvressn 7152 | The value of a function re... |
fvn0fvelrnOLD 7153 | Obsolete version of ~ fvn0... |
fvconst 7154 | The value of a constant fu... |
fnsnr 7155 | If a class belongs to a fu... |
fnsnb 7156 | A function whose domain is... |
fmptsn 7157 | Express a singleton functi... |
fmptsng 7158 | Express a singleton functi... |
fmptsnd 7159 | Express a singleton functi... |
fmptap 7160 | Append an additional value... |
fmptapd 7161 | Append an additional value... |
fmptpr 7162 | Express a pair function in... |
fvresi 7163 | The value of a restricted ... |
fninfp 7164 | Express the class of fixed... |
fnelfp 7165 | Property of a fixed point ... |
fndifnfp 7166 | Express the class of non-f... |
fnelnfp 7167 | Property of a non-fixed po... |
fnnfpeq0 7168 | A function is the identity... |
fvunsn 7169 | Remove an ordered pair not... |
fvsng 7170 | The value of a singleton o... |
fvsn 7171 | The value of a singleton o... |
fvsnun1 7172 | The value of a function wi... |
fvsnun2 7173 | The value of a function wi... |
fnsnsplit 7174 | Split a function into a si... |
fsnunf 7175 | Adjoining a point to a fun... |
fsnunf2 7176 | Adjoining a point to a pun... |
fsnunfv 7177 | Recover the added point fr... |
fsnunres 7178 | Recover the original funct... |
funresdfunsn 7179 | Restricting a function to ... |
fvpr1g 7180 | The value of a function wi... |
fvpr2g 7181 | The value of a function wi... |
fvpr2gOLD 7182 | Obsolete version of ~ fvpr... |
fvpr1 7183 | The value of a function wi... |
fvpr1OLD 7184 | Obsolete version of ~ fvpr... |
fvpr2 7185 | The value of a function wi... |
fvpr2OLD 7186 | Obsolete version of ~ fvpr... |
fprb 7187 | A condition for functionho... |
fvtp1 7188 | The first value of a funct... |
fvtp2 7189 | The second value of a func... |
fvtp3 7190 | The third value of a funct... |
fvtp1g 7191 | The value of a function wi... |
fvtp2g 7192 | The value of a function wi... |
fvtp3g 7193 | The value of a function wi... |
tpres 7194 | An unordered triple of ord... |
fvconst2g 7195 | The value of a constant fu... |
fconst2g 7196 | A constant function expres... |
fvconst2 7197 | The value of a constant fu... |
fconst2 7198 | A constant function expres... |
fconst5 7199 | Two ways to express that a... |
rnmptc 7200 | Range of a constant functi... |
fnprb 7201 | A function whose domain ha... |
fntpb 7202 | A function whose domain ha... |
fnpr2g 7203 | A function whose domain ha... |
fpr2g 7204 | A function that maps a pai... |
fconstfv 7205 | A constant function expres... |
fconst3 7206 | Two ways to express a cons... |
fconst4 7207 | Two ways to express a cons... |
resfunexg 7208 | The restriction of a funct... |
resiexd 7209 | The restriction of the ide... |
fnex 7210 | If the domain of a functio... |
fnexd 7211 | If the domain of a functio... |
funex 7212 | If the domain of a functio... |
opabex 7213 | Existence of a function ex... |
mptexg 7214 | If the domain of a functio... |
mptexgf 7215 | If the domain of a functio... |
mptex 7216 | If the domain of a functio... |
mptexd 7217 | If the domain of a functio... |
mptrabex 7218 | If the domain of a functio... |
fex 7219 | If the domain of a mapping... |
fexd 7220 | If the domain of a mapping... |
mptfvmpt 7221 | A function in maps-to nota... |
eufnfv 7222 | A function is uniquely det... |
funfvima 7223 | A function's value in a pr... |
funfvima2 7224 | A function's value in an i... |
funfvima2d 7225 | A function's value in a pr... |
fnfvima 7226 | The function value of an o... |
fnfvimad 7227 | A function's value belongs... |
resfvresima 7228 | The value of the function ... |
funfvima3 7229 | A class including a functi... |
rexima 7230 | Existential quantification... |
ralima 7231 | Universal quantification u... |
fvclss 7232 | Upper bound for the class ... |
elabrex 7233 | Elementhood in an image se... |
elabrexg 7234 | Elementhood in an image se... |
abrexco 7235 | Composition of two image m... |
imaiun 7236 | The image of an indexed un... |
imauni 7237 | The image of a union is th... |
fniunfv 7238 | The indexed union of a fun... |
funiunfv 7239 | The indexed union of a fun... |
funiunfvf 7240 | The indexed union of a fun... |
eluniima 7241 | Membership in the union of... |
elunirn 7242 | Membership in the union of... |
elunirnALT 7243 | Alternate proof of ~ eluni... |
elunirn2OLD 7244 | Obsolete version of ~ elfv... |
fnunirn 7245 | Membership in a union of s... |
dff13 7246 | A one-to-one function in t... |
dff13f 7247 | A one-to-one function in t... |
f1veqaeq 7248 | If the values of a one-to-... |
f1cofveqaeq 7249 | If the values of a composi... |
f1cofveqaeqALT 7250 | Alternate proof of ~ f1cof... |
2f1fvneq 7251 | If two one-to-one function... |
f1mpt 7252 | Express injection for a ma... |
f1fveq 7253 | Equality of function value... |
f1elima 7254 | Membership in the image of... |
f1imass 7255 | Taking images under a one-... |
f1imaeq 7256 | Taking images under a one-... |
f1imapss 7257 | Taking images under a one-... |
fpropnf1 7258 | A function, given by an un... |
f1dom3fv3dif 7259 | The function values for a ... |
f1dom3el3dif 7260 | The codomain of a 1-1 func... |
dff14a 7261 | A one-to-one function in t... |
dff14b 7262 | A one-to-one function in t... |
f12dfv 7263 | A one-to-one function with... |
f13dfv 7264 | A one-to-one function with... |
dff1o6 7265 | A one-to-one onto function... |
f1ocnvfv1 7266 | The converse value of the ... |
f1ocnvfv2 7267 | The value of the converse ... |
f1ocnvfv 7268 | Relationship between the v... |
f1ocnvfvb 7269 | Relationship between the v... |
nvof1o 7270 | An involution is a bijecti... |
nvocnv 7271 | The converse of an involut... |
f1cdmsn 7272 | If a one-to-one function w... |
fsnex 7273 | Relate a function with a s... |
f1prex 7274 | Relate a one-to-one functi... |
f1ocnvdm 7275 | The value of the converse ... |
f1ocnvfvrneq 7276 | If the values of a one-to-... |
fcof1 7277 | An application is injectiv... |
fcofo 7278 | An application is surjecti... |
cbvfo 7279 | Change bound variable betw... |
cbvexfo 7280 | Change bound variable betw... |
cocan1 7281 | An injection is left-cance... |
cocan2 7282 | A surjection is right-canc... |
fcof1oinvd 7283 | Show that a function is th... |
fcof1od 7284 | A function is bijective if... |
2fcoidinvd 7285 | Show that a function is th... |
fcof1o 7286 | Show that two functions ar... |
2fvcoidd 7287 | Show that the composition ... |
2fvidf1od 7288 | A function is bijective if... |
2fvidinvd 7289 | Show that two functions ar... |
foeqcnvco 7290 | Condition for function equ... |
f1eqcocnv 7291 | Condition for function equ... |
f1eqcocnvOLD 7292 | Obsolete version of ~ f1eq... |
fveqf1o 7293 | Given a bijection ` F ` , ... |
nf1const 7294 | A constant function from a... |
nf1oconst 7295 | A constant function from a... |
f1ofvswap 7296 | Swapping two values in a b... |
fliftrel 7297 | ` F ` , a function lift, i... |
fliftel 7298 | Elementhood in the relatio... |
fliftel1 7299 | Elementhood in the relatio... |
fliftcnv 7300 | Converse of the relation `... |
fliftfun 7301 | The function ` F ` is the ... |
fliftfund 7302 | The function ` F ` is the ... |
fliftfuns 7303 | The function ` F ` is the ... |
fliftf 7304 | The domain and range of th... |
fliftval 7305 | The value of the function ... |
isoeq1 7306 | Equality theorem for isomo... |
isoeq2 7307 | Equality theorem for isomo... |
isoeq3 7308 | Equality theorem for isomo... |
isoeq4 7309 | Equality theorem for isomo... |
isoeq5 7310 | Equality theorem for isomo... |
nfiso 7311 | Bound-variable hypothesis ... |
isof1o 7312 | An isomorphism is a one-to... |
isof1oidb 7313 | A function is a bijection ... |
isof1oopb 7314 | A function is a bijection ... |
isorel 7315 | An isomorphism connects bi... |
soisores 7316 | Express the condition of i... |
soisoi 7317 | Infer isomorphism from one... |
isoid 7318 | Identity law for isomorphi... |
isocnv 7319 | Converse law for isomorphi... |
isocnv2 7320 | Converse law for isomorphi... |
isocnv3 7321 | Complementation law for is... |
isores2 7322 | An isomorphism from one we... |
isores1 7323 | An isomorphism from one we... |
isores3 7324 | Induced isomorphism on a s... |
isotr 7325 | Composition (transitive) l... |
isomin 7326 | Isomorphisms preserve mini... |
isoini 7327 | Isomorphisms preserve init... |
isoini2 7328 | Isomorphisms are isomorphi... |
isofrlem 7329 | Lemma for ~ isofr . (Cont... |
isoselem 7330 | Lemma for ~ isose . (Cont... |
isofr 7331 | An isomorphism preserves w... |
isose 7332 | An isomorphism preserves s... |
isofr2 7333 | A weak form of ~ isofr tha... |
isopolem 7334 | Lemma for ~ isopo . (Cont... |
isopo 7335 | An isomorphism preserves t... |
isosolem 7336 | Lemma for ~ isoso . (Cont... |
isoso 7337 | An isomorphism preserves t... |
isowe 7338 | An isomorphism preserves t... |
isowe2 7339 | A weak form of ~ isowe tha... |
f1oiso 7340 | Any one-to-one onto functi... |
f1oiso2 7341 | Any one-to-one onto functi... |
f1owe 7342 | Well-ordering of isomorphi... |
weniso 7343 | A set-like well-ordering h... |
weisoeq 7344 | Thus, there is at most one... |
weisoeq2 7345 | Thus, there is at most one... |
knatar 7346 | The Knaster-Tarski theorem... |
fvresval 7347 | The value of a restricted ... |
funeldmb 7348 | If ` (/) ` is not part of ... |
eqfunresadj 7349 | Law for adjoining an eleme... |
eqfunressuc 7350 | Law for equality of restri... |
fnssintima 7351 | Condition for subset of an... |
imaeqsexv 7352 | Substitute a function valu... |
imaeqsalv 7353 | Substitute a function valu... |
canth 7354 | No set ` A ` is equinumero... |
ncanth 7355 | Cantor's theorem fails for... |
riotaeqdv 7358 | Formula-building deduction... |
riotabidv 7359 | Formula-building deduction... |
riotaeqbidv 7360 | Equality deduction for res... |
riotaex 7361 | Restricted iota is a set. ... |
riotav 7362 | An iota restricted to the ... |
riotauni 7363 | Restricted iota in terms o... |
nfriota1 7364 | The abstraction variable i... |
nfriotadw 7365 | Deduction version of ~ nfr... |
cbvriotaw 7366 | Change bound variable in a... |
cbvriotavw 7367 | Change bound variable in a... |
cbvriotavwOLD 7368 | Obsolete version of ~ cbvr... |
nfriotad 7369 | Deduction version of ~ nfr... |
nfriota 7370 | A variable not free in a w... |
cbvriota 7371 | Change bound variable in a... |
cbvriotav 7372 | Change bound variable in a... |
csbriota 7373 | Interchange class substitu... |
riotacl2 7374 | Membership law for "the un... |
riotacl 7375 | Closure of restricted iota... |
riotasbc 7376 | Substitution law for descr... |
riotabidva 7377 | Equivalent wff's yield equ... |
riotabiia 7378 | Equivalent wff's yield equ... |
riota1 7379 | Property of restricted iot... |
riota1a 7380 | Property of iota. (Contri... |
riota2df 7381 | A deduction version of ~ r... |
riota2f 7382 | This theorem shows a condi... |
riota2 7383 | This theorem shows a condi... |
riotaeqimp 7384 | If two restricted iota des... |
riotaprop 7385 | Properties of a restricted... |
riota5f 7386 | A method for computing res... |
riota5 7387 | A method for computing res... |
riotass2 7388 | Restriction of a unique el... |
riotass 7389 | Restriction of a unique el... |
moriotass 7390 | Restriction of a unique el... |
snriota 7391 | A restricted class abstrac... |
riotaxfrd 7392 | Change the variable ` x ` ... |
eusvobj2 7393 | Specify the same property ... |
eusvobj1 7394 | Specify the same object in... |
f1ofveu 7395 | There is one domain elemen... |
f1ocnvfv3 7396 | Value of the converse of a... |
riotaund 7397 | Restricted iota equals the... |
riotassuni 7398 | The restricted iota class ... |
riotaclb 7399 | Bidirectional closure of r... |
riotarab 7400 | Restricted iota of a restr... |
oveq 7407 | Equality theorem for opera... |
oveq1 7408 | Equality theorem for opera... |
oveq2 7409 | Equality theorem for opera... |
oveq12 7410 | Equality theorem for opera... |
oveq1i 7411 | Equality inference for ope... |
oveq2i 7412 | Equality inference for ope... |
oveq12i 7413 | Equality inference for ope... |
oveqi 7414 | Equality inference for ope... |
oveq123i 7415 | Equality inference for ope... |
oveq1d 7416 | Equality deduction for ope... |
oveq2d 7417 | Equality deduction for ope... |
oveqd 7418 | Equality deduction for ope... |
oveq12d 7419 | Equality deduction for ope... |
oveqan12d 7420 | Equality deduction for ope... |
oveqan12rd 7421 | Equality deduction for ope... |
oveq123d 7422 | Equality deduction for ope... |
fvoveq1d 7423 | Equality deduction for nes... |
fvoveq1 7424 | Equality theorem for neste... |
ovanraleqv 7425 | Equality theorem for a con... |
imbrov2fvoveq 7426 | Equality theorem for neste... |
ovrspc2v 7427 | If an operation value is e... |
oveqrspc2v 7428 | Restricted specialization ... |
oveqdr 7429 | Equality of two operations... |
nfovd 7430 | Deduction version of bound... |
nfov 7431 | Bound-variable hypothesis ... |
oprabidw 7432 | The law of concretion. Sp... |
oprabid 7433 | The law of concretion. Sp... |
ovex 7434 | The result of an operation... |
ovexi 7435 | The result of an operation... |
ovexd 7436 | The result of an operation... |
ovssunirn 7437 | The result of an operation... |
0ov 7438 | Operation value of the emp... |
ovprc 7439 | The value of an operation ... |
ovprc1 7440 | The value of an operation ... |
ovprc2 7441 | The value of an operation ... |
ovrcl 7442 | Reverse closure for an ope... |
csbov123 7443 | Move class substitution in... |
csbov 7444 | Move class substitution in... |
csbov12g 7445 | Move class substitution in... |
csbov1g 7446 | Move class substitution in... |
csbov2g 7447 | Move class substitution in... |
rspceov 7448 | A frequently used special ... |
elovimad 7449 | Elementhood of the image s... |
fnbrovb 7450 | Value of a binary operatio... |
fnotovb 7451 | Equivalence of operation v... |
opabbrex 7452 | A collection of ordered pa... |
opabresex2 7453 | Restrictions of a collecti... |
opabresex2d 7454 | Obsolete version of ~ opab... |
fvmptopab 7455 | The function value of a ma... |
fvmptopabOLD 7456 | Obsolete version of ~ fvmp... |
f1opr 7457 | Condition for an operation... |
brfvopab 7458 | The classes involved in a ... |
dfoprab2 7459 | Class abstraction for oper... |
reloprab 7460 | An operation class abstrac... |
oprabv 7461 | If a pair and a class are ... |
nfoprab1 7462 | The abstraction variables ... |
nfoprab2 7463 | The abstraction variables ... |
nfoprab3 7464 | The abstraction variables ... |
nfoprab 7465 | Bound-variable hypothesis ... |
oprabbid 7466 | Equivalent wff's yield equ... |
oprabbidv 7467 | Equivalent wff's yield equ... |
oprabbii 7468 | Equivalent wff's yield equ... |
ssoprab2 7469 | Equivalence of ordered pai... |
ssoprab2b 7470 | Equivalence of ordered pai... |
eqoprab2bw 7471 | Equivalence of ordered pai... |
eqoprab2b 7472 | Equivalence of ordered pai... |
mpoeq123 7473 | An equality theorem for th... |
mpoeq12 7474 | An equality theorem for th... |
mpoeq123dva 7475 | An equality deduction for ... |
mpoeq123dv 7476 | An equality deduction for ... |
mpoeq123i 7477 | An equality inference for ... |
mpoeq3dva 7478 | Slightly more general equa... |
mpoeq3ia 7479 | An equality inference for ... |
mpoeq3dv 7480 | An equality deduction for ... |
nfmpo1 7481 | Bound-variable hypothesis ... |
nfmpo2 7482 | Bound-variable hypothesis ... |
nfmpo 7483 | Bound-variable hypothesis ... |
0mpo0 7484 | A mapping operation with e... |
mpo0v 7485 | A mapping operation with e... |
mpo0 7486 | A mapping operation with e... |
oprab4 7487 | Two ways to state the doma... |
cbvoprab1 7488 | Rule used to change first ... |
cbvoprab2 7489 | Change the second bound va... |
cbvoprab12 7490 | Rule used to change first ... |
cbvoprab12v 7491 | Rule used to change first ... |
cbvoprab3 7492 | Rule used to change the th... |
cbvoprab3v 7493 | Rule used to change the th... |
cbvmpox 7494 | Rule to change the bound v... |
cbvmpo 7495 | Rule to change the bound v... |
cbvmpov 7496 | Rule to change the bound v... |
elimdelov 7497 | Eliminate a hypothesis whi... |
ovif 7498 | Move a conditional outside... |
ovif2 7499 | Move a conditional outside... |
ovif12 7500 | Move a conditional outside... |
ifov 7501 | Move a conditional outside... |
dmoprab 7502 | The domain of an operation... |
dmoprabss 7503 | The domain of an operation... |
rnoprab 7504 | The range of an operation ... |
rnoprab2 7505 | The range of a restricted ... |
reldmoprab 7506 | The domain of an operation... |
oprabss 7507 | Structure of an operation ... |
eloprabga 7508 | The law of concretion for ... |
eloprabgaOLD 7509 | Obsolete version of ~ elop... |
eloprabg 7510 | The law of concretion for ... |
ssoprab2i 7511 | Inference of operation cla... |
mpov 7512 | Operation with universal d... |
mpomptx 7513 | Express a two-argument fun... |
mpompt 7514 | Express a two-argument fun... |
mpodifsnif 7515 | A mapping with two argumen... |
mposnif 7516 | A mapping with two argumen... |
fconstmpo 7517 | Representation of a consta... |
resoprab 7518 | Restriction of an operatio... |
resoprab2 7519 | Restriction of an operator... |
resmpo 7520 | Restriction of the mapping... |
funoprabg 7521 | "At most one" is a suffici... |
funoprab 7522 | "At most one" is a suffici... |
fnoprabg 7523 | Functionality and domain o... |
mpofun 7524 | The maps-to notation for a... |
mpofunOLD 7525 | Obsolete version of ~ mpof... |
fnoprab 7526 | Functionality and domain o... |
ffnov 7527 | An operation maps to a cla... |
fovcld 7528 | Closure law for an operati... |
fovcl 7529 | Closure law for an operati... |
eqfnov 7530 | Equality of two operations... |
eqfnov2 7531 | Two operators with the sam... |
fnov 7532 | Representation of a functi... |
mpo2eqb 7533 | Bidirectional equality the... |
rnmpo 7534 | The range of an operation ... |
reldmmpo 7535 | The domain of an operation... |
elrnmpog 7536 | Membership in the range of... |
elrnmpo 7537 | Membership in the range of... |
elrnmpores 7538 | Membership in the range of... |
ralrnmpo 7539 | A restricted quantifier ov... |
rexrnmpo 7540 | A restricted quantifier ov... |
ovid 7541 | The value of an operation ... |
ovidig 7542 | The value of an operation ... |
ovidi 7543 | The value of an operation ... |
ov 7544 | The value of an operation ... |
ovigg 7545 | The value of an operation ... |
ovig 7546 | The value of an operation ... |
ovmpt4g 7547 | Value of a function given ... |
ovmpos 7548 | Value of a function given ... |
ov2gf 7549 | The value of an operation ... |
ovmpodxf 7550 | Value of an operation give... |
ovmpodx 7551 | Value of an operation give... |
ovmpod 7552 | Value of an operation give... |
ovmpox 7553 | The value of an operation ... |
ovmpoga 7554 | Value of an operation give... |
ovmpoa 7555 | Value of an operation give... |
ovmpodf 7556 | Alternate deduction versio... |
ovmpodv 7557 | Alternate deduction versio... |
ovmpodv2 7558 | Alternate deduction versio... |
ovmpog 7559 | Value of an operation give... |
ovmpo 7560 | Value of an operation give... |
ovmpot 7561 | The value of an operation ... |
fvmpopr2d 7562 | Value of an operation give... |
ov3 7563 | The value of an operation ... |
ov6g 7564 | The value of an operation ... |
ovg 7565 | The value of an operation ... |
ovres 7566 | The value of a restricted ... |
ovresd 7567 | Lemma for converting metri... |
oprres 7568 | The restriction of an oper... |
oprssov 7569 | The value of a member of t... |
fovcdm 7570 | An operation's value belon... |
fovcdmda 7571 | An operation's value belon... |
fovcdmd 7572 | An operation's value belon... |
fnrnov 7573 | The range of an operation ... |
foov 7574 | An onto mapping of an oper... |
fnovrn 7575 | An operation's value belon... |
ovelrn 7576 | A member of an operation's... |
funimassov 7577 | Membership relation for th... |
ovelimab 7578 | Operation value in an imag... |
ovima0 7579 | An operation value is a me... |
ovconst2 7580 | The value of a constant op... |
oprssdm 7581 | Domain of closure of an op... |
nssdmovg 7582 | The value of an operation ... |
ndmovg 7583 | The value of an operation ... |
ndmov 7584 | The value of an operation ... |
ndmovcl 7585 | The closure of an operatio... |
ndmovrcl 7586 | Reverse closure law, when ... |
ndmovcom 7587 | Any operation is commutati... |
ndmovass 7588 | Any operation is associati... |
ndmovdistr 7589 | Any operation is distribut... |
ndmovord 7590 | Elimination of redundant a... |
ndmovordi 7591 | Elimination of redundant a... |
caovclg 7592 | Convert an operation closu... |
caovcld 7593 | Convert an operation closu... |
caovcl 7594 | Convert an operation closu... |
caovcomg 7595 | Convert an operation commu... |
caovcomd 7596 | Convert an operation commu... |
caovcom 7597 | Convert an operation commu... |
caovassg 7598 | Convert an operation assoc... |
caovassd 7599 | Convert an operation assoc... |
caovass 7600 | Convert an operation assoc... |
caovcang 7601 | Convert an operation cance... |
caovcand 7602 | Convert an operation cance... |
caovcanrd 7603 | Commute the arguments of a... |
caovcan 7604 | Convert an operation cance... |
caovordig 7605 | Convert an operation order... |
caovordid 7606 | Convert an operation order... |
caovordg 7607 | Convert an operation order... |
caovordd 7608 | Convert an operation order... |
caovord2d 7609 | Operation ordering law wit... |
caovord3d 7610 | Ordering law. (Contribute... |
caovord 7611 | Convert an operation order... |
caovord2 7612 | Operation ordering law wit... |
caovord3 7613 | Ordering law. (Contribute... |
caovdig 7614 | Convert an operation distr... |
caovdid 7615 | Convert an operation distr... |
caovdir2d 7616 | Convert an operation distr... |
caovdirg 7617 | Convert an operation rever... |
caovdird 7618 | Convert an operation distr... |
caovdi 7619 | Convert an operation distr... |
caov32d 7620 | Rearrange arguments in a c... |
caov12d 7621 | Rearrange arguments in a c... |
caov31d 7622 | Rearrange arguments in a c... |
caov13d 7623 | Rearrange arguments in a c... |
caov4d 7624 | Rearrange arguments in a c... |
caov411d 7625 | Rearrange arguments in a c... |
caov42d 7626 | Rearrange arguments in a c... |
caov32 7627 | Rearrange arguments in a c... |
caov12 7628 | Rearrange arguments in a c... |
caov31 7629 | Rearrange arguments in a c... |
caov13 7630 | Rearrange arguments in a c... |
caov4 7631 | Rearrange arguments in a c... |
caov411 7632 | Rearrange arguments in a c... |
caov42 7633 | Rearrange arguments in a c... |
caovdir 7634 | Reverse distributive law. ... |
caovdilem 7635 | Lemma used by real number ... |
caovlem2 7636 | Lemma used in real number ... |
caovmo 7637 | Uniqueness of inverse elem... |
imaeqexov 7638 | Substitute an operation va... |
imaeqalov 7639 | Substitute an operation va... |
mpondm0 7640 | The value of an operation ... |
elmpocl 7641 | If a two-parameter class i... |
elmpocl1 7642 | If a two-parameter class i... |
elmpocl2 7643 | If a two-parameter class i... |
elovmpo 7644 | Utility lemma for two-para... |
elovmporab 7645 | Implications for the value... |
elovmporab1w 7646 | Implications for the value... |
elovmporab1 7647 | Implications for the value... |
2mpo0 7648 | If the operation value of ... |
relmptopab 7649 | Any function to sets of or... |
f1ocnvd 7650 | Describe an implicit one-t... |
f1od 7651 | Describe an implicit one-t... |
f1ocnv2d 7652 | Describe an implicit one-t... |
f1o2d 7653 | Describe an implicit one-t... |
f1opw2 7654 | A one-to-one mapping induc... |
f1opw 7655 | A one-to-one mapping induc... |
elovmpt3imp 7656 | If the value of a function... |
ovmpt3rab1 7657 | The value of an operation ... |
ovmpt3rabdm 7658 | If the value of a function... |
elovmpt3rab1 7659 | Implications for the value... |
elovmpt3rab 7660 | Implications for the value... |
ofeqd 7665 | Equality theorem for funct... |
ofeq 7666 | Equality theorem for funct... |
ofreq 7667 | Equality theorem for funct... |
ofexg 7668 | A function operation restr... |
nfof 7669 | Hypothesis builder for fun... |
nfofr 7670 | Hypothesis builder for fun... |
ofrfvalg 7671 | Value of a relation applie... |
offval 7672 | Value of an operation appl... |
ofrfval 7673 | Value of a relation applie... |
ofval 7674 | Evaluate a function operat... |
ofrval 7675 | Exhibit a function relatio... |
offn 7676 | The function operation pro... |
offun 7677 | The function operation pro... |
offval2f 7678 | The function operation exp... |
ofmresval 7679 | Value of a restriction of ... |
fnfvof 7680 | Function value of a pointw... |
off 7681 | The function operation pro... |
ofres 7682 | Restrict the operands of a... |
offval2 7683 | The function operation exp... |
ofrfval2 7684 | The function relation acti... |
ofmpteq 7685 | Value of a pointwise opera... |
ofco 7686 | The composition of a funct... |
offveq 7687 | Convert an identity of the... |
offveqb 7688 | Equivalent expressions for... |
ofc1 7689 | Left operation by a consta... |
ofc2 7690 | Right operation by a const... |
ofc12 7691 | Function operation on two ... |
caofref 7692 | Transfer a reflexive law t... |
caofinvl 7693 | Transfer a left inverse la... |
caofid0l 7694 | Transfer a left identity l... |
caofid0r 7695 | Transfer a right identity ... |
caofid1 7696 | Transfer a right absorptio... |
caofid2 7697 | Transfer a right absorptio... |
caofcom 7698 | Transfer a commutative law... |
caofrss 7699 | Transfer a relation subset... |
caofass 7700 | Transfer an associative la... |
caoftrn 7701 | Transfer a transitivity la... |
caofdi 7702 | Transfer a distributive la... |
caofdir 7703 | Transfer a reverse distrib... |
caonncan 7704 | Transfer ~ nncan -shaped l... |
relrpss 7707 | The proper subset relation... |
brrpssg 7708 | The proper subset relation... |
brrpss 7709 | The proper subset relation... |
porpss 7710 | Every class is partially o... |
sorpss 7711 | Express strict ordering un... |
sorpssi 7712 | Property of a chain of set... |
sorpssun 7713 | A chain of sets is closed ... |
sorpssin 7714 | A chain of sets is closed ... |
sorpssuni 7715 | In a chain of sets, a maxi... |
sorpssint 7716 | In a chain of sets, a mini... |
sorpsscmpl 7717 | The componentwise compleme... |
zfun 7719 | Axiom of Union expressed w... |
axun2 7720 | A variant of the Axiom of ... |
uniex2 7721 | The Axiom of Union using t... |
vuniex 7722 | The union of a setvar is a... |
uniexg 7723 | The ZF Axiom of Union in c... |
uniex 7724 | The Axiom of Union in clas... |
uniexd 7725 | Deduction version of the Z... |
unex 7726 | The union of two sets is a... |
tpex 7727 | An unordered triple of cla... |
unexb 7728 | Existence of union is equi... |
unexg 7729 | A union of two sets is a s... |
xpexg 7730 | The Cartesian product of t... |
xpexd 7731 | The Cartesian product of t... |
3xpexg 7732 | The Cartesian product of t... |
xpex 7733 | The Cartesian product of t... |
unexd 7734 | The union of two sets is a... |
sqxpexg 7735 | The Cartesian square of a ... |
abnexg 7736 | Sufficient condition for a... |
abnex 7737 | Sufficient condition for a... |
snnex 7738 | The class of all singleton... |
pwnex 7739 | The class of all power set... |
difex2 7740 | If the subtrahend of a cla... |
difsnexi 7741 | If the difference of a cla... |
uniuni 7742 | Expression for double unio... |
uniexr 7743 | Converse of the Axiom of U... |
uniexb 7744 | The Axiom of Union and its... |
pwexr 7745 | Converse of the Axiom of P... |
pwexb 7746 | The Axiom of Power Sets an... |
elpwpwel 7747 | A class belongs to a doubl... |
eldifpw 7748 | Membership in a power clas... |
elpwun 7749 | Membership in the power cl... |
pwuncl 7750 | Power classes are closed u... |
iunpw 7751 | An indexed union of a powe... |
fr3nr 7752 | A well-founded relation ha... |
epne3 7753 | A well-founded class conta... |
dfwe2 7754 | Alternate definition of we... |
epweon 7755 | The membership relation we... |
epweonALT 7756 | Alternate proof of ~ epweo... |
ordon 7757 | The class of all ordinal n... |
onprc 7758 | No set contains all ordina... |
ssorduni 7759 | The union of a class of or... |
ssonuni 7760 | The union of a set of ordi... |
ssonunii 7761 | The union of a set of ordi... |
ordeleqon 7762 | A way to express the ordin... |
ordsson 7763 | Any ordinal class is a sub... |
dford5 7764 | A class is ordinal iff it ... |
onss 7765 | An ordinal number is a sub... |
predon 7766 | The predecessor of an ordi... |
predonOLD 7767 | Obsolete version of ~ pred... |
ssonprc 7768 | Two ways of saying a class... |
onuni 7769 | The union of an ordinal nu... |
orduni 7770 | The union of an ordinal cl... |
onint 7771 | The intersection (infimum)... |
onint0 7772 | The intersection of a clas... |
onssmin 7773 | A nonempty class of ordina... |
onminesb 7774 | If a property is true for ... |
onminsb 7775 | If a property is true for ... |
oninton 7776 | The intersection of a none... |
onintrab 7777 | The intersection of a clas... |
onintrab2 7778 | An existence condition equ... |
onnmin 7779 | No member of a set of ordi... |
onnminsb 7780 | An ordinal number smaller ... |
oneqmin 7781 | A way to show that an ordi... |
uniordint 7782 | The union of a set of ordi... |
onminex 7783 | If a wff is true for an or... |
sucon 7784 | The class of all ordinal n... |
sucexb 7785 | A successor exists iff its... |
sucexg 7786 | The successor of a set is ... |
sucex 7787 | The successor of a set is ... |
onmindif2 7788 | The minimum of a class of ... |
ordsuci 7789 | The successor of an ordina... |
sucexeloni 7790 | If the successor of an ord... |
sucexeloniOLD 7791 | Obsolete version of ~ suce... |
onsuc 7792 | The successor of an ordina... |
suceloniOLD 7793 | Obsolete version of ~ onsu... |
ordsuc 7794 | A class is ordinal if and ... |
ordsucOLD 7795 | Obsolete version of ~ ords... |
ordpwsuc 7796 | The collection of ordinals... |
onpwsuc 7797 | The collection of ordinal ... |
onsucb 7798 | A class is an ordinal numb... |
ordsucss 7799 | The successor of an elemen... |
onpsssuc 7800 | An ordinal number is a pro... |
ordelsuc 7801 | A set belongs to an ordina... |
onsucmin 7802 | The successor of an ordina... |
ordsucelsuc 7803 | Membership is inherited by... |
ordsucsssuc 7804 | The subclass relationship ... |
ordsucuniel 7805 | Given an element ` A ` of ... |
ordsucun 7806 | The successor of the maxim... |
ordunpr 7807 | The maximum of two ordinal... |
ordunel 7808 | The maximum of two ordinal... |
onsucuni 7809 | A class of ordinal numbers... |
ordsucuni 7810 | An ordinal class is a subc... |
orduniorsuc 7811 | An ordinal class is either... |
unon 7812 | The class of all ordinal n... |
ordunisuc 7813 | An ordinal class is equal ... |
orduniss2 7814 | The union of the ordinal s... |
onsucuni2 7815 | A successor ordinal is the... |
0elsuc 7816 | The successor of an ordina... |
limon 7817 | The class of ordinal numbe... |
onuniorsuc 7818 | An ordinal number is eithe... |
onssi 7819 | An ordinal number is a sub... |
onsuci 7820 | The successor of an ordina... |
onuniorsuciOLD 7821 | Obsolete version of ~ onun... |
onuninsuci 7822 | An ordinal is equal to its... |
onsucssi 7823 | A set belongs to an ordina... |
nlimsucg 7824 | A successor is not a limit... |
orduninsuc 7825 | An ordinal class is equal ... |
ordunisuc2 7826 | An ordinal equal to its un... |
ordzsl 7827 | An ordinal is zero, a succ... |
onzsl 7828 | An ordinal number is zero,... |
dflim3 7829 | An alternate definition of... |
dflim4 7830 | An alternate definition of... |
limsuc 7831 | The successor of a member ... |
limsssuc 7832 | A class includes a limit o... |
nlimon 7833 | Two ways to express the cl... |
limuni3 7834 | The union of a nonempty cl... |
tfi 7835 | The Principle of Transfini... |
tfisg 7836 | A closed form of ~ tfis . ... |
tfis 7837 | Transfinite Induction Sche... |
tfis2f 7838 | Transfinite Induction Sche... |
tfis2 7839 | Transfinite Induction Sche... |
tfis3 7840 | Transfinite Induction Sche... |
tfisi 7841 | A transfinite induction sc... |
tfinds 7842 | Principle of Transfinite I... |
tfindsg 7843 | Transfinite Induction (inf... |
tfindsg2 7844 | Transfinite Induction (inf... |
tfindes 7845 | Transfinite Induction with... |
tfinds2 7846 | Transfinite Induction (inf... |
tfinds3 7847 | Principle of Transfinite I... |
dfom2 7850 | An alternate definition of... |
elom 7851 | Membership in omega. The ... |
omsson 7852 | Omega is a subset of ` On ... |
limomss 7853 | The class of natural numbe... |
nnon 7854 | A natural number is an ord... |
nnoni 7855 | A natural number is an ord... |
nnord 7856 | A natural number is ordina... |
trom 7857 | The class of finite ordina... |
ordom 7858 | The class of finite ordina... |
elnn 7859 | A member of a natural numb... |
omon 7860 | The class of natural numbe... |
omelon2 7861 | Omega is an ordinal number... |
nnlim 7862 | A natural number is not a ... |
omssnlim 7863 | The class of natural numbe... |
limom 7864 | Omega is a limit ordinal. ... |
peano2b 7865 | A class belongs to omega i... |
nnsuc 7866 | A nonzero natural number i... |
omsucne 7867 | A natural number is not th... |
ssnlim 7868 | An ordinal subclass of non... |
omsinds 7869 | Strong (or "total") induct... |
omsindsOLD 7870 | Obsolete version of ~ omsi... |
omun 7871 | The union of two finite or... |
peano1 7872 | Zero is a natural number. ... |
peano1OLD 7873 | Obsolete version of ~ pean... |
peano2 7874 | The successor of any natur... |
peano3 7875 | The successor of any natur... |
peano4 7876 | Two natural numbers are eq... |
peano5 7877 | The induction postulate: a... |
peano5OLD 7878 | Obsolete version of ~ pean... |
nn0suc 7879 | A natural number is either... |
find 7880 | The Principle of Finite In... |
findOLD 7881 | Obsolete version of ~ find... |
finds 7882 | Principle of Finite Induct... |
findsg 7883 | Principle of Finite Induct... |
finds2 7884 | Principle of Finite Induct... |
finds1 7885 | Principle of Finite Induct... |
findes 7886 | Finite induction with expl... |
dmexg 7887 | The domain of a set is a s... |
rnexg 7888 | The range of a set is a se... |
dmexd 7889 | The domain of a set is a s... |
fndmexd 7890 | If a function is a set, it... |
dmfex 7891 | If a mapping is a set, its... |
fndmexb 7892 | The domain of a function i... |
fdmexb 7893 | The domain of a function i... |
dmfexALT 7894 | Alternate proof of ~ dmfex... |
dmex 7895 | The domain of a set is a s... |
rnex 7896 | The range of a set is a se... |
iprc 7897 | The identity function is a... |
resiexg 7898 | The existence of a restric... |
imaexg 7899 | The image of a set is a se... |
imaex 7900 | The image of a set is a se... |
exse2 7901 | Any set relation is set-li... |
xpexr 7902 | If a Cartesian product is ... |
xpexr2 7903 | If a nonempty Cartesian pr... |
xpexcnv 7904 | A condition where the conv... |
soex 7905 | If the relation in a stric... |
elxp4 7906 | Membership in a Cartesian ... |
elxp5 7907 | Membership in a Cartesian ... |
cnvexg 7908 | The converse of a set is a... |
cnvex 7909 | The converse of a set is a... |
relcnvexb 7910 | A relation is a set iff it... |
f1oexrnex 7911 | If the range of a 1-1 onto... |
f1oexbi 7912 | There is a one-to-one onto... |
coexg 7913 | The composition of two set... |
coex 7914 | The composition of two set... |
funcnvuni 7915 | The union of a chain (with... |
fun11uni 7916 | The union of a chain (with... |
fex2 7917 | A function with bounded do... |
fabexg 7918 | Existence of a set of func... |
fabex 7919 | Existence of a set of func... |
f1oabexg 7920 | The class of all 1-1-onto ... |
fiunlem 7921 | Lemma for ~ fiun and ~ f1i... |
fiun 7922 | The union of a chain (with... |
f1iun 7923 | The union of a chain (with... |
fviunfun 7924 | The function value of an i... |
ffoss 7925 | Relationship between a map... |
f11o 7926 | Relationship between one-t... |
resfunexgALT 7927 | Alternate proof of ~ resfu... |
cofunexg 7928 | Existence of a composition... |
cofunex2g 7929 | Existence of a composition... |
fnexALT 7930 | Alternate proof of ~ fnex ... |
funexw 7931 | Weak version of ~ funex th... |
mptexw 7932 | Weak version of ~ mptex th... |
funrnex 7933 | If the domain of a functio... |
zfrep6 7934 | A version of the Axiom of ... |
focdmex 7935 | If the domain of an onto f... |
f1dmex 7936 | If the codomain of a one-t... |
f1ovv 7937 | The codomain/range of a 1-... |
fvclex 7938 | Existence of the class of ... |
fvresex 7939 | Existence of the class of ... |
abrexexg 7940 | Existence of a class abstr... |
abrexexgOLD 7941 | Obsolete version of ~ abre... |
abrexex 7942 | Existence of a class abstr... |
iunexg 7943 | The existence of an indexe... |
abrexex2g 7944 | Existence of an existentia... |
opabex3d 7945 | Existence of an ordered pa... |
opabex3rd 7946 | Existence of an ordered pa... |
opabex3 7947 | Existence of an ordered pa... |
iunex 7948 | The existence of an indexe... |
abrexex2 7949 | Existence of an existentia... |
abexssex 7950 | Existence of a class abstr... |
abexex 7951 | A condition where a class ... |
f1oweALT 7952 | Alternate proof of ~ f1owe... |
wemoiso 7953 | Thus, there is at most one... |
wemoiso2 7954 | Thus, there is at most one... |
oprabexd 7955 | Existence of an operator a... |
oprabex 7956 | Existence of an operation ... |
oprabex3 7957 | Existence of an operation ... |
oprabrexex2 7958 | Existence of an existentia... |
ab2rexex 7959 | Existence of a class abstr... |
ab2rexex2 7960 | Existence of an existentia... |
xpexgALT 7961 | Alternate proof of ~ xpexg... |
offval3 7962 | General value of ` ( F oF ... |
offres 7963 | Pointwise combination comm... |
ofmres 7964 | Equivalent expressions for... |
ofmresex 7965 | Existence of a restriction... |
1stval 7970 | The value of the function ... |
2ndval 7971 | The value of the function ... |
1stnpr 7972 | Value of the first-member ... |
2ndnpr 7973 | Value of the second-member... |
1st0 7974 | The value of the first-mem... |
2nd0 7975 | The value of the second-me... |
op1st 7976 | Extract the first member o... |
op2nd 7977 | Extract the second member ... |
op1std 7978 | Extract the first member o... |
op2ndd 7979 | Extract the second member ... |
op1stg 7980 | Extract the first member o... |
op2ndg 7981 | Extract the second member ... |
ot1stg 7982 | Extract the first member o... |
ot2ndg 7983 | Extract the second member ... |
ot3rdg 7984 | Extract the third member o... |
1stval2 7985 | Alternate value of the fun... |
2ndval2 7986 | Alternate value of the fun... |
oteqimp 7987 | The components of an order... |
fo1st 7988 | The ` 1st ` function maps ... |
fo2nd 7989 | The ` 2nd ` function maps ... |
br1steqg 7990 | Uniqueness condition for t... |
br2ndeqg 7991 | Uniqueness condition for t... |
f1stres 7992 | Mapping of a restriction o... |
f2ndres 7993 | Mapping of a restriction o... |
fo1stres 7994 | Onto mapping of a restrict... |
fo2ndres 7995 | Onto mapping of a restrict... |
1st2val 7996 | Value of an alternate defi... |
2nd2val 7997 | Value of an alternate defi... |
1stcof 7998 | Composition of the first m... |
2ndcof 7999 | Composition of the second ... |
xp1st 8000 | Location of the first elem... |
xp2nd 8001 | Location of the second ele... |
elxp6 8002 | Membership in a Cartesian ... |
elxp7 8003 | Membership in a Cartesian ... |
eqopi 8004 | Equality with an ordered p... |
xp2 8005 | Representation of Cartesia... |
unielxp 8006 | The membership relation fo... |
1st2nd2 8007 | Reconstruction of a member... |
1st2ndb 8008 | Reconstruction of an order... |
xpopth 8009 | An ordered pair theorem fo... |
eqop 8010 | Two ways to express equali... |
eqop2 8011 | Two ways to express equali... |
op1steq 8012 | Two ways of expressing tha... |
opreuopreu 8013 | There is a unique ordered ... |
el2xptp 8014 | A member of a nested Carte... |
el2xptp0 8015 | A member of a nested Carte... |
el2xpss 8016 | Version of ~ elrel for tri... |
2nd1st 8017 | Swap the members of an ord... |
1st2nd 8018 | Reconstruction of a member... |
1stdm 8019 | The first ordered pair com... |
2ndrn 8020 | The second ordered pair co... |
1st2ndbr 8021 | Express an element of a re... |
releldm2 8022 | Two ways of expressing mem... |
reldm 8023 | An expression for the doma... |
releldmdifi 8024 | One way of expressing memb... |
funfv1st2nd 8025 | The function value for the... |
funelss 8026 | If the first component of ... |
funeldmdif 8027 | Two ways of expressing mem... |
sbcopeq1a 8028 | Equality theorem for subst... |
csbopeq1a 8029 | Equality theorem for subst... |
sbcoteq1a 8030 | Equality theorem for subst... |
dfopab2 8031 | A way to define an ordered... |
dfoprab3s 8032 | A way to define an operati... |
dfoprab3 8033 | Operation class abstractio... |
dfoprab4 8034 | Operation class abstractio... |
dfoprab4f 8035 | Operation class abstractio... |
opabex2 8036 | Condition for an operation... |
opabn1stprc 8037 | An ordered-pair class abst... |
opiota 8038 | The property of a uniquely... |
cnvoprab 8039 | The converse of a class ab... |
dfxp3 8040 | Define the Cartesian produ... |
elopabi 8041 | A consequence of membershi... |
eloprabi 8042 | A consequence of membershi... |
mpomptsx 8043 | Express a two-argument fun... |
mpompts 8044 | Express a two-argument fun... |
dmmpossx 8045 | The domain of a mapping is... |
fmpox 8046 | Functionality, domain and ... |
fmpo 8047 | Functionality, domain and ... |
fnmpo 8048 | Functionality and domain o... |
fnmpoi 8049 | Functionality and domain o... |
dmmpo 8050 | Domain of a class given by... |
ovmpoelrn 8051 | An operation's value belon... |
dmmpoga 8052 | Domain of an operation giv... |
dmmpogaOLD 8053 | Obsolete version of ~ dmmp... |
dmmpog 8054 | Domain of an operation giv... |
mpoexxg 8055 | Existence of an operation ... |
mpoexg 8056 | Existence of an operation ... |
mpoexga 8057 | If the domain of an operat... |
mpoexw 8058 | Weak version of ~ mpoex th... |
mpoex 8059 | If the domain of an operat... |
mptmpoopabbrd 8060 | The operation value of a f... |
mptmpoopabbrdOLD 8061 | Obsolete version of ~ mptm... |
mptmpoopabovd 8062 | The operation value of a f... |
mptmpoopabbrdOLDOLD 8063 | Obsolete version of ~ mptm... |
mptmpoopabovdOLD 8064 | Obsolete version of ~ mptm... |
el2mpocsbcl 8065 | If the operation value of ... |
el2mpocl 8066 | If the operation value of ... |
fnmpoovd 8067 | A function with a Cartesia... |
offval22 8068 | The function operation exp... |
brovpreldm 8069 | If a binary relation holds... |
bropopvvv 8070 | If a binary relation holds... |
bropfvvvvlem 8071 | Lemma for ~ bropfvvvv . (... |
bropfvvvv 8072 | If a binary relation holds... |
ovmptss 8073 | If all the values of the m... |
relmpoopab 8074 | Any function to sets of or... |
fmpoco 8075 | Composition of two functio... |
oprabco 8076 | Composition of a function ... |
oprab2co 8077 | Composition of operator ab... |
df1st2 8078 | An alternate possible defi... |
df2nd2 8079 | An alternate possible defi... |
1stconst 8080 | The mapping of a restricti... |
2ndconst 8081 | The mapping of a restricti... |
dfmpo 8082 | Alternate definition for t... |
mposn 8083 | An operation (in maps-to n... |
curry1 8084 | Composition with ` ``' ( 2... |
curry1val 8085 | The value of a curried fun... |
curry1f 8086 | Functionality of a curried... |
curry2 8087 | Composition with ` ``' ( 1... |
curry2f 8088 | Functionality of a curried... |
curry2val 8089 | The value of a curried fun... |
cnvf1olem 8090 | Lemma for ~ cnvf1o . (Con... |
cnvf1o 8091 | Describe a function that m... |
fparlem1 8092 | Lemma for ~ fpar . (Contr... |
fparlem2 8093 | Lemma for ~ fpar . (Contr... |
fparlem3 8094 | Lemma for ~ fpar . (Contr... |
fparlem4 8095 | Lemma for ~ fpar . (Contr... |
fpar 8096 | Merge two functions in par... |
fsplit 8097 | A function that can be use... |
fsplitfpar 8098 | Merge two functions with a... |
offsplitfpar 8099 | Express the function opera... |
f2ndf 8100 | The ` 2nd ` (second compon... |
fo2ndf 8101 | The ` 2nd ` (second compon... |
f1o2ndf1 8102 | The ` 2nd ` (second compon... |
opco1 8103 | Value of an operation prec... |
opco2 8104 | Value of an operation prec... |
opco1i 8105 | Inference form of ~ opco1 ... |
frxp 8106 | A lexicographical ordering... |
xporderlem 8107 | Lemma for lexicographical ... |
poxp 8108 | A lexicographical ordering... |
soxp 8109 | A lexicographical ordering... |
wexp 8110 | A lexicographical ordering... |
fnwelem 8111 | Lemma for ~ fnwe . (Contr... |
fnwe 8112 | A variant on lexicographic... |
fnse 8113 | Condition for the well-ord... |
fvproj 8114 | Value of a function on ord... |
fimaproj 8115 | Image of a cartesian produ... |
ralxpes 8116 | A version of ~ ralxp with ... |
ralxp3f 8117 | Restricted for all over a ... |
ralxp3 8118 | Restricted for all over a ... |
ralxp3es 8119 | Restricted for-all over a ... |
frpoins3xpg 8120 | Special case of founded pa... |
frpoins3xp3g 8121 | Special case of founded pa... |
xpord2lem 8122 | Lemma for Cartesian produc... |
poxp2 8123 | Another way of partially o... |
frxp2 8124 | Another way of giving a we... |
xpord2pred 8125 | Calculate the predecessor ... |
sexp2 8126 | Condition for the relation... |
xpord2indlem 8127 | Induction over the Cartesi... |
xpord2ind 8128 | Induction over the Cartesi... |
xpord3lem 8129 | Lemma for triple ordering.... |
poxp3 8130 | Triple Cartesian product p... |
frxp3 8131 | Give well-foundedness over... |
xpord3pred 8132 | Calculate the predecsessor... |
sexp3 8133 | Show that the triple order... |
xpord3inddlem 8134 | Induction over the triple ... |
xpord3indd 8135 | Induction over the triple ... |
xpord3ind 8136 | Induction over the triple ... |
orderseqlem 8137 | Lemma for ~ poseq and ~ so... |
poseq 8138 | A partial ordering of ordi... |
soseq 8139 | A linear ordering of ordin... |
suppval 8142 | The value of the operation... |
supp0prc 8143 | The support of a class is ... |
suppvalbr 8144 | The value of the operation... |
supp0 8145 | The support of the empty s... |
suppval1 8146 | The value of the operation... |
suppvalfng 8147 | The value of the operation... |
suppvalfn 8148 | The value of the operation... |
elsuppfng 8149 | An element of the support ... |
elsuppfn 8150 | An element of the support ... |
cnvimadfsn 8151 | The support of functions "... |
suppimacnvss 8152 | The support of functions "... |
suppimacnv 8153 | Support sets of functions ... |
fsuppeq 8154 | Two ways of writing the su... |
fsuppeqg 8155 | Version of ~ fsuppeq avoid... |
suppssdm 8156 | The support of a function ... |
suppsnop 8157 | The support of a singleton... |
snopsuppss 8158 | The support of a singleton... |
fvn0elsupp 8159 | If the function value for ... |
fvn0elsuppb 8160 | The function value for a g... |
rexsupp 8161 | Existential quantification... |
ressuppss 8162 | The support of the restric... |
suppun 8163 | The support of a class/fun... |
ressuppssdif 8164 | The support of the restric... |
mptsuppdifd 8165 | The support of a function ... |
mptsuppd 8166 | The support of a function ... |
extmptsuppeq 8167 | The support of an extended... |
suppfnss 8168 | The support of a function ... |
funsssuppss 8169 | The support of a function ... |
fnsuppres 8170 | Two ways to express restri... |
fnsuppeq0 8171 | The support of a function ... |
fczsupp0 8172 | The support of a constant ... |
suppss 8173 | Show that the support of a... |
suppssOLD 8174 | Obsolete version of ~ supp... |
suppssr 8175 | A function is zero outside... |
suppssrg 8176 | A function is zero outside... |
suppssov1 8177 | Formula building theorem f... |
suppssov2 8178 | Formula building theorem f... |
suppssof1 8179 | Formula building theorem f... |
suppss2 8180 | Show that the support of a... |
suppsssn 8181 | Show that the support of a... |
suppssfv 8182 | Formula building theorem f... |
suppofssd 8183 | Condition for the support ... |
suppofss1d 8184 | Condition for the support ... |
suppofss2d 8185 | Condition for the support ... |
suppco 8186 | The support of the composi... |
suppcoss 8187 | The support of the composi... |
supp0cosupp0 8188 | The support of the composi... |
imacosupp 8189 | The image of the support o... |
opeliunxp2f 8190 | Membership in a union of C... |
mpoxeldm 8191 | If there is an element of ... |
mpoxneldm 8192 | If the first argument of a... |
mpoxopn0yelv 8193 | If there is an element of ... |
mpoxopynvov0g 8194 | If the second argument of ... |
mpoxopxnop0 8195 | If the first argument of a... |
mpoxopx0ov0 8196 | If the first argument of a... |
mpoxopxprcov0 8197 | If the components of the f... |
mpoxopynvov0 8198 | If the second argument of ... |
mpoxopoveq 8199 | Value of an operation give... |
mpoxopovel 8200 | Element of the value of an... |
mpoxopoveqd 8201 | Value of an operation give... |
brovex 8202 | A binary relation of the v... |
brovmpoex 8203 | A binary relation of the v... |
sprmpod 8204 | The extension of a binary ... |
tposss 8207 | Subset theorem for transpo... |
tposeq 8208 | Equality theorem for trans... |
tposeqd 8209 | Equality theorem for trans... |
tposssxp 8210 | The transposition is a sub... |
reltpos 8211 | The transposition is a rel... |
brtpos2 8212 | Value of the transposition... |
brtpos0 8213 | The behavior of ` tpos ` w... |
reldmtpos 8214 | Necessary and sufficient c... |
brtpos 8215 | The transposition swaps ar... |
ottpos 8216 | The transposition swaps th... |
relbrtpos 8217 | The transposition swaps ar... |
dmtpos 8218 | The domain of ` tpos F ` w... |
rntpos 8219 | The range of ` tpos F ` wh... |
tposexg 8220 | The transposition of a set... |
ovtpos 8221 | The transposition swaps th... |
tposfun 8222 | The transposition of a fun... |
dftpos2 8223 | Alternate definition of ` ... |
dftpos3 8224 | Alternate definition of ` ... |
dftpos4 8225 | Alternate definition of ` ... |
tpostpos 8226 | Value of the double transp... |
tpostpos2 8227 | Value of the double transp... |
tposfn2 8228 | The domain of a transposit... |
tposfo2 8229 | Condition for a surjective... |
tposf2 8230 | The domain and codomain of... |
tposf12 8231 | Condition for an injective... |
tposf1o2 8232 | Condition of a bijective t... |
tposfo 8233 | The domain and codomain/ra... |
tposf 8234 | The domain and codomain of... |
tposfn 8235 | Functionality of a transpo... |
tpos0 8236 | Transposition of the empty... |
tposco 8237 | Transposition of a composi... |
tpossym 8238 | Two ways to say a function... |
tposeqi 8239 | Equality theorem for trans... |
tposex 8240 | A transposition is a set. ... |
nftpos 8241 | Hypothesis builder for tra... |
tposoprab 8242 | Transposition of a class o... |
tposmpo 8243 | Transposition of a two-arg... |
tposconst 8244 | The transposition of a con... |
mpocurryd 8249 | The currying of an operati... |
mpocurryvald 8250 | The value of a curried ope... |
fvmpocurryd 8251 | The value of the value of ... |
pwuninel2 8254 | Direct proof of ~ pwuninel... |
pwuninel 8255 | The power set of the union... |
undefval 8256 | Value of the undefined val... |
undefnel2 8257 | The undefined value genera... |
undefnel 8258 | The undefined value genera... |
undefne0 8259 | The undefined value genera... |
frecseq123 8262 | Equality theorem for the w... |
nffrecs 8263 | Bound-variable hypothesis ... |
csbfrecsg 8264 | Move class substitution in... |
fpr3g 8265 | Functions defined by well-... |
frrlem1 8266 | Lemma for well-founded rec... |
frrlem2 8267 | Lemma for well-founded rec... |
frrlem3 8268 | Lemma for well-founded rec... |
frrlem4 8269 | Lemma for well-founded rec... |
frrlem5 8270 | Lemma for well-founded rec... |
frrlem6 8271 | Lemma for well-founded rec... |
frrlem7 8272 | Lemma for well-founded rec... |
frrlem8 8273 | Lemma for well-founded rec... |
frrlem9 8274 | Lemma for well-founded rec... |
frrlem10 8275 | Lemma for well-founded rec... |
frrlem11 8276 | Lemma for well-founded rec... |
frrlem12 8277 | Lemma for well-founded rec... |
frrlem13 8278 | Lemma for well-founded rec... |
frrlem14 8279 | Lemma for well-founded rec... |
fprlem1 8280 | Lemma for well-founded rec... |
fprlem2 8281 | Lemma for well-founded rec... |
fpr2a 8282 | Weak version of ~ fpr2 whi... |
fpr1 8283 | Law of well-founded recurs... |
fpr2 8284 | Law of well-founded recurs... |
fpr3 8285 | Law of well-founded recurs... |
frrrel 8286 | Show without using the axi... |
frrdmss 8287 | Show without using the axi... |
frrdmcl 8288 | Show without using the axi... |
fprfung 8289 | A "function" defined by we... |
fprresex 8290 | The restriction of a funct... |
dfwrecsOLD 8293 | Obsolete definition of the... |
wrecseq123 8294 | General equality theorem f... |
wrecseq123OLD 8295 | Obsolete version of ~ wrec... |
nfwrecs 8296 | Bound-variable hypothesis ... |
nfwrecsOLD 8297 | Obsolete proof of ~ nfwrec... |
wrecseq1 8298 | Equality theorem for the w... |
wrecseq2 8299 | Equality theorem for the w... |
wrecseq3 8300 | Equality theorem for the w... |
csbwrecsg 8301 | Move class substitution in... |
wfr3g 8302 | Functions defined by well-... |
wfrlem1OLD 8303 | Lemma for well-ordered rec... |
wfrlem2OLD 8304 | Lemma for well-ordered rec... |
wfrlem3OLD 8305 | Lemma for well-ordered rec... |
wfrlem3OLDa 8306 | Lemma for well-ordered rec... |
wfrlem4OLD 8307 | Lemma for well-ordered rec... |
wfrlem5OLD 8308 | Lemma for well-ordered rec... |
wfrrelOLD 8309 | Obsolete proof of ~ wfrrel... |
wfrdmssOLD 8310 | Obsolete proof of ~ wfrdms... |
wfrlem8OLD 8311 | Lemma for well-ordered rec... |
wfrdmclOLD 8312 | Obsolete version of ~ wfrd... |
wfrlem10OLD 8313 | Lemma for well-ordered rec... |
wfrfunOLD 8314 | Obsolete proof of ~ wfrfun... |
wfrlem12OLD 8315 | Lemma for well-ordered rec... |
wfrlem13OLD 8316 | Lemma for well-ordered rec... |
wfrlem14OLD 8317 | Lemma for well-ordered rec... |
wfrlem15OLD 8318 | Lemma for well-ordered rec... |
wfrlem16OLD 8319 | Lemma for well-ordered rec... |
wfrlem17OLD 8320 | Without using ~ ax-rep , s... |
wfr2aOLD 8321 | Obsolete version of ~ wfr2... |
wfr1OLD 8322 | Obsolete version of ~ wfr1... |
wfr2OLD 8323 | Obsolete version of ~ wfr2... |
wfrrel 8324 | The well-ordered recursion... |
wfrdmss 8325 | The domain of the well-ord... |
wfrdmcl 8326 | The predecessor class of a... |
wfrfun 8327 | The "function" generated b... |
wfrresex 8328 | Show without using the axi... |
wfr2a 8329 | A weak version of ~ wfr2 w... |
wfr1 8330 | The Principle of Well-Orde... |
wfr2 8331 | The Principle of Well-Orde... |
wfr3 8332 | The principle of Well-Orde... |
wfr3OLD 8333 | Obsolete form of ~ wfr3 as... |
iunon 8334 | The indexed union of a set... |
iinon 8335 | The nonempty indexed inter... |
onfununi 8336 | A property of functions on... |
onovuni 8337 | A variant of ~ onfununi fo... |
onoviun 8338 | A variant of ~ onovuni wit... |
onnseq 8339 | There are no length ` _om ... |
dfsmo2 8342 | Alternate definition of a ... |
issmo 8343 | Conditions for which ` A `... |
issmo2 8344 | Alternate definition of a ... |
smoeq 8345 | Equality theorem for stric... |
smodm 8346 | The domain of a strictly m... |
smores 8347 | A strictly monotone functi... |
smores3 8348 | A strictly monotone functi... |
smores2 8349 | A strictly monotone ordina... |
smodm2 8350 | The domain of a strictly m... |
smofvon2 8351 | The function values of a s... |
iordsmo 8352 | The identity relation rest... |
smo0 8353 | The null set is a strictly... |
smofvon 8354 | If ` B ` is a strictly mon... |
smoel 8355 | If ` x ` is less than ` y ... |
smoiun 8356 | The value of a strictly mo... |
smoiso 8357 | If ` F ` is an isomorphism... |
smoel2 8358 | A strictly monotone ordina... |
smo11 8359 | A strictly monotone ordina... |
smoord 8360 | A strictly monotone ordina... |
smoword 8361 | A strictly monotone ordina... |
smogt 8362 | A strictly monotone ordina... |
smocdmdom 8363 | The codomain of a strictly... |
smoiso2 8364 | The strictly monotone ordi... |
dfrecs3 8367 | The old definition of tran... |
dfrecs3OLD 8368 | Obsolete version of ~ dfre... |
recseq 8369 | Equality theorem for ` rec... |
nfrecs 8370 | Bound-variable hypothesis ... |
tfrlem1 8371 | A technical lemma for tran... |
tfrlem3a 8372 | Lemma for transfinite recu... |
tfrlem3 8373 | Lemma for transfinite recu... |
tfrlem4 8374 | Lemma for transfinite recu... |
tfrlem5 8375 | Lemma for transfinite recu... |
recsfval 8376 | Lemma for transfinite recu... |
tfrlem6 8377 | Lemma for transfinite recu... |
tfrlem7 8378 | Lemma for transfinite recu... |
tfrlem8 8379 | Lemma for transfinite recu... |
tfrlem9 8380 | Lemma for transfinite recu... |
tfrlem9a 8381 | Lemma for transfinite recu... |
tfrlem10 8382 | Lemma for transfinite recu... |
tfrlem11 8383 | Lemma for transfinite recu... |
tfrlem12 8384 | Lemma for transfinite recu... |
tfrlem13 8385 | Lemma for transfinite recu... |
tfrlem14 8386 | Lemma for transfinite recu... |
tfrlem15 8387 | Lemma for transfinite recu... |
tfrlem16 8388 | Lemma for finite recursion... |
tfr1a 8389 | A weak version of ~ tfr1 w... |
tfr2a 8390 | A weak version of ~ tfr2 w... |
tfr2b 8391 | Without assuming ~ ax-rep ... |
tfr1 8392 | Principle of Transfinite R... |
tfr2 8393 | Principle of Transfinite R... |
tfr3 8394 | Principle of Transfinite R... |
tfr1ALT 8395 | Alternate proof of ~ tfr1 ... |
tfr2ALT 8396 | Alternate proof of ~ tfr2 ... |
tfr3ALT 8397 | Alternate proof of ~ tfr3 ... |
recsfnon 8398 | Strong transfinite recursi... |
recsval 8399 | Strong transfinite recursi... |
tz7.44lem1 8400 | The ordered pair abstracti... |
tz7.44-1 8401 | The value of ` F ` at ` (/... |
tz7.44-2 8402 | The value of ` F ` at a su... |
tz7.44-3 8403 | The value of ` F ` at a li... |
rdgeq1 8406 | Equality theorem for the r... |
rdgeq2 8407 | Equality theorem for the r... |
rdgeq12 8408 | Equality theorem for the r... |
nfrdg 8409 | Bound-variable hypothesis ... |
rdglem1 8410 | Lemma used with the recurs... |
rdgfun 8411 | The recursive definition g... |
rdgdmlim 8412 | The domain of the recursiv... |
rdgfnon 8413 | The recursive definition g... |
rdgvalg 8414 | Value of the recursive def... |
rdgval 8415 | Value of the recursive def... |
rdg0 8416 | The initial value of the r... |
rdgseg 8417 | The initial segments of th... |
rdgsucg 8418 | The value of the recursive... |
rdgsuc 8419 | The value of the recursive... |
rdglimg 8420 | The value of the recursive... |
rdglim 8421 | The value of the recursive... |
rdg0g 8422 | The initial value of the r... |
rdgsucmptf 8423 | The value of the recursive... |
rdgsucmptnf 8424 | The value of the recursive... |
rdgsucmpt2 8425 | This version of ~ rdgsucmp... |
rdgsucmpt 8426 | The value of the recursive... |
rdglim2 8427 | The value of the recursive... |
rdglim2a 8428 | The value of the recursive... |
rdg0n 8429 | If ` A ` is a proper class... |
frfnom 8430 | The function generated by ... |
fr0g 8431 | The initial value resultin... |
frsuc 8432 | The successor value result... |
frsucmpt 8433 | The successor value result... |
frsucmptn 8434 | The value of the finite re... |
frsucmpt2 8435 | The successor value result... |
tz7.48lem 8436 | A way of showing an ordina... |
tz7.48-2 8437 | Proposition 7.48(2) of [Ta... |
tz7.48-1 8438 | Proposition 7.48(1) of [Ta... |
tz7.48-3 8439 | Proposition 7.48(3) of [Ta... |
tz7.49 8440 | Proposition 7.49 of [Takeu... |
tz7.49c 8441 | Corollary of Proposition 7... |
seqomlem0 8444 | Lemma for ` seqom ` . Cha... |
seqomlem1 8445 | Lemma for ` seqom ` . The... |
seqomlem2 8446 | Lemma for ` seqom ` . (Co... |
seqomlem3 8447 | Lemma for ` seqom ` . (Co... |
seqomlem4 8448 | Lemma for ` seqom ` . (Co... |
seqomeq12 8449 | Equality theorem for ` seq... |
fnseqom 8450 | An index-aware recursive d... |
seqom0g 8451 | Value of an index-aware re... |
seqomsuc 8452 | Value of an index-aware re... |
omsucelsucb 8453 | Membership is inherited by... |
df1o2 8468 | Expanded value of the ordi... |
df2o3 8469 | Expanded value of the ordi... |
df2o2 8470 | Expanded value of the ordi... |
1oex 8471 | Ordinal 1 is a set. (Cont... |
2oex 8472 | ` 2o ` is a set. (Contrib... |
1on 8473 | Ordinal 1 is an ordinal nu... |
1onOLD 8474 | Obsolete version of ~ 1on ... |
2on 8475 | Ordinal 2 is an ordinal nu... |
2onOLD 8476 | Obsolete version of ~ 2on ... |
2on0 8477 | Ordinal two is not zero. ... |
ord3 8478 | Ordinal 3 is an ordinal cl... |
3on 8479 | Ordinal 3 is an ordinal nu... |
4on 8480 | Ordinal 4 is an ordinal nu... |
1oexOLD 8481 | Obsolete version of ~ 1oex... |
2oexOLD 8482 | Obsolete version of ~ 2oex... |
1n0 8483 | Ordinal one is not equal t... |
nlim1 8484 | 1 is not a limit ordinal. ... |
nlim2 8485 | 2 is not a limit ordinal. ... |
xp01disj 8486 | Cartesian products with th... |
xp01disjl 8487 | Cartesian products with th... |
ordgt0ge1 8488 | Two ways to express that a... |
ordge1n0 8489 | An ordinal greater than or... |
el1o 8490 | Membership in ordinal one.... |
ord1eln01 8491 | An ordinal that is not 0 o... |
ord2eln012 8492 | An ordinal that is not 0, ... |
1ellim 8493 | A limit ordinal contains 1... |
2ellim 8494 | A limit ordinal contains 2... |
dif1o 8495 | Two ways to say that ` A `... |
ondif1 8496 | Two ways to say that ` A `... |
ondif2 8497 | Two ways to say that ` A `... |
2oconcl 8498 | Closure of the pair swappi... |
0lt1o 8499 | Ordinal zero is less than ... |
dif20el 8500 | An ordinal greater than on... |
0we1 8501 | The empty set is a well-or... |
brwitnlem 8502 | Lemma for relations which ... |
fnoa 8503 | Functionality and domain o... |
fnom 8504 | Functionality and domain o... |
fnoe 8505 | Functionality and domain o... |
oav 8506 | Value of ordinal addition.... |
omv 8507 | Value of ordinal multiplic... |
oe0lem 8508 | A helper lemma for ~ oe0 a... |
oev 8509 | Value of ordinal exponenti... |
oevn0 8510 | Value of ordinal exponenti... |
oa0 8511 | Addition with zero. Propo... |
om0 8512 | Ordinal multiplication wit... |
oe0m 8513 | Value of zero raised to an... |
om0x 8514 | Ordinal multiplication wit... |
oe0m0 8515 | Ordinal exponentiation wit... |
oe0m1 8516 | Ordinal exponentiation wit... |
oe0 8517 | Ordinal exponentiation wit... |
oev2 8518 | Alternate value of ordinal... |
oasuc 8519 | Addition with successor. ... |
oesuclem 8520 | Lemma for ~ oesuc . (Cont... |
omsuc 8521 | Multiplication with succes... |
oesuc 8522 | Ordinal exponentiation wit... |
onasuc 8523 | Addition with successor. ... |
onmsuc 8524 | Multiplication with succes... |
onesuc 8525 | Exponentiation with a succ... |
oa1suc 8526 | Addition with 1 is same as... |
oalim 8527 | Ordinal addition with a li... |
omlim 8528 | Ordinal multiplication wit... |
oelim 8529 | Ordinal exponentiation wit... |
oacl 8530 | Closure law for ordinal ad... |
omcl 8531 | Closure law for ordinal mu... |
oecl 8532 | Closure law for ordinal ex... |
oa0r 8533 | Ordinal addition with zero... |
om0r 8534 | Ordinal multiplication wit... |
o1p1e2 8535 | 1 + 1 = 2 for ordinal numb... |
o2p2e4 8536 | 2 + 2 = 4 for ordinal numb... |
om1 8537 | Ordinal multiplication wit... |
om1r 8538 | Ordinal multiplication wit... |
oe1 8539 | Ordinal exponentiation wit... |
oe1m 8540 | Ordinal exponentiation wit... |
oaordi 8541 | Ordering property of ordin... |
oaord 8542 | Ordering property of ordin... |
oacan 8543 | Left cancellation law for ... |
oaword 8544 | Weak ordering property of ... |
oawordri 8545 | Weak ordering property of ... |
oaord1 8546 | An ordinal is less than it... |
oaword1 8547 | An ordinal is less than or... |
oaword2 8548 | An ordinal is less than or... |
oawordeulem 8549 | Lemma for ~ oawordex . (C... |
oawordeu 8550 | Existence theorem for weak... |
oawordexr 8551 | Existence theorem for weak... |
oawordex 8552 | Existence theorem for weak... |
oaordex 8553 | Existence theorem for orde... |
oa00 8554 | An ordinal sum is zero iff... |
oalimcl 8555 | The ordinal sum with a lim... |
oaass 8556 | Ordinal addition is associ... |
oarec 8557 | Recursive definition of or... |
oaf1o 8558 | Left addition by a constan... |
oacomf1olem 8559 | Lemma for ~ oacomf1o . (C... |
oacomf1o 8560 | Define a bijection from ` ... |
omordi 8561 | Ordering property of ordin... |
omord2 8562 | Ordering property of ordin... |
omord 8563 | Ordering property of ordin... |
omcan 8564 | Left cancellation law for ... |
omword 8565 | Weak ordering property of ... |
omwordi 8566 | Weak ordering property of ... |
omwordri 8567 | Weak ordering property of ... |
omword1 8568 | An ordinal is less than or... |
omword2 8569 | An ordinal is less than or... |
om00 8570 | The product of two ordinal... |
om00el 8571 | The product of two nonzero... |
omordlim 8572 | Ordering involving the pro... |
omlimcl 8573 | The product of any nonzero... |
odi 8574 | Distributive law for ordin... |
omass 8575 | Multiplication of ordinal ... |
oneo 8576 | If an ordinal number is ev... |
omeulem1 8577 | Lemma for ~ omeu : existen... |
omeulem2 8578 | Lemma for ~ omeu : uniquen... |
omopth2 8579 | An ordered pair-like theor... |
omeu 8580 | The division algorithm for... |
oen0 8581 | Ordinal exponentiation wit... |
oeordi 8582 | Ordering law for ordinal e... |
oeord 8583 | Ordering property of ordin... |
oecan 8584 | Left cancellation law for ... |
oeword 8585 | Weak ordering property of ... |
oewordi 8586 | Weak ordering property of ... |
oewordri 8587 | Weak ordering property of ... |
oeworde 8588 | Ordinal exponentiation com... |
oeordsuc 8589 | Ordering property of ordin... |
oelim2 8590 | Ordinal exponentiation wit... |
oeoalem 8591 | Lemma for ~ oeoa . (Contr... |
oeoa 8592 | Sum of exponents law for o... |
oeoelem 8593 | Lemma for ~ oeoe . (Contr... |
oeoe 8594 | Product of exponents law f... |
oelimcl 8595 | The ordinal exponential wi... |
oeeulem 8596 | Lemma for ~ oeeu . (Contr... |
oeeui 8597 | The division algorithm for... |
oeeu 8598 | The division algorithm for... |
nna0 8599 | Addition with zero. Theor... |
nnm0 8600 | Multiplication with zero. ... |
nnasuc 8601 | Addition with successor. ... |
nnmsuc 8602 | Multiplication with succes... |
nnesuc 8603 | Exponentiation with a succ... |
nna0r 8604 | Addition to zero. Remark ... |
nnm0r 8605 | Multiplication with zero. ... |
nnacl 8606 | Closure of addition of nat... |
nnmcl 8607 | Closure of multiplication ... |
nnecl 8608 | Closure of exponentiation ... |
nnacli 8609 | ` _om ` is closed under ad... |
nnmcli 8610 | ` _om ` is closed under mu... |
nnarcl 8611 | Reverse closure law for ad... |
nnacom 8612 | Addition of natural number... |
nnaordi 8613 | Ordering property of addit... |
nnaord 8614 | Ordering property of addit... |
nnaordr 8615 | Ordering property of addit... |
nnawordi 8616 | Adding to both sides of an... |
nnaass 8617 | Addition of natural number... |
nndi 8618 | Distributive law for natur... |
nnmass 8619 | Multiplication of natural ... |
nnmsucr 8620 | Multiplication with succes... |
nnmcom 8621 | Multiplication of natural ... |
nnaword 8622 | Weak ordering property of ... |
nnacan 8623 | Cancellation law for addit... |
nnaword1 8624 | Weak ordering property of ... |
nnaword2 8625 | Weak ordering property of ... |
nnmordi 8626 | Ordering property of multi... |
nnmord 8627 | Ordering property of multi... |
nnmword 8628 | Weak ordering property of ... |
nnmcan 8629 | Cancellation law for multi... |
nnmwordi 8630 | Weak ordering property of ... |
nnmwordri 8631 | Weak ordering property of ... |
nnawordex 8632 | Equivalence for weak order... |
nnaordex 8633 | Equivalence for ordering. ... |
nnaordex2 8634 | Equivalence for ordering. ... |
1onn 8635 | The ordinal 1 is a natural... |
1onnALT 8636 | Shorter proof of ~ 1onn us... |
2onn 8637 | The ordinal 2 is a natural... |
2onnALT 8638 | Shorter proof of ~ 2onn us... |
3onn 8639 | The ordinal 3 is a natural... |
4onn 8640 | The ordinal 4 is a natural... |
1one2o 8641 | Ordinal one is not ordinal... |
oaabslem 8642 | Lemma for ~ oaabs . (Cont... |
oaabs 8643 | Ordinal addition absorbs a... |
oaabs2 8644 | The absorption law ~ oaabs... |
omabslem 8645 | Lemma for ~ omabs . (Cont... |
omabs 8646 | Ordinal multiplication is ... |
nnm1 8647 | Multiply an element of ` _... |
nnm2 8648 | Multiply an element of ` _... |
nn2m 8649 | Multiply an element of ` _... |
nnneo 8650 | If a natural number is eve... |
nneob 8651 | A natural number is even i... |
omsmolem 8652 | Lemma for ~ omsmo . (Cont... |
omsmo 8653 | A strictly monotonic ordin... |
omopthlem1 8654 | Lemma for ~ omopthi . (Co... |
omopthlem2 8655 | Lemma for ~ omopthi . (Co... |
omopthi 8656 | An ordered pair theorem fo... |
omopth 8657 | An ordered pair theorem fo... |
nnasmo 8658 | There is at most one left ... |
eldifsucnn 8659 | Condition for membership i... |
on2recsfn 8662 | Show that double recursion... |
on2recsov 8663 | Calculate the value of the... |
on2ind 8664 | Double induction over ordi... |
on3ind 8665 | Triple induction over ordi... |
coflton 8666 | Cofinality theorem for ord... |
cofon1 8667 | Cofinality theorem for ord... |
cofon2 8668 | Cofinality theorem for ord... |
cofonr 8669 | Inverse cofinality law for... |
naddfn 8670 | Natural addition is a func... |
naddcllem 8671 | Lemma for ordinal addition... |
naddcl 8672 | Closure law for natural ad... |
naddov 8673 | The value of natural addit... |
naddov2 8674 | Alternate expression for n... |
naddov3 8675 | Alternate expression for n... |
naddf 8676 | Function statement for nat... |
naddcom 8677 | Natural addition commutes.... |
naddrid 8678 | Ordinal zero is the additi... |
naddlid 8679 | Ordinal zero is the additi... |
naddssim 8680 | Ordinal less-than-or-equal... |
naddelim 8681 | Ordinal less-than is prese... |
naddel1 8682 | Ordinal less-than is not a... |
naddel2 8683 | Ordinal less-than is not a... |
naddss1 8684 | Ordinal less-than-or-equal... |
naddss2 8685 | Ordinal less-than-or-equal... |
naddword1 8686 | Weak-ordering principle fo... |
naddword2 8687 | Weak-ordering principle fo... |
naddunif 8688 | Uniformity theorem for nat... |
naddasslem1 8689 | Lemma for ~ naddass . Exp... |
naddasslem2 8690 | Lemma for ~ naddass . Exp... |
naddass 8691 | Natural ordinal addition i... |
nadd32 8692 | Commutative/associative la... |
nadd4 8693 | Rearragement of terms in a... |
nadd42 8694 | Rearragement of terms in a... |
naddel12 8695 | Natural addition to both s... |
dfer2 8700 | Alternate definition of eq... |
dfec2 8702 | Alternate definition of ` ... |
ecexg 8703 | An equivalence class modul... |
ecexr 8704 | A nonempty equivalence cla... |
ereq1 8706 | Equality theorem for equiv... |
ereq2 8707 | Equality theorem for equiv... |
errel 8708 | An equivalence relation is... |
erdm 8709 | The domain of an equivalen... |
ercl 8710 | Elementhood in the field o... |
ersym 8711 | An equivalence relation is... |
ercl2 8712 | Elementhood in the field o... |
ersymb 8713 | An equivalence relation is... |
ertr 8714 | An equivalence relation is... |
ertrd 8715 | A transitivity relation fo... |
ertr2d 8716 | A transitivity relation fo... |
ertr3d 8717 | A transitivity relation fo... |
ertr4d 8718 | A transitivity relation fo... |
erref 8719 | An equivalence relation is... |
ercnv 8720 | The converse of an equival... |
errn 8721 | The range and domain of an... |
erssxp 8722 | An equivalence relation is... |
erex 8723 | An equivalence relation is... |
erexb 8724 | An equivalence relation is... |
iserd 8725 | A reflexive, symmetric, tr... |
iseri 8726 | A reflexive, symmetric, tr... |
iseriALT 8727 | Alternate proof of ~ iseri... |
brdifun 8728 | Evaluate the incomparabili... |
swoer 8729 | Incomparability under a st... |
swoord1 8730 | The incomparability equiva... |
swoord2 8731 | The incomparability equiva... |
swoso 8732 | If the incomparability rel... |
eqerlem 8733 | Lemma for ~ eqer . (Contr... |
eqer 8734 | Equivalence relation invol... |
ider 8735 | The identity relation is a... |
0er 8736 | The empty set is an equiva... |
eceq1 8737 | Equality theorem for equiv... |
eceq1d 8738 | Equality theorem for equiv... |
eceq2 8739 | Equality theorem for equiv... |
eceq2i 8740 | Equality theorem for the `... |
eceq2d 8741 | Equality theorem for the `... |
elecg 8742 | Membership in an equivalen... |
elec 8743 | Membership in an equivalen... |
relelec 8744 | Membership in an equivalen... |
ecss 8745 | An equivalence class is a ... |
ecdmn0 8746 | A representative of a none... |
ereldm 8747 | Equality of equivalence cl... |
erth 8748 | Basic property of equivale... |
erth2 8749 | Basic property of equivale... |
erthi 8750 | Basic property of equivale... |
erdisj 8751 | Equivalence classes do not... |
ecidsn 8752 | An equivalence class modul... |
qseq1 8753 | Equality theorem for quoti... |
qseq2 8754 | Equality theorem for quoti... |
qseq2i 8755 | Equality theorem for quoti... |
qseq2d 8756 | Equality theorem for quoti... |
qseq12 8757 | Equality theorem for quoti... |
elqsg 8758 | Closed form of ~ elqs . (... |
elqs 8759 | Membership in a quotient s... |
elqsi 8760 | Membership in a quotient s... |
elqsecl 8761 | Membership in a quotient s... |
ecelqsg 8762 | Membership of an equivalen... |
ecelqsi 8763 | Membership of an equivalen... |
ecopqsi 8764 | "Closure" law for equivale... |
qsexg 8765 | A quotient set exists. (C... |
qsex 8766 | A quotient set exists. (C... |
uniqs 8767 | The union of a quotient se... |
qsss 8768 | A quotient set is a set of... |
uniqs2 8769 | The union of a quotient se... |
snec 8770 | The singleton of an equiva... |
ecqs 8771 | Equivalence class in terms... |
ecid 8772 | A set is equal to its cose... |
qsid 8773 | A set is equal to its quot... |
ectocld 8774 | Implicit substitution of c... |
ectocl 8775 | Implicit substitution of c... |
elqsn0 8776 | A quotient set does not co... |
ecelqsdm 8777 | Membership of an equivalen... |
xpider 8778 | A Cartesian square is an e... |
iiner 8779 | The intersection of a none... |
riiner 8780 | The relative intersection ... |
erinxp 8781 | A restricted equivalence r... |
ecinxp 8782 | Restrict the relation in a... |
qsinxp 8783 | Restrict the equivalence r... |
qsdisj 8784 | Members of a quotient set ... |
qsdisj2 8785 | A quotient set is a disjoi... |
qsel 8786 | If an element of a quotien... |
uniinqs 8787 | Class union distributes ov... |
qliftlem 8788 | Lemma for theorems about a... |
qliftrel 8789 | ` F ` , a function lift, i... |
qliftel 8790 | Elementhood in the relatio... |
qliftel1 8791 | Elementhood in the relatio... |
qliftfun 8792 | The function ` F ` is the ... |
qliftfund 8793 | The function ` F ` is the ... |
qliftfuns 8794 | The function ` F ` is the ... |
qliftf 8795 | The domain and codomain of... |
qliftval 8796 | The value of the function ... |
ecoptocl 8797 | Implicit substitution of c... |
2ecoptocl 8798 | Implicit substitution of c... |
3ecoptocl 8799 | Implicit substitution of c... |
brecop 8800 | Binary relation on a quoti... |
brecop2 8801 | Binary relation on a quoti... |
eroveu 8802 | Lemma for ~ erov and ~ ero... |
erovlem 8803 | Lemma for ~ erov and ~ ero... |
erov 8804 | The value of an operation ... |
eroprf 8805 | Functionality of an operat... |
erov2 8806 | The value of an operation ... |
eroprf2 8807 | Functionality of an operat... |
ecopoveq 8808 | This is the first of sever... |
ecopovsym 8809 | Assuming the operation ` F... |
ecopovtrn 8810 | Assuming that operation ` ... |
ecopover 8811 | Assuming that operation ` ... |
eceqoveq 8812 | Equality of equivalence re... |
ecovcom 8813 | Lemma used to transfer a c... |
ecovass 8814 | Lemma used to transfer an ... |
ecovdi 8815 | Lemma used to transfer a d... |
mapprc 8820 | When ` A ` is a proper cla... |
pmex 8821 | The class of all partial f... |
mapex 8822 | The class of all functions... |
fnmap 8823 | Set exponentiation has a u... |
fnpm 8824 | Partial function exponenti... |
reldmmap 8825 | Set exponentiation is a we... |
mapvalg 8826 | The value of set exponenti... |
pmvalg 8827 | The value of the partial m... |
mapval 8828 | The value of set exponenti... |
elmapg 8829 | Membership relation for se... |
elmapd 8830 | Deduction form of ~ elmapg... |
elmapdd 8831 | Deduction associated with ... |
mapdm0 8832 | The empty set is the only ... |
elpmg 8833 | The predicate "is a partia... |
elpm2g 8834 | The predicate "is a partia... |
elpm2r 8835 | Sufficient condition for b... |
elpmi 8836 | A partial function is a fu... |
pmfun 8837 | A partial function is a fu... |
elmapex 8838 | Eliminate antecedent for m... |
elmapi 8839 | A mapping is a function, f... |
mapfset 8840 | If ` B ` is a set, the val... |
mapssfset 8841 | The value of the set expon... |
mapfoss 8842 | The value of the set expon... |
fsetsspwxp 8843 | The class of all functions... |
fset0 8844 | The set of functions from ... |
fsetdmprc0 8845 | The set of functions with ... |
fsetex 8846 | The set of functions betwe... |
f1setex 8847 | The set of injections betw... |
fosetex 8848 | The set of surjections bet... |
f1osetex 8849 | The set of bijections betw... |
fsetfcdm 8850 | The class of functions wit... |
fsetfocdm 8851 | The class of functions wit... |
fsetprcnex 8852 | The class of all functions... |
fsetcdmex 8853 | The class of all functions... |
fsetexb 8854 | The class of all functions... |
elmapfn 8855 | A mapping is a function wi... |
elmapfun 8856 | A mapping is always a func... |
elmapssres 8857 | A restricted mapping is a ... |
fpmg 8858 | A total function is a part... |
pmss12g 8859 | Subset relation for the se... |
pmresg 8860 | Elementhood of a restricte... |
elmap 8861 | Membership relation for se... |
mapval2 8862 | Alternate expression for t... |
elpm 8863 | The predicate "is a partia... |
elpm2 8864 | The predicate "is a partia... |
fpm 8865 | A total function is a part... |
mapsspm 8866 | Set exponentiation is a su... |
pmsspw 8867 | Partial maps are a subset ... |
mapsspw 8868 | Set exponentiation is a su... |
mapfvd 8869 | The value of a function th... |
elmapresaun 8870 | ~ fresaun transposed to ma... |
fvmptmap 8871 | Special case of ~ fvmpt fo... |
map0e 8872 | Set exponentiation with an... |
map0b 8873 | Set exponentiation with an... |
map0g 8874 | Set exponentiation is empt... |
0map0sn0 8875 | The set of mappings of the... |
mapsnd 8876 | The value of set exponenti... |
map0 8877 | Set exponentiation is empt... |
mapsn 8878 | The value of set exponenti... |
mapss 8879 | Subset inheritance for set... |
fdiagfn 8880 | Functionality of the diago... |
fvdiagfn 8881 | Functionality of the diago... |
mapsnconst 8882 | Every singleton map is a c... |
mapsncnv 8883 | Expression for the inverse... |
mapsnf1o2 8884 | Explicit bijection between... |
mapsnf1o3 8885 | Explicit bijection in the ... |
ralxpmap 8886 | Quantification over functi... |
dfixp 8889 | Eliminate the expression `... |
ixpsnval 8890 | The value of an infinite C... |
elixp2 8891 | Membership in an infinite ... |
fvixp 8892 | Projection of a factor of ... |
ixpfn 8893 | A nuple is a function. (C... |
elixp 8894 | Membership in an infinite ... |
elixpconst 8895 | Membership in an infinite ... |
ixpconstg 8896 | Infinite Cartesian product... |
ixpconst 8897 | Infinite Cartesian product... |
ixpeq1 8898 | Equality theorem for infin... |
ixpeq1d 8899 | Equality theorem for infin... |
ss2ixp 8900 | Subclass theorem for infin... |
ixpeq2 8901 | Equality theorem for infin... |
ixpeq2dva 8902 | Equality theorem for infin... |
ixpeq2dv 8903 | Equality theorem for infin... |
cbvixp 8904 | Change bound variable in a... |
cbvixpv 8905 | Change bound variable in a... |
nfixpw 8906 | Bound-variable hypothesis ... |
nfixp 8907 | Bound-variable hypothesis ... |
nfixp1 8908 | The index variable in an i... |
ixpprc 8909 | A cartesian product of pro... |
ixpf 8910 | A member of an infinite Ca... |
uniixp 8911 | The union of an infinite C... |
ixpexg 8912 | The existence of an infini... |
ixpin 8913 | The intersection of two in... |
ixpiin 8914 | The indexed intersection o... |
ixpint 8915 | The intersection of a coll... |
ixp0x 8916 | An infinite Cartesian prod... |
ixpssmap2g 8917 | An infinite Cartesian prod... |
ixpssmapg 8918 | An infinite Cartesian prod... |
0elixp 8919 | Membership of the empty se... |
ixpn0 8920 | The infinite Cartesian pro... |
ixp0 8921 | The infinite Cartesian pro... |
ixpssmap 8922 | An infinite Cartesian prod... |
resixp 8923 | Restriction of an element ... |
undifixp 8924 | Union of two projections o... |
mptelixpg 8925 | Condition for an explicit ... |
resixpfo 8926 | Restriction of elements of... |
elixpsn 8927 | Membership in a class of s... |
ixpsnf1o 8928 | A bijection between a clas... |
mapsnf1o 8929 | A bijection between a set ... |
boxriin 8930 | A rectangular subset of a ... |
boxcutc 8931 | The relative complement of... |
relen 8940 | Equinumerosity is a relati... |
reldom 8941 | Dominance is a relation. ... |
relsdom 8942 | Strict dominance is a rela... |
encv 8943 | If two classes are equinum... |
breng 8944 | Equinumerosity relation. ... |
bren 8945 | Equinumerosity relation. ... |
brenOLD 8946 | Obsolete version of ~ bren... |
brdom2g 8947 | Dominance relation. This ... |
brdomg 8948 | Dominance relation. (Cont... |
brdomgOLD 8949 | Obsolete version of ~ brdo... |
brdomi 8950 | Dominance relation. (Cont... |
brdomiOLD 8951 | Obsolete version of ~ brdo... |
brdom 8952 | Dominance relation. (Cont... |
domen 8953 | Dominance in terms of equi... |
domeng 8954 | Dominance in terms of equi... |
ctex 8955 | A countable set is a set. ... |
f1oen4g 8956 | The domain and range of a ... |
f1dom4g 8957 | The domain of a one-to-one... |
f1oen3g 8958 | The domain and range of a ... |
f1dom3g 8959 | The domain of a one-to-one... |
f1oen2g 8960 | The domain and range of a ... |
f1dom2g 8961 | The domain of a one-to-one... |
f1dom2gOLD 8962 | Obsolete version of ~ f1do... |
f1oeng 8963 | The domain and range of a ... |
f1domg 8964 | The domain of a one-to-one... |
f1oen 8965 | The domain and range of a ... |
f1dom 8966 | The domain of a one-to-one... |
brsdom 8967 | Strict dominance relation,... |
isfi 8968 | Express " ` A ` is finite"... |
enssdom 8969 | Equinumerosity implies dom... |
dfdom2 8970 | Alternate definition of do... |
endom 8971 | Equinumerosity implies dom... |
sdomdom 8972 | Strict dominance implies d... |
sdomnen 8973 | Strict dominance implies n... |
brdom2 8974 | Dominance in terms of stri... |
bren2 8975 | Equinumerosity expressed i... |
enrefg 8976 | Equinumerosity is reflexiv... |
enref 8977 | Equinumerosity is reflexiv... |
eqeng 8978 | Equality implies equinumer... |
domrefg 8979 | Dominance is reflexive. (... |
en2d 8980 | Equinumerosity inference f... |
en3d 8981 | Equinumerosity inference f... |
en2i 8982 | Equinumerosity inference f... |
en3i 8983 | Equinumerosity inference f... |
dom2lem 8984 | A mapping (first hypothesi... |
dom2d 8985 | A mapping (first hypothesi... |
dom3d 8986 | A mapping (first hypothesi... |
dom2 8987 | A mapping (first hypothesi... |
dom3 8988 | A mapping (first hypothesi... |
idssen 8989 | Equality implies equinumer... |
domssl 8990 | If ` A ` is a subset of ` ... |
domssr 8991 | If ` C ` is a superset of ... |
ssdomg 8992 | A set dominates its subset... |
ener 8993 | Equinumerosity is an equiv... |
ensymb 8994 | Symmetry of equinumerosity... |
ensym 8995 | Symmetry of equinumerosity... |
ensymi 8996 | Symmetry of equinumerosity... |
ensymd 8997 | Symmetry of equinumerosity... |
entr 8998 | Transitivity of equinumero... |
domtr 8999 | Transitivity of dominance ... |
entri 9000 | A chained equinumerosity i... |
entr2i 9001 | A chained equinumerosity i... |
entr3i 9002 | A chained equinumerosity i... |
entr4i 9003 | A chained equinumerosity i... |
endomtr 9004 | Transitivity of equinumero... |
domentr 9005 | Transitivity of dominance ... |
f1imaeng 9006 | If a function is one-to-on... |
f1imaen2g 9007 | If a function is one-to-on... |
f1imaen 9008 | If a function is one-to-on... |
en0 9009 | The empty set is equinumer... |
en0OLD 9010 | Obsolete version of ~ en0 ... |
en0ALT 9011 | Shorter proof of ~ en0 , d... |
en0r 9012 | The empty set is equinumer... |
ensn1 9013 | A singleton is equinumerou... |
ensn1OLD 9014 | Obsolete version of ~ ensn... |
ensn1g 9015 | A singleton is equinumerou... |
enpr1g 9016 | ` { A , A } ` has only one... |
en1 9017 | A set is equinumerous to o... |
en1OLD 9018 | Obsolete version of ~ en1 ... |
en1b 9019 | A set is equinumerous to o... |
en1bOLD 9020 | Obsolete version of ~ en1b... |
reuen1 9021 | Two ways to express "exact... |
euen1 9022 | Two ways to express "exact... |
euen1b 9023 | Two ways to express " ` A ... |
en1uniel 9024 | A singleton contains its s... |
en1unielOLD 9025 | Obsolete version of ~ en1u... |
2dom 9026 | A set that dominates ordin... |
fundmen 9027 | A function is equinumerous... |
fundmeng 9028 | A function is equinumerous... |
cnven 9029 | A relational set is equinu... |
cnvct 9030 | If a set is countable, so ... |
fndmeng 9031 | A function is equinumerate... |
mapsnend 9032 | Set exponentiation to a si... |
mapsnen 9033 | Set exponentiation to a si... |
snmapen 9034 | Set exponentiation: a sing... |
snmapen1 9035 | Set exponentiation: a sing... |
map1 9036 | Set exponentiation: ordina... |
en2sn 9037 | Two singletons are equinum... |
en2snOLD 9038 | Obsolete version of ~ en2s... |
en2snOLDOLD 9039 | Obsolete version of ~ en2s... |
snfi 9040 | A singleton is finite. (C... |
fiprc 9041 | The class of finite sets i... |
unen 9042 | Equinumerosity of union of... |
enrefnn 9043 | Equinumerosity is reflexiv... |
en2prd 9044 | Two unordered pairs are eq... |
enpr2d 9045 | A pair with distinct eleme... |
enpr2dOLD 9046 | Obsolete version of ~ enpr... |
ssct 9047 | Any subset of a countable ... |
ssctOLD 9048 | Obsolete version of ~ ssct... |
difsnen 9049 | All decrements of a set ar... |
domdifsn 9050 | Dominance over a set with ... |
xpsnen 9051 | A set is equinumerous to i... |
xpsneng 9052 | A set is equinumerous to i... |
xp1en 9053 | One times a cardinal numbe... |
endisj 9054 | Any two sets are equinumer... |
undom 9055 | Dominance law for union. ... |
undomOLD 9056 | Obsolete version of ~ undo... |
xpcomf1o 9057 | The canonical bijection fr... |
xpcomco 9058 | Composition with the bijec... |
xpcomen 9059 | Commutative law for equinu... |
xpcomeng 9060 | Commutative law for equinu... |
xpsnen2g 9061 | A set is equinumerous to i... |
xpassen 9062 | Associative law for equinu... |
xpdom2 9063 | Dominance law for Cartesia... |
xpdom2g 9064 | Dominance law for Cartesia... |
xpdom1g 9065 | Dominance law for Cartesia... |
xpdom3 9066 | A set is dominated by its ... |
xpdom1 9067 | Dominance law for Cartesia... |
domunsncan 9068 | A singleton cancellation l... |
omxpenlem 9069 | Lemma for ~ omxpen . (Con... |
omxpen 9070 | The cardinal and ordinal p... |
omf1o 9071 | Construct an explicit bije... |
pw2f1olem 9072 | Lemma for ~ pw2f1o . (Con... |
pw2f1o 9073 | The power set of a set is ... |
pw2eng 9074 | The power set of a set is ... |
pw2en 9075 | The power set of a set is ... |
fopwdom 9076 | Covering implies injection... |
enfixsn 9077 | Given two equipollent sets... |
sucdom2OLD 9078 | Obsolete version of ~ sucd... |
sbthlem1 9079 | Lemma for ~ sbth . (Contr... |
sbthlem2 9080 | Lemma for ~ sbth . (Contr... |
sbthlem3 9081 | Lemma for ~ sbth . (Contr... |
sbthlem4 9082 | Lemma for ~ sbth . (Contr... |
sbthlem5 9083 | Lemma for ~ sbth . (Contr... |
sbthlem6 9084 | Lemma for ~ sbth . (Contr... |
sbthlem7 9085 | Lemma for ~ sbth . (Contr... |
sbthlem8 9086 | Lemma for ~ sbth . (Contr... |
sbthlem9 9087 | Lemma for ~ sbth . (Contr... |
sbthlem10 9088 | Lemma for ~ sbth . (Contr... |
sbth 9089 | Schroeder-Bernstein Theore... |
sbthb 9090 | Schroeder-Bernstein Theore... |
sbthcl 9091 | Schroeder-Bernstein Theore... |
dfsdom2 9092 | Alternate definition of st... |
brsdom2 9093 | Alternate definition of st... |
sdomnsym 9094 | Strict dominance is asymme... |
domnsym 9095 | Theorem 22(i) of [Suppes] ... |
0domg 9096 | Any set dominates the empt... |
0domgOLD 9097 | Obsolete version of ~ 0dom... |
dom0 9098 | A set dominated by the emp... |
dom0OLD 9099 | Obsolete version of ~ dom0... |
0sdomg 9100 | A set strictly dominates t... |
0sdomgOLD 9101 | Obsolete version of ~ 0sdo... |
0dom 9102 | Any set dominates the empt... |
0sdom 9103 | A set strictly dominates t... |
sdom0 9104 | The empty set does not str... |
sdom0OLD 9105 | Obsolete version of ~ sdom... |
sdomdomtr 9106 | Transitivity of strict dom... |
sdomentr 9107 | Transitivity of strict dom... |
domsdomtr 9108 | Transitivity of dominance ... |
ensdomtr 9109 | Transitivity of equinumero... |
sdomirr 9110 | Strict dominance is irrefl... |
sdomtr 9111 | Strict dominance is transi... |
sdomn2lp 9112 | Strict dominance has no 2-... |
enen1 9113 | Equality-like theorem for ... |
enen2 9114 | Equality-like theorem for ... |
domen1 9115 | Equality-like theorem for ... |
domen2 9116 | Equality-like theorem for ... |
sdomen1 9117 | Equality-like theorem for ... |
sdomen2 9118 | Equality-like theorem for ... |
domtriord 9119 | Dominance is trichotomous ... |
sdomel 9120 | For ordinals, strict domin... |
sdomdif 9121 | The difference of a set fr... |
onsdominel 9122 | An ordinal with more eleme... |
domunsn 9123 | Dominance over a set with ... |
fodomr 9124 | There exists a mapping fro... |
pwdom 9125 | Injection of sets implies ... |
canth2 9126 | Cantor's Theorem. No set ... |
canth2g 9127 | Cantor's theorem with the ... |
2pwuninel 9128 | The power set of the power... |
2pwne 9129 | No set equals the power se... |
disjen 9130 | A stronger form of ~ pwuni... |
disjenex 9131 | Existence version of ~ dis... |
domss2 9132 | A corollary of ~ disjenex ... |
domssex2 9133 | A corollary of ~ disjenex ... |
domssex 9134 | Weakening of ~ domssex2 to... |
xpf1o 9135 | Construct a bijection on a... |
xpen 9136 | Equinumerosity law for Car... |
mapen 9137 | Two set exponentiations ar... |
mapdom1 9138 | Order-preserving property ... |
mapxpen 9139 | Equinumerosity law for dou... |
xpmapenlem 9140 | Lemma for ~ xpmapen . (Co... |
xpmapen 9141 | Equinumerosity law for set... |
mapunen 9142 | Equinumerosity law for set... |
map2xp 9143 | A cardinal power with expo... |
mapdom2 9144 | Order-preserving property ... |
mapdom3 9145 | Set exponentiation dominat... |
pwen 9146 | If two sets are equinumero... |
ssenen 9147 | Equinumerosity of equinume... |
limenpsi 9148 | A limit ordinal is equinum... |
limensuci 9149 | A limit ordinal is equinum... |
limensuc 9150 | A limit ordinal is equinum... |
infensuc 9151 | Any infinite ordinal is eq... |
dif1enlem 9152 | Lemma for ~ rexdif1en and ... |
dif1enlemOLD 9153 | Obsolete version of ~ dif1... |
rexdif1en 9154 | If a set is equinumerous t... |
rexdif1enOLD 9155 | Obsolete version of ~ rexd... |
dif1en 9156 | If a set ` A ` is equinume... |
dif1ennn 9157 | If a set ` A ` is equinume... |
dif1enOLD 9158 | Obsolete version of ~ dif1... |
findcard 9159 | Schema for induction on th... |
findcard2 9160 | Schema for induction on th... |
findcard2s 9161 | Variation of ~ findcard2 r... |
findcard2d 9162 | Deduction version of ~ fin... |
nnfi 9163 | Natural numbers are finite... |
pssnn 9164 | A proper subset of a natur... |
ssnnfi 9165 | A subset of a natural numb... |
ssnnfiOLD 9166 | Obsolete version of ~ ssnn... |
0fin 9167 | The empty set is finite. ... |
unfi 9168 | The union of two finite se... |
ssfi 9169 | A subset of a finite set i... |
ssfiALT 9170 | Shorter proof of ~ ssfi us... |
imafi 9171 | Images of finite sets are ... |
pwfir 9172 | If the power set of a set ... |
pwfilem 9173 | Lemma for ~ pwfi . (Contr... |
pwfi 9174 | The power set of a finite ... |
diffi 9175 | If ` A ` is finite, ` ( A ... |
cnvfi 9176 | If a set is finite, its co... |
fnfi 9177 | A version of ~ fnex for fi... |
f1oenfi 9178 | If the domain of a one-to-... |
f1oenfirn 9179 | If the range of a one-to-o... |
f1domfi 9180 | If the codomain of a one-t... |
f1domfi2 9181 | If the domain of a one-to-... |
enreffi 9182 | Equinumerosity is reflexiv... |
ensymfib 9183 | Symmetry of equinumerosity... |
entrfil 9184 | Transitivity of equinumero... |
enfii 9185 | A set equinumerous to a fi... |
enfi 9186 | Equinumerous sets have the... |
enfiALT 9187 | Shorter proof of ~ enfi us... |
domfi 9188 | A set dominated by a finit... |
entrfi 9189 | Transitivity of equinumero... |
entrfir 9190 | Transitivity of equinumero... |
domtrfil 9191 | Transitivity of dominance ... |
domtrfi 9192 | Transitivity of dominance ... |
domtrfir 9193 | Transitivity of dominance ... |
f1imaenfi 9194 | If a function is one-to-on... |
ssdomfi 9195 | A finite set dominates its... |
ssdomfi2 9196 | A set dominates its finite... |
sbthfilem 9197 | Lemma for ~ sbthfi . (Con... |
sbthfi 9198 | Schroeder-Bernstein Theore... |
domnsymfi 9199 | If a set dominates a finit... |
sdomdomtrfi 9200 | Transitivity of strict dom... |
domsdomtrfi 9201 | Transitivity of dominance ... |
sucdom2 9202 | Strict dominance of a set ... |
phplem1 9203 | Lemma for Pigeonhole Princ... |
phplem2 9204 | Lemma for Pigeonhole Princ... |
nneneq 9205 | Two equinumerous natural n... |
php 9206 | Pigeonhole Principle. A n... |
php2 9207 | Corollary of Pigeonhole Pr... |
php3 9208 | Corollary of Pigeonhole Pr... |
php4 9209 | Corollary of the Pigeonhol... |
php5 9210 | Corollary of the Pigeonhol... |
phpeqd 9211 | Corollary of the Pigeonhol... |
nndomog 9212 | Cardinal ordering agrees w... |
phplem1OLD 9213 | Obsolete lemma for ~ php a... |
phplem2OLD 9214 | Obsolete lemma for ~ php a... |
phplem3OLD 9215 | Obsolete version of ~ phpl... |
phplem4OLD 9216 | Obsolete version of ~ phpl... |
nneneqOLD 9217 | Obsolete version of ~ nnen... |
phpOLD 9218 | Obsolete version of ~ php ... |
php2OLD 9219 | Obsolete version of ~ php2... |
php3OLD 9220 | Obsolete version of ~ php3... |
phpeqdOLD 9221 | Obsolete version of ~ phpe... |
nndomogOLD 9222 | Obsolete version of ~ nndo... |
snnen2oOLD 9223 | Obsolete version of ~ snne... |
onomeneq 9224 | An ordinal number equinume... |
onomeneqOLD 9225 | Obsolete version of ~ onom... |
onfin 9226 | An ordinal number is finit... |
onfin2 9227 | A set is a natural number ... |
nnfiOLD 9228 | Obsolete version of ~ nnfi... |
nndomo 9229 | Cardinal ordering agrees w... |
nnsdomo 9230 | Cardinal ordering agrees w... |
sucdom 9231 | Strict dominance of a set ... |
sucdomOLD 9232 | Obsolete version of ~ sucd... |
snnen2o 9233 | A singleton ` { A } ` is n... |
0sdom1dom 9234 | Strict dominance over 0 is... |
0sdom1domALT 9235 | Alternate proof of ~ 0sdom... |
1sdom2 9236 | Ordinal 1 is strictly domi... |
1sdom2ALT 9237 | Alternate proof of ~ 1sdom... |
sdom1 9238 | A set has less than one me... |
sdom1OLD 9239 | Obsolete version of ~ sdom... |
modom 9240 | Two ways to express "at mo... |
modom2 9241 | Two ways to express "at mo... |
rex2dom 9242 | A set that has at least 2 ... |
1sdom2dom 9243 | Strict dominance over 1 is... |
1sdom 9244 | A set that strictly domina... |
1sdomOLD 9245 | Obsolete version of ~ 1sdo... |
unxpdomlem1 9246 | Lemma for ~ unxpdom . (Tr... |
unxpdomlem2 9247 | Lemma for ~ unxpdom . (Co... |
unxpdomlem3 9248 | Lemma for ~ unxpdom . (Co... |
unxpdom 9249 | Cartesian product dominate... |
unxpdom2 9250 | Corollary of ~ unxpdom . ... |
sucxpdom 9251 | Cartesian product dominate... |
pssinf 9252 | A set equinumerous to a pr... |
fisseneq 9253 | A finite set is equal to i... |
ominf 9254 | The set of natural numbers... |
ominfOLD 9255 | Obsolete version of ~ omin... |
isinf 9256 | Any set that is not finite... |
isinfOLD 9257 | Obsolete version of ~ isin... |
fineqvlem 9258 | Lemma for ~ fineqv . (Con... |
fineqv 9259 | If the Axiom of Infinity i... |
enfiiOLD 9260 | Obsolete version of ~ enfi... |
pssnnOLD 9261 | Obsolete version of ~ pssn... |
xpfir 9262 | The components of a nonemp... |
ssfid 9263 | A subset of a finite set i... |
infi 9264 | The intersection of two se... |
rabfi 9265 | A restricted class built f... |
finresfin 9266 | The restriction of a finit... |
f1finf1o 9267 | Any injection from one fin... |
f1finf1oOLD 9268 | Obsolete version of ~ f1fi... |
nfielex 9269 | If a class is not finite, ... |
en1eqsn 9270 | A set with one element is ... |
en1eqsnOLD 9271 | Obsolete version of ~ en1e... |
en1eqsnbi 9272 | A set containing an elemen... |
dif1ennnALT 9273 | Alternate proof of ~ dif1e... |
enp1ilem 9274 | Lemma for uses of ~ enp1i ... |
enp1i 9275 | Proof induction for ~ en2 ... |
enp1iOLD 9276 | Obsolete version of ~ enp1... |
en2 9277 | A set equinumerous to ordi... |
en3 9278 | A set equinumerous to ordi... |
en4 9279 | A set equinumerous to ordi... |
findcard2OLD 9280 | Obsolete version of ~ find... |
findcard3 9281 | Schema for strong inductio... |
findcard3OLD 9282 | Obsolete version of ~ find... |
ac6sfi 9283 | A version of ~ ac6s for fi... |
frfi 9284 | A partial order is well-fo... |
fimax2g 9285 | A finite set has a maximum... |
fimaxg 9286 | A finite set has a maximum... |
fisupg 9287 | Lemma showing existence an... |
wofi 9288 | A total order on a finite ... |
ordunifi 9289 | The maximum of a finite co... |
nnunifi 9290 | The union (supremum) of a ... |
unblem1 9291 | Lemma for ~ unbnn . After... |
unblem2 9292 | Lemma for ~ unbnn . The v... |
unblem3 9293 | Lemma for ~ unbnn . The v... |
unblem4 9294 | Lemma for ~ unbnn . The f... |
unbnn 9295 | Any unbounded subset of na... |
unbnn2 9296 | Version of ~ unbnn that do... |
isfinite2 9297 | Any set strictly dominated... |
nnsdomg 9298 | Omega strictly dominates a... |
nnsdomgOLD 9299 | Obsolete version of ~ nnsd... |
isfiniteg 9300 | A set is finite iff it is ... |
infsdomnn 9301 | An infinite set strictly d... |
infsdomnnOLD 9302 | Obsolete version of ~ infs... |
infn0 9303 | An infinite set is not emp... |
infn0ALT 9304 | Shorter proof of ~ infn0 u... |
fin2inf 9305 | This (useless) theorem, wh... |
unfilem1 9306 | Lemma for proving that the... |
unfilem2 9307 | Lemma for proving that the... |
unfilem3 9308 | Lemma for proving that the... |
unfiOLD 9309 | Obsolete version of ~ unfi... |
unfir 9310 | If a union is finite, the ... |
unfi2 9311 | The union of two finite se... |
difinf 9312 | An infinite set ` A ` minu... |
xpfi 9313 | The Cartesian product of t... |
xpfiOLD 9314 | Obsolete version of ~ xpfi... |
3xpfi 9315 | The Cartesian product of t... |
domunfican 9316 | A finite set union cancell... |
infcntss 9317 | Every infinite set has a d... |
prfi 9318 | An unordered pair is finit... |
tpfi 9319 | An unordered triple is fin... |
fiint 9320 | Equivalent ways of stating... |
fodomfi 9321 | An onto function implies d... |
fodomfib 9322 | Equivalence of an onto map... |
fofinf1o 9323 | Any surjection from one fi... |
rneqdmfinf1o 9324 | Any function from a finite... |
fidomdm 9325 | Any finite set dominates i... |
dmfi 9326 | The domain of a finite set... |
fundmfibi 9327 | A function is finite if an... |
resfnfinfin 9328 | The restriction of a funct... |
residfi 9329 | A restricted identity func... |
cnvfiALT 9330 | Shorter proof of ~ cnvfi u... |
rnfi 9331 | The range of a finite set ... |
f1dmvrnfibi 9332 | A one-to-one function whos... |
f1vrnfibi 9333 | A one-to-one function whic... |
fofi 9334 | If an onto function has a ... |
f1fi 9335 | If a 1-to-1 function has a... |
iunfi 9336 | The finite union of finite... |
unifi 9337 | The finite union of finite... |
unifi2 9338 | The finite union of finite... |
infssuni 9339 | If an infinite set ` A ` i... |
unirnffid 9340 | The union of the range of ... |
imafiALT 9341 | Shorter proof of ~ imafi u... |
pwfilemOLD 9342 | Obsolete version of ~ pwfi... |
pwfiOLD 9343 | Obsolete version of ~ pwfi... |
mapfi 9344 | Set exponentiation of fini... |
ixpfi 9345 | A Cartesian product of fin... |
ixpfi2 9346 | A Cartesian product of fin... |
mptfi 9347 | A finite mapping set is fi... |
abrexfi 9348 | An image set from a finite... |
cnvimamptfin 9349 | A preimage of a mapping wi... |
elfpw 9350 | Membership in a class of f... |
unifpw 9351 | A set is the union of its ... |
f1opwfi 9352 | A one-to-one mapping induc... |
fissuni 9353 | A finite subset of a union... |
fipreima 9354 | Given a finite subset ` A ... |
finsschain 9355 | A finite subset of the uni... |
indexfi 9356 | If for every element of a ... |
relfsupp 9359 | The property of a function... |
relprcnfsupp 9360 | A proper class is never fi... |
isfsupp 9361 | The property of a class to... |
isfsuppd 9362 | Deduction form of ~ isfsup... |
funisfsupp 9363 | The property of a function... |
fsuppimp 9364 | Implications of a class be... |
fsuppimpd 9365 | A finitely supported funct... |
fisuppfi 9366 | A function on a finite set... |
fidmfisupp 9367 | A function with a finite d... |
fdmfisuppfi 9368 | The support of a function ... |
fdmfifsupp 9369 | A function with a finite d... |
fsuppmptdm 9370 | A mapping with a finite do... |
fndmfisuppfi 9371 | The support of a function ... |
fndmfifsupp 9372 | A function with a finite d... |
suppeqfsuppbi 9373 | If two functions have the ... |
suppssfifsupp 9374 | If the support of a functi... |
fsuppsssupp 9375 | If the support of a functi... |
fsuppxpfi 9376 | The cartesian product of t... |
fczfsuppd 9377 | A constant function with v... |
fsuppun 9378 | The union of two finitely ... |
fsuppunfi 9379 | The union of the support o... |
fsuppunbi 9380 | If the union of two classe... |
0fsupp 9381 | The empty set is a finitel... |
snopfsupp 9382 | A singleton containing an ... |
funsnfsupp 9383 | Finite support for a funct... |
fsuppres 9384 | The restriction of a finit... |
fmptssfisupp 9385 | The restriction of a mappi... |
ressuppfi 9386 | If the support of the rest... |
resfsupp 9387 | If the restriction of a fu... |
resfifsupp 9388 | The restriction of a funct... |
ffsuppbi 9389 | Two ways of saying that a ... |
fsuppmptif 9390 | A function mapping an argu... |
sniffsupp 9391 | A function mapping all but... |
fsuppcolem 9392 | Lemma for ~ fsuppco . For... |
fsuppco 9393 | The composition of a 1-1 f... |
fsuppco2 9394 | The composition of a funct... |
fsuppcor 9395 | The composition of a funct... |
mapfienlem1 9396 | Lemma 1 for ~ mapfien . (... |
mapfienlem2 9397 | Lemma 2 for ~ mapfien . (... |
mapfienlem3 9398 | Lemma 3 for ~ mapfien . (... |
mapfien 9399 | A bijection of the base se... |
mapfien2 9400 | Equinumerousity relation f... |
fival 9403 | The set of all the finite ... |
elfi 9404 | Specific properties of an ... |
elfi2 9405 | The empty intersection nee... |
elfir 9406 | Sufficient condition for a... |
intrnfi 9407 | Sufficient condition for t... |
iinfi 9408 | An indexed intersection of... |
inelfi 9409 | The intersection of two se... |
ssfii 9410 | Any element of a set ` A `... |
fi0 9411 | The set of finite intersec... |
fieq0 9412 | A set is empty iff the cla... |
fiin 9413 | The elements of ` ( fi `` ... |
dffi2 9414 | The set of finite intersec... |
fiss 9415 | Subset relationship for fu... |
inficl 9416 | A set which is closed unde... |
fipwuni 9417 | The set of finite intersec... |
fisn 9418 | A singleton is closed unde... |
fiuni 9419 | The union of the finite in... |
fipwss 9420 | If a set is a family of su... |
elfiun 9421 | A finite intersection of e... |
dffi3 9422 | The set of finite intersec... |
fifo 9423 | Describe a surjection from... |
marypha1lem 9424 | Core induction for Philip ... |
marypha1 9425 | (Philip) Hall's marriage t... |
marypha2lem1 9426 | Lemma for ~ marypha2 . Pr... |
marypha2lem2 9427 | Lemma for ~ marypha2 . Pr... |
marypha2lem3 9428 | Lemma for ~ marypha2 . Pr... |
marypha2lem4 9429 | Lemma for ~ marypha2 . Pr... |
marypha2 9430 | Version of ~ marypha1 usin... |
dfsup2 9435 | Quantifier-free definition... |
supeq1 9436 | Equality theorem for supre... |
supeq1d 9437 | Equality deduction for sup... |
supeq1i 9438 | Equality inference for sup... |
supeq2 9439 | Equality theorem for supre... |
supeq3 9440 | Equality theorem for supre... |
supeq123d 9441 | Equality deduction for sup... |
nfsup 9442 | Hypothesis builder for sup... |
supmo 9443 | Any class ` B ` has at mos... |
supexd 9444 | A supremum is a set. (Con... |
supeu 9445 | A supremum is unique. Sim... |
supval2 9446 | Alternate expression for t... |
eqsup 9447 | Sufficient condition for a... |
eqsupd 9448 | Sufficient condition for a... |
supcl 9449 | A supremum belongs to its ... |
supub 9450 | A supremum is an upper bou... |
suplub 9451 | A supremum is the least up... |
suplub2 9452 | Bidirectional form of ~ su... |
supnub 9453 | An upper bound is not less... |
supex 9454 | A supremum is a set. (Con... |
sup00 9455 | The supremum under an empt... |
sup0riota 9456 | The supremum of an empty s... |
sup0 9457 | The supremum of an empty s... |
supmax 9458 | The greatest element of a ... |
fisup2g 9459 | A finite set satisfies the... |
fisupcl 9460 | A nonempty finite set cont... |
supgtoreq 9461 | The supremum of a finite s... |
suppr 9462 | The supremum of a pair. (... |
supsn 9463 | The supremum of a singleto... |
supisolem 9464 | Lemma for ~ supiso . (Con... |
supisoex 9465 | Lemma for ~ supiso . (Con... |
supiso 9466 | Image of a supremum under ... |
infeq1 9467 | Equality theorem for infim... |
infeq1d 9468 | Equality deduction for inf... |
infeq1i 9469 | Equality inference for inf... |
infeq2 9470 | Equality theorem for infim... |
infeq3 9471 | Equality theorem for infim... |
infeq123d 9472 | Equality deduction for inf... |
nfinf 9473 | Hypothesis builder for inf... |
infexd 9474 | An infimum is a set. (Con... |
eqinf 9475 | Sufficient condition for a... |
eqinfd 9476 | Sufficient condition for a... |
infval 9477 | Alternate expression for t... |
infcllem 9478 | Lemma for ~ infcl , ~ infl... |
infcl 9479 | An infimum belongs to its ... |
inflb 9480 | An infimum is a lower boun... |
infglb 9481 | An infimum is the greatest... |
infglbb 9482 | Bidirectional form of ~ in... |
infnlb 9483 | A lower bound is not great... |
infex 9484 | An infimum is a set. (Con... |
infmin 9485 | The smallest element of a ... |
infmo 9486 | Any class ` B ` has at mos... |
infeu 9487 | An infimum is unique. (Co... |
fimin2g 9488 | A finite set has a minimum... |
fiming 9489 | A finite set has a minimum... |
fiinfg 9490 | Lemma showing existence an... |
fiinf2g 9491 | A finite set satisfies the... |
fiinfcl 9492 | A nonempty finite set cont... |
infltoreq 9493 | The infimum of a finite se... |
infpr 9494 | The infimum of a pair. (C... |
infsupprpr 9495 | The infimum of a proper pa... |
infsn 9496 | The infimum of a singleton... |
inf00 9497 | The infimum regarding an e... |
infempty 9498 | The infimum of an empty se... |
infiso 9499 | Image of an infimum under ... |
dfoi 9502 | Rewrite ~ df-oi with abbre... |
oieq1 9503 | Equality theorem for ordin... |
oieq2 9504 | Equality theorem for ordin... |
nfoi 9505 | Hypothesis builder for ord... |
ordiso2 9506 | Generalize ~ ordiso to pro... |
ordiso 9507 | Order-isomorphic ordinal n... |
ordtypecbv 9508 | Lemma for ~ ordtype . (Co... |
ordtypelem1 9509 | Lemma for ~ ordtype . (Co... |
ordtypelem2 9510 | Lemma for ~ ordtype . (Co... |
ordtypelem3 9511 | Lemma for ~ ordtype . (Co... |
ordtypelem4 9512 | Lemma for ~ ordtype . (Co... |
ordtypelem5 9513 | Lemma for ~ ordtype . (Co... |
ordtypelem6 9514 | Lemma for ~ ordtype . (Co... |
ordtypelem7 9515 | Lemma for ~ ordtype . ` ra... |
ordtypelem8 9516 | Lemma for ~ ordtype . (Co... |
ordtypelem9 9517 | Lemma for ~ ordtype . Eit... |
ordtypelem10 9518 | Lemma for ~ ordtype . Usi... |
oi0 9519 | Definition of the ordinal ... |
oicl 9520 | The order type of the well... |
oif 9521 | The order isomorphism of t... |
oiiso2 9522 | The order isomorphism of t... |
ordtype 9523 | For any set-like well-orde... |
oiiniseg 9524 | ` ran F ` is an initial se... |
ordtype2 9525 | For any set-like well-orde... |
oiexg 9526 | The order isomorphism on a... |
oion 9527 | The order type of the well... |
oiiso 9528 | The order isomorphism of t... |
oien 9529 | The order type of a well-o... |
oieu 9530 | Uniqueness of the unique o... |
oismo 9531 | When ` A ` is a subclass o... |
oiid 9532 | The order type of an ordin... |
hartogslem1 9533 | Lemma for ~ hartogs . (Co... |
hartogslem2 9534 | Lemma for ~ hartogs . (Co... |
hartogs 9535 | The class of ordinals domi... |
wofib 9536 | The only sets which are we... |
wemaplem1 9537 | Value of the lexicographic... |
wemaplem2 9538 | Lemma for ~ wemapso . Tra... |
wemaplem3 9539 | Lemma for ~ wemapso . Tra... |
wemappo 9540 | Construct lexicographic or... |
wemapsolem 9541 | Lemma for ~ wemapso . (Co... |
wemapso 9542 | Construct lexicographic or... |
wemapso2lem 9543 | Lemma for ~ wemapso2 . (C... |
wemapso2 9544 | An alternative to having a... |
card2on 9545 | The alternate definition o... |
card2inf 9546 | The alternate definition o... |
harf 9549 | Functionality of the Harto... |
harcl 9550 | Values of the Hartogs func... |
harval 9551 | Function value of the Hart... |
elharval 9552 | The Hartogs number of a se... |
harndom 9553 | The Hartogs number of a se... |
harword 9554 | Weak ordering property of ... |
relwdom 9557 | Weak dominance is a relati... |
brwdom 9558 | Property of weak dominance... |
brwdomi 9559 | Property of weak dominance... |
brwdomn0 9560 | Weak dominance over nonemp... |
0wdom 9561 | Any set weakly dominates t... |
fowdom 9562 | An onto function implies w... |
wdomref 9563 | Reflexivity of weak domina... |
brwdom2 9564 | Alternate characterization... |
domwdom 9565 | Weak dominance is implied ... |
wdomtr 9566 | Transitivity of weak domin... |
wdomen1 9567 | Equality-like theorem for ... |
wdomen2 9568 | Equality-like theorem for ... |
wdompwdom 9569 | Weak dominance strengthens... |
canthwdom 9570 | Cantor's Theorem, stated u... |
wdom2d 9571 | Deduce weak dominance from... |
wdomd 9572 | Deduce weak dominance from... |
brwdom3 9573 | Condition for weak dominan... |
brwdom3i 9574 | Weak dominance implies exi... |
unwdomg 9575 | Weak dominance of a (disjo... |
xpwdomg 9576 | Weak dominance of a Cartes... |
wdomima2g 9577 | A set is weakly dominant o... |
wdomimag 9578 | A set is weakly dominant o... |
unxpwdom2 9579 | Lemma for ~ unxpwdom . (C... |
unxpwdom 9580 | If a Cartesian product is ... |
ixpiunwdom 9581 | Describe an onto function ... |
harwdom 9582 | The value of the Hartogs f... |
axreg2 9584 | Axiom of Regularity expres... |
zfregcl 9585 | The Axiom of Regularity wi... |
zfreg 9586 | The Axiom of Regularity us... |
elirrv 9587 | The membership relation is... |
elirr 9588 | No class is a member of it... |
elneq 9589 | A class is not equal to an... |
nelaneq 9590 | A class is not an element ... |
epinid0 9591 | The membership relation an... |
sucprcreg 9592 | A class is equal to its su... |
ruv 9593 | The Russell class is equal... |
ruALT 9594 | Alternate proof of ~ ru , ... |
disjcsn 9595 | A class is disjoint from i... |
zfregfr 9596 | The membership relation is... |
en2lp 9597 | No class has 2-cycle membe... |
elnanel 9598 | Two classes are not elemen... |
cnvepnep 9599 | The membership (epsilon) r... |
epnsym 9600 | The membership (epsilon) r... |
elnotel 9601 | A class cannot be an eleme... |
elnel 9602 | A class cannot be an eleme... |
en3lplem1 9603 | Lemma for ~ en3lp . (Cont... |
en3lplem2 9604 | Lemma for ~ en3lp . (Cont... |
en3lp 9605 | No class has 3-cycle membe... |
preleqg 9606 | Equality of two unordered ... |
preleq 9607 | Equality of two unordered ... |
preleqALT 9608 | Alternate proof of ~ prele... |
opthreg 9609 | Theorem for alternate repr... |
suc11reg 9610 | The successor operation be... |
dford2 9611 | Assuming ~ ax-reg , an ord... |
inf0 9612 | Existence of ` _om ` impli... |
inf1 9613 | Variation of Axiom of Infi... |
inf2 9614 | Variation of Axiom of Infi... |
inf3lema 9615 | Lemma for our Axiom of Inf... |
inf3lemb 9616 | Lemma for our Axiom of Inf... |
inf3lemc 9617 | Lemma for our Axiom of Inf... |
inf3lemd 9618 | Lemma for our Axiom of Inf... |
inf3lem1 9619 | Lemma for our Axiom of Inf... |
inf3lem2 9620 | Lemma for our Axiom of Inf... |
inf3lem3 9621 | Lemma for our Axiom of Inf... |
inf3lem4 9622 | Lemma for our Axiom of Inf... |
inf3lem5 9623 | Lemma for our Axiom of Inf... |
inf3lem6 9624 | Lemma for our Axiom of Inf... |
inf3lem7 9625 | Lemma for our Axiom of Inf... |
inf3 9626 | Our Axiom of Infinity ~ ax... |
infeq5i 9627 | Half of ~ infeq5 . (Contr... |
infeq5 9628 | The statement "there exist... |
zfinf 9630 | Axiom of Infinity expresse... |
axinf2 9631 | A standard version of Axio... |
zfinf2 9633 | A standard version of the ... |
omex 9634 | The existence of omega (th... |
axinf 9635 | The first version of the A... |
inf5 9636 | The statement "there exist... |
omelon 9637 | Omega is an ordinal number... |
dfom3 9638 | The class of natural numbe... |
elom3 9639 | A simplification of ~ elom... |
dfom4 9640 | A simplification of ~ df-o... |
dfom5 9641 | ` _om ` is the smallest li... |
oancom 9642 | Ordinal addition is not co... |
isfinite 9643 | A set is finite iff it is ... |
fict 9644 | A finite set is countable ... |
nnsdom 9645 | A natural number is strict... |
omenps 9646 | Omega is equinumerous to a... |
omensuc 9647 | The set of natural numbers... |
infdifsn 9648 | Removing a singleton from ... |
infdiffi 9649 | Removing a finite set from... |
unbnn3 9650 | Any unbounded subset of na... |
noinfep 9651 | Using the Axiom of Regular... |
cantnffval 9654 | The value of the Cantor no... |
cantnfdm 9655 | The domain of the Cantor n... |
cantnfvalf 9656 | Lemma for ~ cantnf . The ... |
cantnfs 9657 | Elementhood in the set of ... |
cantnfcl 9658 | Basic properties of the or... |
cantnfval 9659 | The value of the Cantor no... |
cantnfval2 9660 | Alternate expression for t... |
cantnfsuc 9661 | The value of the recursive... |
cantnfle 9662 | A lower bound on the ` CNF... |
cantnflt 9663 | An upper bound on the part... |
cantnflt2 9664 | An upper bound on the ` CN... |
cantnff 9665 | The ` CNF ` function is a ... |
cantnf0 9666 | The value of the zero func... |
cantnfrescl 9667 | A function is finitely sup... |
cantnfres 9668 | The ` CNF ` function respe... |
cantnfp1lem1 9669 | Lemma for ~ cantnfp1 . (C... |
cantnfp1lem2 9670 | Lemma for ~ cantnfp1 . (C... |
cantnfp1lem3 9671 | Lemma for ~ cantnfp1 . (C... |
cantnfp1 9672 | If ` F ` is created by add... |
oemapso 9673 | The relation ` T ` is a st... |
oemapval 9674 | Value of the relation ` T ... |
oemapvali 9675 | If ` F < G ` , then there ... |
cantnflem1a 9676 | Lemma for ~ cantnf . (Con... |
cantnflem1b 9677 | Lemma for ~ cantnf . (Con... |
cantnflem1c 9678 | Lemma for ~ cantnf . (Con... |
cantnflem1d 9679 | Lemma for ~ cantnf . (Con... |
cantnflem1 9680 | Lemma for ~ cantnf . This... |
cantnflem2 9681 | Lemma for ~ cantnf . (Con... |
cantnflem3 9682 | Lemma for ~ cantnf . Here... |
cantnflem4 9683 | Lemma for ~ cantnf . Comp... |
cantnf 9684 | The Cantor Normal Form the... |
oemapwe 9685 | The lexicographic order on... |
cantnffval2 9686 | An alternate definition of... |
cantnff1o 9687 | Simplify the isomorphism o... |
wemapwe 9688 | Construct lexicographic or... |
oef1o 9689 | A bijection of the base se... |
cnfcomlem 9690 | Lemma for ~ cnfcom . (Con... |
cnfcom 9691 | Any ordinal ` B ` is equin... |
cnfcom2lem 9692 | Lemma for ~ cnfcom2 . (Co... |
cnfcom2 9693 | Any nonzero ordinal ` B ` ... |
cnfcom3lem 9694 | Lemma for ~ cnfcom3 . (Co... |
cnfcom3 9695 | Any infinite ordinal ` B `... |
cnfcom3clem 9696 | Lemma for ~ cnfcom3c . (C... |
cnfcom3c 9697 | Wrap the construction of ~... |
ttrcleq 9700 | Equality theorem for trans... |
nfttrcld 9701 | Bound variable hypothesis ... |
nfttrcl 9702 | Bound variable hypothesis ... |
relttrcl 9703 | The transitive closure of ... |
brttrcl 9704 | Characterization of elemen... |
brttrcl2 9705 | Characterization of elemen... |
ssttrcl 9706 | If ` R ` is a relation, th... |
ttrcltr 9707 | The transitive closure of ... |
ttrclresv 9708 | The transitive closure of ... |
ttrclco 9709 | Composition law for the tr... |
cottrcl 9710 | Composition law for the tr... |
ttrclss 9711 | If ` R ` is a subclass of ... |
dmttrcl 9712 | The domain of a transitive... |
rnttrcl 9713 | The range of a transitive ... |
ttrclexg 9714 | If ` R ` is a set, then so... |
dfttrcl2 9715 | When ` R ` is a set and a ... |
ttrclselem1 9716 | Lemma for ~ ttrclse . Sho... |
ttrclselem2 9717 | Lemma for ~ ttrclse . Sho... |
ttrclse 9718 | If ` R ` is set-like over ... |
trcl 9719 | For any set ` A ` , show t... |
tz9.1 9720 | Every set has a transitive... |
tz9.1c 9721 | Alternate expression for t... |
epfrs 9722 | The strong form of the Axi... |
zfregs 9723 | The strong form of the Axi... |
zfregs2 9724 | Alternate strong form of t... |
setind 9725 | Set (epsilon) induction. ... |
setind2 9726 | Set (epsilon) induction, s... |
tcvalg 9729 | Value of the transitive cl... |
tcid 9730 | Defining property of the t... |
tctr 9731 | Defining property of the t... |
tcmin 9732 | Defining property of the t... |
tc2 9733 | A variant of the definitio... |
tcsni 9734 | The transitive closure of ... |
tcss 9735 | The transitive closure fun... |
tcel 9736 | The transitive closure fun... |
tcidm 9737 | The transitive closure fun... |
tc0 9738 | The transitive closure of ... |
tc00 9739 | The transitive closure is ... |
frmin 9740 | Every (possibly proper) su... |
frind 9741 | A subclass of a well-found... |
frinsg 9742 | Well-Founded Induction Sch... |
frins 9743 | Well-Founded Induction Sch... |
frins2f 9744 | Well-Founded Induction sch... |
frins2 9745 | Well-Founded Induction sch... |
frins3 9746 | Well-Founded Induction sch... |
frr3g 9747 | Functions defined by well-... |
frrlem15 9748 | Lemma for general well-fou... |
frrlem16 9749 | Lemma for general well-fou... |
frr1 9750 | Law of general well-founde... |
frr2 9751 | Law of general well-founde... |
frr3 9752 | Law of general well-founde... |
r1funlim 9757 | The cumulative hierarchy o... |
r1fnon 9758 | The cumulative hierarchy o... |
r10 9759 | Value of the cumulative hi... |
r1sucg 9760 | Value of the cumulative hi... |
r1suc 9761 | Value of the cumulative hi... |
r1limg 9762 | Value of the cumulative hi... |
r1lim 9763 | Value of the cumulative hi... |
r1fin 9764 | The first ` _om ` levels o... |
r1sdom 9765 | Each stage in the cumulati... |
r111 9766 | The cumulative hierarchy i... |
r1tr 9767 | The cumulative hierarchy o... |
r1tr2 9768 | The union of a cumulative ... |
r1ordg 9769 | Ordering relation for the ... |
r1ord3g 9770 | Ordering relation for the ... |
r1ord 9771 | Ordering relation for the ... |
r1ord2 9772 | Ordering relation for the ... |
r1ord3 9773 | Ordering relation for the ... |
r1sssuc 9774 | The value of the cumulativ... |
r1pwss 9775 | Each set of the cumulative... |
r1sscl 9776 | Each set of the cumulative... |
r1val1 9777 | The value of the cumulativ... |
tz9.12lem1 9778 | Lemma for ~ tz9.12 . (Con... |
tz9.12lem2 9779 | Lemma for ~ tz9.12 . (Con... |
tz9.12lem3 9780 | Lemma for ~ tz9.12 . (Con... |
tz9.12 9781 | A set is well-founded if a... |
tz9.13 9782 | Every set is well-founded,... |
tz9.13g 9783 | Every set is well-founded,... |
rankwflemb 9784 | Two ways of saying a set i... |
rankf 9785 | The domain and codomain of... |
rankon 9786 | The rank of a set is an or... |
r1elwf 9787 | Any member of the cumulati... |
rankvalb 9788 | Value of the rank function... |
rankr1ai 9789 | One direction of ~ rankr1a... |
rankvaln 9790 | Value of the rank function... |
rankidb 9791 | Identity law for the rank ... |
rankdmr1 9792 | A rank is a member of the ... |
rankr1ag 9793 | A version of ~ rankr1a tha... |
rankr1bg 9794 | A relationship between ran... |
r1rankidb 9795 | Any set is a subset of the... |
r1elssi 9796 | The range of the ` R1 ` fu... |
r1elss 9797 | The range of the ` R1 ` fu... |
pwwf 9798 | A power set is well-founde... |
sswf 9799 | A subset of a well-founded... |
snwf 9800 | A singleton is well-founde... |
unwf 9801 | A binary union is well-fou... |
prwf 9802 | An unordered pair is well-... |
opwf 9803 | An ordered pair is well-fo... |
unir1 9804 | The cumulative hierarchy o... |
jech9.3 9805 | Every set belongs to some ... |
rankwflem 9806 | Every set is well-founded,... |
rankval 9807 | Value of the rank function... |
rankvalg 9808 | Value of the rank function... |
rankval2 9809 | Value of an alternate defi... |
uniwf 9810 | A union is well-founded if... |
rankr1clem 9811 | Lemma for ~ rankr1c . (Co... |
rankr1c 9812 | A relationship between the... |
rankidn 9813 | A relationship between the... |
rankpwi 9814 | The rank of a power set. ... |
rankelb 9815 | The membership relation is... |
wfelirr 9816 | A well-founded set is not ... |
rankval3b 9817 | The value of the rank func... |
ranksnb 9818 | The rank of a singleton. ... |
rankonidlem 9819 | Lemma for ~ rankonid . (C... |
rankonid 9820 | The rank of an ordinal num... |
onwf 9821 | The ordinals are all well-... |
onssr1 9822 | Initial segments of the or... |
rankr1g 9823 | A relationship between the... |
rankid 9824 | Identity law for the rank ... |
rankr1 9825 | A relationship between the... |
ssrankr1 9826 | A relationship between an ... |
rankr1a 9827 | A relationship between ran... |
r1val2 9828 | The value of the cumulativ... |
r1val3 9829 | The value of the cumulativ... |
rankel 9830 | The membership relation is... |
rankval3 9831 | The value of the rank func... |
bndrank 9832 | Any class whose elements h... |
unbndrank 9833 | The elements of a proper c... |
rankpw 9834 | The rank of a power set. ... |
ranklim 9835 | The rank of a set belongs ... |
r1pw 9836 | A stronger property of ` R... |
r1pwALT 9837 | Alternate shorter proof of... |
r1pwcl 9838 | The cumulative hierarchy o... |
rankssb 9839 | The subset relation is inh... |
rankss 9840 | The subset relation is inh... |
rankunb 9841 | The rank of the union of t... |
rankprb 9842 | The rank of an unordered p... |
rankopb 9843 | The rank of an ordered pai... |
rankuni2b 9844 | The value of the rank func... |
ranksn 9845 | The rank of a singleton. ... |
rankuni2 9846 | The rank of a union. Part... |
rankun 9847 | The rank of the union of t... |
rankpr 9848 | The rank of an unordered p... |
rankop 9849 | The rank of an ordered pai... |
r1rankid 9850 | Any set is a subset of the... |
rankeq0b 9851 | A set is empty iff its ran... |
rankeq0 9852 | A set is empty iff its ran... |
rankr1id 9853 | The rank of the hierarchy ... |
rankuni 9854 | The rank of a union. Part... |
rankr1b 9855 | A relationship between ran... |
ranksuc 9856 | The rank of a successor. ... |
rankuniss 9857 | Upper bound of the rank of... |
rankval4 9858 | The rank of a set is the s... |
rankbnd 9859 | The rank of a set is bound... |
rankbnd2 9860 | The rank of a set is bound... |
rankc1 9861 | A relationship that can be... |
rankc2 9862 | A relationship that can be... |
rankelun 9863 | Rank membership is inherit... |
rankelpr 9864 | Rank membership is inherit... |
rankelop 9865 | Rank membership is inherit... |
rankxpl 9866 | A lower bound on the rank ... |
rankxpu 9867 | An upper bound on the rank... |
rankfu 9868 | An upper bound on the rank... |
rankmapu 9869 | An upper bound on the rank... |
rankxplim 9870 | The rank of a Cartesian pr... |
rankxplim2 9871 | If the rank of a Cartesian... |
rankxplim3 9872 | The rank of a Cartesian pr... |
rankxpsuc 9873 | The rank of a Cartesian pr... |
tcwf 9874 | The transitive closure fun... |
tcrank 9875 | This theorem expresses two... |
scottex 9876 | Scott's trick collects all... |
scott0 9877 | Scott's trick collects all... |
scottexs 9878 | Theorem scheme version of ... |
scott0s 9879 | Theorem scheme version of ... |
cplem1 9880 | Lemma for the Collection P... |
cplem2 9881 | Lemma for the Collection P... |
cp 9882 | Collection Principle. Thi... |
bnd 9883 | A very strong generalizati... |
bnd2 9884 | A variant of the Boundedne... |
kardex 9885 | The collection of all sets... |
karden 9886 | If we allow the Axiom of R... |
htalem 9887 | Lemma for defining an emul... |
hta 9888 | A ZFC emulation of Hilbert... |
djueq12 9895 | Equality theorem for disjo... |
djueq1 9896 | Equality theorem for disjo... |
djueq2 9897 | Equality theorem for disjo... |
nfdju 9898 | Bound-variable hypothesis ... |
djuex 9899 | The disjoint union of sets... |
djuexb 9900 | The disjoint union of two ... |
djulcl 9901 | Left closure of disjoint u... |
djurcl 9902 | Right closure of disjoint ... |
djulf1o 9903 | The left injection functio... |
djurf1o 9904 | The right injection functi... |
inlresf 9905 | The left injection restric... |
inlresf1 9906 | The left injection restric... |
inrresf 9907 | The right injection restri... |
inrresf1 9908 | The right injection restri... |
djuin 9909 | The images of any classes ... |
djur 9910 | A member of a disjoint uni... |
djuss 9911 | A disjoint union is a subc... |
djuunxp 9912 | The union of a disjoint un... |
djuexALT 9913 | Alternate proof of ~ djuex... |
eldju1st 9914 | The first component of an ... |
eldju2ndl 9915 | The second component of an... |
eldju2ndr 9916 | The second component of an... |
djuun 9917 | The disjoint union of two ... |
1stinl 9918 | The first component of the... |
2ndinl 9919 | The second component of th... |
1stinr 9920 | The first component of the... |
2ndinr 9921 | The second component of th... |
updjudhf 9922 | The mapping of an element ... |
updjudhcoinlf 9923 | The composition of the map... |
updjudhcoinrg 9924 | The composition of the map... |
updjud 9925 | Universal property of the ... |
cardf2 9934 | The cardinality function i... |
cardon 9935 | The cardinal number of a s... |
isnum2 9936 | A way to express well-orde... |
isnumi 9937 | A set equinumerous to an o... |
ennum 9938 | Equinumerous sets are equi... |
finnum 9939 | Every finite set is numera... |
onenon 9940 | Every ordinal number is nu... |
tskwe 9941 | A Tarski set is well-order... |
xpnum 9942 | The cartesian product of n... |
cardval3 9943 | An alternate definition of... |
cardid2 9944 | Any numerable set is equin... |
isnum3 9945 | A set is numerable iff it ... |
oncardval 9946 | The value of the cardinal ... |
oncardid 9947 | Any ordinal number is equi... |
cardonle 9948 | The cardinal of an ordinal... |
card0 9949 | The cardinality of the emp... |
cardidm 9950 | The cardinality function i... |
oncard 9951 | A set is a cardinal number... |
ficardom 9952 | The cardinal number of a f... |
ficardid 9953 | A finite set is equinumero... |
cardnn 9954 | The cardinality of a natur... |
cardnueq0 9955 | The empty set is the only ... |
cardne 9956 | No member of a cardinal nu... |
carden2a 9957 | If two sets have equal non... |
carden2b 9958 | If two sets are equinumero... |
card1 9959 | A set has cardinality one ... |
cardsn 9960 | A singleton has cardinalit... |
carddomi2 9961 | Two sets have the dominanc... |
sdomsdomcardi 9962 | A set strictly dominates i... |
cardlim 9963 | An infinite cardinal is a ... |
cardsdomelir 9964 | A cardinal strictly domina... |
cardsdomel 9965 | A cardinal strictly domina... |
iscard 9966 | Two ways to express the pr... |
iscard2 9967 | Two ways to express the pr... |
carddom2 9968 | Two numerable sets have th... |
harcard 9969 | The class of ordinal numbe... |
cardprclem 9970 | Lemma for ~ cardprc . (Co... |
cardprc 9971 | The class of all cardinal ... |
carduni 9972 | The union of a set of card... |
cardiun 9973 | The indexed union of a set... |
cardennn 9974 | If ` A ` is equinumerous t... |
cardsucinf 9975 | The cardinality of the suc... |
cardsucnn 9976 | The cardinality of the suc... |
cardom 9977 | The set of natural numbers... |
carden2 9978 | Two numerable sets are equ... |
cardsdom2 9979 | A numerable set is strictl... |
domtri2 9980 | Trichotomy of dominance fo... |
nnsdomel 9981 | Strict dominance and eleme... |
cardval2 9982 | An alternate version of th... |
isinffi 9983 | An infinite set contains s... |
fidomtri 9984 | Trichotomy of dominance wi... |
fidomtri2 9985 | Trichotomy of dominance wi... |
harsdom 9986 | The Hartogs number of a we... |
onsdom 9987 | Any well-orderable set is ... |
harval2 9988 | An alternate expression fo... |
harsucnn 9989 | The next cardinal after a ... |
cardmin2 9990 | The smallest ordinal that ... |
pm54.43lem 9991 | In Theorem *54.43 of [Whit... |
pm54.43 9992 | Theorem *54.43 of [Whitehe... |
enpr2 9993 | An unordered pair with dis... |
pr2nelemOLD 9994 | Obsolete version of ~ enpr... |
pr2ne 9995 | If an unordered pair has t... |
pr2neOLD 9996 | Obsolete version of ~ pr2n... |
prdom2 9997 | An unordered pair has at m... |
en2eqpr 9998 | Building a set with two el... |
en2eleq 9999 | Express a set of pair card... |
en2other2 10000 | Taking the other element t... |
dif1card 10001 | The cardinality of a nonem... |
leweon 10002 | Lexicographical order is a... |
r0weon 10003 | A set-like well-ordering o... |
infxpenlem 10004 | Lemma for ~ infxpen . (Co... |
infxpen 10005 | Every infinite ordinal is ... |
xpomen 10006 | The Cartesian product of o... |
xpct 10007 | The cartesian product of t... |
infxpidm2 10008 | Every infinite well-ordera... |
infxpenc 10009 | A canonical version of ~ i... |
infxpenc2lem1 10010 | Lemma for ~ infxpenc2 . (... |
infxpenc2lem2 10011 | Lemma for ~ infxpenc2 . (... |
infxpenc2lem3 10012 | Lemma for ~ infxpenc2 . (... |
infxpenc2 10013 | Existence form of ~ infxpe... |
iunmapdisj 10014 | The union ` U_ n e. C ( A ... |
fseqenlem1 10015 | Lemma for ~ fseqen . (Con... |
fseqenlem2 10016 | Lemma for ~ fseqen . (Con... |
fseqdom 10017 | One half of ~ fseqen . (C... |
fseqen 10018 | A set that is equinumerous... |
infpwfidom 10019 | The collection of finite s... |
dfac8alem 10020 | Lemma for ~ dfac8a . If t... |
dfac8a 10021 | Numeration theorem: every ... |
dfac8b 10022 | The well-ordering theorem:... |
dfac8clem 10023 | Lemma for ~ dfac8c . (Con... |
dfac8c 10024 | If the union of a set is w... |
ac10ct 10025 | A proof of the well-orderi... |
ween 10026 | A set is numerable iff it ... |
ac5num 10027 | A version of ~ ac5b with t... |
ondomen 10028 | If a set is dominated by a... |
numdom 10029 | A set dominated by a numer... |
ssnum 10030 | A subset of a numerable se... |
onssnum 10031 | All subsets of the ordinal... |
indcardi 10032 | Indirect strong induction ... |
acnrcl 10033 | Reverse closure for the ch... |
acneq 10034 | Equality theorem for the c... |
isacn 10035 | The property of being a ch... |
acni 10036 | The property of being a ch... |
acni2 10037 | The property of being a ch... |
acni3 10038 | The property of being a ch... |
acnlem 10039 | Construct a mapping satisf... |
numacn 10040 | A well-orderable set has c... |
finacn 10041 | Every set has finite choic... |
acndom 10042 | A set with long choice seq... |
acnnum 10043 | A set ` X ` which has choi... |
acnen 10044 | The class of choice sets o... |
acndom2 10045 | A set smaller than one wit... |
acnen2 10046 | The class of sets with cho... |
fodomacn 10047 | A version of ~ fodom that ... |
fodomnum 10048 | A version of ~ fodom that ... |
fonum 10049 | A surjection maps numerabl... |
numwdom 10050 | A surjection maps numerabl... |
fodomfi2 10051 | Onto functions define domi... |
wdomfil 10052 | Weak dominance agrees with... |
infpwfien 10053 | Any infinite well-orderabl... |
inffien 10054 | The set of finite intersec... |
wdomnumr 10055 | Weak dominance agrees with... |
alephfnon 10056 | The aleph function is a fu... |
aleph0 10057 | The first infinite cardina... |
alephlim 10058 | Value of the aleph functio... |
alephsuc 10059 | Value of the aleph functio... |
alephon 10060 | An aleph is an ordinal num... |
alephcard 10061 | Every aleph is a cardinal ... |
alephnbtwn 10062 | No cardinal can be sandwic... |
alephnbtwn2 10063 | No set has equinumerosity ... |
alephordilem1 10064 | Lemma for ~ alephordi . (... |
alephordi 10065 | Strict ordering property o... |
alephord 10066 | Ordering property of the a... |
alephord2 10067 | Ordering property of the a... |
alephord2i 10068 | Ordering property of the a... |
alephord3 10069 | Ordering property of the a... |
alephsucdom 10070 | A set dominated by an alep... |
alephsuc2 10071 | An alternate representatio... |
alephdom 10072 | Relationship between inclu... |
alephgeom 10073 | Every aleph is greater tha... |
alephislim 10074 | Every aleph is a limit ord... |
aleph11 10075 | The aleph function is one-... |
alephf1 10076 | The aleph function is a on... |
alephsdom 10077 | If an ordinal is smaller t... |
alephdom2 10078 | A dominated initial ordina... |
alephle 10079 | The argument of the aleph ... |
cardaleph 10080 | Given any transfinite card... |
cardalephex 10081 | Every transfinite cardinal... |
infenaleph 10082 | An infinite numerable set ... |
isinfcard 10083 | Two ways to express the pr... |
iscard3 10084 | Two ways to express the pr... |
cardnum 10085 | Two ways to express the cl... |
alephinit 10086 | An infinite initial ordina... |
carduniima 10087 | The union of the image of ... |
cardinfima 10088 | If a mapping to cardinals ... |
alephiso 10089 | Aleph is an order isomorph... |
alephprc 10090 | The class of all transfini... |
alephsson 10091 | The class of transfinite c... |
unialeph 10092 | The union of the class of ... |
alephsmo 10093 | The aleph function is stri... |
alephf1ALT 10094 | Alternate proof of ~ aleph... |
alephfplem1 10095 | Lemma for ~ alephfp . (Co... |
alephfplem2 10096 | Lemma for ~ alephfp . (Co... |
alephfplem3 10097 | Lemma for ~ alephfp . (Co... |
alephfplem4 10098 | Lemma for ~ alephfp . (Co... |
alephfp 10099 | The aleph function has a f... |
alephfp2 10100 | The aleph function has at ... |
alephval3 10101 | An alternate way to expres... |
alephsucpw2 10102 | The power set of an aleph ... |
mappwen 10103 | Power rule for cardinal ar... |
finnisoeu 10104 | A finite totally ordered s... |
iunfictbso 10105 | Countability of a countabl... |
aceq1 10108 | Equivalence of two version... |
aceq0 10109 | Equivalence of two version... |
aceq2 10110 | Equivalence of two version... |
aceq3lem 10111 | Lemma for ~ dfac3 . (Cont... |
dfac3 10112 | Equivalence of two version... |
dfac4 10113 | Equivalence of two version... |
dfac5lem1 10114 | Lemma for ~ dfac5 . (Cont... |
dfac5lem2 10115 | Lemma for ~ dfac5 . (Cont... |
dfac5lem3 10116 | Lemma for ~ dfac5 . (Cont... |
dfac5lem4 10117 | Lemma for ~ dfac5 . (Cont... |
dfac5lem5 10118 | Lemma for ~ dfac5 . (Cont... |
dfac5 10119 | Equivalence of two version... |
dfac2a 10120 | Our Axiom of Choice (in th... |
dfac2b 10121 | Axiom of Choice (first for... |
dfac2 10122 | Axiom of Choice (first for... |
dfac7 10123 | Equivalence of the Axiom o... |
dfac0 10124 | Equivalence of two version... |
dfac1 10125 | Equivalence of two version... |
dfac8 10126 | A proof of the equivalency... |
dfac9 10127 | Equivalence of the axiom o... |
dfac10 10128 | Axiom of Choice equivalent... |
dfac10c 10129 | Axiom of Choice equivalent... |
dfac10b 10130 | Axiom of Choice equivalent... |
acacni 10131 | A choice equivalent: every... |
dfacacn 10132 | A choice equivalent: every... |
dfac13 10133 | The axiom of choice holds ... |
dfac12lem1 10134 | Lemma for ~ dfac12 . (Con... |
dfac12lem2 10135 | Lemma for ~ dfac12 . (Con... |
dfac12lem3 10136 | Lemma for ~ dfac12 . (Con... |
dfac12r 10137 | The axiom of choice holds ... |
dfac12k 10138 | Equivalence of ~ dfac12 an... |
dfac12a 10139 | The axiom of choice holds ... |
dfac12 10140 | The axiom of choice holds ... |
kmlem1 10141 | Lemma for 5-quantifier AC ... |
kmlem2 10142 | Lemma for 5-quantifier AC ... |
kmlem3 10143 | Lemma for 5-quantifier AC ... |
kmlem4 10144 | Lemma for 5-quantifier AC ... |
kmlem5 10145 | Lemma for 5-quantifier AC ... |
kmlem6 10146 | Lemma for 5-quantifier AC ... |
kmlem7 10147 | Lemma for 5-quantifier AC ... |
kmlem8 10148 | Lemma for 5-quantifier AC ... |
kmlem9 10149 | Lemma for 5-quantifier AC ... |
kmlem10 10150 | Lemma for 5-quantifier AC ... |
kmlem11 10151 | Lemma for 5-quantifier AC ... |
kmlem12 10152 | Lemma for 5-quantifier AC ... |
kmlem13 10153 | Lemma for 5-quantifier AC ... |
kmlem14 10154 | Lemma for 5-quantifier AC ... |
kmlem15 10155 | Lemma for 5-quantifier AC ... |
kmlem16 10156 | Lemma for 5-quantifier AC ... |
dfackm 10157 | Equivalence of the Axiom o... |
undjudom 10158 | Cardinal addition dominate... |
endjudisj 10159 | Equinumerosity of a disjoi... |
djuen 10160 | Disjoint unions of equinum... |
djuenun 10161 | Disjoint union is equinume... |
dju1en 10162 | Cardinal addition with car... |
dju1dif 10163 | Adding and subtracting one... |
dju1p1e2 10164 | 1+1=2 for cardinal number ... |
dju1p1e2ALT 10165 | Alternate proof of ~ dju1p... |
dju0en 10166 | Cardinal addition with car... |
xp2dju 10167 | Two times a cardinal numbe... |
djucomen 10168 | Commutative law for cardin... |
djuassen 10169 | Associative law for cardin... |
xpdjuen 10170 | Cardinal multiplication di... |
mapdjuen 10171 | Sum of exponents law for c... |
pwdjuen 10172 | Sum of exponents law for c... |
djudom1 10173 | Ordering law for cardinal ... |
djudom2 10174 | Ordering law for cardinal ... |
djudoml 10175 | A set is dominated by its ... |
djuxpdom 10176 | Cartesian product dominate... |
djufi 10177 | The disjoint union of two ... |
cdainflem 10178 | Any partition of omega int... |
djuinf 10179 | A set is infinite iff the ... |
infdju1 10180 | An infinite set is equinum... |
pwdju1 10181 | The sum of a powerset with... |
pwdjuidm 10182 | If the natural numbers inj... |
djulepw 10183 | If ` A ` is idempotent und... |
onadju 10184 | The cardinal and ordinal s... |
cardadju 10185 | The cardinal sum is equinu... |
djunum 10186 | The disjoint union of two ... |
unnum 10187 | The union of two numerable... |
nnadju 10188 | The cardinal and ordinal s... |
nnadjuALT 10189 | Shorter proof of ~ nnadju ... |
ficardadju 10190 | The disjoint union of fini... |
ficardun 10191 | The cardinality of the uni... |
ficardunOLD 10192 | Obsolete version of ~ fica... |
ficardun2 10193 | The cardinality of the uni... |
ficardun2OLD 10194 | Obsolete version of ~ fica... |
pwsdompw 10195 | Lemma for ~ domtriom . Th... |
unctb 10196 | The union of two countable... |
infdjuabs 10197 | Absorption law for additio... |
infunabs 10198 | An infinite set is equinum... |
infdju 10199 | The sum of two cardinal nu... |
infdif 10200 | The cardinality of an infi... |
infdif2 10201 | Cardinality ordering for a... |
infxpdom 10202 | Dominance law for multipli... |
infxpabs 10203 | Absorption law for multipl... |
infunsdom1 10204 | The union of two sets that... |
infunsdom 10205 | The union of two sets that... |
infxp 10206 | Absorption law for multipl... |
pwdjudom 10207 | A property of dominance ov... |
infpss 10208 | Every infinite set has an ... |
infmap2 10209 | An exponentiation law for ... |
ackbij2lem1 10210 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem1 10211 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem2 10212 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem3 10213 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem4 10214 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem5 10215 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem6 10216 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem7 10217 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem8 10218 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem9 10219 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem10 10220 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem11 10221 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem12 10222 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem13 10223 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem14 10224 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem15 10225 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem16 10226 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem17 10227 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem18 10228 | Lemma for ~ ackbij1 . (Co... |
ackbij1 10229 | The Ackermann bijection, p... |
ackbij1b 10230 | The Ackermann bijection, p... |
ackbij2lem2 10231 | Lemma for ~ ackbij2 . (Co... |
ackbij2lem3 10232 | Lemma for ~ ackbij2 . (Co... |
ackbij2lem4 10233 | Lemma for ~ ackbij2 . (Co... |
ackbij2 10234 | The Ackermann bijection, p... |
r1om 10235 | The set of hereditarily fi... |
fictb 10236 | A set is countable iff its... |
cflem 10237 | A lemma used to simplify c... |
cfval 10238 | Value of the cofinality fu... |
cff 10239 | Cofinality is a function o... |
cfub 10240 | An upper bound on cofinali... |
cflm 10241 | Value of the cofinality fu... |
cf0 10242 | Value of the cofinality fu... |
cardcf 10243 | Cofinality is a cardinal n... |
cflecard 10244 | Cofinality is bounded by t... |
cfle 10245 | Cofinality is bounded by i... |
cfon 10246 | The cofinality of any set ... |
cfeq0 10247 | Only the ordinal zero has ... |
cfsuc 10248 | Value of the cofinality fu... |
cff1 10249 | There is always a map from... |
cfflb 10250 | If there is a cofinal map ... |
cfval2 10251 | Another expression for the... |
coflim 10252 | A simpler expression for t... |
cflim3 10253 | Another expression for the... |
cflim2 10254 | The cofinality function is... |
cfom 10255 | Value of the cofinality fu... |
cfss 10256 | There is a cofinal subset ... |
cfslb 10257 | Any cofinal subset of ` A ... |
cfslbn 10258 | Any subset of ` A ` smalle... |
cfslb2n 10259 | Any small collection of sm... |
cofsmo 10260 | Any cofinal map implies th... |
cfsmolem 10261 | Lemma for ~ cfsmo . (Cont... |
cfsmo 10262 | The map in ~ cff1 can be a... |
cfcoflem 10263 | Lemma for ~ cfcof , showin... |
coftr 10264 | If there is a cofinal map ... |
cfcof 10265 | If there is a cofinal map ... |
cfidm 10266 | The cofinality function is... |
alephsing 10267 | The cofinality of a limit ... |
sornom 10268 | The range of a single-step... |
isfin1a 10283 | Definition of a Ia-finite ... |
fin1ai 10284 | Property of a Ia-finite se... |
isfin2 10285 | Definition of a II-finite ... |
fin2i 10286 | Property of a II-finite se... |
isfin3 10287 | Definition of a III-finite... |
isfin4 10288 | Definition of a IV-finite ... |
fin4i 10289 | Infer that a set is IV-inf... |
isfin5 10290 | Definition of a V-finite s... |
isfin6 10291 | Definition of a VI-finite ... |
isfin7 10292 | Definition of a VII-finite... |
sdom2en01 10293 | A set with less than two e... |
infpssrlem1 10294 | Lemma for ~ infpssr . (Co... |
infpssrlem2 10295 | Lemma for ~ infpssr . (Co... |
infpssrlem3 10296 | Lemma for ~ infpssr . (Co... |
infpssrlem4 10297 | Lemma for ~ infpssr . (Co... |
infpssrlem5 10298 | Lemma for ~ infpssr . (Co... |
infpssr 10299 | Dedekind infinity implies ... |
fin4en1 10300 | Dedekind finite is a cardi... |
ssfin4 10301 | Dedekind finite sets have ... |
domfin4 10302 | A set dominated by a Dedek... |
ominf4 10303 | ` _om ` is Dedekind infini... |
infpssALT 10304 | Alternate proof of ~ infps... |
isfin4-2 10305 | Alternate definition of IV... |
isfin4p1 10306 | Alternate definition of IV... |
fin23lem7 10307 | Lemma for ~ isfin2-2 . Th... |
fin23lem11 10308 | Lemma for ~ isfin2-2 . (C... |
fin2i2 10309 | A II-finite set contains m... |
isfin2-2 10310 | ` Fin2 ` expressed in term... |
ssfin2 10311 | A subset of a II-finite se... |
enfin2i 10312 | II-finiteness is a cardina... |
fin23lem24 10313 | Lemma for ~ fin23 . In a ... |
fincssdom 10314 | In a chain of finite sets,... |
fin23lem25 10315 | Lemma for ~ fin23 . In a ... |
fin23lem26 10316 | Lemma for ~ fin23lem22 . ... |
fin23lem23 10317 | Lemma for ~ fin23lem22 . ... |
fin23lem22 10318 | Lemma for ~ fin23 but coul... |
fin23lem27 10319 | The mapping constructed in... |
isfin3ds 10320 | Property of a III-finite s... |
ssfin3ds 10321 | A subset of a III-finite s... |
fin23lem12 10322 | The beginning of the proof... |
fin23lem13 10323 | Lemma for ~ fin23 . Each ... |
fin23lem14 10324 | Lemma for ~ fin23 . ` U ` ... |
fin23lem15 10325 | Lemma for ~ fin23 . ` U ` ... |
fin23lem16 10326 | Lemma for ~ fin23 . ` U ` ... |
fin23lem19 10327 | Lemma for ~ fin23 . The f... |
fin23lem20 10328 | Lemma for ~ fin23 . ` X ` ... |
fin23lem17 10329 | Lemma for ~ fin23 . By ? ... |
fin23lem21 10330 | Lemma for ~ fin23 . ` X ` ... |
fin23lem28 10331 | Lemma for ~ fin23 . The r... |
fin23lem29 10332 | Lemma for ~ fin23 . The r... |
fin23lem30 10333 | Lemma for ~ fin23 . The r... |
fin23lem31 10334 | Lemma for ~ fin23 . The r... |
fin23lem32 10335 | Lemma for ~ fin23 . Wrap ... |
fin23lem33 10336 | Lemma for ~ fin23 . Disch... |
fin23lem34 10337 | Lemma for ~ fin23 . Estab... |
fin23lem35 10338 | Lemma for ~ fin23 . Stric... |
fin23lem36 10339 | Lemma for ~ fin23 . Weak ... |
fin23lem38 10340 | Lemma for ~ fin23 . The c... |
fin23lem39 10341 | Lemma for ~ fin23 . Thus,... |
fin23lem40 10342 | Lemma for ~ fin23 . ` Fin2... |
fin23lem41 10343 | Lemma for ~ fin23 . A set... |
isf32lem1 10344 | Lemma for ~ isfin3-2 . De... |
isf32lem2 10345 | Lemma for ~ isfin3-2 . No... |
isf32lem3 10346 | Lemma for ~ isfin3-2 . Be... |
isf32lem4 10347 | Lemma for ~ isfin3-2 . Be... |
isf32lem5 10348 | Lemma for ~ isfin3-2 . Th... |
isf32lem6 10349 | Lemma for ~ isfin3-2 . Ea... |
isf32lem7 10350 | Lemma for ~ isfin3-2 . Di... |
isf32lem8 10351 | Lemma for ~ isfin3-2 . K ... |
isf32lem9 10352 | Lemma for ~ isfin3-2 . Co... |
isf32lem10 10353 | Lemma for isfin3-2 . Writ... |
isf32lem11 10354 | Lemma for ~ isfin3-2 . Re... |
isf32lem12 10355 | Lemma for ~ isfin3-2 . (C... |
isfin32i 10356 | One half of ~ isfin3-2 . ... |
isf33lem 10357 | Lemma for ~ isfin3-3 . (C... |
isfin3-2 10358 | Weakly Dedekind-infinite s... |
isfin3-3 10359 | Weakly Dedekind-infinite s... |
fin33i 10360 | Inference from ~ isfin3-3 ... |
compsscnvlem 10361 | Lemma for ~ compsscnv . (... |
compsscnv 10362 | Complementation on a power... |
isf34lem1 10363 | Lemma for ~ isfin3-4 . (C... |
isf34lem2 10364 | Lemma for ~ isfin3-4 . (C... |
compssiso 10365 | Complementation is an anti... |
isf34lem3 10366 | Lemma for ~ isfin3-4 . (C... |
compss 10367 | Express image under of the... |
isf34lem4 10368 | Lemma for ~ isfin3-4 . (C... |
isf34lem5 10369 | Lemma for ~ isfin3-4 . (C... |
isf34lem7 10370 | Lemma for ~ isfin3-4 . (C... |
isf34lem6 10371 | Lemma for ~ isfin3-4 . (C... |
fin34i 10372 | Inference from ~ isfin3-4 ... |
isfin3-4 10373 | Weakly Dedekind-infinite s... |
fin11a 10374 | Every I-finite set is Ia-f... |
enfin1ai 10375 | Ia-finiteness is a cardina... |
isfin1-2 10376 | A set is finite in the usu... |
isfin1-3 10377 | A set is I-finite iff ever... |
isfin1-4 10378 | A set is I-finite iff ever... |
dffin1-5 10379 | Compact quantifier-free ve... |
fin23 10380 | Every II-finite set (every... |
fin34 10381 | Every III-finite set is IV... |
isfin5-2 10382 | Alternate definition of V-... |
fin45 10383 | Every IV-finite set is V-f... |
fin56 10384 | Every V-finite set is VI-f... |
fin17 10385 | Every I-finite set is VII-... |
fin67 10386 | Every VI-finite set is VII... |
isfin7-2 10387 | A set is VII-finite iff it... |
fin71num 10388 | A well-orderable set is VI... |
dffin7-2 10389 | Class form of ~ isfin7-2 .... |
dfacfin7 10390 | Axiom of Choice equivalent... |
fin1a2lem1 10391 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem2 10392 | Lemma for ~ fin1a2 . The ... |
fin1a2lem3 10393 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem4 10394 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem5 10395 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem6 10396 | Lemma for ~ fin1a2 . Esta... |
fin1a2lem7 10397 | Lemma for ~ fin1a2 . Spli... |
fin1a2lem8 10398 | Lemma for ~ fin1a2 . Spli... |
fin1a2lem9 10399 | Lemma for ~ fin1a2 . In a... |
fin1a2lem10 10400 | Lemma for ~ fin1a2 . A no... |
fin1a2lem11 10401 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem12 10402 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem13 10403 | Lemma for ~ fin1a2 . (Con... |
fin12 10404 | Weak theorem which skips I... |
fin1a2s 10405 | An II-infinite set can hav... |
fin1a2 10406 | Every Ia-finite set is II-... |
itunifval 10407 | Function value of iterated... |
itunifn 10408 | Functionality of the itera... |
ituni0 10409 | A zero-fold iterated union... |
itunisuc 10410 | Successor iterated union. ... |
itunitc1 10411 | Each union iterate is a me... |
itunitc 10412 | The union of all union ite... |
ituniiun 10413 | Unwrap an iterated union f... |
hsmexlem7 10414 | Lemma for ~ hsmex . Prope... |
hsmexlem8 10415 | Lemma for ~ hsmex . Prope... |
hsmexlem9 10416 | Lemma for ~ hsmex . Prope... |
hsmexlem1 10417 | Lemma for ~ hsmex . Bound... |
hsmexlem2 10418 | Lemma for ~ hsmex . Bound... |
hsmexlem3 10419 | Lemma for ~ hsmex . Clear... |
hsmexlem4 10420 | Lemma for ~ hsmex . The c... |
hsmexlem5 10421 | Lemma for ~ hsmex . Combi... |
hsmexlem6 10422 | Lemma for ~ hsmex . (Cont... |
hsmex 10423 | The collection of heredita... |
hsmex2 10424 | The set of hereditary size... |
hsmex3 10425 | The set of hereditary size... |
axcc2lem 10427 | Lemma for ~ axcc2 . (Cont... |
axcc2 10428 | A possibly more useful ver... |
axcc3 10429 | A possibly more useful ver... |
axcc4 10430 | A version of ~ axcc3 that ... |
acncc 10431 | An ~ ax-cc equivalent: eve... |
axcc4dom 10432 | Relax the constraint on ~ ... |
domtriomlem 10433 | Lemma for ~ domtriom . (C... |
domtriom 10434 | Trichotomy of equinumerosi... |
fin41 10435 | Under countable choice, th... |
dominf 10436 | A nonempty set that is a s... |
dcomex 10438 | The Axiom of Dependent Cho... |
axdc2lem 10439 | Lemma for ~ axdc2 . We co... |
axdc2 10440 | An apparent strengthening ... |
axdc3lem 10441 | The class ` S ` of finite ... |
axdc3lem2 10442 | Lemma for ~ axdc3 . We ha... |
axdc3lem3 10443 | Simple substitution lemma ... |
axdc3lem4 10444 | Lemma for ~ axdc3 . We ha... |
axdc3 10445 | Dependent Choice. Axiom D... |
axdc4lem 10446 | Lemma for ~ axdc4 . (Cont... |
axdc4 10447 | A more general version of ... |
axcclem 10448 | Lemma for ~ axcc . (Contr... |
axcc 10449 | Although CC can be proven ... |
zfac 10451 | Axiom of Choice expressed ... |
ac2 10452 | Axiom of Choice equivalent... |
ac3 10453 | Axiom of Choice using abbr... |
axac3 10455 | This theorem asserts that ... |
ackm 10456 | A remarkable equivalent to... |
axac2 10457 | Derive ~ ax-ac2 from ~ ax-... |
axac 10458 | Derive ~ ax-ac from ~ ax-a... |
axaci 10459 | Apply a choice equivalent.... |
cardeqv 10460 | All sets are well-orderabl... |
numth3 10461 | All sets are well-orderabl... |
numth2 10462 | Numeration theorem: any se... |
numth 10463 | Numeration theorem: every ... |
ac7 10464 | An Axiom of Choice equival... |
ac7g 10465 | An Axiom of Choice equival... |
ac4 10466 | Equivalent of Axiom of Cho... |
ac4c 10467 | Equivalent of Axiom of Cho... |
ac5 10468 | An Axiom of Choice equival... |
ac5b 10469 | Equivalent of Axiom of Cho... |
ac6num 10470 | A version of ~ ac6 which t... |
ac6 10471 | Equivalent of Axiom of Cho... |
ac6c4 10472 | Equivalent of Axiom of Cho... |
ac6c5 10473 | Equivalent of Axiom of Cho... |
ac9 10474 | An Axiom of Choice equival... |
ac6s 10475 | Equivalent of Axiom of Cho... |
ac6n 10476 | Equivalent of Axiom of Cho... |
ac6s2 10477 | Generalization of the Axio... |
ac6s3 10478 | Generalization of the Axio... |
ac6sg 10479 | ~ ac6s with sethood as ant... |
ac6sf 10480 | Version of ~ ac6 with boun... |
ac6s4 10481 | Generalization of the Axio... |
ac6s5 10482 | Generalization of the Axio... |
ac8 10483 | An Axiom of Choice equival... |
ac9s 10484 | An Axiom of Choice equival... |
numthcor 10485 | Any set is strictly domina... |
weth 10486 | Well-ordering theorem: any... |
zorn2lem1 10487 | Lemma for ~ zorn2 . (Cont... |
zorn2lem2 10488 | Lemma for ~ zorn2 . (Cont... |
zorn2lem3 10489 | Lemma for ~ zorn2 . (Cont... |
zorn2lem4 10490 | Lemma for ~ zorn2 . (Cont... |
zorn2lem5 10491 | Lemma for ~ zorn2 . (Cont... |
zorn2lem6 10492 | Lemma for ~ zorn2 . (Cont... |
zorn2lem7 10493 | Lemma for ~ zorn2 . (Cont... |
zorn2g 10494 | Zorn's Lemma of [Monk1] p.... |
zorng 10495 | Zorn's Lemma. If the unio... |
zornn0g 10496 | Variant of Zorn's lemma ~ ... |
zorn2 10497 | Zorn's Lemma of [Monk1] p.... |
zorn 10498 | Zorn's Lemma. If the unio... |
zornn0 10499 | Variant of Zorn's lemma ~ ... |
ttukeylem1 10500 | Lemma for ~ ttukey . Expa... |
ttukeylem2 10501 | Lemma for ~ ttukey . A pr... |
ttukeylem3 10502 | Lemma for ~ ttukey . (Con... |
ttukeylem4 10503 | Lemma for ~ ttukey . (Con... |
ttukeylem5 10504 | Lemma for ~ ttukey . The ... |
ttukeylem6 10505 | Lemma for ~ ttukey . (Con... |
ttukeylem7 10506 | Lemma for ~ ttukey . (Con... |
ttukey2g 10507 | The Teichmüller-Tukey... |
ttukeyg 10508 | The Teichmüller-Tukey... |
ttukey 10509 | The Teichmüller-Tukey... |
axdclem 10510 | Lemma for ~ axdc . (Contr... |
axdclem2 10511 | Lemma for ~ axdc . Using ... |
axdc 10512 | This theorem derives ~ ax-... |
fodomg 10513 | An onto function implies d... |
fodom 10514 | An onto function implies d... |
dmct 10515 | The domain of a countable ... |
rnct 10516 | The range of a countable s... |
fodomb 10517 | Equivalence of an onto map... |
wdomac 10518 | When assuming AC, weak and... |
brdom3 10519 | Equivalence to a dominance... |
brdom5 10520 | An equivalence to a domina... |
brdom4 10521 | An equivalence to a domina... |
brdom7disj 10522 | An equivalence to a domina... |
brdom6disj 10523 | An equivalence to a domina... |
fin71ac 10524 | Once we allow AC, the "str... |
imadomg 10525 | An image of a function und... |
fimact 10526 | The image by a function of... |
fnrndomg 10527 | The range of a function is... |
fnct 10528 | If the domain of a functio... |
mptct 10529 | A countable mapping set is... |
iunfo 10530 | Existence of an onto funct... |
iundom2g 10531 | An upper bound for the car... |
iundomg 10532 | An upper bound for the car... |
iundom 10533 | An upper bound for the car... |
unidom 10534 | An upper bound for the car... |
uniimadom 10535 | An upper bound for the car... |
uniimadomf 10536 | An upper bound for the car... |
cardval 10537 | The value of the cardinal ... |
cardid 10538 | Any set is equinumerous to... |
cardidg 10539 | Any set is equinumerous to... |
cardidd 10540 | Any set is equinumerous to... |
cardf 10541 | The cardinality function i... |
carden 10542 | Two sets are equinumerous ... |
cardeq0 10543 | Only the empty set has car... |
unsnen 10544 | Equinumerosity of a set wi... |
carddom 10545 | Two sets have the dominanc... |
cardsdom 10546 | Two sets have the strict d... |
domtri 10547 | Trichotomy law for dominan... |
entric 10548 | Trichotomy of equinumerosi... |
entri2 10549 | Trichotomy of dominance an... |
entri3 10550 | Trichotomy of dominance. ... |
sdomsdomcard 10551 | A set strictly dominates i... |
canth3 10552 | Cantor's theorem in terms ... |
infxpidm 10553 | Every infinite class is eq... |
ondomon 10554 | The class of ordinals domi... |
cardmin 10555 | The smallest ordinal that ... |
ficard 10556 | A set is finite iff its ca... |
infinf 10557 | Equivalence between two in... |
unirnfdomd 10558 | The union of the range of ... |
konigthlem 10559 | Lemma for ~ konigth . (Co... |
konigth 10560 | Konig's Theorem. If ` m (... |
alephsucpw 10561 | The power set of an aleph ... |
aleph1 10562 | The set exponentiation of ... |
alephval2 10563 | An alternate way to expres... |
dominfac 10564 | A nonempty set that is a s... |
iunctb 10565 | The countable union of cou... |
unictb 10566 | The countable union of cou... |
infmap 10567 | An exponentiation law for ... |
alephadd 10568 | The sum of two alephs is t... |
alephmul 10569 | The product of two alephs ... |
alephexp1 10570 | An exponentiation law for ... |
alephsuc3 10571 | An alternate representatio... |
alephexp2 10572 | An expression equinumerous... |
alephreg 10573 | A successor aleph is regul... |
pwcfsdom 10574 | A corollary of Konig's The... |
cfpwsdom 10575 | A corollary of Konig's The... |
alephom 10576 | From ~ canth2 , we know th... |
smobeth 10577 | The beth function is stric... |
nd1 10578 | A lemma for proving condit... |
nd2 10579 | A lemma for proving condit... |
nd3 10580 | A lemma for proving condit... |
nd4 10581 | A lemma for proving condit... |
axextnd 10582 | A version of the Axiom of ... |
axrepndlem1 10583 | Lemma for the Axiom of Rep... |
axrepndlem2 10584 | Lemma for the Axiom of Rep... |
axrepnd 10585 | A version of the Axiom of ... |
axunndlem1 10586 | Lemma for the Axiom of Uni... |
axunnd 10587 | A version of the Axiom of ... |
axpowndlem1 10588 | Lemma for the Axiom of Pow... |
axpowndlem2 10589 | Lemma for the Axiom of Pow... |
axpowndlem3 10590 | Lemma for the Axiom of Pow... |
axpowndlem4 10591 | Lemma for the Axiom of Pow... |
axpownd 10592 | A version of the Axiom of ... |
axregndlem1 10593 | Lemma for the Axiom of Reg... |
axregndlem2 10594 | Lemma for the Axiom of Reg... |
axregnd 10595 | A version of the Axiom of ... |
axinfndlem1 10596 | Lemma for the Axiom of Inf... |
axinfnd 10597 | A version of the Axiom of ... |
axacndlem1 10598 | Lemma for the Axiom of Cho... |
axacndlem2 10599 | Lemma for the Axiom of Cho... |
axacndlem3 10600 | Lemma for the Axiom of Cho... |
axacndlem4 10601 | Lemma for the Axiom of Cho... |
axacndlem5 10602 | Lemma for the Axiom of Cho... |
axacnd 10603 | A version of the Axiom of ... |
zfcndext 10604 | Axiom of Extensionality ~ ... |
zfcndrep 10605 | Axiom of Replacement ~ ax-... |
zfcndun 10606 | Axiom of Union ~ ax-un , r... |
zfcndpow 10607 | Axiom of Power Sets ~ ax-p... |
zfcndreg 10608 | Axiom of Regularity ~ ax-r... |
zfcndinf 10609 | Axiom of Infinity ~ ax-inf... |
zfcndac 10610 | Axiom of Choice ~ ax-ac , ... |
elgch 10613 | Elementhood in the collect... |
fingch 10614 | A finite set is a GCH-set.... |
gchi 10615 | The only GCH-sets which ha... |
gchen1 10616 | If ` A <_ B < ~P A ` , and... |
gchen2 10617 | If ` A < B <_ ~P A ` , and... |
gchor 10618 | If ` A <_ B <_ ~P A ` , an... |
engch 10619 | The property of being a GC... |
gchdomtri 10620 | Under certain conditions, ... |
fpwwe2cbv 10621 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem1 10622 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem2 10623 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem3 10624 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem4 10625 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem5 10626 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem6 10627 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem7 10628 | Lemma for ~ fpwwe2 . Show... |
fpwwe2lem8 10629 | Lemma for ~ fpwwe2 . Give... |
fpwwe2lem9 10630 | Lemma for ~ fpwwe2 . Give... |
fpwwe2lem10 10631 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem11 10632 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem12 10633 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2 10634 | Given any function ` F ` f... |
fpwwecbv 10635 | Lemma for ~ fpwwe . (Cont... |
fpwwelem 10636 | Lemma for ~ fpwwe . (Cont... |
fpwwe 10637 | Given any function ` F ` f... |
canth4 10638 | An "effective" form of Can... |
canthnumlem 10639 | Lemma for ~ canthnum . (C... |
canthnum 10640 | The set of well-orderable ... |
canthwelem 10641 | Lemma for ~ canthwe . (Co... |
canthwe 10642 | The set of well-orders of ... |
canthp1lem1 10643 | Lemma for ~ canthp1 . (Co... |
canthp1lem2 10644 | Lemma for ~ canthp1 . (Co... |
canthp1 10645 | A slightly stronger form o... |
finngch 10646 | The exclusion of finite se... |
gchdju1 10647 | An infinite GCH-set is ide... |
gchinf 10648 | An infinite GCH-set is Ded... |
pwfseqlem1 10649 | Lemma for ~ pwfseq . Deri... |
pwfseqlem2 10650 | Lemma for ~ pwfseq . (Con... |
pwfseqlem3 10651 | Lemma for ~ pwfseq . Usin... |
pwfseqlem4a 10652 | Lemma for ~ pwfseqlem4 . ... |
pwfseqlem4 10653 | Lemma for ~ pwfseq . Deri... |
pwfseqlem5 10654 | Lemma for ~ pwfseq . Alth... |
pwfseq 10655 | The powerset of a Dedekind... |
pwxpndom2 10656 | The powerset of a Dedekind... |
pwxpndom 10657 | The powerset of a Dedekind... |
pwdjundom 10658 | The powerset of a Dedekind... |
gchdjuidm 10659 | An infinite GCH-set is ide... |
gchxpidm 10660 | An infinite GCH-set is ide... |
gchpwdom 10661 | A relationship between dom... |
gchaleph 10662 | If ` ( aleph `` A ) ` is a... |
gchaleph2 10663 | If ` ( aleph `` A ) ` and ... |
hargch 10664 | If ` A + ~~ ~P A ` , then ... |
alephgch 10665 | If ` ( aleph `` suc A ) ` ... |
gch2 10666 | It is sufficient to requir... |
gch3 10667 | An equivalent formulation ... |
gch-kn 10668 | The equivalence of two ver... |
gchaclem 10669 | Lemma for ~ gchac (obsolet... |
gchhar 10670 | A "local" form of ~ gchac ... |
gchacg 10671 | A "local" form of ~ gchac ... |
gchac 10672 | The Generalized Continuum ... |
elwina 10677 | Conditions of weak inacces... |
elina 10678 | Conditions of strong inacc... |
winaon 10679 | A weakly inaccessible card... |
inawinalem 10680 | Lemma for ~ inawina . (Co... |
inawina 10681 | Every strongly inaccessibl... |
omina 10682 | ` _om ` is a strongly inac... |
winacard 10683 | A weakly inaccessible card... |
winainflem 10684 | A weakly inaccessible card... |
winainf 10685 | A weakly inaccessible card... |
winalim 10686 | A weakly inaccessible card... |
winalim2 10687 | A nontrivial weakly inacce... |
winafp 10688 | A nontrivial weakly inacce... |
winafpi 10689 | This theorem, which states... |
gchina 10690 | Assuming the GCH, weakly a... |
iswun 10695 | Properties of a weak unive... |
wuntr 10696 | A weak universe is transit... |
wununi 10697 | A weak universe is closed ... |
wunpw 10698 | A weak universe is closed ... |
wunelss 10699 | The elements of a weak uni... |
wunpr 10700 | A weak universe is closed ... |
wunun 10701 | A weak universe is closed ... |
wuntp 10702 | A weak universe is closed ... |
wunss 10703 | A weak universe is closed ... |
wunin 10704 | A weak universe is closed ... |
wundif 10705 | A weak universe is closed ... |
wunint 10706 | A weak universe is closed ... |
wunsn 10707 | A weak universe is closed ... |
wunsuc 10708 | A weak universe is closed ... |
wun0 10709 | A weak universe contains t... |
wunr1om 10710 | A weak universe is infinit... |
wunom 10711 | A weak universe contains a... |
wunfi 10712 | A weak universe contains a... |
wunop 10713 | A weak universe is closed ... |
wunot 10714 | A weak universe is closed ... |
wunxp 10715 | A weak universe is closed ... |
wunpm 10716 | A weak universe is closed ... |
wunmap 10717 | A weak universe is closed ... |
wunf 10718 | A weak universe is closed ... |
wundm 10719 | A weak universe is closed ... |
wunrn 10720 | A weak universe is closed ... |
wuncnv 10721 | A weak universe is closed ... |
wunres 10722 | A weak universe is closed ... |
wunfv 10723 | A weak universe is closed ... |
wunco 10724 | A weak universe is closed ... |
wuntpos 10725 | A weak universe is closed ... |
intwun 10726 | The intersection of a coll... |
r1limwun 10727 | Each limit stage in the cu... |
r1wunlim 10728 | The weak universes in the ... |
wunex2 10729 | Construct a weak universe ... |
wunex 10730 | Construct a weak universe ... |
uniwun 10731 | Every set is contained in ... |
wunex3 10732 | Construct a weak universe ... |
wuncval 10733 | Value of the weak universe... |
wuncid 10734 | The weak universe closure ... |
wunccl 10735 | The weak universe closure ... |
wuncss 10736 | The weak universe closure ... |
wuncidm 10737 | The weak universe closure ... |
wuncval2 10738 | Our earlier expression for... |
eltskg 10741 | Properties of a Tarski cla... |
eltsk2g 10742 | Properties of a Tarski cla... |
tskpwss 10743 | First axiom of a Tarski cl... |
tskpw 10744 | Second axiom of a Tarski c... |
tsken 10745 | Third axiom of a Tarski cl... |
0tsk 10746 | The empty set is a (transi... |
tsksdom 10747 | An element of a Tarski cla... |
tskssel 10748 | A part of a Tarski class s... |
tskss 10749 | The subsets of an element ... |
tskin 10750 | The intersection of two el... |
tsksn 10751 | A singleton of an element ... |
tsktrss 10752 | A transitive element of a ... |
tsksuc 10753 | If an element of a Tarski ... |
tsk0 10754 | A nonempty Tarski class co... |
tsk1 10755 | One is an element of a non... |
tsk2 10756 | Two is an element of a non... |
2domtsk 10757 | If a Tarski class is not e... |
tskr1om 10758 | A nonempty Tarski class is... |
tskr1om2 10759 | A nonempty Tarski class co... |
tskinf 10760 | A nonempty Tarski class is... |
tskpr 10761 | If ` A ` and ` B ` are mem... |
tskop 10762 | If ` A ` and ` B ` are mem... |
tskxpss 10763 | A Cartesian product of two... |
tskwe2 10764 | A Tarski class is well-ord... |
inttsk 10765 | The intersection of a coll... |
inar1 10766 | ` ( R1 `` A ) ` for ` A ` ... |
r1omALT 10767 | Alternate proof of ~ r1om ... |
rankcf 10768 | Any set must be at least a... |
inatsk 10769 | ` ( R1 `` A ) ` for ` A ` ... |
r1omtsk 10770 | The set of hereditarily fi... |
tskord 10771 | A Tarski class contains al... |
tskcard 10772 | An even more direct relati... |
r1tskina 10773 | There is a direct relation... |
tskuni 10774 | The union of an element of... |
tskwun 10775 | A nonempty transitive Tars... |
tskint 10776 | The intersection of an ele... |
tskun 10777 | The union of two elements ... |
tskxp 10778 | The Cartesian product of t... |
tskmap 10779 | Set exponentiation is an e... |
tskurn 10780 | A transitive Tarski class ... |
elgrug 10783 | Properties of a Grothendie... |
grutr 10784 | A Grothendieck universe is... |
gruelss 10785 | A Grothendieck universe is... |
grupw 10786 | A Grothendieck universe co... |
gruss 10787 | Any subset of an element o... |
grupr 10788 | A Grothendieck universe co... |
gruurn 10789 | A Grothendieck universe co... |
gruiun 10790 | If ` B ( x ) ` is a family... |
gruuni 10791 | A Grothendieck universe co... |
grurn 10792 | A Grothendieck universe co... |
gruima 10793 | A Grothendieck universe co... |
gruel 10794 | Any element of an element ... |
grusn 10795 | A Grothendieck universe co... |
gruop 10796 | A Grothendieck universe co... |
gruun 10797 | A Grothendieck universe co... |
gruxp 10798 | A Grothendieck universe co... |
grumap 10799 | A Grothendieck universe co... |
gruixp 10800 | A Grothendieck universe co... |
gruiin 10801 | A Grothendieck universe co... |
gruf 10802 | A Grothendieck universe co... |
gruen 10803 | A Grothendieck universe co... |
gruwun 10804 | A nonempty Grothendieck un... |
intgru 10805 | The intersection of a fami... |
ingru 10806 | The intersection of a univ... |
wfgru 10807 | The wellfounded part of a ... |
grudomon 10808 | Each ordinal that is compa... |
gruina 10809 | If a Grothendieck universe... |
grur1a 10810 | A characterization of Grot... |
grur1 10811 | A characterization of Grot... |
grutsk1 10812 | Grothendieck universes are... |
grutsk 10813 | Grothendieck universes are... |
axgroth5 10815 | The Tarski-Grothendieck ax... |
axgroth2 10816 | Alternate version of the T... |
grothpw 10817 | Derive the Axiom of Power ... |
grothpwex 10818 | Derive the Axiom of Power ... |
axgroth6 10819 | The Tarski-Grothendieck ax... |
grothomex 10820 | The Tarski-Grothendieck Ax... |
grothac 10821 | The Tarski-Grothendieck Ax... |
axgroth3 10822 | Alternate version of the T... |
axgroth4 10823 | Alternate version of the T... |
grothprimlem 10824 | Lemma for ~ grothprim . E... |
grothprim 10825 | The Tarski-Grothendieck Ax... |
grothtsk 10826 | The Tarski-Grothendieck Ax... |
inaprc 10827 | An equivalent to the Tarsk... |
tskmval 10830 | Value of our tarski map. ... |
tskmid 10831 | The set ` A ` is an elemen... |
tskmcl 10832 | A Tarski class that contai... |
sstskm 10833 | Being a part of ` ( tarski... |
eltskm 10834 | Belonging to ` ( tarskiMap... |
elni 10867 | Membership in the class of... |
elni2 10868 | Membership in the class of... |
pinn 10869 | A positive integer is a na... |
pion 10870 | A positive integer is an o... |
piord 10871 | A positive integer is ordi... |
niex 10872 | The class of positive inte... |
0npi 10873 | The empty set is not a pos... |
1pi 10874 | Ordinal 'one' is a positiv... |
addpiord 10875 | Positive integer addition ... |
mulpiord 10876 | Positive integer multiplic... |
mulidpi 10877 | 1 is an identity element f... |
ltpiord 10878 | Positive integer 'less tha... |
ltsopi 10879 | Positive integer 'less tha... |
ltrelpi 10880 | Positive integer 'less tha... |
dmaddpi 10881 | Domain of addition on posi... |
dmmulpi 10882 | Domain of multiplication o... |
addclpi 10883 | Closure of addition of pos... |
mulclpi 10884 | Closure of multiplication ... |
addcompi 10885 | Addition of positive integ... |
addasspi 10886 | Addition of positive integ... |
mulcompi 10887 | Multiplication of positive... |
mulasspi 10888 | Multiplication of positive... |
distrpi 10889 | Multiplication of positive... |
addcanpi 10890 | Addition cancellation law ... |
mulcanpi 10891 | Multiplication cancellatio... |
addnidpi 10892 | There is no identity eleme... |
ltexpi 10893 | Ordering on positive integ... |
ltapi 10894 | Ordering property of addit... |
ltmpi 10895 | Ordering property of multi... |
1lt2pi 10896 | One is less than two (one ... |
nlt1pi 10897 | No positive integer is les... |
indpi 10898 | Principle of Finite Induct... |
enqbreq 10910 | Equivalence relation for p... |
enqbreq2 10911 | Equivalence relation for p... |
enqer 10912 | The equivalence relation f... |
enqex 10913 | The equivalence relation f... |
nqex 10914 | The class of positive frac... |
0nnq 10915 | The empty set is not a pos... |
elpqn 10916 | Each positive fraction is ... |
ltrelnq 10917 | Positive fraction 'less th... |
pinq 10918 | The representatives of pos... |
1nq 10919 | The positive fraction 'one... |
nqereu 10920 | There is a unique element ... |
nqerf 10921 | Corollary of ~ nqereu : th... |
nqercl 10922 | Corollary of ~ nqereu : cl... |
nqerrel 10923 | Any member of ` ( N. X. N.... |
nqerid 10924 | Corollary of ~ nqereu : th... |
enqeq 10925 | Corollary of ~ nqereu : if... |
nqereq 10926 | The function ` /Q ` acts a... |
addpipq2 10927 | Addition of positive fract... |
addpipq 10928 | Addition of positive fract... |
addpqnq 10929 | Addition of positive fract... |
mulpipq2 10930 | Multiplication of positive... |
mulpipq 10931 | Multiplication of positive... |
mulpqnq 10932 | Multiplication of positive... |
ordpipq 10933 | Ordering of positive fract... |
ordpinq 10934 | Ordering of positive fract... |
addpqf 10935 | Closure of addition on pos... |
addclnq 10936 | Closure of addition on pos... |
mulpqf 10937 | Closure of multiplication ... |
mulclnq 10938 | Closure of multiplication ... |
addnqf 10939 | Domain of addition on posi... |
mulnqf 10940 | Domain of multiplication o... |
addcompq 10941 | Addition of positive fract... |
addcomnq 10942 | Addition of positive fract... |
mulcompq 10943 | Multiplication of positive... |
mulcomnq 10944 | Multiplication of positive... |
adderpqlem 10945 | Lemma for ~ adderpq . (Co... |
mulerpqlem 10946 | Lemma for ~ mulerpq . (Co... |
adderpq 10947 | Addition is compatible wit... |
mulerpq 10948 | Multiplication is compatib... |
addassnq 10949 | Addition of positive fract... |
mulassnq 10950 | Multiplication of positive... |
mulcanenq 10951 | Lemma for distributive law... |
distrnq 10952 | Multiplication of positive... |
1nqenq 10953 | The equivalence class of r... |
mulidnq 10954 | Multiplication identity el... |
recmulnq 10955 | Relationship between recip... |
recidnq 10956 | A positive fraction times ... |
recclnq 10957 | Closure law for positive f... |
recrecnq 10958 | Reciprocal of reciprocal o... |
dmrecnq 10959 | Domain of reciprocal on po... |
ltsonq 10960 | 'Less than' is a strict or... |
lterpq 10961 | Compatibility of ordering ... |
ltanq 10962 | Ordering property of addit... |
ltmnq 10963 | Ordering property of multi... |
1lt2nq 10964 | One is less than two (one ... |
ltaddnq 10965 | The sum of two fractions i... |
ltexnq 10966 | Ordering on positive fract... |
halfnq 10967 | One-half of any positive f... |
nsmallnq 10968 | The is no smallest positiv... |
ltbtwnnq 10969 | There exists a number betw... |
ltrnq 10970 | Ordering property of recip... |
archnq 10971 | For any fraction, there is... |
npex 10977 | The class of positive real... |
elnp 10978 | Membership in positive rea... |
elnpi 10979 | Membership in positive rea... |
prn0 10980 | A positive real is not emp... |
prpssnq 10981 | A positive real is a subse... |
elprnq 10982 | A positive real is a set o... |
0npr 10983 | The empty set is not a pos... |
prcdnq 10984 | A positive real is closed ... |
prub 10985 | A positive fraction not in... |
prnmax 10986 | A positive real has no lar... |
npomex 10987 | A simplifying observation,... |
prnmadd 10988 | A positive real has no lar... |
ltrelpr 10989 | Positive real 'less than' ... |
genpv 10990 | Value of general operation... |
genpelv 10991 | Membership in value of gen... |
genpprecl 10992 | Pre-closure law for genera... |
genpdm 10993 | Domain of general operatio... |
genpn0 10994 | The result of an operation... |
genpss 10995 | The result of an operation... |
genpnnp 10996 | The result of an operation... |
genpcd 10997 | Downward closure of an ope... |
genpnmax 10998 | An operation on positive r... |
genpcl 10999 | Closure of an operation on... |
genpass 11000 | Associativity of an operat... |
plpv 11001 | Value of addition on posit... |
mpv 11002 | Value of multiplication on... |
dmplp 11003 | Domain of addition on posi... |
dmmp 11004 | Domain of multiplication o... |
nqpr 11005 | The canonical embedding of... |
1pr 11006 | The positive real number '... |
addclprlem1 11007 | Lemma to prove downward cl... |
addclprlem2 11008 | Lemma to prove downward cl... |
addclpr 11009 | Closure of addition on pos... |
mulclprlem 11010 | Lemma to prove downward cl... |
mulclpr 11011 | Closure of multiplication ... |
addcompr 11012 | Addition of positive reals... |
addasspr 11013 | Addition of positive reals... |
mulcompr 11014 | Multiplication of positive... |
mulasspr 11015 | Multiplication of positive... |
distrlem1pr 11016 | Lemma for distributive law... |
distrlem4pr 11017 | Lemma for distributive law... |
distrlem5pr 11018 | Lemma for distributive law... |
distrpr 11019 | Multiplication of positive... |
1idpr 11020 | 1 is an identity element f... |
ltprord 11021 | Positive real 'less than' ... |
psslinpr 11022 | Proper subset is a linear ... |
ltsopr 11023 | Positive real 'less than' ... |
prlem934 11024 | Lemma 9-3.4 of [Gleason] p... |
ltaddpr 11025 | The sum of two positive re... |
ltaddpr2 11026 | The sum of two positive re... |
ltexprlem1 11027 | Lemma for Proposition 9-3.... |
ltexprlem2 11028 | Lemma for Proposition 9-3.... |
ltexprlem3 11029 | Lemma for Proposition 9-3.... |
ltexprlem4 11030 | Lemma for Proposition 9-3.... |
ltexprlem5 11031 | Lemma for Proposition 9-3.... |
ltexprlem6 11032 | Lemma for Proposition 9-3.... |
ltexprlem7 11033 | Lemma for Proposition 9-3.... |
ltexpri 11034 | Proposition 9-3.5(iv) of [... |
ltaprlem 11035 | Lemma for Proposition 9-3.... |
ltapr 11036 | Ordering property of addit... |
addcanpr 11037 | Addition cancellation law ... |
prlem936 11038 | Lemma 9-3.6 of [Gleason] p... |
reclem2pr 11039 | Lemma for Proposition 9-3.... |
reclem3pr 11040 | Lemma for Proposition 9-3.... |
reclem4pr 11041 | Lemma for Proposition 9-3.... |
recexpr 11042 | The reciprocal of a positi... |
suplem1pr 11043 | The union of a nonempty, b... |
suplem2pr 11044 | The union of a set of posi... |
supexpr 11045 | The union of a nonempty, b... |
enrer 11054 | The equivalence relation f... |
nrex1 11055 | The class of signed reals ... |
enrbreq 11056 | Equivalence relation for s... |
enreceq 11057 | Equivalence class equality... |
enrex 11058 | The equivalence relation f... |
ltrelsr 11059 | Signed real 'less than' is... |
addcmpblnr 11060 | Lemma showing compatibilit... |
mulcmpblnrlem 11061 | Lemma used in lemma showin... |
mulcmpblnr 11062 | Lemma showing compatibilit... |
prsrlem1 11063 | Decomposing signed reals i... |
addsrmo 11064 | There is at most one resul... |
mulsrmo 11065 | There is at most one resul... |
addsrpr 11066 | Addition of signed reals i... |
mulsrpr 11067 | Multiplication of signed r... |
ltsrpr 11068 | Ordering of signed reals i... |
gt0srpr 11069 | Greater than zero in terms... |
0nsr 11070 | The empty set is not a sig... |
0r 11071 | The constant ` 0R ` is a s... |
1sr 11072 | The constant ` 1R ` is a s... |
m1r 11073 | The constant ` -1R ` is a ... |
addclsr 11074 | Closure of addition on sig... |
mulclsr 11075 | Closure of multiplication ... |
dmaddsr 11076 | Domain of addition on sign... |
dmmulsr 11077 | Domain of multiplication o... |
addcomsr 11078 | Addition of signed reals i... |
addasssr 11079 | Addition of signed reals i... |
mulcomsr 11080 | Multiplication of signed r... |
mulasssr 11081 | Multiplication of signed r... |
distrsr 11082 | Multiplication of signed r... |
m1p1sr 11083 | Minus one plus one is zero... |
m1m1sr 11084 | Minus one times minus one ... |
ltsosr 11085 | Signed real 'less than' is... |
0lt1sr 11086 | 0 is less than 1 for signe... |
1ne0sr 11087 | 1 and 0 are distinct for s... |
0idsr 11088 | The signed real number 0 i... |
1idsr 11089 | 1 is an identity element f... |
00sr 11090 | A signed real times 0 is 0... |
ltasr 11091 | Ordering property of addit... |
pn0sr 11092 | A signed real plus its neg... |
negexsr 11093 | Existence of negative sign... |
recexsrlem 11094 | The reciprocal of a positi... |
addgt0sr 11095 | The sum of two positive si... |
mulgt0sr 11096 | The product of two positiv... |
sqgt0sr 11097 | The square of a nonzero si... |
recexsr 11098 | The reciprocal of a nonzer... |
mappsrpr 11099 | Mapping from positive sign... |
ltpsrpr 11100 | Mapping of order from posi... |
map2psrpr 11101 | Equivalence for positive s... |
supsrlem 11102 | Lemma for supremum theorem... |
supsr 11103 | A nonempty, bounded set of... |
opelcn 11120 | Ordered pair membership in... |
opelreal 11121 | Ordered pair membership in... |
elreal 11122 | Membership in class of rea... |
elreal2 11123 | Ordered pair membership in... |
0ncn 11124 | The empty set is not a com... |
ltrelre 11125 | 'Less than' is a relation ... |
addcnsr 11126 | Addition of complex number... |
mulcnsr 11127 | Multiplication of complex ... |
eqresr 11128 | Equality of real numbers i... |
addresr 11129 | Addition of real numbers i... |
mulresr 11130 | Multiplication of real num... |
ltresr 11131 | Ordering of real subset of... |
ltresr2 11132 | Ordering of real subset of... |
dfcnqs 11133 | Technical trick to permit ... |
addcnsrec 11134 | Technical trick to permit ... |
mulcnsrec 11135 | Technical trick to permit ... |
axaddf 11136 | Addition is an operation o... |
axmulf 11137 | Multiplication is an opera... |
axcnex 11138 | The complex numbers form a... |
axresscn 11139 | The real numbers are a sub... |
ax1cn 11140 | 1 is a complex number. Ax... |
axicn 11141 | ` _i ` is a complex number... |
axaddcl 11142 | Closure law for addition o... |
axaddrcl 11143 | Closure law for addition i... |
axmulcl 11144 | Closure law for multiplica... |
axmulrcl 11145 | Closure law for multiplica... |
axmulcom 11146 | Multiplication of complex ... |
axaddass 11147 | Addition of complex number... |
axmulass 11148 | Multiplication of complex ... |
axdistr 11149 | Distributive law for compl... |
axi2m1 11150 | i-squared equals -1 (expre... |
ax1ne0 11151 | 1 and 0 are distinct. Axi... |
ax1rid 11152 | ` 1 ` is an identity eleme... |
axrnegex 11153 | Existence of negative of r... |
axrrecex 11154 | Existence of reciprocal of... |
axcnre 11155 | A complex number can be ex... |
axpre-lttri 11156 | Ordering on reals satisfie... |
axpre-lttrn 11157 | Ordering on reals is trans... |
axpre-ltadd 11158 | Ordering property of addit... |
axpre-mulgt0 11159 | The product of two positiv... |
axpre-sup 11160 | A nonempty, bounded-above ... |
wuncn 11161 | A weak universe containing... |
cnex 11187 | Alias for ~ ax-cnex . See... |
addcl 11188 | Alias for ~ ax-addcl , for... |
readdcl 11189 | Alias for ~ ax-addrcl , fo... |
mulcl 11190 | Alias for ~ ax-mulcl , for... |
remulcl 11191 | Alias for ~ ax-mulrcl , fo... |
mulcom 11192 | Alias for ~ ax-mulcom , fo... |
addass 11193 | Alias for ~ ax-addass , fo... |
mulass 11194 | Alias for ~ ax-mulass , fo... |
adddi 11195 | Alias for ~ ax-distr , for... |
recn 11196 | A real number is a complex... |
reex 11197 | The real numbers form a se... |
reelprrecn 11198 | Reals are a subset of the ... |
cnelprrecn 11199 | Complex numbers are a subs... |
mpomulf 11200 | Multiplication is an opera... |
elimne0 11201 | Hypothesis for weak deduct... |
adddir 11202 | Distributive law for compl... |
0cn 11203 | Zero is a complex number. ... |
0cnd 11204 | Zero is a complex number, ... |
c0ex 11205 | Zero is a set. (Contribut... |
1cnd 11206 | One is a complex number, d... |
1ex 11207 | One is a set. (Contribute... |
cnre 11208 | Alias for ~ ax-cnre , for ... |
mulrid 11209 | The number 1 is an identit... |
mullid 11210 | Identity law for multiplic... |
1re 11211 | The number 1 is real. Thi... |
1red 11212 | The number 1 is real, dedu... |
0re 11213 | The number 0 is real. Rem... |
0red 11214 | The number 0 is real, dedu... |
mulridi 11215 | Identity law for multiplic... |
mullidi 11216 | Identity law for multiplic... |
addcli 11217 | Closure law for addition. ... |
mulcli 11218 | Closure law for multiplica... |
mulcomi 11219 | Commutative law for multip... |
mulcomli 11220 | Commutative law for multip... |
addassi 11221 | Associative law for additi... |
mulassi 11222 | Associative law for multip... |
adddii 11223 | Distributive law (left-dis... |
adddiri 11224 | Distributive law (right-di... |
recni 11225 | A real number is a complex... |
readdcli 11226 | Closure law for addition o... |
remulcli 11227 | Closure law for multiplica... |
mulridd 11228 | Identity law for multiplic... |
mullidd 11229 | Identity law for multiplic... |
addcld 11230 | Closure law for addition. ... |
mulcld 11231 | Closure law for multiplica... |
mulcomd 11232 | Commutative law for multip... |
addassd 11233 | Associative law for additi... |
mulassd 11234 | Associative law for multip... |
adddid 11235 | Distributive law (left-dis... |
adddird 11236 | Distributive law (right-di... |
adddirp1d 11237 | Distributive law, plus 1 v... |
joinlmuladdmuld 11238 | Join AB+CB into (A+C) on L... |
recnd 11239 | Deduction from real number... |
readdcld 11240 | Closure law for addition o... |
remulcld 11241 | Closure law for multiplica... |
pnfnre 11252 | Plus infinity is not a rea... |
pnfnre2 11253 | Plus infinity is not a rea... |
mnfnre 11254 | Minus infinity is not a re... |
ressxr 11255 | The standard reals are a s... |
rexpssxrxp 11256 | The Cartesian product of s... |
rexr 11257 | A standard real is an exte... |
0xr 11258 | Zero is an extended real. ... |
renepnf 11259 | No (finite) real equals pl... |
renemnf 11260 | No real equals minus infin... |
rexrd 11261 | A standard real is an exte... |
renepnfd 11262 | No (finite) real equals pl... |
renemnfd 11263 | No real equals minus infin... |
pnfex 11264 | Plus infinity exists. (Co... |
pnfxr 11265 | Plus infinity belongs to t... |
pnfnemnf 11266 | Plus and minus infinity ar... |
mnfnepnf 11267 | Minus and plus infinity ar... |
mnfxr 11268 | Minus infinity belongs to ... |
rexri 11269 | A standard real is an exte... |
1xr 11270 | ` 1 ` is an extended real ... |
renfdisj 11271 | The reals and the infiniti... |
ltrelxr 11272 | "Less than" is a relation ... |
ltrel 11273 | "Less than" is a relation.... |
lerelxr 11274 | "Less than or equal to" is... |
lerel 11275 | "Less than or equal to" is... |
xrlenlt 11276 | "Less than or equal to" ex... |
xrlenltd 11277 | "Less than or equal to" ex... |
xrltnle 11278 | "Less than" expressed in t... |
xrnltled 11279 | "Not less than" implies "l... |
ssxr 11280 | The three (non-exclusive) ... |
ltxrlt 11281 | The standard less-than ` <... |
axlttri 11282 | Ordering on reals satisfie... |
axlttrn 11283 | Ordering on reals is trans... |
axltadd 11284 | Ordering property of addit... |
axmulgt0 11285 | The product of two positiv... |
axsup 11286 | A nonempty, bounded-above ... |
lttr 11287 | Alias for ~ axlttrn , for ... |
mulgt0 11288 | The product of two positiv... |
lenlt 11289 | 'Less than or equal to' ex... |
ltnle 11290 | 'Less than' expressed in t... |
ltso 11291 | 'Less than' is a strict or... |
gtso 11292 | 'Greater than' is a strict... |
lttri2 11293 | Consequence of trichotomy.... |
lttri3 11294 | Trichotomy law for 'less t... |
lttri4 11295 | Trichotomy law for 'less t... |
letri3 11296 | Trichotomy law. (Contribu... |
leloe 11297 | 'Less than or equal to' ex... |
eqlelt 11298 | Equality in terms of 'less... |
ltle 11299 | 'Less than' implies 'less ... |
leltne 11300 | 'Less than or equal to' im... |
lelttr 11301 | Transitive law. (Contribu... |
leltletr 11302 | Transitive law, weaker for... |
ltletr 11303 | Transitive law. (Contribu... |
ltleletr 11304 | Transitive law, weaker for... |
letr 11305 | Transitive law. (Contribu... |
ltnr 11306 | 'Less than' is irreflexive... |
leid 11307 | 'Less than or equal to' is... |
ltne 11308 | 'Less than' implies not eq... |
ltnsym 11309 | 'Less than' is not symmetr... |
ltnsym2 11310 | 'Less than' is antisymmetr... |
letric 11311 | Trichotomy law. (Contribu... |
ltlen 11312 | 'Less than' expressed in t... |
eqle 11313 | Equality implies 'less tha... |
eqled 11314 | Equality implies 'less tha... |
ltadd2 11315 | Addition to both sides of ... |
ne0gt0 11316 | A nonzero nonnegative numb... |
lecasei 11317 | Ordering elimination by ca... |
lelttric 11318 | Trichotomy law. (Contribu... |
ltlecasei 11319 | Ordering elimination by ca... |
ltnri 11320 | 'Less than' is irreflexive... |
eqlei 11321 | Equality implies 'less tha... |
eqlei2 11322 | Equality implies 'less tha... |
gtneii 11323 | 'Less than' implies not eq... |
ltneii 11324 | 'Greater than' implies not... |
lttri2i 11325 | Consequence of trichotomy.... |
lttri3i 11326 | Consequence of trichotomy.... |
letri3i 11327 | Consequence of trichotomy.... |
leloei 11328 | 'Less than or equal to' in... |
ltleni 11329 | 'Less than' expressed in t... |
ltnsymi 11330 | 'Less than' is not symmetr... |
lenlti 11331 | 'Less than or equal to' in... |
ltnlei 11332 | 'Less than' in terms of 'l... |
ltlei 11333 | 'Less than' implies 'less ... |
ltleii 11334 | 'Less than' implies 'less ... |
ltnei 11335 | 'Less than' implies not eq... |
letrii 11336 | Trichotomy law for 'less t... |
lttri 11337 | 'Less than' is transitive.... |
lelttri 11338 | 'Less than or equal to', '... |
ltletri 11339 | 'Less than', 'less than or... |
letri 11340 | 'Less than or equal to' is... |
le2tri3i 11341 | Extended trichotomy law fo... |
ltadd2i 11342 | Addition to both sides of ... |
mulgt0i 11343 | The product of two positiv... |
mulgt0ii 11344 | The product of two positiv... |
ltnrd 11345 | 'Less than' is irreflexive... |
gtned 11346 | 'Less than' implies not eq... |
ltned 11347 | 'Greater than' implies not... |
ne0gt0d 11348 | A nonzero nonnegative numb... |
lttrid 11349 | Ordering on reals satisfie... |
lttri2d 11350 | Consequence of trichotomy.... |
lttri3d 11351 | Consequence of trichotomy.... |
lttri4d 11352 | Trichotomy law for 'less t... |
letri3d 11353 | Consequence of trichotomy.... |
leloed 11354 | 'Less than or equal to' in... |
eqleltd 11355 | Equality in terms of 'less... |
ltlend 11356 | 'Less than' expressed in t... |
lenltd 11357 | 'Less than or equal to' in... |
ltnled 11358 | 'Less than' in terms of 'l... |
ltled 11359 | 'Less than' implies 'less ... |
ltnsymd 11360 | 'Less than' implies 'less ... |
nltled 11361 | 'Not less than ' implies '... |
lensymd 11362 | 'Less than or equal to' im... |
letrid 11363 | Trichotomy law for 'less t... |
leltned 11364 | 'Less than or equal to' im... |
leneltd 11365 | 'Less than or equal to' an... |
mulgt0d 11366 | The product of two positiv... |
ltadd2d 11367 | Addition to both sides of ... |
letrd 11368 | Transitive law deduction f... |
lelttrd 11369 | Transitive law deduction f... |
ltadd2dd 11370 | Addition to both sides of ... |
ltletrd 11371 | Transitive law deduction f... |
lttrd 11372 | Transitive law deduction f... |
lelttrdi 11373 | If a number is less than a... |
dedekind 11374 | The Dedekind cut theorem. ... |
dedekindle 11375 | The Dedekind cut theorem, ... |
mul12 11376 | Commutative/associative la... |
mul32 11377 | Commutative/associative la... |
mul31 11378 | Commutative/associative la... |
mul4 11379 | Rearrangement of 4 factors... |
mul4r 11380 | Rearrangement of 4 factors... |
muladd11 11381 | A simple product of sums e... |
1p1times 11382 | Two times a number. (Cont... |
peano2cn 11383 | A theorem for complex numb... |
peano2re 11384 | A theorem for reals analog... |
readdcan 11385 | Cancellation law for addit... |
00id 11386 | ` 0 ` is its own additive ... |
mul02lem1 11387 | Lemma for ~ mul02 . If an... |
mul02lem2 11388 | Lemma for ~ mul02 . Zero ... |
mul02 11389 | Multiplication by ` 0 ` . ... |
mul01 11390 | Multiplication by ` 0 ` . ... |
addrid 11391 | ` 0 ` is an additive ident... |
cnegex 11392 | Existence of the negative ... |
cnegex2 11393 | Existence of a left invers... |
addlid 11394 | ` 0 ` is a left identity f... |
addcan 11395 | Cancellation law for addit... |
addcan2 11396 | Cancellation law for addit... |
addcom 11397 | Addition commutes. This u... |
addridi 11398 | ` 0 ` is an additive ident... |
addlidi 11399 | ` 0 ` is a left identity f... |
mul02i 11400 | Multiplication by 0. Theo... |
mul01i 11401 | Multiplication by ` 0 ` . ... |
addcomi 11402 | Addition commutes. Based ... |
addcomli 11403 | Addition commutes. (Contr... |
addcani 11404 | Cancellation law for addit... |
addcan2i 11405 | Cancellation law for addit... |
mul12i 11406 | Commutative/associative la... |
mul32i 11407 | Commutative/associative la... |
mul4i 11408 | Rearrangement of 4 factors... |
mul02d 11409 | Multiplication by 0. Theo... |
mul01d 11410 | Multiplication by ` 0 ` . ... |
addridd 11411 | ` 0 ` is an additive ident... |
addlidd 11412 | ` 0 ` is a left identity f... |
addcomd 11413 | Addition commutes. Based ... |
addcand 11414 | Cancellation law for addit... |
addcan2d 11415 | Cancellation law for addit... |
addcanad 11416 | Cancelling a term on the l... |
addcan2ad 11417 | Cancelling a term on the r... |
addneintrd 11418 | Introducing a term on the ... |
addneintr2d 11419 | Introducing a term on the ... |
mul12d 11420 | Commutative/associative la... |
mul32d 11421 | Commutative/associative la... |
mul31d 11422 | Commutative/associative la... |
mul4d 11423 | Rearrangement of 4 factors... |
muladd11r 11424 | A simple product of sums e... |
comraddd 11425 | Commute RHS addition, in d... |
ltaddneg 11426 | Adding a negative number t... |
ltaddnegr 11427 | Adding a negative number t... |
add12 11428 | Commutative/associative la... |
add32 11429 | Commutative/associative la... |
add32r 11430 | Commutative/associative la... |
add4 11431 | Rearrangement of 4 terms i... |
add42 11432 | Rearrangement of 4 terms i... |
add12i 11433 | Commutative/associative la... |
add32i 11434 | Commutative/associative la... |
add4i 11435 | Rearrangement of 4 terms i... |
add42i 11436 | Rearrangement of 4 terms i... |
add12d 11437 | Commutative/associative la... |
add32d 11438 | Commutative/associative la... |
add4d 11439 | Rearrangement of 4 terms i... |
add42d 11440 | Rearrangement of 4 terms i... |
0cnALT 11445 | Alternate proof of ~ 0cn w... |
0cnALT2 11446 | Alternate proof of ~ 0cnAL... |
negeu 11447 | Existential uniqueness of ... |
subval 11448 | Value of subtraction, whic... |
negeq 11449 | Equality theorem for negat... |
negeqi 11450 | Equality inference for neg... |
negeqd 11451 | Equality deduction for neg... |
nfnegd 11452 | Deduction version of ~ nfn... |
nfneg 11453 | Bound-variable hypothesis ... |
csbnegg 11454 | Move class substitution in... |
negex 11455 | A negative is a set. (Con... |
subcl 11456 | Closure law for subtractio... |
negcl 11457 | Closure law for negative. ... |
negicn 11458 | ` -u _i ` is a complex num... |
subf 11459 | Subtraction is an operatio... |
subadd 11460 | Relationship between subtr... |
subadd2 11461 | Relationship between subtr... |
subsub23 11462 | Swap subtrahend and result... |
pncan 11463 | Cancellation law for subtr... |
pncan2 11464 | Cancellation law for subtr... |
pncan3 11465 | Subtraction and addition o... |
npcan 11466 | Cancellation law for subtr... |
addsubass 11467 | Associative-type law for a... |
addsub 11468 | Law for addition and subtr... |
subadd23 11469 | Commutative/associative la... |
addsub12 11470 | Commutative/associative la... |
2addsub 11471 | Law for subtraction and ad... |
addsubeq4 11472 | Relation between sums and ... |
pncan3oi 11473 | Subtraction and addition o... |
mvrraddi 11474 | Move the right term in a s... |
mvlladdi 11475 | Move the left term in a su... |
subid 11476 | Subtraction of a number fr... |
subid1 11477 | Identity law for subtracti... |
npncan 11478 | Cancellation law for subtr... |
nppcan 11479 | Cancellation law for subtr... |
nnpcan 11480 | Cancellation law for subtr... |
nppcan3 11481 | Cancellation law for subtr... |
subcan2 11482 | Cancellation law for subtr... |
subeq0 11483 | If the difference between ... |
npncan2 11484 | Cancellation law for subtr... |
subsub2 11485 | Law for double subtraction... |
nncan 11486 | Cancellation law for subtr... |
subsub 11487 | Law for double subtraction... |
nppcan2 11488 | Cancellation law for subtr... |
subsub3 11489 | Law for double subtraction... |
subsub4 11490 | Law for double subtraction... |
sub32 11491 | Swap the second and third ... |
nnncan 11492 | Cancellation law for subtr... |
nnncan1 11493 | Cancellation law for subtr... |
nnncan2 11494 | Cancellation law for subtr... |
npncan3 11495 | Cancellation law for subtr... |
pnpcan 11496 | Cancellation law for mixed... |
pnpcan2 11497 | Cancellation law for mixed... |
pnncan 11498 | Cancellation law for mixed... |
ppncan 11499 | Cancellation law for mixed... |
addsub4 11500 | Rearrangement of 4 terms i... |
subadd4 11501 | Rearrangement of 4 terms i... |
sub4 11502 | Rearrangement of 4 terms i... |
neg0 11503 | Minus 0 equals 0. (Contri... |
negid 11504 | Addition of a number and i... |
negsub 11505 | Relationship between subtr... |
subneg 11506 | Relationship between subtr... |
negneg 11507 | A number is equal to the n... |
neg11 11508 | Negative is one-to-one. (... |
negcon1 11509 | Negative contraposition la... |
negcon2 11510 | Negative contraposition la... |
negeq0 11511 | A number is zero iff its n... |
subcan 11512 | Cancellation law for subtr... |
negsubdi 11513 | Distribution of negative o... |
negdi 11514 | Distribution of negative o... |
negdi2 11515 | Distribution of negative o... |
negsubdi2 11516 | Distribution of negative o... |
neg2sub 11517 | Relationship between subtr... |
renegcli 11518 | Closure law for negative o... |
resubcli 11519 | Closure law for subtractio... |
renegcl 11520 | Closure law for negative o... |
resubcl 11521 | Closure law for subtractio... |
negreb 11522 | The negative of a real is ... |
peano2cnm 11523 | "Reverse" second Peano pos... |
peano2rem 11524 | "Reverse" second Peano pos... |
negcli 11525 | Closure law for negative. ... |
negidi 11526 | Addition of a number and i... |
negnegi 11527 | A number is equal to the n... |
subidi 11528 | Subtraction of a number fr... |
subid1i 11529 | Identity law for subtracti... |
negne0bi 11530 | A number is nonzero iff it... |
negrebi 11531 | The negative of a real is ... |
negne0i 11532 | The negative of a nonzero ... |
subcli 11533 | Closure law for subtractio... |
pncan3i 11534 | Subtraction and addition o... |
negsubi 11535 | Relationship between subtr... |
subnegi 11536 | Relationship between subtr... |
subeq0i 11537 | If the difference between ... |
neg11i 11538 | Negative is one-to-one. (... |
negcon1i 11539 | Negative contraposition la... |
negcon2i 11540 | Negative contraposition la... |
negdii 11541 | Distribution of negative o... |
negsubdii 11542 | Distribution of negative o... |
negsubdi2i 11543 | Distribution of negative o... |
subaddi 11544 | Relationship between subtr... |
subadd2i 11545 | Relationship between subtr... |
subaddrii 11546 | Relationship between subtr... |
subsub23i 11547 | Swap subtrahend and result... |
addsubassi 11548 | Associative-type law for s... |
addsubi 11549 | Law for subtraction and ad... |
subcani 11550 | Cancellation law for subtr... |
subcan2i 11551 | Cancellation law for subtr... |
pnncani 11552 | Cancellation law for mixed... |
addsub4i 11553 | Rearrangement of 4 terms i... |
0reALT 11554 | Alternate proof of ~ 0re .... |
negcld 11555 | Closure law for negative. ... |
subidd 11556 | Subtraction of a number fr... |
subid1d 11557 | Identity law for subtracti... |
negidd 11558 | Addition of a number and i... |
negnegd 11559 | A number is equal to the n... |
negeq0d 11560 | A number is zero iff its n... |
negne0bd 11561 | A number is nonzero iff it... |
negcon1d 11562 | Contraposition law for una... |
negcon1ad 11563 | Contraposition law for una... |
neg11ad 11564 | The negatives of two compl... |
negned 11565 | If two complex numbers are... |
negne0d 11566 | The negative of a nonzero ... |
negrebd 11567 | The negative of a real is ... |
subcld 11568 | Closure law for subtractio... |
pncand 11569 | Cancellation law for subtr... |
pncan2d 11570 | Cancellation law for subtr... |
pncan3d 11571 | Subtraction and addition o... |
npcand 11572 | Cancellation law for subtr... |
nncand 11573 | Cancellation law for subtr... |
negsubd 11574 | Relationship between subtr... |
subnegd 11575 | Relationship between subtr... |
subeq0d 11576 | If the difference between ... |
subne0d 11577 | Two unequal numbers have n... |
subeq0ad 11578 | The difference of two comp... |
subne0ad 11579 | If the difference of two c... |
neg11d 11580 | If the difference between ... |
negdid 11581 | Distribution of negative o... |
negdi2d 11582 | Distribution of negative o... |
negsubdid 11583 | Distribution of negative o... |
negsubdi2d 11584 | Distribution of negative o... |
neg2subd 11585 | Relationship between subtr... |
subaddd 11586 | Relationship between subtr... |
subadd2d 11587 | Relationship between subtr... |
addsubassd 11588 | Associative-type law for s... |
addsubd 11589 | Law for subtraction and ad... |
subadd23d 11590 | Commutative/associative la... |
addsub12d 11591 | Commutative/associative la... |
npncand 11592 | Cancellation law for subtr... |
nppcand 11593 | Cancellation law for subtr... |
nppcan2d 11594 | Cancellation law for subtr... |
nppcan3d 11595 | Cancellation law for subtr... |
subsubd 11596 | Law for double subtraction... |
subsub2d 11597 | Law for double subtraction... |
subsub3d 11598 | Law for double subtraction... |
subsub4d 11599 | Law for double subtraction... |
sub32d 11600 | Swap the second and third ... |
nnncand 11601 | Cancellation law for subtr... |
nnncan1d 11602 | Cancellation law for subtr... |
nnncan2d 11603 | Cancellation law for subtr... |
npncan3d 11604 | Cancellation law for subtr... |
pnpcand 11605 | Cancellation law for mixed... |
pnpcan2d 11606 | Cancellation law for mixed... |
pnncand 11607 | Cancellation law for mixed... |
ppncand 11608 | Cancellation law for mixed... |
subcand 11609 | Cancellation law for subtr... |
subcan2d 11610 | Cancellation law for subtr... |
subcanad 11611 | Cancellation law for subtr... |
subneintrd 11612 | Introducing subtraction on... |
subcan2ad 11613 | Cancellation law for subtr... |
subneintr2d 11614 | Introducing subtraction on... |
addsub4d 11615 | Rearrangement of 4 terms i... |
subadd4d 11616 | Rearrangement of 4 terms i... |
sub4d 11617 | Rearrangement of 4 terms i... |
2addsubd 11618 | Law for subtraction and ad... |
addsubeq4d 11619 | Relation between sums and ... |
subeqxfrd 11620 | Transfer two terms of a su... |
mvlraddd 11621 | Move the right term in a s... |
mvlladdd 11622 | Move the left term in a su... |
mvrraddd 11623 | Move the right term in a s... |
mvrladdd 11624 | Move the left term in a su... |
assraddsubd 11625 | Associate RHS addition-sub... |
subaddeqd 11626 | Transfer two terms of a su... |
addlsub 11627 | Left-subtraction: Subtrac... |
addrsub 11628 | Right-subtraction: Subtra... |
subexsub 11629 | A subtraction law: Exchan... |
addid0 11630 | If adding a number to a an... |
addn0nid 11631 | Adding a nonzero number to... |
pnpncand 11632 | Addition/subtraction cance... |
subeqrev 11633 | Reverse the order of subtr... |
addeq0 11634 | Two complex numbers add up... |
pncan1 11635 | Cancellation law for addit... |
npcan1 11636 | Cancellation law for subtr... |
subeq0bd 11637 | If two complex numbers are... |
renegcld 11638 | Closure law for negative o... |
resubcld 11639 | Closure law for subtractio... |
negn0 11640 | The image under negation o... |
negf1o 11641 | Negation is an isomorphism... |
kcnktkm1cn 11642 | k times k minus 1 is a com... |
muladd 11643 | Product of two sums. (Con... |
subdi 11644 | Distribution of multiplica... |
subdir 11645 | Distribution of multiplica... |
ine0 11646 | The imaginary unit ` _i ` ... |
mulneg1 11647 | Product with negative is n... |
mulneg2 11648 | The product with a negativ... |
mulneg12 11649 | Swap the negative sign in ... |
mul2neg 11650 | Product of two negatives. ... |
submul2 11651 | Convert a subtraction to a... |
mulm1 11652 | Product with minus one is ... |
addneg1mul 11653 | Addition with product with... |
mulsub 11654 | Product of two differences... |
mulsub2 11655 | Swap the order of subtract... |
mulm1i 11656 | Product with minus one is ... |
mulneg1i 11657 | Product with negative is n... |
mulneg2i 11658 | Product with negative is n... |
mul2negi 11659 | Product of two negatives. ... |
subdii 11660 | Distribution of multiplica... |
subdiri 11661 | Distribution of multiplica... |
muladdi 11662 | Product of two sums. (Con... |
mulm1d 11663 | Product with minus one is ... |
mulneg1d 11664 | Product with negative is n... |
mulneg2d 11665 | Product with negative is n... |
mul2negd 11666 | Product of two negatives. ... |
subdid 11667 | Distribution of multiplica... |
subdird 11668 | Distribution of multiplica... |
muladdd 11669 | Product of two sums. (Con... |
mulsubd 11670 | Product of two differences... |
muls1d 11671 | Multiplication by one minu... |
mulsubfacd 11672 | Multiplication followed by... |
addmulsub 11673 | The product of a sum and a... |
subaddmulsub 11674 | The difference with a prod... |
mulsubaddmulsub 11675 | A special difference of a ... |
gt0ne0 11676 | Positive implies nonzero. ... |
lt0ne0 11677 | A number which is less tha... |
ltadd1 11678 | Addition to both sides of ... |
leadd1 11679 | Addition to both sides of ... |
leadd2 11680 | Addition to both sides of ... |
ltsubadd 11681 | 'Less than' relationship b... |
ltsubadd2 11682 | 'Less than' relationship b... |
lesubadd 11683 | 'Less than or equal to' re... |
lesubadd2 11684 | 'Less than or equal to' re... |
ltaddsub 11685 | 'Less than' relationship b... |
ltaddsub2 11686 | 'Less than' relationship b... |
leaddsub 11687 | 'Less than or equal to' re... |
leaddsub2 11688 | 'Less than or equal to' re... |
suble 11689 | Swap subtrahends in an ine... |
lesub 11690 | Swap subtrahends in an ine... |
ltsub23 11691 | 'Less than' relationship b... |
ltsub13 11692 | 'Less than' relationship b... |
le2add 11693 | Adding both sides of two '... |
ltleadd 11694 | Adding both sides of two o... |
leltadd 11695 | Adding both sides of two o... |
lt2add 11696 | Adding both sides of two '... |
addgt0 11697 | The sum of 2 positive numb... |
addgegt0 11698 | The sum of nonnegative and... |
addgtge0 11699 | The sum of nonnegative and... |
addge0 11700 | The sum of 2 nonnegative n... |
ltaddpos 11701 | Adding a positive number t... |
ltaddpos2 11702 | Adding a positive number t... |
ltsubpos 11703 | Subtracting a positive num... |
posdif 11704 | Comparison of two numbers ... |
lesub1 11705 | Subtraction from both side... |
lesub2 11706 | Subtraction of both sides ... |
ltsub1 11707 | Subtraction from both side... |
ltsub2 11708 | Subtraction of both sides ... |
lt2sub 11709 | Subtracting both sides of ... |
le2sub 11710 | Subtracting both sides of ... |
ltneg 11711 | Negative of both sides of ... |
ltnegcon1 11712 | Contraposition of negative... |
ltnegcon2 11713 | Contraposition of negative... |
leneg 11714 | Negative of both sides of ... |
lenegcon1 11715 | Contraposition of negative... |
lenegcon2 11716 | Contraposition of negative... |
lt0neg1 11717 | Comparison of a number and... |
lt0neg2 11718 | Comparison of a number and... |
le0neg1 11719 | Comparison of a number and... |
le0neg2 11720 | Comparison of a number and... |
addge01 11721 | A number is less than or e... |
addge02 11722 | A number is less than or e... |
add20 11723 | Two nonnegative numbers ar... |
subge0 11724 | Nonnegative subtraction. ... |
suble0 11725 | Nonpositive subtraction. ... |
leaddle0 11726 | The sum of a real number a... |
subge02 11727 | Nonnegative subtraction. ... |
lesub0 11728 | Lemma to show a nonnegativ... |
mulge0 11729 | The product of two nonnega... |
mullt0 11730 | The product of two negativ... |
msqgt0 11731 | A nonzero square is positi... |
msqge0 11732 | A square is nonnegative. ... |
0lt1 11733 | 0 is less than 1. Theorem... |
0le1 11734 | 0 is less than or equal to... |
relin01 11735 | An interval law for less t... |
ltordlem 11736 | Lemma for ~ ltord1 . (Con... |
ltord1 11737 | Infer an ordering relation... |
leord1 11738 | Infer an ordering relation... |
eqord1 11739 | A strictly increasing real... |
ltord2 11740 | Infer an ordering relation... |
leord2 11741 | Infer an ordering relation... |
eqord2 11742 | A strictly decreasing real... |
wloglei 11743 | Form of ~ wlogle where bot... |
wlogle 11744 | If the predicate ` ch ( x ... |
leidi 11745 | 'Less than or equal to' is... |
gt0ne0i 11746 | Positive means nonzero (us... |
gt0ne0ii 11747 | Positive implies nonzero. ... |
msqgt0i 11748 | A nonzero square is positi... |
msqge0i 11749 | A square is nonnegative. ... |
addgt0i 11750 | Addition of 2 positive num... |
addge0i 11751 | Addition of 2 nonnegative ... |
addgegt0i 11752 | Addition of nonnegative an... |
addgt0ii 11753 | Addition of 2 positive num... |
add20i 11754 | Two nonnegative numbers ar... |
ltnegi 11755 | Negative of both sides of ... |
lenegi 11756 | Negative of both sides of ... |
ltnegcon2i 11757 | Contraposition of negative... |
mulge0i 11758 | The product of two nonnega... |
lesub0i 11759 | Lemma to show a nonnegativ... |
ltaddposi 11760 | Adding a positive number t... |
posdifi 11761 | Comparison of two numbers ... |
ltnegcon1i 11762 | Contraposition of negative... |
lenegcon1i 11763 | Contraposition of negative... |
subge0i 11764 | Nonnegative subtraction. ... |
ltadd1i 11765 | Addition to both sides of ... |
leadd1i 11766 | Addition to both sides of ... |
leadd2i 11767 | Addition to both sides of ... |
ltsubaddi 11768 | 'Less than' relationship b... |
lesubaddi 11769 | 'Less than or equal to' re... |
ltsubadd2i 11770 | 'Less than' relationship b... |
lesubadd2i 11771 | 'Less than or equal to' re... |
ltaddsubi 11772 | 'Less than' relationship b... |
lt2addi 11773 | Adding both side of two in... |
le2addi 11774 | Adding both side of two in... |
gt0ne0d 11775 | Positive implies nonzero. ... |
lt0ne0d 11776 | Something less than zero i... |
leidd 11777 | 'Less than or equal to' is... |
msqgt0d 11778 | A nonzero square is positi... |
msqge0d 11779 | A square is nonnegative. ... |
lt0neg1d 11780 | Comparison of a number and... |
lt0neg2d 11781 | Comparison of a number and... |
le0neg1d 11782 | Comparison of a number and... |
le0neg2d 11783 | Comparison of a number and... |
addgegt0d 11784 | Addition of nonnegative an... |
addgtge0d 11785 | Addition of positive and n... |
addgt0d 11786 | Addition of 2 positive num... |
addge0d 11787 | Addition of 2 nonnegative ... |
mulge0d 11788 | The product of two nonnega... |
ltnegd 11789 | Negative of both sides of ... |
lenegd 11790 | Negative of both sides of ... |
ltnegcon1d 11791 | Contraposition of negative... |
ltnegcon2d 11792 | Contraposition of negative... |
lenegcon1d 11793 | Contraposition of negative... |
lenegcon2d 11794 | Contraposition of negative... |
ltaddposd 11795 | Adding a positive number t... |
ltaddpos2d 11796 | Adding a positive number t... |
ltsubposd 11797 | Subtracting a positive num... |
posdifd 11798 | Comparison of two numbers ... |
addge01d 11799 | A number is less than or e... |
addge02d 11800 | A number is less than or e... |
subge0d 11801 | Nonnegative subtraction. ... |
suble0d 11802 | Nonpositive subtraction. ... |
subge02d 11803 | Nonnegative subtraction. ... |
ltadd1d 11804 | Addition to both sides of ... |
leadd1d 11805 | Addition to both sides of ... |
leadd2d 11806 | Addition to both sides of ... |
ltsubaddd 11807 | 'Less than' relationship b... |
lesubaddd 11808 | 'Less than or equal to' re... |
ltsubadd2d 11809 | 'Less than' relationship b... |
lesubadd2d 11810 | 'Less than or equal to' re... |
ltaddsubd 11811 | 'Less than' relationship b... |
ltaddsub2d 11812 | 'Less than' relationship b... |
leaddsub2d 11813 | 'Less than or equal to' re... |
subled 11814 | Swap subtrahends in an ine... |
lesubd 11815 | Swap subtrahends in an ine... |
ltsub23d 11816 | 'Less than' relationship b... |
ltsub13d 11817 | 'Less than' relationship b... |
lesub1d 11818 | Subtraction from both side... |
lesub2d 11819 | Subtraction of both sides ... |
ltsub1d 11820 | Subtraction from both side... |
ltsub2d 11821 | Subtraction of both sides ... |
ltadd1dd 11822 | Addition to both sides of ... |
ltsub1dd 11823 | Subtraction from both side... |
ltsub2dd 11824 | Subtraction of both sides ... |
leadd1dd 11825 | Addition to both sides of ... |
leadd2dd 11826 | Addition to both sides of ... |
lesub1dd 11827 | Subtraction from both side... |
lesub2dd 11828 | Subtraction of both sides ... |
lesub3d 11829 | The result of subtracting ... |
le2addd 11830 | Adding both side of two in... |
le2subd 11831 | Subtracting both sides of ... |
ltleaddd 11832 | Adding both sides of two o... |
leltaddd 11833 | Adding both sides of two o... |
lt2addd 11834 | Adding both side of two in... |
lt2subd 11835 | Subtracting both sides of ... |
possumd 11836 | Condition for a positive s... |
sublt0d 11837 | When a subtraction gives a... |
ltaddsublt 11838 | Addition and subtraction o... |
1le1 11839 | One is less than or equal ... |
ixi 11840 | ` _i ` times itself is min... |
recextlem1 11841 | Lemma for ~ recex . (Cont... |
recextlem2 11842 | Lemma for ~ recex . (Cont... |
recex 11843 | Existence of reciprocal of... |
mulcand 11844 | Cancellation law for multi... |
mulcan2d 11845 | Cancellation law for multi... |
mulcanad 11846 | Cancellation of a nonzero ... |
mulcan2ad 11847 | Cancellation of a nonzero ... |
mulcan 11848 | Cancellation law for multi... |
mulcan2 11849 | Cancellation law for multi... |
mulcani 11850 | Cancellation law for multi... |
mul0or 11851 | If a product is zero, one ... |
mulne0b 11852 | The product of two nonzero... |
mulne0 11853 | The product of two nonzero... |
mulne0i 11854 | The product of two nonzero... |
muleqadd 11855 | Property of numbers whose ... |
receu 11856 | Existential uniqueness of ... |
mulnzcnf 11857 | Multiplication maps nonzer... |
msq0i 11858 | A number is zero iff its s... |
mul0ori 11859 | If a product is zero, one ... |
msq0d 11860 | A number is zero iff its s... |
mul0ord 11861 | If a product is zero, one ... |
mulne0bd 11862 | The product of two nonzero... |
mulne0d 11863 | The product of two nonzero... |
mulcan1g 11864 | A generalized form of the ... |
mulcan2g 11865 | A generalized form of the ... |
mulne0bad 11866 | A factor of a nonzero comp... |
mulne0bbd 11867 | A factor of a nonzero comp... |
1div0 11870 | You can't divide by zero, ... |
divval 11871 | Value of division: if ` A ... |
divmul 11872 | Relationship between divis... |
divmul2 11873 | Relationship between divis... |
divmul3 11874 | Relationship between divis... |
divcl 11875 | Closure law for division. ... |
reccl 11876 | Closure law for reciprocal... |
divcan2 11877 | A cancellation law for div... |
divcan1 11878 | A cancellation law for div... |
diveq0 11879 | A ratio is zero iff the nu... |
divne0b 11880 | The ratio of nonzero numbe... |
divne0 11881 | The ratio of nonzero numbe... |
recne0 11882 | The reciprocal of a nonzer... |
recid 11883 | Multiplication of a number... |
recid2 11884 | Multiplication of a number... |
divrec 11885 | Relationship between divis... |
divrec2 11886 | Relationship between divis... |
divass 11887 | An associative law for div... |
div23 11888 | A commutative/associative ... |
div32 11889 | A commutative/associative ... |
div13 11890 | A commutative/associative ... |
div12 11891 | A commutative/associative ... |
divmulass 11892 | An associative law for div... |
divmulasscom 11893 | An associative/commutative... |
divdir 11894 | Distribution of division o... |
divcan3 11895 | A cancellation law for div... |
divcan4 11896 | A cancellation law for div... |
div11 11897 | One-to-one relationship fo... |
divid 11898 | A number divided by itself... |
div0 11899 | Division into zero is zero... |
div1 11900 | A number divided by 1 is i... |
1div1e1 11901 | 1 divided by 1 is 1. (Con... |
diveq1 11902 | Equality in terms of unit ... |
divneg 11903 | Move negative sign inside ... |
muldivdir 11904 | Distribution of division o... |
divsubdir 11905 | Distribution of division o... |
subdivcomb1 11906 | Bring a term in a subtract... |
subdivcomb2 11907 | Bring a term in a subtract... |
recrec 11908 | A number is equal to the r... |
rec11 11909 | Reciprocal is one-to-one. ... |
rec11r 11910 | Mutual reciprocals. (Cont... |
divmuldiv 11911 | Multiplication of two rati... |
divdivdiv 11912 | Division of two ratios. T... |
divcan5 11913 | Cancellation of common fac... |
divmul13 11914 | Swap the denominators in t... |
divmul24 11915 | Swap the numerators in the... |
divmuleq 11916 | Cross-multiply in an equal... |
recdiv 11917 | The reciprocal of a ratio.... |
divcan6 11918 | Cancellation of inverted f... |
divdiv32 11919 | Swap denominators in a div... |
divcan7 11920 | Cancel equal divisors in a... |
dmdcan 11921 | Cancellation law for divis... |
divdiv1 11922 | Division into a fraction. ... |
divdiv2 11923 | Division by a fraction. (... |
recdiv2 11924 | Division into a reciprocal... |
ddcan 11925 | Cancellation in a double d... |
divadddiv 11926 | Addition of two ratios. T... |
divsubdiv 11927 | Subtraction of two ratios.... |
conjmul 11928 | Two numbers whose reciproc... |
rereccl 11929 | Closure law for reciprocal... |
redivcl 11930 | Closure law for division o... |
eqneg 11931 | A number equal to its nega... |
eqnegd 11932 | A complex number equals it... |
eqnegad 11933 | If a complex number equals... |
div2neg 11934 | Quotient of two negatives.... |
divneg2 11935 | Move negative sign inside ... |
recclzi 11936 | Closure law for reciprocal... |
recne0zi 11937 | The reciprocal of a nonzer... |
recidzi 11938 | Multiplication of a number... |
div1i 11939 | A number divided by 1 is i... |
eqnegi 11940 | A number equal to its nega... |
reccli 11941 | Closure law for reciprocal... |
recidi 11942 | Multiplication of a number... |
recreci 11943 | A number is equal to the r... |
dividi 11944 | A number divided by itself... |
div0i 11945 | Division into zero is zero... |
divclzi 11946 | Closure law for division. ... |
divcan1zi 11947 | A cancellation law for div... |
divcan2zi 11948 | A cancellation law for div... |
divreczi 11949 | Relationship between divis... |
divcan3zi 11950 | A cancellation law for div... |
divcan4zi 11951 | A cancellation law for div... |
rec11i 11952 | Reciprocal is one-to-one. ... |
divcli 11953 | Closure law for division. ... |
divcan2i 11954 | A cancellation law for div... |
divcan1i 11955 | A cancellation law for div... |
divreci 11956 | Relationship between divis... |
divcan3i 11957 | A cancellation law for div... |
divcan4i 11958 | A cancellation law for div... |
divne0i 11959 | The ratio of nonzero numbe... |
rec11ii 11960 | Reciprocal is one-to-one. ... |
divasszi 11961 | An associative law for div... |
divmulzi 11962 | Relationship between divis... |
divdirzi 11963 | Distribution of division o... |
divdiv23zi 11964 | Swap denominators in a div... |
divmuli 11965 | Relationship between divis... |
divdiv32i 11966 | Swap denominators in a div... |
divassi 11967 | An associative law for div... |
divdiri 11968 | Distribution of division o... |
div23i 11969 | A commutative/associative ... |
div11i 11970 | One-to-one relationship fo... |
divmuldivi 11971 | Multiplication of two rati... |
divmul13i 11972 | Swap denominators of two r... |
divadddivi 11973 | Addition of two ratios. T... |
divdivdivi 11974 | Division of two ratios. T... |
rerecclzi 11975 | Closure law for reciprocal... |
rereccli 11976 | Closure law for reciprocal... |
redivclzi 11977 | Closure law for division o... |
redivcli 11978 | Closure law for division o... |
div1d 11979 | A number divided by 1 is i... |
reccld 11980 | Closure law for reciprocal... |
recne0d 11981 | The reciprocal of a nonzer... |
recidd 11982 | Multiplication of a number... |
recid2d 11983 | Multiplication of a number... |
recrecd 11984 | A number is equal to the r... |
dividd 11985 | A number divided by itself... |
div0d 11986 | Division into zero is zero... |
divcld 11987 | Closure law for division. ... |
divcan1d 11988 | A cancellation law for div... |
divcan2d 11989 | A cancellation law for div... |
divrecd 11990 | Relationship between divis... |
divrec2d 11991 | Relationship between divis... |
divcan3d 11992 | A cancellation law for div... |
divcan4d 11993 | A cancellation law for div... |
diveq0d 11994 | A ratio is zero iff the nu... |
diveq1d 11995 | Equality in terms of unit ... |
diveq1ad 11996 | The quotient of two comple... |
diveq0ad 11997 | A fraction of complex numb... |
divne1d 11998 | If two complex numbers are... |
divne0bd 11999 | A ratio is zero iff the nu... |
divnegd 12000 | Move negative sign inside ... |
divneg2d 12001 | Move negative sign inside ... |
div2negd 12002 | Quotient of two negatives.... |
divne0d 12003 | The ratio of nonzero numbe... |
recdivd 12004 | The reciprocal of a ratio.... |
recdiv2d 12005 | Division into a reciprocal... |
divcan6d 12006 | Cancellation of inverted f... |
ddcand 12007 | Cancellation in a double d... |
rec11d 12008 | Reciprocal is one-to-one. ... |
divmuld 12009 | Relationship between divis... |
div32d 12010 | A commutative/associative ... |
div13d 12011 | A commutative/associative ... |
divdiv32d 12012 | Swap denominators in a div... |
divcan5d 12013 | Cancellation of common fac... |
divcan5rd 12014 | Cancellation of common fac... |
divcan7d 12015 | Cancel equal divisors in a... |
dmdcand 12016 | Cancellation law for divis... |
dmdcan2d 12017 | Cancellation law for divis... |
divdiv1d 12018 | Division into a fraction. ... |
divdiv2d 12019 | Division by a fraction. (... |
divmul2d 12020 | Relationship between divis... |
divmul3d 12021 | Relationship between divis... |
divassd 12022 | An associative law for div... |
div12d 12023 | A commutative/associative ... |
div23d 12024 | A commutative/associative ... |
divdird 12025 | Distribution of division o... |
divsubdird 12026 | Distribution of division o... |
div11d 12027 | One-to-one relationship fo... |
divmuldivd 12028 | Multiplication of two rati... |
divmul13d 12029 | Swap denominators of two r... |
divmul24d 12030 | Swap the numerators in the... |
divadddivd 12031 | Addition of two ratios. T... |
divsubdivd 12032 | Subtraction of two ratios.... |
divmuleqd 12033 | Cross-multiply in an equal... |
divdivdivd 12034 | Division of two ratios. T... |
diveq1bd 12035 | If two complex numbers are... |
div2sub 12036 | Swap the order of subtract... |
div2subd 12037 | Swap subtrahend and minuen... |
rereccld 12038 | Closure law for reciprocal... |
redivcld 12039 | Closure law for division o... |
subrec 12040 | Subtraction of reciprocals... |
subreci 12041 | Subtraction of reciprocals... |
subrecd 12042 | Subtraction of reciprocals... |
mvllmuld 12043 | Move the left term in a pr... |
mvllmuli 12044 | Move the left term in a pr... |
ldiv 12045 | Left-division. (Contribut... |
rdiv 12046 | Right-division. (Contribu... |
mdiv 12047 | A division law. (Contribu... |
lineq 12048 | Solution of a (scalar) lin... |
elimgt0 12049 | Hypothesis for weak deduct... |
elimge0 12050 | Hypothesis for weak deduct... |
ltp1 12051 | A number is less than itse... |
lep1 12052 | A number is less than or e... |
ltm1 12053 | A number minus 1 is less t... |
lem1 12054 | A number minus 1 is less t... |
letrp1 12055 | A transitive property of '... |
p1le 12056 | A transitive property of p... |
recgt0 12057 | The reciprocal of a positi... |
prodgt0 12058 | Infer that a multiplicand ... |
prodgt02 12059 | Infer that a multiplier is... |
ltmul1a 12060 | Lemma for ~ ltmul1 . Mult... |
ltmul1 12061 | Multiplication of both sid... |
ltmul2 12062 | Multiplication of both sid... |
lemul1 12063 | Multiplication of both sid... |
lemul2 12064 | Multiplication of both sid... |
lemul1a 12065 | Multiplication of both sid... |
lemul2a 12066 | Multiplication of both sid... |
ltmul12a 12067 | Comparison of product of t... |
lemul12b 12068 | Comparison of product of t... |
lemul12a 12069 | Comparison of product of t... |
mulgt1 12070 | The product of two numbers... |
ltmulgt11 12071 | Multiplication by a number... |
ltmulgt12 12072 | Multiplication by a number... |
lemulge11 12073 | Multiplication by a number... |
lemulge12 12074 | Multiplication by a number... |
ltdiv1 12075 | Division of both sides of ... |
lediv1 12076 | Division of both sides of ... |
gt0div 12077 | Division of a positive num... |
ge0div 12078 | Division of a nonnegative ... |
divgt0 12079 | The ratio of two positive ... |
divge0 12080 | The ratio of nonnegative a... |
mulge0b 12081 | A condition for multiplica... |
mulle0b 12082 | A condition for multiplica... |
mulsuble0b 12083 | A condition for multiplica... |
ltmuldiv 12084 | 'Less than' relationship b... |
ltmuldiv2 12085 | 'Less than' relationship b... |
ltdivmul 12086 | 'Less than' relationship b... |
ledivmul 12087 | 'Less than or equal to' re... |
ltdivmul2 12088 | 'Less than' relationship b... |
lt2mul2div 12089 | 'Less than' relationship b... |
ledivmul2 12090 | 'Less than or equal to' re... |
lemuldiv 12091 | 'Less than or equal' relat... |
lemuldiv2 12092 | 'Less than or equal' relat... |
ltrec 12093 | The reciprocal of both sid... |
lerec 12094 | The reciprocal of both sid... |
lt2msq1 12095 | Lemma for ~ lt2msq . (Con... |
lt2msq 12096 | Two nonnegative numbers co... |
ltdiv2 12097 | Division of a positive num... |
ltrec1 12098 | Reciprocal swap in a 'less... |
lerec2 12099 | Reciprocal swap in a 'less... |
ledivdiv 12100 | Invert ratios of positive ... |
lediv2 12101 | Division of a positive num... |
ltdiv23 12102 | Swap denominator with othe... |
lediv23 12103 | Swap denominator with othe... |
lediv12a 12104 | Comparison of ratio of two... |
lediv2a 12105 | Division of both sides of ... |
reclt1 12106 | The reciprocal of a positi... |
recgt1 12107 | The reciprocal of a positi... |
recgt1i 12108 | The reciprocal of a number... |
recp1lt1 12109 | Construct a number less th... |
recreclt 12110 | Given a positive number ` ... |
le2msq 12111 | The square function on non... |
msq11 12112 | The square of a nonnegativ... |
ledivp1 12113 | "Less than or equal to" an... |
squeeze0 12114 | If a nonnegative number is... |
ltp1i 12115 | A number is less than itse... |
recgt0i 12116 | The reciprocal of a positi... |
recgt0ii 12117 | The reciprocal of a positi... |
prodgt0i 12118 | Infer that a multiplicand ... |
divgt0i 12119 | The ratio of two positive ... |
divge0i 12120 | The ratio of nonnegative a... |
ltreci 12121 | The reciprocal of both sid... |
lereci 12122 | The reciprocal of both sid... |
lt2msqi 12123 | The square function on non... |
le2msqi 12124 | The square function on non... |
msq11i 12125 | The square of a nonnegativ... |
divgt0i2i 12126 | The ratio of two positive ... |
ltrecii 12127 | The reciprocal of both sid... |
divgt0ii 12128 | The ratio of two positive ... |
ltmul1i 12129 | Multiplication of both sid... |
ltdiv1i 12130 | Division of both sides of ... |
ltmuldivi 12131 | 'Less than' relationship b... |
ltmul2i 12132 | Multiplication of both sid... |
lemul1i 12133 | Multiplication of both sid... |
lemul2i 12134 | Multiplication of both sid... |
ltdiv23i 12135 | Swap denominator with othe... |
ledivp1i 12136 | "Less than or equal to" an... |
ltdivp1i 12137 | Less-than and division rel... |
ltdiv23ii 12138 | Swap denominator with othe... |
ltmul1ii 12139 | Multiplication of both sid... |
ltdiv1ii 12140 | Division of both sides of ... |
ltp1d 12141 | A number is less than itse... |
lep1d 12142 | A number is less than or e... |
ltm1d 12143 | A number minus 1 is less t... |
lem1d 12144 | A number minus 1 is less t... |
recgt0d 12145 | The reciprocal of a positi... |
divgt0d 12146 | The ratio of two positive ... |
mulgt1d 12147 | The product of two numbers... |
lemulge11d 12148 | Multiplication by a number... |
lemulge12d 12149 | Multiplication by a number... |
lemul1ad 12150 | Multiplication of both sid... |
lemul2ad 12151 | Multiplication of both sid... |
ltmul12ad 12152 | Comparison of product of t... |
lemul12ad 12153 | Comparison of product of t... |
lemul12bd 12154 | Comparison of product of t... |
fimaxre 12155 | A finite set of real numbe... |
fimaxre2 12156 | A nonempty finite set of r... |
fimaxre3 12157 | A nonempty finite set of r... |
fiminre 12158 | A nonempty finite set of r... |
fiminre2 12159 | A nonempty finite set of r... |
negfi 12160 | The negation of a finite s... |
lbreu 12161 | If a set of reals contains... |
lbcl 12162 | If a set of reals contains... |
lble 12163 | If a set of reals contains... |
lbinf 12164 | If a set of reals contains... |
lbinfcl 12165 | If a set of reals contains... |
lbinfle 12166 | If a set of reals contains... |
sup2 12167 | A nonempty, bounded-above ... |
sup3 12168 | A version of the completen... |
infm3lem 12169 | Lemma for ~ infm3 . (Cont... |
infm3 12170 | The completeness axiom for... |
suprcl 12171 | Closure of supremum of a n... |
suprub 12172 | A member of a nonempty bou... |
suprubd 12173 | Natural deduction form of ... |
suprcld 12174 | Natural deduction form of ... |
suprlub 12175 | The supremum of a nonempty... |
suprnub 12176 | An upper bound is not less... |
suprleub 12177 | The supremum of a nonempty... |
supaddc 12178 | The supremum function dist... |
supadd 12179 | The supremum function dist... |
supmul1 12180 | The supremum function dist... |
supmullem1 12181 | Lemma for ~ supmul . (Con... |
supmullem2 12182 | Lemma for ~ supmul . (Con... |
supmul 12183 | The supremum function dist... |
sup3ii 12184 | A version of the completen... |
suprclii 12185 | Closure of supremum of a n... |
suprubii 12186 | A member of a nonempty bou... |
suprlubii 12187 | The supremum of a nonempty... |
suprnubii 12188 | An upper bound is not less... |
suprleubii 12189 | The supremum of a nonempty... |
riotaneg 12190 | The negative of the unique... |
negiso 12191 | Negation is an order anti-... |
dfinfre 12192 | The infimum of a set of re... |
infrecl 12193 | Closure of infimum of a no... |
infrenegsup 12194 | The infimum of a set of re... |
infregelb 12195 | Any lower bound of a nonem... |
infrelb 12196 | If a nonempty set of real ... |
infrefilb 12197 | The infimum of a finite se... |
supfirege 12198 | The supremum of a finite s... |
inelr 12199 | The imaginary unit ` _i ` ... |
rimul 12200 | A real number times the im... |
cru 12201 | The representation of comp... |
crne0 12202 | The real representation of... |
creur 12203 | The real part of a complex... |
creui 12204 | The imaginary part of a co... |
cju 12205 | The complex conjugate of a... |
ofsubeq0 12206 | Function analogue of ~ sub... |
ofnegsub 12207 | Function analogue of ~ neg... |
ofsubge0 12208 | Function analogue of ~ sub... |
nnexALT 12211 | Alternate proof of ~ nnex ... |
peano5nni 12212 | Peano's inductive postulat... |
nnssre 12213 | The positive integers are ... |
nnsscn 12214 | The positive integers are ... |
nnex 12215 | The set of positive intege... |
nnre 12216 | A positive integer is a re... |
nncn 12217 | A positive integer is a co... |
nnrei 12218 | A positive integer is a re... |
nncni 12219 | A positive integer is a co... |
1nn 12220 | Peano postulate: 1 is a po... |
peano2nn 12221 | Peano postulate: a success... |
dfnn2 12222 | Alternate definition of th... |
dfnn3 12223 | Alternate definition of th... |
nnred 12224 | A positive integer is a re... |
nncnd 12225 | A positive integer is a co... |
peano2nnd 12226 | Peano postulate: a success... |
nnind 12227 | Principle of Mathematical ... |
nnindALT 12228 | Principle of Mathematical ... |
nnindd 12229 | Principle of Mathematical ... |
nn1m1nn 12230 | Every positive integer is ... |
nn1suc 12231 | If a statement holds for 1... |
nnaddcl 12232 | Closure of addition of pos... |
nnmulcl 12233 | Closure of multiplication ... |
nnmulcli 12234 | Closure of multiplication ... |
nnmtmip 12235 | "Minus times minus is plus... |
nn2ge 12236 | There exists a positive in... |
nnge1 12237 | A positive integer is one ... |
nngt1ne1 12238 | A positive integer is grea... |
nnle1eq1 12239 | A positive integer is less... |
nngt0 12240 | A positive integer is posi... |
nnnlt1 12241 | A positive integer is not ... |
nnnle0 12242 | A positive integer is not ... |
nnne0 12243 | A positive integer is nonz... |
nnneneg 12244 | No positive integer is equ... |
0nnn 12245 | Zero is not a positive int... |
0nnnALT 12246 | Alternate proof of ~ 0nnn ... |
nnne0ALT 12247 | Alternate version of ~ nnn... |
nngt0i 12248 | A positive integer is posi... |
nnne0i 12249 | A positive integer is nonz... |
nndivre 12250 | The quotient of a real and... |
nnrecre 12251 | The reciprocal of a positi... |
nnrecgt0 12252 | The reciprocal of a positi... |
nnsub 12253 | Subtraction of positive in... |
nnsubi 12254 | Subtraction of positive in... |
nndiv 12255 | Two ways to express " ` A ... |
nndivtr 12256 | Transitive property of div... |
nnge1d 12257 | A positive integer is one ... |
nngt0d 12258 | A positive integer is posi... |
nnne0d 12259 | A positive integer is nonz... |
nnrecred 12260 | The reciprocal of a positi... |
nnaddcld 12261 | Closure of addition of pos... |
nnmulcld 12262 | Closure of multiplication ... |
nndivred 12263 | A positive integer is one ... |
0ne1 12280 | Zero is different from one... |
1m1e0 12281 | One minus one equals zero.... |
2nn 12282 | 2 is a positive integer. ... |
2re 12283 | The number 2 is real. (Co... |
2cn 12284 | The number 2 is a complex ... |
2cnALT 12285 | Alternate proof of ~ 2cn .... |
2ex 12286 | The number 2 is a set. (C... |
2cnd 12287 | The number 2 is a complex ... |
3nn 12288 | 3 is a positive integer. ... |
3re 12289 | The number 3 is real. (Co... |
3cn 12290 | The number 3 is a complex ... |
3ex 12291 | The number 3 is a set. (C... |
4nn 12292 | 4 is a positive integer. ... |
4re 12293 | The number 4 is real. (Co... |
4cn 12294 | The number 4 is a complex ... |
5nn 12295 | 5 is a positive integer. ... |
5re 12296 | The number 5 is real. (Co... |
5cn 12297 | The number 5 is a complex ... |
6nn 12298 | 6 is a positive integer. ... |
6re 12299 | The number 6 is real. (Co... |
6cn 12300 | The number 6 is a complex ... |
7nn 12301 | 7 is a positive integer. ... |
7re 12302 | The number 7 is real. (Co... |
7cn 12303 | The number 7 is a complex ... |
8nn 12304 | 8 is a positive integer. ... |
8re 12305 | The number 8 is real. (Co... |
8cn 12306 | The number 8 is a complex ... |
9nn 12307 | 9 is a positive integer. ... |
9re 12308 | The number 9 is real. (Co... |
9cn 12309 | The number 9 is a complex ... |
0le0 12310 | Zero is nonnegative. (Con... |
0le2 12311 | The number 0 is less than ... |
2pos 12312 | The number 2 is positive. ... |
2ne0 12313 | The number 2 is nonzero. ... |
3pos 12314 | The number 3 is positive. ... |
3ne0 12315 | The number 3 is nonzero. ... |
4pos 12316 | The number 4 is positive. ... |
4ne0 12317 | The number 4 is nonzero. ... |
5pos 12318 | The number 5 is positive. ... |
6pos 12319 | The number 6 is positive. ... |
7pos 12320 | The number 7 is positive. ... |
8pos 12321 | The number 8 is positive. ... |
9pos 12322 | The number 9 is positive. ... |
neg1cn 12323 | -1 is a complex number. (... |
neg1rr 12324 | -1 is a real number. (Con... |
neg1ne0 12325 | -1 is nonzero. (Contribut... |
neg1lt0 12326 | -1 is less than 0. (Contr... |
negneg1e1 12327 | ` -u -u 1 ` is 1. (Contri... |
1pneg1e0 12328 | ` 1 + -u 1 ` is 0. (Contr... |
0m0e0 12329 | 0 minus 0 equals 0. (Cont... |
1m0e1 12330 | 1 - 0 = 1. (Contributed b... |
0p1e1 12331 | 0 + 1 = 1. (Contributed b... |
fv0p1e1 12332 | Function value at ` N + 1 ... |
1p0e1 12333 | 1 + 0 = 1. (Contributed b... |
1p1e2 12334 | 1 + 1 = 2. (Contributed b... |
2m1e1 12335 | 2 - 1 = 1. The result is ... |
1e2m1 12336 | 1 = 2 - 1. (Contributed b... |
3m1e2 12337 | 3 - 1 = 2. (Contributed b... |
4m1e3 12338 | 4 - 1 = 3. (Contributed b... |
5m1e4 12339 | 5 - 1 = 4. (Contributed b... |
6m1e5 12340 | 6 - 1 = 5. (Contributed b... |
7m1e6 12341 | 7 - 1 = 6. (Contributed b... |
8m1e7 12342 | 8 - 1 = 7. (Contributed b... |
9m1e8 12343 | 9 - 1 = 8. (Contributed b... |
2p2e4 12344 | Two plus two equals four. ... |
2times 12345 | Two times a number. (Cont... |
times2 12346 | A number times 2. (Contri... |
2timesi 12347 | Two times a number. (Cont... |
times2i 12348 | A number times 2. (Contri... |
2txmxeqx 12349 | Two times a complex number... |
2div2e1 12350 | 2 divided by 2 is 1. (Con... |
2p1e3 12351 | 2 + 1 = 3. (Contributed b... |
1p2e3 12352 | 1 + 2 = 3. For a shorter ... |
1p2e3ALT 12353 | Alternate proof of ~ 1p2e3... |
3p1e4 12354 | 3 + 1 = 4. (Contributed b... |
4p1e5 12355 | 4 + 1 = 5. (Contributed b... |
5p1e6 12356 | 5 + 1 = 6. (Contributed b... |
6p1e7 12357 | 6 + 1 = 7. (Contributed b... |
7p1e8 12358 | 7 + 1 = 8. (Contributed b... |
8p1e9 12359 | 8 + 1 = 9. (Contributed b... |
3p2e5 12360 | 3 + 2 = 5. (Contributed b... |
3p3e6 12361 | 3 + 3 = 6. (Contributed b... |
4p2e6 12362 | 4 + 2 = 6. (Contributed b... |
4p3e7 12363 | 4 + 3 = 7. (Contributed b... |
4p4e8 12364 | 4 + 4 = 8. (Contributed b... |
5p2e7 12365 | 5 + 2 = 7. (Contributed b... |
5p3e8 12366 | 5 + 3 = 8. (Contributed b... |
5p4e9 12367 | 5 + 4 = 9. (Contributed b... |
6p2e8 12368 | 6 + 2 = 8. (Contributed b... |
6p3e9 12369 | 6 + 3 = 9. (Contributed b... |
7p2e9 12370 | 7 + 2 = 9. (Contributed b... |
1t1e1 12371 | 1 times 1 equals 1. (Cont... |
2t1e2 12372 | 2 times 1 equals 2. (Cont... |
2t2e4 12373 | 2 times 2 equals 4. (Cont... |
3t1e3 12374 | 3 times 1 equals 3. (Cont... |
3t2e6 12375 | 3 times 2 equals 6. (Cont... |
3t3e9 12376 | 3 times 3 equals 9. (Cont... |
4t2e8 12377 | 4 times 2 equals 8. (Cont... |
2t0e0 12378 | 2 times 0 equals 0. (Cont... |
4d2e2 12379 | One half of four is two. ... |
1lt2 12380 | 1 is less than 2. (Contri... |
2lt3 12381 | 2 is less than 3. (Contri... |
1lt3 12382 | 1 is less than 3. (Contri... |
3lt4 12383 | 3 is less than 4. (Contri... |
2lt4 12384 | 2 is less than 4. (Contri... |
1lt4 12385 | 1 is less than 4. (Contri... |
4lt5 12386 | 4 is less than 5. (Contri... |
3lt5 12387 | 3 is less than 5. (Contri... |
2lt5 12388 | 2 is less than 5. (Contri... |
1lt5 12389 | 1 is less than 5. (Contri... |
5lt6 12390 | 5 is less than 6. (Contri... |
4lt6 12391 | 4 is less than 6. (Contri... |
3lt6 12392 | 3 is less than 6. (Contri... |
2lt6 12393 | 2 is less than 6. (Contri... |
1lt6 12394 | 1 is less than 6. (Contri... |
6lt7 12395 | 6 is less than 7. (Contri... |
5lt7 12396 | 5 is less than 7. (Contri... |
4lt7 12397 | 4 is less than 7. (Contri... |
3lt7 12398 | 3 is less than 7. (Contri... |
2lt7 12399 | 2 is less than 7. (Contri... |
1lt7 12400 | 1 is less than 7. (Contri... |
7lt8 12401 | 7 is less than 8. (Contri... |
6lt8 12402 | 6 is less than 8. (Contri... |
5lt8 12403 | 5 is less than 8. (Contri... |
4lt8 12404 | 4 is less than 8. (Contri... |
3lt8 12405 | 3 is less than 8. (Contri... |
2lt8 12406 | 2 is less than 8. (Contri... |
1lt8 12407 | 1 is less than 8. (Contri... |
8lt9 12408 | 8 is less than 9. (Contri... |
7lt9 12409 | 7 is less than 9. (Contri... |
6lt9 12410 | 6 is less than 9. (Contri... |
5lt9 12411 | 5 is less than 9. (Contri... |
4lt9 12412 | 4 is less than 9. (Contri... |
3lt9 12413 | 3 is less than 9. (Contri... |
2lt9 12414 | 2 is less than 9. (Contri... |
1lt9 12415 | 1 is less than 9. (Contri... |
0ne2 12416 | 0 is not equal to 2. (Con... |
1ne2 12417 | 1 is not equal to 2. (Con... |
1le2 12418 | 1 is less than or equal to... |
2cnne0 12419 | 2 is a nonzero complex num... |
2rene0 12420 | 2 is a nonzero real number... |
1le3 12421 | 1 is less than or equal to... |
neg1mulneg1e1 12422 | ` -u 1 x. -u 1 ` is 1. (C... |
halfre 12423 | One-half is real. (Contri... |
halfcn 12424 | One-half is a complex numb... |
halfgt0 12425 | One-half is greater than z... |
halfge0 12426 | One-half is not negative. ... |
halflt1 12427 | One-half is less than one.... |
1mhlfehlf 12428 | Prove that 1 - 1/2 = 1/2. ... |
8th4div3 12429 | An eighth of four thirds i... |
halfpm6th 12430 | One half plus or minus one... |
it0e0 12431 | i times 0 equals 0. (Cont... |
2mulicn 12432 | ` ( 2 x. _i ) e. CC ` . (... |
2muline0 12433 | ` ( 2 x. _i ) =/= 0 ` . (... |
halfcl 12434 | Closure of half of a numbe... |
rehalfcl 12435 | Real closure of half. (Co... |
half0 12436 | Half of a number is zero i... |
2halves 12437 | Two halves make a whole. ... |
halfpos2 12438 | A number is positive iff i... |
halfpos 12439 | A positive number is great... |
halfnneg2 12440 | A number is nonnegative if... |
halfaddsubcl 12441 | Closure of half-sum and ha... |
halfaddsub 12442 | Sum and difference of half... |
subhalfhalf 12443 | Subtracting the half of a ... |
lt2halves 12444 | A sum is less than the who... |
addltmul 12445 | Sum is less than product f... |
nominpos 12446 | There is no smallest posit... |
avglt1 12447 | Ordering property for aver... |
avglt2 12448 | Ordering property for aver... |
avgle1 12449 | Ordering property for aver... |
avgle2 12450 | Ordering property for aver... |
avgle 12451 | The average of two numbers... |
2timesd 12452 | Two times a number. (Cont... |
times2d 12453 | A number times 2. (Contri... |
halfcld 12454 | Closure of half of a numbe... |
2halvesd 12455 | Two halves make a whole. ... |
rehalfcld 12456 | Real closure of half. (Co... |
lt2halvesd 12457 | A sum is less than the who... |
rehalfcli 12458 | Half a real number is real... |
lt2addmuld 12459 | If two real numbers are le... |
add1p1 12460 | Adding two times 1 to a nu... |
sub1m1 12461 | Subtracting two times 1 fr... |
cnm2m1cnm3 12462 | Subtracting 2 and afterwar... |
xp1d2m1eqxm1d2 12463 | A complex number increased... |
div4p1lem1div2 12464 | An integer greater than 5,... |
nnunb 12465 | The set of positive intege... |
arch 12466 | Archimedean property of re... |
nnrecl 12467 | There exists a positive in... |
bndndx 12468 | A bounded real sequence ` ... |
elnn0 12471 | Nonnegative integers expre... |
nnssnn0 12472 | Positive naturals are a su... |
nn0ssre 12473 | Nonnegative integers are a... |
nn0sscn 12474 | Nonnegative integers are a... |
nn0ex 12475 | The set of nonnegative int... |
nnnn0 12476 | A positive integer is a no... |
nnnn0i 12477 | A positive integer is a no... |
nn0re 12478 | A nonnegative integer is a... |
nn0cn 12479 | A nonnegative integer is a... |
nn0rei 12480 | A nonnegative integer is a... |
nn0cni 12481 | A nonnegative integer is a... |
dfn2 12482 | The set of positive intege... |
elnnne0 12483 | The positive integer prope... |
0nn0 12484 | 0 is a nonnegative integer... |
1nn0 12485 | 1 is a nonnegative integer... |
2nn0 12486 | 2 is a nonnegative integer... |
3nn0 12487 | 3 is a nonnegative integer... |
4nn0 12488 | 4 is a nonnegative integer... |
5nn0 12489 | 5 is a nonnegative integer... |
6nn0 12490 | 6 is a nonnegative integer... |
7nn0 12491 | 7 is a nonnegative integer... |
8nn0 12492 | 8 is a nonnegative integer... |
9nn0 12493 | 9 is a nonnegative integer... |
nn0ge0 12494 | A nonnegative integer is g... |
nn0nlt0 12495 | A nonnegative integer is n... |
nn0ge0i 12496 | Nonnegative integers are n... |
nn0le0eq0 12497 | A nonnegative integer is l... |
nn0p1gt0 12498 | A nonnegative integer incr... |
nnnn0addcl 12499 | A positive integer plus a ... |
nn0nnaddcl 12500 | A nonnegative integer plus... |
0mnnnnn0 12501 | The result of subtracting ... |
un0addcl 12502 | If ` S ` is closed under a... |
un0mulcl 12503 | If ` S ` is closed under m... |
nn0addcl 12504 | Closure of addition of non... |
nn0mulcl 12505 | Closure of multiplication ... |
nn0addcli 12506 | Closure of addition of non... |
nn0mulcli 12507 | Closure of multiplication ... |
nn0p1nn 12508 | A nonnegative integer plus... |
peano2nn0 12509 | Second Peano postulate for... |
nnm1nn0 12510 | A positive integer minus 1... |
elnn0nn 12511 | The nonnegative integer pr... |
elnnnn0 12512 | The positive integer prope... |
elnnnn0b 12513 | The positive integer prope... |
elnnnn0c 12514 | The positive integer prope... |
nn0addge1 12515 | A number is less than or e... |
nn0addge2 12516 | A number is less than or e... |
nn0addge1i 12517 | A number is less than or e... |
nn0addge2i 12518 | A number is less than or e... |
nn0sub 12519 | Subtraction of nonnegative... |
ltsubnn0 12520 | Subtracting a nonnegative ... |
nn0negleid 12521 | A nonnegative integer is g... |
difgtsumgt 12522 | If the difference of a rea... |
nn0le2xi 12523 | A nonnegative integer is l... |
nn0lele2xi 12524 | 'Less than or equal to' im... |
fcdmnn0supp 12525 | Two ways to write the supp... |
fcdmnn0fsupp 12526 | A function into ` NN0 ` is... |
fcdmnn0suppg 12527 | Version of ~ fcdmnn0supp a... |
fcdmnn0fsuppg 12528 | Version of ~ fcdmnn0fsupp ... |
nnnn0d 12529 | A positive integer is a no... |
nn0red 12530 | A nonnegative integer is a... |
nn0cnd 12531 | A nonnegative integer is a... |
nn0ge0d 12532 | A nonnegative integer is g... |
nn0addcld 12533 | Closure of addition of non... |
nn0mulcld 12534 | Closure of multiplication ... |
nn0readdcl 12535 | Closure law for addition o... |
nn0n0n1ge2 12536 | A nonnegative integer whic... |
nn0n0n1ge2b 12537 | A nonnegative integer is n... |
nn0ge2m1nn 12538 | If a nonnegative integer i... |
nn0ge2m1nn0 12539 | If a nonnegative integer i... |
nn0nndivcl 12540 | Closure law for dividing o... |
elxnn0 12543 | An extended nonnegative in... |
nn0ssxnn0 12544 | The standard nonnegative i... |
nn0xnn0 12545 | A standard nonnegative int... |
xnn0xr 12546 | An extended nonnegative in... |
0xnn0 12547 | Zero is an extended nonneg... |
pnf0xnn0 12548 | Positive infinity is an ex... |
nn0nepnf 12549 | No standard nonnegative in... |
nn0xnn0d 12550 | A standard nonnegative int... |
nn0nepnfd 12551 | No standard nonnegative in... |
xnn0nemnf 12552 | No extended nonnegative in... |
xnn0xrnemnf 12553 | The extended nonnegative i... |
xnn0nnn0pnf 12554 | An extended nonnegative in... |
elz 12557 | Membership in the set of i... |
nnnegz 12558 | The negative of a positive... |
zre 12559 | An integer is a real. (Co... |
zcn 12560 | An integer is a complex nu... |
zrei 12561 | An integer is a real numbe... |
zssre 12562 | The integers are a subset ... |
zsscn 12563 | The integers are a subset ... |
zex 12564 | The set of integers exists... |
elnnz 12565 | Positive integer property ... |
0z 12566 | Zero is an integer. (Cont... |
0zd 12567 | Zero is an integer, deduct... |
elnn0z 12568 | Nonnegative integer proper... |
elznn0nn 12569 | Integer property expressed... |
elznn0 12570 | Integer property expressed... |
elznn 12571 | Integer property expressed... |
zle0orge1 12572 | There is no integer in the... |
elz2 12573 | Membership in the set of i... |
dfz2 12574 | Alternative definition of ... |
zexALT 12575 | Alternate proof of ~ zex .... |
nnz 12576 | A positive integer is an i... |
nnssz 12577 | Positive integers are a su... |
nn0ssz 12578 | Nonnegative integers are a... |
nnzOLD 12579 | Obsolete version of ~ nnz ... |
nn0z 12580 | A nonnegative integer is a... |
nn0zd 12581 | A nonnegative integer is a... |
nnzd 12582 | A positive integer is an i... |
nnzi 12583 | A positive integer is an i... |
nn0zi 12584 | A nonnegative integer is a... |
elnnz1 12585 | Positive integer property ... |
znnnlt1 12586 | An integer is not a positi... |
nnzrab 12587 | Positive integers expresse... |
nn0zrab 12588 | Nonnegative integers expre... |
1z 12589 | One is an integer. (Contr... |
1zzd 12590 | One is an integer, deducti... |
2z 12591 | 2 is an integer. (Contrib... |
3z 12592 | 3 is an integer. (Contrib... |
4z 12593 | 4 is an integer. (Contrib... |
znegcl 12594 | Closure law for negative i... |
neg1z 12595 | -1 is an integer. (Contri... |
znegclb 12596 | A complex number is an int... |
nn0negz 12597 | The negative of a nonnegat... |
nn0negzi 12598 | The negative of a nonnegat... |
zaddcl 12599 | Closure of addition of int... |
peano2z 12600 | Second Peano postulate gen... |
zsubcl 12601 | Closure of subtraction of ... |
peano2zm 12602 | "Reverse" second Peano pos... |
zletr 12603 | Transitive law of ordering... |
zrevaddcl 12604 | Reverse closure law for ad... |
znnsub 12605 | The positive difference of... |
znn0sub 12606 | The nonnegative difference... |
nzadd 12607 | The sum of a real number n... |
zmulcl 12608 | Closure of multiplication ... |
zltp1le 12609 | Integer ordering relation.... |
zleltp1 12610 | Integer ordering relation.... |
zlem1lt 12611 | Integer ordering relation.... |
zltlem1 12612 | Integer ordering relation.... |
zgt0ge1 12613 | An integer greater than ` ... |
nnleltp1 12614 | Positive integer ordering ... |
nnltp1le 12615 | Positive integer ordering ... |
nnaddm1cl 12616 | Closure of addition of pos... |
nn0ltp1le 12617 | Nonnegative integer orderi... |
nn0leltp1 12618 | Nonnegative integer orderi... |
nn0ltlem1 12619 | Nonnegative integer orderi... |
nn0sub2 12620 | Subtraction of nonnegative... |
nn0lt10b 12621 | A nonnegative integer less... |
nn0lt2 12622 | A nonnegative integer less... |
nn0le2is012 12623 | A nonnegative integer whic... |
nn0lem1lt 12624 | Nonnegative integer orderi... |
nnlem1lt 12625 | Positive integer ordering ... |
nnltlem1 12626 | Positive integer ordering ... |
nnm1ge0 12627 | A positive integer decreas... |
nn0ge0div 12628 | Division of a nonnegative ... |
zdiv 12629 | Two ways to express " ` M ... |
zdivadd 12630 | Property of divisibility: ... |
zdivmul 12631 | Property of divisibility: ... |
zextle 12632 | An extensionality-like pro... |
zextlt 12633 | An extensionality-like pro... |
recnz 12634 | The reciprocal of a number... |
btwnnz 12635 | A number between an intege... |
gtndiv 12636 | A larger number does not d... |
halfnz 12637 | One-half is not an integer... |
3halfnz 12638 | Three halves is not an int... |
suprzcl 12639 | The supremum of a bounded-... |
prime 12640 | Two ways to express " ` A ... |
msqznn 12641 | The square of a nonzero in... |
zneo 12642 | No even integer equals an ... |
nneo 12643 | A positive integer is even... |
nneoi 12644 | A positive integer is even... |
zeo 12645 | An integer is even or odd.... |
zeo2 12646 | An integer is even or odd ... |
peano2uz2 12647 | Second Peano postulate for... |
peano5uzi 12648 | Peano's inductive postulat... |
peano5uzti 12649 | Peano's inductive postulat... |
dfuzi 12650 | An expression for the uppe... |
uzind 12651 | Induction on the upper int... |
uzind2 12652 | Induction on the upper int... |
uzind3 12653 | Induction on the upper int... |
nn0ind 12654 | Principle of Mathematical ... |
nn0indALT 12655 | Principle of Mathematical ... |
nn0indd 12656 | Principle of Mathematical ... |
fzind 12657 | Induction on the integers ... |
fnn0ind 12658 | Induction on the integers ... |
nn0ind-raph 12659 | Principle of Mathematical ... |
zindd 12660 | Principle of Mathematical ... |
fzindd 12661 | Induction on the integers ... |
btwnz 12662 | Any real number can be san... |
zred 12663 | An integer is a real numbe... |
zcnd 12664 | An integer is a complex nu... |
znegcld 12665 | Closure law for negative i... |
peano2zd 12666 | Deduction from second Pean... |
zaddcld 12667 | Closure of addition of int... |
zsubcld 12668 | Closure of subtraction of ... |
zmulcld 12669 | Closure of multiplication ... |
znnn0nn 12670 | The negative of a negative... |
zadd2cl 12671 | Increasing an integer by 2... |
zriotaneg 12672 | The negative of the unique... |
suprfinzcl 12673 | The supremum of a nonempty... |
9p1e10 12676 | 9 + 1 = 10. (Contributed ... |
dfdec10 12677 | Version of the definition ... |
decex 12678 | A decimal number is a set.... |
deceq1 12679 | Equality theorem for the d... |
deceq2 12680 | Equality theorem for the d... |
deceq1i 12681 | Equality theorem for the d... |
deceq2i 12682 | Equality theorem for the d... |
deceq12i 12683 | Equality theorem for the d... |
numnncl 12684 | Closure for a numeral (wit... |
num0u 12685 | Add a zero in the units pl... |
num0h 12686 | Add a zero in the higher p... |
numcl 12687 | Closure for a decimal inte... |
numsuc 12688 | The successor of a decimal... |
deccl 12689 | Closure for a numeral. (C... |
10nn 12690 | 10 is a positive integer. ... |
10pos 12691 | The number 10 is positive.... |
10nn0 12692 | 10 is a nonnegative intege... |
10re 12693 | The number 10 is real. (C... |
decnncl 12694 | Closure for a numeral. (C... |
dec0u 12695 | Add a zero in the units pl... |
dec0h 12696 | Add a zero in the higher p... |
numnncl2 12697 | Closure for a decimal inte... |
decnncl2 12698 | Closure for a decimal inte... |
numlt 12699 | Comparing two decimal inte... |
numltc 12700 | Comparing two decimal inte... |
le9lt10 12701 | A "decimal digit" (i.e. a ... |
declt 12702 | Comparing two decimal inte... |
decltc 12703 | Comparing two decimal inte... |
declth 12704 | Comparing two decimal inte... |
decsuc 12705 | The successor of a decimal... |
3declth 12706 | Comparing two decimal inte... |
3decltc 12707 | Comparing two decimal inte... |
decle 12708 | Comparing two decimal inte... |
decleh 12709 | Comparing two decimal inte... |
declei 12710 | Comparing a digit to a dec... |
numlti 12711 | Comparing a digit to a dec... |
declti 12712 | Comparing a digit to a dec... |
decltdi 12713 | Comparing a digit to a dec... |
numsucc 12714 | The successor of a decimal... |
decsucc 12715 | The successor of a decimal... |
1e0p1 12716 | The successor of zero. (C... |
dec10p 12717 | Ten plus an integer. (Con... |
numma 12718 | Perform a multiply-add of ... |
nummac 12719 | Perform a multiply-add of ... |
numma2c 12720 | Perform a multiply-add of ... |
numadd 12721 | Add two decimal integers `... |
numaddc 12722 | Add two decimal integers `... |
nummul1c 12723 | The product of a decimal i... |
nummul2c 12724 | The product of a decimal i... |
decma 12725 | Perform a multiply-add of ... |
decmac 12726 | Perform a multiply-add of ... |
decma2c 12727 | Perform a multiply-add of ... |
decadd 12728 | Add two numerals ` M ` and... |
decaddc 12729 | Add two numerals ` M ` and... |
decaddc2 12730 | Add two numerals ` M ` and... |
decrmanc 12731 | Perform a multiply-add of ... |
decrmac 12732 | Perform a multiply-add of ... |
decaddm10 12733 | The sum of two multiples o... |
decaddi 12734 | Add two numerals ` M ` and... |
decaddci 12735 | Add two numerals ` M ` and... |
decaddci2 12736 | Add two numerals ` M ` and... |
decsubi 12737 | Difference between a numer... |
decmul1 12738 | The product of a numeral w... |
decmul1c 12739 | The product of a numeral w... |
decmul2c 12740 | The product of a numeral w... |
decmulnc 12741 | The product of a numeral w... |
11multnc 12742 | The product of 11 (as nume... |
decmul10add 12743 | A multiplication of a numb... |
6p5lem 12744 | Lemma for ~ 6p5e11 and rel... |
5p5e10 12745 | 5 + 5 = 10. (Contributed ... |
6p4e10 12746 | 6 + 4 = 10. (Contributed ... |
6p5e11 12747 | 6 + 5 = 11. (Contributed ... |
6p6e12 12748 | 6 + 6 = 12. (Contributed ... |
7p3e10 12749 | 7 + 3 = 10. (Contributed ... |
7p4e11 12750 | 7 + 4 = 11. (Contributed ... |
7p5e12 12751 | 7 + 5 = 12. (Contributed ... |
7p6e13 12752 | 7 + 6 = 13. (Contributed ... |
7p7e14 12753 | 7 + 7 = 14. (Contributed ... |
8p2e10 12754 | 8 + 2 = 10. (Contributed ... |
8p3e11 12755 | 8 + 3 = 11. (Contributed ... |
8p4e12 12756 | 8 + 4 = 12. (Contributed ... |
8p5e13 12757 | 8 + 5 = 13. (Contributed ... |
8p6e14 12758 | 8 + 6 = 14. (Contributed ... |
8p7e15 12759 | 8 + 7 = 15. (Contributed ... |
8p8e16 12760 | 8 + 8 = 16. (Contributed ... |
9p2e11 12761 | 9 + 2 = 11. (Contributed ... |
9p3e12 12762 | 9 + 3 = 12. (Contributed ... |
9p4e13 12763 | 9 + 4 = 13. (Contributed ... |
9p5e14 12764 | 9 + 5 = 14. (Contributed ... |
9p6e15 12765 | 9 + 6 = 15. (Contributed ... |
9p7e16 12766 | 9 + 7 = 16. (Contributed ... |
9p8e17 12767 | 9 + 8 = 17. (Contributed ... |
9p9e18 12768 | 9 + 9 = 18. (Contributed ... |
10p10e20 12769 | 10 + 10 = 20. (Contribute... |
10m1e9 12770 | 10 - 1 = 9. (Contributed ... |
4t3lem 12771 | Lemma for ~ 4t3e12 and rel... |
4t3e12 12772 | 4 times 3 equals 12. (Con... |
4t4e16 12773 | 4 times 4 equals 16. (Con... |
5t2e10 12774 | 5 times 2 equals 10. (Con... |
5t3e15 12775 | 5 times 3 equals 15. (Con... |
5t4e20 12776 | 5 times 4 equals 20. (Con... |
5t5e25 12777 | 5 times 5 equals 25. (Con... |
6t2e12 12778 | 6 times 2 equals 12. (Con... |
6t3e18 12779 | 6 times 3 equals 18. (Con... |
6t4e24 12780 | 6 times 4 equals 24. (Con... |
6t5e30 12781 | 6 times 5 equals 30. (Con... |
6t6e36 12782 | 6 times 6 equals 36. (Con... |
7t2e14 12783 | 7 times 2 equals 14. (Con... |
7t3e21 12784 | 7 times 3 equals 21. (Con... |
7t4e28 12785 | 7 times 4 equals 28. (Con... |
7t5e35 12786 | 7 times 5 equals 35. (Con... |
7t6e42 12787 | 7 times 6 equals 42. (Con... |
7t7e49 12788 | 7 times 7 equals 49. (Con... |
8t2e16 12789 | 8 times 2 equals 16. (Con... |
8t3e24 12790 | 8 times 3 equals 24. (Con... |
8t4e32 12791 | 8 times 4 equals 32. (Con... |
8t5e40 12792 | 8 times 5 equals 40. (Con... |
8t6e48 12793 | 8 times 6 equals 48. (Con... |
8t7e56 12794 | 8 times 7 equals 56. (Con... |
8t8e64 12795 | 8 times 8 equals 64. (Con... |
9t2e18 12796 | 9 times 2 equals 18. (Con... |
9t3e27 12797 | 9 times 3 equals 27. (Con... |
9t4e36 12798 | 9 times 4 equals 36. (Con... |
9t5e45 12799 | 9 times 5 equals 45. (Con... |
9t6e54 12800 | 9 times 6 equals 54. (Con... |
9t7e63 12801 | 9 times 7 equals 63. (Con... |
9t8e72 12802 | 9 times 8 equals 72. (Con... |
9t9e81 12803 | 9 times 9 equals 81. (Con... |
9t11e99 12804 | 9 times 11 equals 99. (Co... |
9lt10 12805 | 9 is less than 10. (Contr... |
8lt10 12806 | 8 is less than 10. (Contr... |
7lt10 12807 | 7 is less than 10. (Contr... |
6lt10 12808 | 6 is less than 10. (Contr... |
5lt10 12809 | 5 is less than 10. (Contr... |
4lt10 12810 | 4 is less than 10. (Contr... |
3lt10 12811 | 3 is less than 10. (Contr... |
2lt10 12812 | 2 is less than 10. (Contr... |
1lt10 12813 | 1 is less than 10. (Contr... |
decbin0 12814 | Decompose base 4 into base... |
decbin2 12815 | Decompose base 4 into base... |
decbin3 12816 | Decompose base 4 into base... |
halfthird 12817 | Half minus a third. (Cont... |
5recm6rec 12818 | One fifth minus one sixth.... |
uzval 12821 | The value of the upper int... |
uzf 12822 | The domain and codomain of... |
eluz1 12823 | Membership in the upper se... |
eluzel2 12824 | Implication of membership ... |
eluz2 12825 | Membership in an upper set... |
eluzmn 12826 | Membership in an earlier u... |
eluz1i 12827 | Membership in an upper set... |
eluzuzle 12828 | An integer in an upper set... |
eluzelz 12829 | A member of an upper set o... |
eluzelre 12830 | A member of an upper set o... |
eluzelcn 12831 | A member of an upper set o... |
eluzle 12832 | Implication of membership ... |
eluz 12833 | Membership in an upper set... |
uzid 12834 | Membership of the least me... |
uzidd 12835 | Membership of the least me... |
uzn0 12836 | The upper integers are all... |
uztrn 12837 | Transitive law for sets of... |
uztrn2 12838 | Transitive law for sets of... |
uzneg 12839 | Contraposition law for upp... |
uzssz 12840 | An upper set of integers i... |
uzssre 12841 | An upper set of integers i... |
uzss 12842 | Subset relationship for tw... |
uztric 12843 | Totality of the ordering r... |
uz11 12844 | The upper integers functio... |
eluzp1m1 12845 | Membership in the next upp... |
eluzp1l 12846 | Strict ordering implied by... |
eluzp1p1 12847 | Membership in the next upp... |
eluzadd 12848 | Membership in a later uppe... |
eluzsub 12849 | Membership in an earlier u... |
eluzaddi 12850 | Membership in a later uppe... |
eluzaddiOLD 12851 | Obsolete version of ~ eluz... |
eluzsubi 12852 | Membership in an earlier u... |
eluzsubiOLD 12853 | Obsolete version of ~ eluz... |
eluzaddOLD 12854 | Obsolete version of ~ eluz... |
eluzsubOLD 12855 | Obsolete version of ~ eluz... |
subeluzsub 12856 | Membership of a difference... |
uzm1 12857 | Choices for an element of ... |
uznn0sub 12858 | The nonnegative difference... |
uzin 12859 | Intersection of two upper ... |
uzp1 12860 | Choices for an element of ... |
nn0uz 12861 | Nonnegative integers expre... |
nnuz 12862 | Positive integers expresse... |
elnnuz 12863 | A positive integer express... |
elnn0uz 12864 | A nonnegative integer expr... |
eluz2nn 12865 | An integer greater than or... |
eluz4eluz2 12866 | An integer greater than or... |
eluz4nn 12867 | An integer greater than or... |
eluzge2nn0 12868 | If an integer is greater t... |
eluz2n0 12869 | An integer greater than or... |
uzuzle23 12870 | An integer in the upper se... |
eluzge3nn 12871 | If an integer is greater t... |
uz3m2nn 12872 | An integer greater than or... |
1eluzge0 12873 | 1 is an integer greater th... |
2eluzge0 12874 | 2 is an integer greater th... |
2eluzge1 12875 | 2 is an integer greater th... |
uznnssnn 12876 | The upper integers startin... |
raluz 12877 | Restricted universal quant... |
raluz2 12878 | Restricted universal quant... |
rexuz 12879 | Restricted existential qua... |
rexuz2 12880 | Restricted existential qua... |
2rexuz 12881 | Double existential quantif... |
peano2uz 12882 | Second Peano postulate for... |
peano2uzs 12883 | Second Peano postulate for... |
peano2uzr 12884 | Reversed second Peano axio... |
uzaddcl 12885 | Addition closure law for a... |
nn0pzuz 12886 | The sum of a nonnegative i... |
uzind4 12887 | Induction on the upper set... |
uzind4ALT 12888 | Induction on the upper set... |
uzind4s 12889 | Induction on the upper set... |
uzind4s2 12890 | Induction on the upper set... |
uzind4i 12891 | Induction on the upper int... |
uzwo 12892 | Well-ordering principle: a... |
uzwo2 12893 | Well-ordering principle: a... |
nnwo 12894 | Well-ordering principle: a... |
nnwof 12895 | Well-ordering principle: a... |
nnwos 12896 | Well-ordering principle: a... |
indstr 12897 | Strong Mathematical Induct... |
eluznn0 12898 | Membership in a nonnegativ... |
eluznn 12899 | Membership in a positive u... |
eluz2b1 12900 | Two ways to say "an intege... |
eluz2gt1 12901 | An integer greater than or... |
eluz2b2 12902 | Two ways to say "an intege... |
eluz2b3 12903 | Two ways to say "an intege... |
uz2m1nn 12904 | One less than an integer g... |
1nuz2 12905 | 1 is not in ` ( ZZ>= `` 2 ... |
elnn1uz2 12906 | A positive integer is eith... |
uz2mulcl 12907 | Closure of multiplication ... |
indstr2 12908 | Strong Mathematical Induct... |
uzinfi 12909 | Extract the lower bound of... |
nninf 12910 | The infimum of the set of ... |
nn0inf 12911 | The infimum of the set of ... |
infssuzle 12912 | The infimum of a subset of... |
infssuzcl 12913 | The infimum of a subset of... |
ublbneg 12914 | The image under negation o... |
eqreznegel 12915 | Two ways to express the im... |
supminf 12916 | The supremum of a bounded-... |
lbzbi 12917 | If a set of reals is bound... |
zsupss 12918 | Any nonempty bounded subse... |
suprzcl2 12919 | The supremum of a bounded-... |
suprzub 12920 | The supremum of a bounded-... |
uzsupss 12921 | Any bounded subset of an u... |
nn01to3 12922 | A (nonnegative) integer be... |
nn0ge2m1nnALT 12923 | Alternate proof of ~ nn0ge... |
uzwo3 12924 | Well-ordering principle: a... |
zmin 12925 | There is a unique smallest... |
zmax 12926 | There is a unique largest ... |
zbtwnre 12927 | There is a unique integer ... |
rebtwnz 12928 | There is a unique greatest... |
elq 12931 | Membership in the set of r... |
qmulz 12932 | If ` A ` is rational, then... |
znq 12933 | The ratio of an integer an... |
qre 12934 | A rational number is a rea... |
zq 12935 | An integer is a rational n... |
qred 12936 | A rational number is a rea... |
zssq 12937 | The integers are a subset ... |
nn0ssq 12938 | The nonnegative integers a... |
nnssq 12939 | The positive integers are ... |
qssre 12940 | The rationals are a subset... |
qsscn 12941 | The rationals are a subset... |
qex 12942 | The set of rational number... |
nnq 12943 | A positive integer is rati... |
qcn 12944 | A rational number is a com... |
qexALT 12945 | Alternate proof of ~ qex .... |
qaddcl 12946 | Closure of addition of rat... |
qnegcl 12947 | Closure law for the negati... |
qmulcl 12948 | Closure of multiplication ... |
qsubcl 12949 | Closure of subtraction of ... |
qreccl 12950 | Closure of reciprocal of r... |
qdivcl 12951 | Closure of division of rat... |
qrevaddcl 12952 | Reverse closure law for ad... |
nnrecq 12953 | The reciprocal of a positi... |
irradd 12954 | The sum of an irrational n... |
irrmul 12955 | The product of an irration... |
elpq 12956 | A positive rational is the... |
elpqb 12957 | A class is a positive rati... |
rpnnen1lem2 12958 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem1 12959 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem3 12960 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem4 12961 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem5 12962 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem6 12963 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1 12964 | One half of ~ rpnnen , whe... |
reexALT 12965 | Alternate proof of ~ reex ... |
cnref1o 12966 | There is a natural one-to-... |
cnexALT 12967 | The set of complex numbers... |
xrex 12968 | The set of extended reals ... |
addex 12969 | The addition operation is ... |
mulex 12970 | The multiplication operati... |
elrp 12973 | Membership in the set of p... |
elrpii 12974 | Membership in the set of p... |
1rp 12975 | 1 is a positive real. (Co... |
2rp 12976 | 2 is a positive real. (Co... |
3rp 12977 | 3 is a positive real. (Co... |
rpssre 12978 | The positive reals are a s... |
rpre 12979 | A positive real is a real.... |
rpxr 12980 | A positive real is an exte... |
rpcn 12981 | A positive real is a compl... |
nnrp 12982 | A positive integer is a po... |
rpgt0 12983 | A positive real is greater... |
rpge0 12984 | A positive real is greater... |
rpregt0 12985 | A positive real is a posit... |
rprege0 12986 | A positive real is a nonne... |
rpne0 12987 | A positive real is nonzero... |
rprene0 12988 | A positive real is a nonze... |
rpcnne0 12989 | A positive real is a nonze... |
rpcndif0 12990 | A positive real number is ... |
ralrp 12991 | Quantification over positi... |
rexrp 12992 | Quantification over positi... |
rpaddcl 12993 | Closure law for addition o... |
rpmulcl 12994 | Closure law for multiplica... |
rpmtmip 12995 | "Minus times minus is plus... |
rpdivcl 12996 | Closure law for division o... |
rpreccl 12997 | Closure law for reciprocat... |
rphalfcl 12998 | Closure law for half of a ... |
rpgecl 12999 | A number greater than or e... |
rphalflt 13000 | Half of a positive real is... |
rerpdivcl 13001 | Closure law for division o... |
ge0p1rp 13002 | A nonnegative number plus ... |
rpneg 13003 | Either a nonzero real or i... |
negelrp 13004 | Elementhood of a negation ... |
negelrpd 13005 | The negation of a negative... |
0nrp 13006 | Zero is not a positive rea... |
ltsubrp 13007 | Subtracting a positive rea... |
ltaddrp 13008 | Adding a positive number t... |
difrp 13009 | Two ways to say one number... |
elrpd 13010 | Membership in the set of p... |
nnrpd 13011 | A positive integer is a po... |
zgt1rpn0n1 13012 | An integer greater than 1 ... |
rpred 13013 | A positive real is a real.... |
rpxrd 13014 | A positive real is an exte... |
rpcnd 13015 | A positive real is a compl... |
rpgt0d 13016 | A positive real is greater... |
rpge0d 13017 | A positive real is greater... |
rpne0d 13018 | A positive real is nonzero... |
rpregt0d 13019 | A positive real is real an... |
rprege0d 13020 | A positive real is real an... |
rprene0d 13021 | A positive real is a nonze... |
rpcnne0d 13022 | A positive real is a nonze... |
rpreccld 13023 | Closure law for reciprocat... |
rprecred 13024 | Closure law for reciprocat... |
rphalfcld 13025 | Closure law for half of a ... |
reclt1d 13026 | The reciprocal of a positi... |
recgt1d 13027 | The reciprocal of a positi... |
rpaddcld 13028 | Closure law for addition o... |
rpmulcld 13029 | Closure law for multiplica... |
rpdivcld 13030 | Closure law for division o... |
ltrecd 13031 | The reciprocal of both sid... |
lerecd 13032 | The reciprocal of both sid... |
ltrec1d 13033 | Reciprocal swap in a 'less... |
lerec2d 13034 | Reciprocal swap in a 'less... |
lediv2ad 13035 | Division of both sides of ... |
ltdiv2d 13036 | Division of a positive num... |
lediv2d 13037 | Division of a positive num... |
ledivdivd 13038 | Invert ratios of positive ... |
divge1 13039 | The ratio of a number over... |
divlt1lt 13040 | A real number divided by a... |
divle1le 13041 | A real number divided by a... |
ledivge1le 13042 | If a number is less than o... |
ge0p1rpd 13043 | A nonnegative number plus ... |
rerpdivcld 13044 | Closure law for division o... |
ltsubrpd 13045 | Subtracting a positive rea... |
ltaddrpd 13046 | Adding a positive number t... |
ltaddrp2d 13047 | Adding a positive number t... |
ltmulgt11d 13048 | Multiplication by a number... |
ltmulgt12d 13049 | Multiplication by a number... |
gt0divd 13050 | Division of a positive num... |
ge0divd 13051 | Division of a nonnegative ... |
rpgecld 13052 | A number greater than or e... |
divge0d 13053 | The ratio of nonnegative a... |
ltmul1d 13054 | The ratio of nonnegative a... |
ltmul2d 13055 | Multiplication of both sid... |
lemul1d 13056 | Multiplication of both sid... |
lemul2d 13057 | Multiplication of both sid... |
ltdiv1d 13058 | Division of both sides of ... |
lediv1d 13059 | Division of both sides of ... |
ltmuldivd 13060 | 'Less than' relationship b... |
ltmuldiv2d 13061 | 'Less than' relationship b... |
lemuldivd 13062 | 'Less than or equal to' re... |
lemuldiv2d 13063 | 'Less than or equal to' re... |
ltdivmuld 13064 | 'Less than' relationship b... |
ltdivmul2d 13065 | 'Less than' relationship b... |
ledivmuld 13066 | 'Less than or equal to' re... |
ledivmul2d 13067 | 'Less than or equal to' re... |
ltmul1dd 13068 | The ratio of nonnegative a... |
ltmul2dd 13069 | Multiplication of both sid... |
ltdiv1dd 13070 | Division of both sides of ... |
lediv1dd 13071 | Division of both sides of ... |
lediv12ad 13072 | Comparison of ratio of two... |
mul2lt0rlt0 13073 | If the result of a multipl... |
mul2lt0rgt0 13074 | If the result of a multipl... |
mul2lt0llt0 13075 | If the result of a multipl... |
mul2lt0lgt0 13076 | If the result of a multipl... |
mul2lt0bi 13077 | If the result of a multipl... |
prodge0rd 13078 | Infer that a multiplicand ... |
prodge0ld 13079 | Infer that a multiplier is... |
ltdiv23d 13080 | Swap denominator with othe... |
lediv23d 13081 | Swap denominator with othe... |
lt2mul2divd 13082 | The ratio of nonnegative a... |
nnledivrp 13083 | Division of a positive int... |
nn0ledivnn 13084 | Division of a nonnegative ... |
addlelt 13085 | If the sum of a real numbe... |
ltxr 13092 | The 'less than' binary rel... |
elxr 13093 | Membership in the set of e... |
xrnemnf 13094 | An extended real other tha... |
xrnepnf 13095 | An extended real other tha... |
xrltnr 13096 | The extended real 'less th... |
ltpnf 13097 | Any (finite) real is less ... |
ltpnfd 13098 | Any (finite) real is less ... |
0ltpnf 13099 | Zero is less than plus inf... |
mnflt 13100 | Minus infinity is less tha... |
mnfltd 13101 | Minus infinity is less tha... |
mnflt0 13102 | Minus infinity is less tha... |
mnfltpnf 13103 | Minus infinity is less tha... |
mnfltxr 13104 | Minus infinity is less tha... |
pnfnlt 13105 | No extended real is greate... |
nltmnf 13106 | No extended real is less t... |
pnfge 13107 | Plus infinity is an upper ... |
xnn0n0n1ge2b 13108 | An extended nonnegative in... |
0lepnf 13109 | 0 less than or equal to po... |
xnn0ge0 13110 | An extended nonnegative in... |
mnfle 13111 | Minus infinity is less tha... |
mnfled 13112 | Minus infinity is less tha... |
xrltnsym 13113 | Ordering on the extended r... |
xrltnsym2 13114 | 'Less than' is antisymmetr... |
xrlttri 13115 | Ordering on the extended r... |
xrlttr 13116 | Ordering on the extended r... |
xrltso 13117 | 'Less than' is a strict or... |
xrlttri2 13118 | Trichotomy law for 'less t... |
xrlttri3 13119 | Trichotomy law for 'less t... |
xrleloe 13120 | 'Less than or equal' expre... |
xrleltne 13121 | 'Less than or equal to' im... |
xrltlen 13122 | 'Less than' expressed in t... |
dfle2 13123 | Alternative definition of ... |
dflt2 13124 | Alternative definition of ... |
xrltle 13125 | 'Less than' implies 'less ... |
xrltled 13126 | 'Less than' implies 'less ... |
xrleid 13127 | 'Less than or equal to' is... |
xrleidd 13128 | 'Less than or equal to' is... |
xrletri 13129 | Trichotomy law for extende... |
xrletri3 13130 | Trichotomy law for extende... |
xrletrid 13131 | Trichotomy law for extende... |
xrlelttr 13132 | Transitive law for orderin... |
xrltletr 13133 | Transitive law for orderin... |
xrletr 13134 | Transitive law for orderin... |
xrlttrd 13135 | Transitive law for orderin... |
xrlelttrd 13136 | Transitive law for orderin... |
xrltletrd 13137 | Transitive law for orderin... |
xrletrd 13138 | Transitive law for orderin... |
xrltne 13139 | 'Less than' implies not eq... |
nltpnft 13140 | An extended real is not le... |
xgepnf 13141 | An extended real which is ... |
ngtmnft 13142 | An extended real is not gr... |
xlemnf 13143 | An extended real which is ... |
xrrebnd 13144 | An extended real is real i... |
xrre 13145 | A way of proving that an e... |
xrre2 13146 | An extended real between t... |
xrre3 13147 | A way of proving that an e... |
ge0gtmnf 13148 | A nonnegative extended rea... |
ge0nemnf 13149 | A nonnegative extended rea... |
xrrege0 13150 | A nonnegative extended rea... |
xrmax1 13151 | An extended real is less t... |
xrmax2 13152 | An extended real is less t... |
xrmin1 13153 | The minimum of two extende... |
xrmin2 13154 | The minimum of two extende... |
xrmaxeq 13155 | The maximum of two extende... |
xrmineq 13156 | The minimum of two extende... |
xrmaxlt 13157 | Two ways of saying the max... |
xrltmin 13158 | Two ways of saying an exte... |
xrmaxle 13159 | Two ways of saying the max... |
xrlemin 13160 | Two ways of saying a numbe... |
max1 13161 | A number is less than or e... |
max1ALT 13162 | A number is less than or e... |
max2 13163 | A number is less than or e... |
2resupmax 13164 | The supremum of two real n... |
min1 13165 | The minimum of two numbers... |
min2 13166 | The minimum of two numbers... |
maxle 13167 | Two ways of saying the max... |
lemin 13168 | Two ways of saying a numbe... |
maxlt 13169 | Two ways of saying the max... |
ltmin 13170 | Two ways of saying a numbe... |
lemaxle 13171 | A real number which is les... |
max0sub 13172 | Decompose a real number in... |
ifle 13173 | An if statement transforms... |
z2ge 13174 | There exists an integer gr... |
qbtwnre 13175 | The rational numbers are d... |
qbtwnxr 13176 | The rational numbers are d... |
qsqueeze 13177 | If a nonnegative real is l... |
qextltlem 13178 | Lemma for ~ qextlt and qex... |
qextlt 13179 | An extensionality-like pro... |
qextle 13180 | An extensionality-like pro... |
xralrple 13181 | Show that ` A ` is less th... |
alrple 13182 | Show that ` A ` is less th... |
xnegeq 13183 | Equality of two extended n... |
xnegex 13184 | A negative extended real e... |
xnegpnf 13185 | Minus ` +oo ` . Remark of... |
xnegmnf 13186 | Minus ` -oo ` . Remark of... |
rexneg 13187 | Minus a real number. Rema... |
xneg0 13188 | The negative of zero. (Co... |
xnegcl 13189 | Closure of extended real n... |
xnegneg 13190 | Extended real version of ~... |
xneg11 13191 | Extended real version of ~... |
xltnegi 13192 | Forward direction of ~ xlt... |
xltneg 13193 | Extended real version of ~... |
xleneg 13194 | Extended real version of ~... |
xlt0neg1 13195 | Extended real version of ~... |
xlt0neg2 13196 | Extended real version of ~... |
xle0neg1 13197 | Extended real version of ~... |
xle0neg2 13198 | Extended real version of ~... |
xaddval 13199 | Value of the extended real... |
xaddf 13200 | The extended real addition... |
xmulval 13201 | Value of the extended real... |
xaddpnf1 13202 | Addition of positive infin... |
xaddpnf2 13203 | Addition of positive infin... |
xaddmnf1 13204 | Addition of negative infin... |
xaddmnf2 13205 | Addition of negative infin... |
pnfaddmnf 13206 | Addition of positive and n... |
mnfaddpnf 13207 | Addition of negative and p... |
rexadd 13208 | The extended real addition... |
rexsub 13209 | Extended real subtraction ... |
rexaddd 13210 | The extended real addition... |
xnn0xaddcl 13211 | The extended nonnegative i... |
xaddnemnf 13212 | Closure of extended real a... |
xaddnepnf 13213 | Closure of extended real a... |
xnegid 13214 | Extended real version of ~... |
xaddcl 13215 | The extended real addition... |
xaddcom 13216 | The extended real addition... |
xaddrid 13217 | Extended real version of ~... |
xaddlid 13218 | Extended real version of ~... |
xaddridd 13219 | ` 0 ` is a right identity ... |
xnn0lem1lt 13220 | Extended nonnegative integ... |
xnn0lenn0nn0 13221 | An extended nonnegative in... |
xnn0le2is012 13222 | An extended nonnegative in... |
xnn0xadd0 13223 | The sum of two extended no... |
xnegdi 13224 | Extended real version of ~... |
xaddass 13225 | Associativity of extended ... |
xaddass2 13226 | Associativity of extended ... |
xpncan 13227 | Extended real version of ~... |
xnpcan 13228 | Extended real version of ~... |
xleadd1a 13229 | Extended real version of ~... |
xleadd2a 13230 | Commuted form of ~ xleadd1... |
xleadd1 13231 | Weakened version of ~ xlea... |
xltadd1 13232 | Extended real version of ~... |
xltadd2 13233 | Extended real version of ~... |
xaddge0 13234 | The sum of nonnegative ext... |
xle2add 13235 | Extended real version of ~... |
xlt2add 13236 | Extended real version of ~... |
xsubge0 13237 | Extended real version of ~... |
xposdif 13238 | Extended real version of ~... |
xlesubadd 13239 | Under certain conditions, ... |
xmullem 13240 | Lemma for ~ rexmul . (Con... |
xmullem2 13241 | Lemma for ~ xmulneg1 . (C... |
xmulcom 13242 | Extended real multiplicati... |
xmul01 13243 | Extended real version of ~... |
xmul02 13244 | Extended real version of ~... |
xmulneg1 13245 | Extended real version of ~... |
xmulneg2 13246 | Extended real version of ~... |
rexmul 13247 | The extended real multipli... |
xmulf 13248 | The extended real multipli... |
xmulcl 13249 | Closure of extended real m... |
xmulpnf1 13250 | Multiplication by plus inf... |
xmulpnf2 13251 | Multiplication by plus inf... |
xmulmnf1 13252 | Multiplication by minus in... |
xmulmnf2 13253 | Multiplication by minus in... |
xmulpnf1n 13254 | Multiplication by plus inf... |
xmulrid 13255 | Extended real version of ~... |
xmullid 13256 | Extended real version of ~... |
xmulm1 13257 | Extended real version of ~... |
xmulasslem2 13258 | Lemma for ~ xmulass . (Co... |
xmulgt0 13259 | Extended real version of ~... |
xmulge0 13260 | Extended real version of ~... |
xmulasslem 13261 | Lemma for ~ xmulass . (Co... |
xmulasslem3 13262 | Lemma for ~ xmulass . (Co... |
xmulass 13263 | Associativity of the exten... |
xlemul1a 13264 | Extended real version of ~... |
xlemul2a 13265 | Extended real version of ~... |
xlemul1 13266 | Extended real version of ~... |
xlemul2 13267 | Extended real version of ~... |
xltmul1 13268 | Extended real version of ~... |
xltmul2 13269 | Extended real version of ~... |
xadddilem 13270 | Lemma for ~ xadddi . (Con... |
xadddi 13271 | Distributive property for ... |
xadddir 13272 | Commuted version of ~ xadd... |
xadddi2 13273 | The assumption that the mu... |
xadddi2r 13274 | Commuted version of ~ xadd... |
x2times 13275 | Extended real version of ~... |
xnegcld 13276 | Closure of extended real n... |
xaddcld 13277 | The extended real addition... |
xmulcld 13278 | Closure of extended real m... |
xadd4d 13279 | Rearrangement of 4 terms i... |
xnn0add4d 13280 | Rearrangement of 4 terms i... |
xrsupexmnf 13281 | Adding minus infinity to a... |
xrinfmexpnf 13282 | Adding plus infinity to a ... |
xrsupsslem 13283 | Lemma for ~ xrsupss . (Co... |
xrinfmsslem 13284 | Lemma for ~ xrinfmss . (C... |
xrsupss 13285 | Any subset of extended rea... |
xrinfmss 13286 | Any subset of extended rea... |
xrinfmss2 13287 | Any subset of extended rea... |
xrub 13288 | By quantifying only over r... |
supxr 13289 | The supremum of a set of e... |
supxr2 13290 | The supremum of a set of e... |
supxrcl 13291 | The supremum of an arbitra... |
supxrun 13292 | The supremum of the union ... |
supxrmnf 13293 | Adding minus infinity to a... |
supxrpnf 13294 | The supremum of a set of e... |
supxrunb1 13295 | The supremum of an unbound... |
supxrunb2 13296 | The supremum of an unbound... |
supxrbnd1 13297 | The supremum of a bounded-... |
supxrbnd2 13298 | The supremum of a bounded-... |
xrsup0 13299 | The supremum of an empty s... |
supxrub 13300 | A member of a set of exten... |
supxrlub 13301 | The supremum of a set of e... |
supxrleub 13302 | The supremum of a set of e... |
supxrre 13303 | The real and extended real... |
supxrbnd 13304 | The supremum of a bounded-... |
supxrgtmnf 13305 | The supremum of a nonempty... |
supxrre1 13306 | The supremum of a nonempty... |
supxrre2 13307 | The supremum of a nonempty... |
supxrss 13308 | Smaller sets of extended r... |
infxrcl 13309 | The infimum of an arbitrar... |
infxrlb 13310 | A member of a set of exten... |
infxrgelb 13311 | The infimum of a set of ex... |
infxrre 13312 | The real and extended real... |
infxrmnf 13313 | The infinimum of a set of ... |
xrinf0 13314 | The infimum of the empty s... |
infxrss 13315 | Larger sets of extended re... |
reltre 13316 | For all real numbers there... |
rpltrp 13317 | For all positive real numb... |
reltxrnmnf 13318 | For all extended real numb... |
infmremnf 13319 | The infimum of the reals i... |
infmrp1 13320 | The infimum of the positiv... |
ixxval 13329 | Value of the interval func... |
elixx1 13330 | Membership in an interval ... |
ixxf 13331 | The set of intervals of ex... |
ixxex 13332 | The set of intervals of ex... |
ixxssxr 13333 | The set of intervals of ex... |
elixx3g 13334 | Membership in a set of ope... |
ixxssixx 13335 | An interval is a subset of... |
ixxdisj 13336 | Split an interval into dis... |
ixxun 13337 | Split an interval into two... |
ixxin 13338 | Intersection of two interv... |
ixxss1 13339 | Subset relationship for in... |
ixxss2 13340 | Subset relationship for in... |
ixxss12 13341 | Subset relationship for in... |
ixxub 13342 | Extract the upper bound of... |
ixxlb 13343 | Extract the lower bound of... |
iooex 13344 | The set of open intervals ... |
iooval 13345 | Value of the open interval... |
ioo0 13346 | An empty open interval of ... |
ioon0 13347 | An open interval of extend... |
ndmioo 13348 | The open interval function... |
iooid 13349 | An open interval with iden... |
elioo3g 13350 | Membership in a set of ope... |
elioore 13351 | A member of an open interv... |
lbioo 13352 | An open interval does not ... |
ubioo 13353 | An open interval does not ... |
iooval2 13354 | Value of the open interval... |
iooin 13355 | Intersection of two open i... |
iooss1 13356 | Subset relationship for op... |
iooss2 13357 | Subset relationship for op... |
iocval 13358 | Value of the open-below, c... |
icoval 13359 | Value of the closed-below,... |
iccval 13360 | Value of the closed interv... |
elioo1 13361 | Membership in an open inte... |
elioo2 13362 | Membership in an open inte... |
elioc1 13363 | Membership in an open-belo... |
elico1 13364 | Membership in a closed-bel... |
elicc1 13365 | Membership in a closed int... |
iccid 13366 | A closed interval with ide... |
ico0 13367 | An empty open interval of ... |
ioc0 13368 | An empty open interval of ... |
icc0 13369 | An empty closed interval o... |
dfrp2 13370 | Alternate definition of th... |
elicod 13371 | Membership in a left-close... |
icogelb 13372 | An element of a left-close... |
elicore 13373 | A member of a left-closed ... |
ubioc1 13374 | The upper bound belongs to... |
lbico1 13375 | The lower bound belongs to... |
iccleub 13376 | An element of a closed int... |
iccgelb 13377 | An element of a closed int... |
elioo5 13378 | Membership in an open inte... |
eliooxr 13379 | A nonempty open interval s... |
eliooord 13380 | Ordering implied by a memb... |
elioo4g 13381 | Membership in an open inte... |
ioossre 13382 | An open interval is a set ... |
ioosscn 13383 | An open interval is a set ... |
elioc2 13384 | Membership in an open-belo... |
elico2 13385 | Membership in a closed-bel... |
elicc2 13386 | Membership in a closed rea... |
elicc2i 13387 | Inference for membership i... |
elicc4 13388 | Membership in a closed rea... |
iccss 13389 | Condition for a closed int... |
iccssioo 13390 | Condition for a closed int... |
icossico 13391 | Condition for a closed-bel... |
iccss2 13392 | Condition for a closed int... |
iccssico 13393 | Condition for a closed int... |
iccssioo2 13394 | Condition for a closed int... |
iccssico2 13395 | Condition for a closed int... |
ioomax 13396 | The open interval from min... |
iccmax 13397 | The closed interval from m... |
ioopos 13398 | The set of positive reals ... |
ioorp 13399 | The set of positive reals ... |
iooshf 13400 | Shift the arguments of the... |
iocssre 13401 | A closed-above interval wi... |
icossre 13402 | A closed-below interval wi... |
iccssre 13403 | A closed real interval is ... |
iccssxr 13404 | A closed interval is a set... |
iocssxr 13405 | An open-below, closed-abov... |
icossxr 13406 | A closed-below, open-above... |
ioossicc 13407 | An open interval is a subs... |
iccssred 13408 | A closed real interval is ... |
eliccxr 13409 | A member of a closed inter... |
icossicc 13410 | A closed-below, open-above... |
iocssicc 13411 | A closed-above, open-below... |
ioossico 13412 | An open interval is a subs... |
iocssioo 13413 | Condition for a closed int... |
icossioo 13414 | Condition for a closed int... |
ioossioo 13415 | Condition for an open inte... |
iccsupr 13416 | A nonempty subset of a clo... |
elioopnf 13417 | Membership in an unbounded... |
elioomnf 13418 | Membership in an unbounded... |
elicopnf 13419 | Membership in a closed unb... |
repos 13420 | Two ways of saying that a ... |
ioof 13421 | The set of open intervals ... |
iccf 13422 | The set of closed interval... |
unirnioo 13423 | The union of the range of ... |
dfioo2 13424 | Alternate definition of th... |
ioorebas 13425 | Open intervals are element... |
xrge0neqmnf 13426 | A nonnegative extended rea... |
xrge0nre 13427 | An extended real which is ... |
elrege0 13428 | The predicate "is a nonneg... |
nn0rp0 13429 | A nonnegative integer is a... |
rge0ssre 13430 | Nonnegative real numbers a... |
elxrge0 13431 | Elementhood in the set of ... |
0e0icopnf 13432 | 0 is a member of ` ( 0 [,)... |
0e0iccpnf 13433 | 0 is a member of ` ( 0 [,]... |
ge0addcl 13434 | The nonnegative reals are ... |
ge0mulcl 13435 | The nonnegative reals are ... |
ge0xaddcl 13436 | The nonnegative reals are ... |
ge0xmulcl 13437 | The nonnegative extended r... |
lbicc2 13438 | The lower bound of a close... |
ubicc2 13439 | The upper bound of a close... |
elicc01 13440 | Membership in the closed r... |
elunitrn 13441 | The closed unit interval i... |
elunitcn 13442 | The closed unit interval i... |
0elunit 13443 | Zero is an element of the ... |
1elunit 13444 | One is an element of the c... |
iooneg 13445 | Membership in a negated op... |
iccneg 13446 | Membership in a negated cl... |
icoshft 13447 | A shifted real is a member... |
icoshftf1o 13448 | Shifting a closed-below, o... |
icoun 13449 | The union of two adjacent ... |
icodisj 13450 | Adjacent left-closed right... |
ioounsn 13451 | The union of an open inter... |
snunioo 13452 | The closure of one end of ... |
snunico 13453 | The closure of the open en... |
snunioc 13454 | The closure of the open en... |
prunioo 13455 | The closure of an open rea... |
ioodisj 13456 | If the upper bound of one ... |
ioojoin 13457 | Join two open intervals to... |
difreicc 13458 | The class difference of ` ... |
iccsplit 13459 | Split a closed interval in... |
iccshftr 13460 | Membership in a shifted in... |
iccshftri 13461 | Membership in a shifted in... |
iccshftl 13462 | Membership in a shifted in... |
iccshftli 13463 | Membership in a shifted in... |
iccdil 13464 | Membership in a dilated in... |
iccdili 13465 | Membership in a dilated in... |
icccntr 13466 | Membership in a contracted... |
icccntri 13467 | Membership in a contracted... |
divelunit 13468 | A condition for a ratio to... |
lincmb01cmp 13469 | A linear combination of tw... |
iccf1o 13470 | Describe a bijection from ... |
iccen 13471 | Any nontrivial closed inte... |
xov1plusxeqvd 13472 | A complex number ` X ` is ... |
unitssre 13473 | ` ( 0 [,] 1 ) ` is a subse... |
unitsscn 13474 | The closed unit interval i... |
supicc 13475 | Supremum of a bounded set ... |
supiccub 13476 | The supremum of a bounded ... |
supicclub 13477 | The supremum of a bounded ... |
supicclub2 13478 | The supremum of a bounded ... |
zltaddlt1le 13479 | The sum of an integer and ... |
xnn0xrge0 13480 | An extended nonnegative in... |
fzval 13483 | The value of a finite set ... |
fzval2 13484 | An alternative way of expr... |
fzf 13485 | Establish the domain and c... |
elfz1 13486 | Membership in a finite set... |
elfz 13487 | Membership in a finite set... |
elfz2 13488 | Membership in a finite set... |
elfzd 13489 | Membership in a finite set... |
elfz5 13490 | Membership in a finite set... |
elfz4 13491 | Membership in a finite set... |
elfzuzb 13492 | Membership in a finite set... |
eluzfz 13493 | Membership in a finite set... |
elfzuz 13494 | A member of a finite set o... |
elfzuz3 13495 | Membership in a finite set... |
elfzel2 13496 | Membership in a finite set... |
elfzel1 13497 | Membership in a finite set... |
elfzelz 13498 | A member of a finite set o... |
elfzelzd 13499 | A member of a finite set o... |
fzssz 13500 | A finite sequence of integ... |
elfzle1 13501 | A member of a finite set o... |
elfzle2 13502 | A member of a finite set o... |
elfzuz2 13503 | Implication of membership ... |
elfzle3 13504 | Membership in a finite set... |
eluzfz1 13505 | Membership in a finite set... |
eluzfz2 13506 | Membership in a finite set... |
eluzfz2b 13507 | Membership in a finite set... |
elfz3 13508 | Membership in a finite set... |
elfz1eq 13509 | Membership in a finite set... |
elfzubelfz 13510 | If there is a member in a ... |
peano2fzr 13511 | A Peano-postulate-like the... |
fzn0 13512 | Properties of a finite int... |
fz0 13513 | A finite set of sequential... |
fzn 13514 | A finite set of sequential... |
fzen 13515 | A shifted finite set of se... |
fz1n 13516 | A 1-based finite set of se... |
0nelfz1 13517 | 0 is not an element of a f... |
0fz1 13518 | Two ways to say a finite 1... |
fz10 13519 | There are no integers betw... |
uzsubsubfz 13520 | Membership of an integer g... |
uzsubsubfz1 13521 | Membership of an integer g... |
ige3m2fz 13522 | Membership of an integer g... |
fzsplit2 13523 | Split a finite interval of... |
fzsplit 13524 | Split a finite interval of... |
fzdisj 13525 | Condition for two finite i... |
fz01en 13526 | 0-based and 1-based finite... |
elfznn 13527 | A member of a finite set o... |
elfz1end 13528 | A nonempty finite range of... |
fz1ssnn 13529 | A finite set of positive i... |
fznn0sub 13530 | Subtraction closure for a ... |
fzmmmeqm 13531 | Subtracting the difference... |
fzaddel 13532 | Membership of a sum in a f... |
fzadd2 13533 | Membership of a sum in a f... |
fzsubel 13534 | Membership of a difference... |
fzopth 13535 | A finite set of sequential... |
fzass4 13536 | Two ways to express a nond... |
fzss1 13537 | Subset relationship for fi... |
fzss2 13538 | Subset relationship for fi... |
fzssuz 13539 | A finite set of sequential... |
fzsn 13540 | A finite interval of integ... |
fzssp1 13541 | Subset relationship for fi... |
fzssnn 13542 | Finite sets of sequential ... |
ssfzunsnext 13543 | A subset of a finite seque... |
ssfzunsn 13544 | A subset of a finite seque... |
fzsuc 13545 | Join a successor to the en... |
fzpred 13546 | Join a predecessor to the ... |
fzpreddisj 13547 | A finite set of sequential... |
elfzp1 13548 | Append an element to a fin... |
fzp1ss 13549 | Subset relationship for fi... |
fzelp1 13550 | Membership in a set of seq... |
fzp1elp1 13551 | Add one to an element of a... |
fznatpl1 13552 | Shift membership in a fini... |
fzpr 13553 | A finite interval of integ... |
fztp 13554 | A finite interval of integ... |
fz12pr 13555 | An integer range between 1... |
fzsuc2 13556 | Join a successor to the en... |
fzp1disj 13557 | ` ( M ... ( N + 1 ) ) ` is... |
fzdifsuc 13558 | Remove a successor from th... |
fzprval 13559 | Two ways of defining the f... |
fztpval 13560 | Two ways of defining the f... |
fzrev 13561 | Reversal of start and end ... |
fzrev2 13562 | Reversal of start and end ... |
fzrev2i 13563 | Reversal of start and end ... |
fzrev3 13564 | The "complement" of a memb... |
fzrev3i 13565 | The "complement" of a memb... |
fznn 13566 | Finite set of sequential i... |
elfz1b 13567 | Membership in a 1-based fi... |
elfz1uz 13568 | Membership in a 1-based fi... |
elfzm11 13569 | Membership in a finite set... |
uzsplit 13570 | Express an upper integer s... |
uzdisj 13571 | The first ` N ` elements o... |
fseq1p1m1 13572 | Add/remove an item to/from... |
fseq1m1p1 13573 | Add/remove an item to/from... |
fz1sbc 13574 | Quantification over a one-... |
elfzp1b 13575 | An integer is a member of ... |
elfzm1b 13576 | An integer is a member of ... |
elfzp12 13577 | Options for membership in ... |
fzm1 13578 | Choices for an element of ... |
fzneuz 13579 | No finite set of sequentia... |
fznuz 13580 | Disjointness of the upper ... |
uznfz 13581 | Disjointness of the upper ... |
fzp1nel 13582 | One plus the upper bound o... |
fzrevral 13583 | Reversal of scanning order... |
fzrevral2 13584 | Reversal of scanning order... |
fzrevral3 13585 | Reversal of scanning order... |
fzshftral 13586 | Shift the scanning order i... |
ige2m1fz1 13587 | Membership of an integer g... |
ige2m1fz 13588 | Membership in a 0-based fi... |
elfz2nn0 13589 | Membership in a finite set... |
fznn0 13590 | Characterization of a fini... |
elfznn0 13591 | A member of a finite set o... |
elfz3nn0 13592 | The upper bound of a nonem... |
fz0ssnn0 13593 | Finite sets of sequential ... |
fz1ssfz0 13594 | Subset relationship for fi... |
0elfz 13595 | 0 is an element of a finit... |
nn0fz0 13596 | A nonnegative integer is a... |
elfz0add 13597 | An element of a finite set... |
fz0sn 13598 | An integer range from 0 to... |
fz0tp 13599 | An integer range from 0 to... |
fz0to3un2pr 13600 | An integer range from 0 to... |
fz0to4untppr 13601 | An integer range from 0 to... |
elfz0ubfz0 13602 | An element of a finite set... |
elfz0fzfz0 13603 | A member of a finite set o... |
fz0fzelfz0 13604 | If a member of a finite se... |
fznn0sub2 13605 | Subtraction closure for a ... |
uzsubfz0 13606 | Membership of an integer g... |
fz0fzdiffz0 13607 | The difference of an integ... |
elfzmlbm 13608 | Subtracting the lower boun... |
elfzmlbp 13609 | Subtracting the lower boun... |
fzctr 13610 | Lemma for theorems about t... |
difelfzle 13611 | The difference of two inte... |
difelfznle 13612 | The difference of two inte... |
nn0split 13613 | Express the set of nonnega... |
nn0disj 13614 | The first ` N + 1 ` elemen... |
fz0sn0fz1 13615 | A finite set of sequential... |
fvffz0 13616 | The function value of a fu... |
1fv 13617 | A function on a singleton.... |
4fvwrd4 13618 | The first four function va... |
2ffzeq 13619 | Two functions over 0-based... |
preduz 13620 | The value of the predecess... |
prednn 13621 | The value of the predecess... |
prednn0 13622 | The value of the predecess... |
predfz 13623 | Calculate the predecessor ... |
fzof 13626 | Functionality of the half-... |
elfzoel1 13627 | Reverse closure for half-o... |
elfzoel2 13628 | Reverse closure for half-o... |
elfzoelz 13629 | Reverse closure for half-o... |
fzoval 13630 | Value of the half-open int... |
elfzo 13631 | Membership in a half-open ... |
elfzo2 13632 | Membership in a half-open ... |
elfzouz 13633 | Membership in a half-open ... |
nelfzo 13634 | An integer not being a mem... |
fzolb 13635 | The left endpoint of a hal... |
fzolb2 13636 | The left endpoint of a hal... |
elfzole1 13637 | A member in a half-open in... |
elfzolt2 13638 | A member in a half-open in... |
elfzolt3 13639 | Membership in a half-open ... |
elfzolt2b 13640 | A member in a half-open in... |
elfzolt3b 13641 | Membership in a half-open ... |
elfzop1le2 13642 | A member in a half-open in... |
fzonel 13643 | A half-open range does not... |
elfzouz2 13644 | The upper bound of a half-... |
elfzofz 13645 | A half-open range is conta... |
elfzo3 13646 | Express membership in a ha... |
fzon0 13647 | A half-open integer interv... |
fzossfz 13648 | A half-open range is conta... |
fzossz 13649 | A half-open integer interv... |
fzon 13650 | A half-open set of sequent... |
fzo0n 13651 | A half-open range of nonne... |
fzonlt0 13652 | A half-open integer range ... |
fzo0 13653 | Half-open sets with equal ... |
fzonnsub 13654 | If ` K < N ` then ` N - K ... |
fzonnsub2 13655 | If ` M < N ` then ` N - M ... |
fzoss1 13656 | Subset relationship for ha... |
fzoss2 13657 | Subset relationship for ha... |
fzossrbm1 13658 | Subset of a half-open rang... |
fzo0ss1 13659 | Subset relationship for ha... |
fzossnn0 13660 | A half-open integer range ... |
fzospliti 13661 | One direction of splitting... |
fzosplit 13662 | Split a half-open integer ... |
fzodisj 13663 | Abutting half-open integer... |
fzouzsplit 13664 | Split an upper integer set... |
fzouzdisj 13665 | A half-open integer range ... |
fzoun 13666 | A half-open integer range ... |
fzodisjsn 13667 | A half-open integer range ... |
prinfzo0 13668 | The intersection of a half... |
lbfzo0 13669 | An integer is strictly gre... |
elfzo0 13670 | Membership in a half-open ... |
elfzo0z 13671 | Membership in a half-open ... |
nn0p1elfzo 13672 | A nonnegative integer incr... |
elfzo0le 13673 | A member in a half-open ra... |
elfzonn0 13674 | A member of a half-open ra... |
fzonmapblen 13675 | The result of subtracting ... |
fzofzim 13676 | If a nonnegative integer i... |
fz1fzo0m1 13677 | Translation of one between... |
fzossnn 13678 | Half-open integer ranges s... |
elfzo1 13679 | Membership in a half-open ... |
fzo1fzo0n0 13680 | An integer between 1 and a... |
fzo0n0 13681 | A half-open integer range ... |
fzoaddel 13682 | Translate membership in a ... |
fzo0addel 13683 | Translate membership in a ... |
fzo0addelr 13684 | Translate membership in a ... |
fzoaddel2 13685 | Translate membership in a ... |
elfzoext 13686 | Membership of an integer i... |
elincfzoext 13687 | Membership of an increased... |
fzosubel 13688 | Translate membership in a ... |
fzosubel2 13689 | Membership in a translated... |
fzosubel3 13690 | Membership in a translated... |
eluzgtdifelfzo 13691 | Membership of the differen... |
ige2m2fzo 13692 | Membership of an integer g... |
fzocatel 13693 | Translate membership in a ... |
ubmelfzo 13694 | If an integer in a 1-based... |
elfzodifsumelfzo 13695 | If an integer is in a half... |
elfzom1elp1fzo 13696 | Membership of an integer i... |
elfzom1elfzo 13697 | Membership in a half-open ... |
fzval3 13698 | Expressing a closed intege... |
fz0add1fz1 13699 | Translate membership in a ... |
fzosn 13700 | Expressing a singleton as ... |
elfzomin 13701 | Membership of an integer i... |
zpnn0elfzo 13702 | Membership of an integer i... |
zpnn0elfzo1 13703 | Membership of an integer i... |
fzosplitsnm1 13704 | Removing a singleton from ... |
elfzonlteqm1 13705 | If an element of a half-op... |
fzonn0p1 13706 | A nonnegative integer is e... |
fzossfzop1 13707 | A half-open range of nonne... |
fzonn0p1p1 13708 | If a nonnegative integer i... |
elfzom1p1elfzo 13709 | Increasing an element of a... |
fzo0ssnn0 13710 | Half-open integer ranges s... |
fzo01 13711 | Expressing the singleton o... |
fzo12sn 13712 | A 1-based half-open intege... |
fzo13pr 13713 | A 1-based half-open intege... |
fzo0to2pr 13714 | A half-open integer range ... |
fzo0to3tp 13715 | A half-open integer range ... |
fzo0to42pr 13716 | A half-open integer range ... |
fzo1to4tp 13717 | A half-open integer range ... |
fzo0sn0fzo1 13718 | A half-open range of nonne... |
elfzo0l 13719 | A member of a half-open ra... |
fzoend 13720 | The endpoint of a half-ope... |
fzo0end 13721 | The endpoint of a zero-bas... |
ssfzo12 13722 | Subset relationship for ha... |
ssfzoulel 13723 | If a half-open integer ran... |
ssfzo12bi 13724 | Subset relationship for ha... |
ubmelm1fzo 13725 | The result of subtracting ... |
fzofzp1 13726 | If a point is in a half-op... |
fzofzp1b 13727 | If a point is in a half-op... |
elfzom1b 13728 | An integer is a member of ... |
elfzom1elp1fzo1 13729 | Membership of a nonnegativ... |
elfzo1elm1fzo0 13730 | Membership of a positive i... |
elfzonelfzo 13731 | If an element of a half-op... |
fzonfzoufzol 13732 | If an element of a half-op... |
elfzomelpfzo 13733 | An integer increased by an... |
elfznelfzo 13734 | A value in a finite set of... |
elfznelfzob 13735 | A value in a finite set of... |
peano2fzor 13736 | A Peano-postulate-like the... |
fzosplitsn 13737 | Extending a half-open rang... |
fzosplitpr 13738 | Extending a half-open inte... |
fzosplitprm1 13739 | Extending a half-open inte... |
fzosplitsni 13740 | Membership in a half-open ... |
fzisfzounsn 13741 | A finite interval of integ... |
elfzr 13742 | A member of a finite inter... |
elfzlmr 13743 | A member of a finite inter... |
elfz0lmr 13744 | A member of a finite inter... |
fzostep1 13745 | Two possibilities for a nu... |
fzoshftral 13746 | Shift the scanning order i... |
fzind2 13747 | Induction on the integers ... |
fvinim0ffz 13748 | The function values for th... |
injresinjlem 13749 | Lemma for ~ injresinj . (... |
injresinj 13750 | A function whose restricti... |
subfzo0 13751 | The difference between two... |
flval 13756 | Value of the floor (greate... |
flcl 13757 | The floor (greatest intege... |
reflcl 13758 | The floor (greatest intege... |
fllelt 13759 | A basic property of the fl... |
flcld 13760 | The floor (greatest intege... |
flle 13761 | A basic property of the fl... |
flltp1 13762 | A basic property of the fl... |
fllep1 13763 | A basic property of the fl... |
fraclt1 13764 | The fractional part of a r... |
fracle1 13765 | The fractional part of a r... |
fracge0 13766 | The fractional part of a r... |
flge 13767 | The floor function value i... |
fllt 13768 | The floor function value i... |
flflp1 13769 | Move floor function betwee... |
flid 13770 | An integer is its own floo... |
flidm 13771 | The floor function is idem... |
flidz 13772 | A real number equals its f... |
flltnz 13773 | The floor of a non-integer... |
flwordi 13774 | Ordering relation for the ... |
flword2 13775 | Ordering relation for the ... |
flval2 13776 | An alternate way to define... |
flval3 13777 | An alternate way to define... |
flbi 13778 | A condition equivalent to ... |
flbi2 13779 | A condition equivalent to ... |
adddivflid 13780 | The floor of a sum of an i... |
ico01fl0 13781 | The floor of a real number... |
flge0nn0 13782 | The floor of a number grea... |
flge1nn 13783 | The floor of a number grea... |
fldivnn0 13784 | The floor function of a di... |
refldivcl 13785 | The floor function of a di... |
divfl0 13786 | The floor of a fraction is... |
fladdz 13787 | An integer can be moved in... |
flzadd 13788 | An integer can be moved in... |
flmulnn0 13789 | Move a nonnegative integer... |
btwnzge0 13790 | A real bounded between an ... |
2tnp1ge0ge0 13791 | Two times an integer plus ... |
flhalf 13792 | Ordering relation for the ... |
fldivle 13793 | The floor function of a di... |
fldivnn0le 13794 | The floor function of a di... |
flltdivnn0lt 13795 | The floor function of a di... |
ltdifltdiv 13796 | If the dividend of a divis... |
fldiv4p1lem1div2 13797 | The floor of an integer eq... |
fldiv4lem1div2uz2 13798 | The floor of an integer gr... |
fldiv4lem1div2 13799 | The floor of a positive in... |
ceilval 13800 | The value of the ceiling f... |
dfceil2 13801 | Alternative definition of ... |
ceilval2 13802 | The value of the ceiling f... |
ceicl 13803 | The ceiling function retur... |
ceilcl 13804 | Closure of the ceiling fun... |
ceilcld 13805 | Closure of the ceiling fun... |
ceige 13806 | The ceiling of a real numb... |
ceilge 13807 | The ceiling of a real numb... |
ceilged 13808 | The ceiling of a real numb... |
ceim1l 13809 | One less than the ceiling ... |
ceilm1lt 13810 | One less than the ceiling ... |
ceile 13811 | The ceiling of a real numb... |
ceille 13812 | The ceiling of a real numb... |
ceilid 13813 | An integer is its own ceil... |
ceilidz 13814 | A real number equals its c... |
flleceil 13815 | The floor of a real number... |
fleqceilz 13816 | A real number is an intege... |
quoremz 13817 | Quotient and remainder of ... |
quoremnn0 13818 | Quotient and remainder of ... |
quoremnn0ALT 13819 | Alternate proof of ~ quore... |
intfrac2 13820 | Decompose a real into inte... |
intfracq 13821 | Decompose a rational numbe... |
fldiv 13822 | Cancellation of the embedd... |
fldiv2 13823 | Cancellation of an embedde... |
fznnfl 13824 | Finite set of sequential i... |
uzsup 13825 | An upper set of integers i... |
ioopnfsup 13826 | An upper set of reals is u... |
icopnfsup 13827 | An upper set of reals is u... |
rpsup 13828 | The positive reals are unb... |
resup 13829 | The real numbers are unbou... |
xrsup 13830 | The extended real numbers ... |
modval 13833 | The value of the modulo op... |
modvalr 13834 | The value of the modulo op... |
modcl 13835 | Closure law for the modulo... |
flpmodeq 13836 | Partition of a division in... |
modcld 13837 | Closure law for the modulo... |
mod0 13838 | ` A mod B ` is zero iff ` ... |
mulmod0 13839 | The product of an integer ... |
negmod0 13840 | ` A ` is divisible by ` B ... |
modge0 13841 | The modulo operation is no... |
modlt 13842 | The modulo operation is le... |
modelico 13843 | Modular reduction produces... |
moddiffl 13844 | Value of the modulo operat... |
moddifz 13845 | The modulo operation diffe... |
modfrac 13846 | The fractional part of a n... |
flmod 13847 | The floor function express... |
intfrac 13848 | Break a number into its in... |
zmod10 13849 | An integer modulo 1 is 0. ... |
zmod1congr 13850 | Two arbitrary integers are... |
modmulnn 13851 | Move a positive integer in... |
modvalp1 13852 | The value of the modulo op... |
zmodcl 13853 | Closure law for the modulo... |
zmodcld 13854 | Closure law for the modulo... |
zmodfz 13855 | An integer mod ` B ` lies ... |
zmodfzo 13856 | An integer mod ` B ` lies ... |
zmodfzp1 13857 | An integer mod ` B ` lies ... |
modid 13858 | Identity law for modulo. ... |
modid0 13859 | A positive real number mod... |
modid2 13860 | Identity law for modulo. ... |
zmodid2 13861 | Identity law for modulo re... |
zmodidfzo 13862 | Identity law for modulo re... |
zmodidfzoimp 13863 | Identity law for modulo re... |
0mod 13864 | Special case: 0 modulo a p... |
1mod 13865 | Special case: 1 modulo a r... |
modabs 13866 | Absorption law for modulo.... |
modabs2 13867 | Absorption law for modulo.... |
modcyc 13868 | The modulo operation is pe... |
modcyc2 13869 | The modulo operation is pe... |
modadd1 13870 | Addition property of the m... |
modaddabs 13871 | Absorption law for modulo.... |
modaddmod 13872 | The sum of a real number m... |
muladdmodid 13873 | The sum of a positive real... |
mulp1mod1 13874 | The product of an integer ... |
modmuladd 13875 | Decomposition of an intege... |
modmuladdim 13876 | Implication of a decomposi... |
modmuladdnn0 13877 | Implication of a decomposi... |
negmod 13878 | The negation of a number m... |
m1modnnsub1 13879 | Minus one modulo a positiv... |
m1modge3gt1 13880 | Minus one modulo an intege... |
addmodid 13881 | The sum of a positive inte... |
addmodidr 13882 | The sum of a positive inte... |
modadd2mod 13883 | The sum of a real number m... |
modm1p1mod0 13884 | If a real number modulo a ... |
modltm1p1mod 13885 | If a real number modulo a ... |
modmul1 13886 | Multiplication property of... |
modmul12d 13887 | Multiplication property of... |
modnegd 13888 | Negation property of the m... |
modadd12d 13889 | Additive property of the m... |
modsub12d 13890 | Subtraction property of th... |
modsubmod 13891 | The difference of a real n... |
modsubmodmod 13892 | The difference of a real n... |
2txmodxeq0 13893 | Two times a positive real ... |
2submod 13894 | If a real number is betwee... |
modifeq2int 13895 | If a nonnegative integer i... |
modaddmodup 13896 | The sum of an integer modu... |
modaddmodlo 13897 | The sum of an integer modu... |
modmulmod 13898 | The product of a real numb... |
modmulmodr 13899 | The product of an integer ... |
modaddmulmod 13900 | The sum of a real number a... |
moddi 13901 | Distribute multiplication ... |
modsubdir 13902 | Distribute the modulo oper... |
modeqmodmin 13903 | A real number equals the d... |
modirr 13904 | A number modulo an irratio... |
modfzo0difsn 13905 | For a number within a half... |
modsumfzodifsn 13906 | The sum of a number within... |
modlteq 13907 | Two nonnegative integers l... |
addmodlteq 13908 | Two nonnegative integers l... |
om2uz0i 13909 | The mapping ` G ` is a one... |
om2uzsuci 13910 | The value of ` G ` (see ~ ... |
om2uzuzi 13911 | The value ` G ` (see ~ om2... |
om2uzlti 13912 | Less-than relation for ` G... |
om2uzlt2i 13913 | The mapping ` G ` (see ~ o... |
om2uzrani 13914 | Range of ` G ` (see ~ om2u... |
om2uzf1oi 13915 | ` G ` (see ~ om2uz0i ) is ... |
om2uzisoi 13916 | ` G ` (see ~ om2uz0i ) is ... |
om2uzoi 13917 | An alternative definition ... |
om2uzrdg 13918 | A helper lemma for the val... |
uzrdglem 13919 | A helper lemma for the val... |
uzrdgfni 13920 | The recursive definition g... |
uzrdg0i 13921 | Initial value of a recursi... |
uzrdgsuci 13922 | Successor value of a recur... |
ltweuz 13923 | ` < ` is a well-founded re... |
ltwenn 13924 | Less than well-orders the ... |
ltwefz 13925 | Less than well-orders a se... |
uzenom 13926 | An upper integer set is de... |
uzinf 13927 | An upper integer set is in... |
nnnfi 13928 | The set of positive intege... |
uzrdgxfr 13929 | Transfer the value of the ... |
fzennn 13930 | The cardinality of a finit... |
fzen2 13931 | The cardinality of a finit... |
cardfz 13932 | The cardinality of a finit... |
hashgf1o 13933 | ` G ` maps ` _om ` one-to-... |
fzfi 13934 | A finite interval of integ... |
fzfid 13935 | Commonly used special case... |
fzofi 13936 | Half-open integer sets are... |
fsequb 13937 | The values of a finite rea... |
fsequb2 13938 | The values of a finite rea... |
fseqsupcl 13939 | The values of a finite rea... |
fseqsupubi 13940 | The values of a finite rea... |
nn0ennn 13941 | The nonnegative integers a... |
nnenom 13942 | The set of positive intege... |
nnct 13943 | ` NN ` is countable. (Con... |
uzindi 13944 | Indirect strong induction ... |
axdc4uzlem 13945 | Lemma for ~ axdc4uz . (Co... |
axdc4uz 13946 | A version of ~ axdc4 that ... |
ssnn0fi 13947 | A subset of the nonnegativ... |
rabssnn0fi 13948 | A subset of the nonnegativ... |
uzsinds 13949 | Strong (or "total") induct... |
nnsinds 13950 | Strong (or "total") induct... |
nn0sinds 13951 | Strong (or "total") induct... |
fsuppmapnn0fiublem 13952 | Lemma for ~ fsuppmapnn0fiu... |
fsuppmapnn0fiub 13953 | If all functions of a fini... |
fsuppmapnn0fiubex 13954 | If all functions of a fini... |
fsuppmapnn0fiub0 13955 | If all functions of a fini... |
suppssfz 13956 | Condition for a function o... |
fsuppmapnn0ub 13957 | If a function over the non... |
fsuppmapnn0fz 13958 | If a function over the non... |
mptnn0fsupp 13959 | A mapping from the nonnega... |
mptnn0fsuppd 13960 | A mapping from the nonnega... |
mptnn0fsuppr 13961 | A finitely supported mappi... |
f13idfv 13962 | A one-to-one function with... |
seqex 13965 | Existence of the sequence ... |
seqeq1 13966 | Equality theorem for the s... |
seqeq2 13967 | Equality theorem for the s... |
seqeq3 13968 | Equality theorem for the s... |
seqeq1d 13969 | Equality deduction for the... |
seqeq2d 13970 | Equality deduction for the... |
seqeq3d 13971 | Equality deduction for the... |
seqeq123d 13972 | Equality deduction for the... |
nfseq 13973 | Hypothesis builder for the... |
seqval 13974 | Value of the sequence buil... |
seqfn 13975 | The sequence builder funct... |
seq1 13976 | Value of the sequence buil... |
seq1i 13977 | Value of the sequence buil... |
seqp1 13978 | Value of the sequence buil... |
seqexw 13979 | Weak version of ~ seqex th... |
seqp1d 13980 | Value of the sequence buil... |
seqp1iOLD 13981 | Obsolete version of ~ seqp... |
seqm1 13982 | Value of the sequence buil... |
seqcl2 13983 | Closure properties of the ... |
seqf2 13984 | Range of the recursive seq... |
seqcl 13985 | Closure properties of the ... |
seqf 13986 | Range of the recursive seq... |
seqfveq2 13987 | Equality of sequences. (C... |
seqfeq2 13988 | Equality of sequences. (C... |
seqfveq 13989 | Equality of sequences. (C... |
seqfeq 13990 | Equality of sequences. (C... |
seqshft2 13991 | Shifting the index set of ... |
seqres 13992 | Restricting its characteri... |
serf 13993 | An infinite series of comp... |
serfre 13994 | An infinite series of real... |
monoord 13995 | Ordering relation for a mo... |
monoord2 13996 | Ordering relation for a mo... |
sermono 13997 | The partial sums in an inf... |
seqsplit 13998 | Split a sequence into two ... |
seq1p 13999 | Removing the first term fr... |
seqcaopr3 14000 | Lemma for ~ seqcaopr2 . (... |
seqcaopr2 14001 | The sum of two infinite se... |
seqcaopr 14002 | The sum of two infinite se... |
seqf1olem2a 14003 | Lemma for ~ seqf1o . (Con... |
seqf1olem1 14004 | Lemma for ~ seqf1o . (Con... |
seqf1olem2 14005 | Lemma for ~ seqf1o . (Con... |
seqf1o 14006 | Rearrange a sum via an arb... |
seradd 14007 | The sum of two infinite se... |
sersub 14008 | The difference of two infi... |
seqid3 14009 | A sequence that consists e... |
seqid 14010 | Discarding the first few t... |
seqid2 14011 | The last few partial sums ... |
seqhomo 14012 | Apply a homomorphism to a ... |
seqz 14013 | If the operation ` .+ ` ha... |
seqfeq4 14014 | Equality of series under d... |
seqfeq3 14015 | Equality of series under d... |
seqdistr 14016 | The distributive property ... |
ser0 14017 | The value of the partial s... |
ser0f 14018 | A zero-valued infinite ser... |
serge0 14019 | A finite sum of nonnegativ... |
serle 14020 | Comparison of partial sums... |
ser1const 14021 | Value of the partial serie... |
seqof 14022 | Distribute function operat... |
seqof2 14023 | Distribute function operat... |
expval 14026 | Value of exponentiation to... |
expnnval 14027 | Value of exponentiation to... |
exp0 14028 | Value of a complex number ... |
0exp0e1 14029 | The zeroth power of zero e... |
exp1 14030 | Value of a complex number ... |
expp1 14031 | Value of a complex number ... |
expneg 14032 | Value of a complex number ... |
expneg2 14033 | Value of a complex number ... |
expn1 14034 | A complex number raised to... |
expcllem 14035 | Lemma for proving nonnegat... |
expcl2lem 14036 | Lemma for proving integer ... |
nnexpcl 14037 | Closure of exponentiation ... |
nn0expcl 14038 | Closure of exponentiation ... |
zexpcl 14039 | Closure of exponentiation ... |
qexpcl 14040 | Closure of exponentiation ... |
reexpcl 14041 | Closure of exponentiation ... |
expcl 14042 | Closure law for nonnegativ... |
rpexpcl 14043 | Closure law for integer ex... |
qexpclz 14044 | Closure of integer exponen... |
reexpclz 14045 | Closure of integer exponen... |
expclzlem 14046 | Lemma for ~ expclz . (Con... |
expclz 14047 | Closure law for integer ex... |
m1expcl2 14048 | Closure of integer exponen... |
m1expcl 14049 | Closure of exponentiation ... |
zexpcld 14050 | Closure of exponentiation ... |
nn0expcli 14051 | Closure of exponentiation ... |
nn0sqcl 14052 | The square of a nonnegativ... |
expm1t 14053 | Exponentiation in terms of... |
1exp 14054 | Value of 1 raised to an in... |
expeq0 14055 | A positive integer power i... |
expne0 14056 | A positive integer power i... |
expne0i 14057 | An integer power is nonzer... |
expgt0 14058 | A positive real raised to ... |
expnegz 14059 | Value of a nonzero complex... |
0exp 14060 | Value of zero raised to a ... |
expge0 14061 | A nonnegative real raised ... |
expge1 14062 | A real greater than or equ... |
expgt1 14063 | A real greater than 1 rais... |
mulexp 14064 | Nonnegative integer expone... |
mulexpz 14065 | Integer exponentiation of ... |
exprec 14066 | Integer exponentiation of ... |
expadd 14067 | Sum of exponents law for n... |
expaddzlem 14068 | Lemma for ~ expaddz . (Co... |
expaddz 14069 | Sum of exponents law for i... |
expmul 14070 | Product of exponents law f... |
expmulz 14071 | Product of exponents law f... |
m1expeven 14072 | Exponentiation of negative... |
expsub 14073 | Exponent subtraction law f... |
expp1z 14074 | Value of a nonzero complex... |
expm1 14075 | Value of a nonzero complex... |
expdiv 14076 | Nonnegative integer expone... |
sqval 14077 | Value of the square of a c... |
sqneg 14078 | The square of the negative... |
sqsubswap 14079 | Swap the order of subtract... |
sqcl 14080 | Closure of square. (Contr... |
sqmul 14081 | Distribution of squaring o... |
sqeq0 14082 | A complex number is zero i... |
sqdiv 14083 | Distribution of squaring o... |
sqdivid 14084 | The square of a nonzero co... |
sqne0 14085 | A complex number is nonzer... |
resqcl 14086 | Closure of squaring in rea... |
resqcld 14087 | Closure of squaring in rea... |
sqgt0 14088 | The square of a nonzero re... |
sqn0rp 14089 | The square of a nonzero re... |
nnsqcl 14090 | The positive naturals are ... |
zsqcl 14091 | Integers are closed under ... |
qsqcl 14092 | The square of a rational i... |
sq11 14093 | The square function is one... |
nn0sq11 14094 | The square function is one... |
lt2sq 14095 | The square function is inc... |
le2sq 14096 | The square function is non... |
le2sq2 14097 | The square function is non... |
sqge0 14098 | The square of a real is no... |
sqge0d 14099 | The square of a real is no... |
zsqcl2 14100 | The square of an integer i... |
0expd 14101 | Value of zero raised to a ... |
exp0d 14102 | Value of a complex number ... |
exp1d 14103 | Value of a complex number ... |
expeq0d 14104 | If a positive integer powe... |
sqvald 14105 | Value of square. Inferenc... |
sqcld 14106 | Closure of square. (Contr... |
sqeq0d 14107 | A number is zero iff its s... |
expcld 14108 | Closure law for nonnegativ... |
expp1d 14109 | Value of a complex number ... |
expaddd 14110 | Sum of exponents law for n... |
expmuld 14111 | Product of exponents law f... |
sqrecd 14112 | Square of reciprocal is re... |
expclzd 14113 | Closure law for integer ex... |
expne0d 14114 | A nonnegative integer powe... |
expnegd 14115 | Value of a nonzero complex... |
exprecd 14116 | An integer power of a reci... |
expp1zd 14117 | Value of a nonzero complex... |
expm1d 14118 | Value of a nonzero complex... |
expsubd 14119 | Exponent subtraction law f... |
sqmuld 14120 | Distribution of squaring o... |
sqdivd 14121 | Distribution of squaring o... |
expdivd 14122 | Nonnegative integer expone... |
mulexpd 14123 | Nonnegative integer expone... |
znsqcld 14124 | The square of a nonzero in... |
reexpcld 14125 | Closure of exponentiation ... |
expge0d 14126 | A nonnegative real raised ... |
expge1d 14127 | A real greater than or equ... |
ltexp2a 14128 | Exponent ordering relation... |
expmordi 14129 | Base ordering relationship... |
rpexpmord 14130 | Base ordering relationship... |
expcan 14131 | Cancellation law for integ... |
ltexp2 14132 | Strict ordering law for ex... |
leexp2 14133 | Ordering law for exponenti... |
leexp2a 14134 | Weak ordering relationship... |
ltexp2r 14135 | The integer powers of a fi... |
leexp2r 14136 | Weak ordering relationship... |
leexp1a 14137 | Weak base ordering relatio... |
exple1 14138 | A real between 0 and 1 inc... |
expubnd 14139 | An upper bound on ` A ^ N ... |
sumsqeq0 14140 | The sum of two squres of r... |
sqvali 14141 | Value of square. Inferenc... |
sqcli 14142 | Closure of square. (Contr... |
sqeq0i 14143 | A complex number is zero i... |
sqrecii 14144 | The square of a reciprocal... |
sqmuli 14145 | Distribution of squaring o... |
sqdivi 14146 | Distribution of squaring o... |
resqcli 14147 | Closure of square in reals... |
sqgt0i 14148 | The square of a nonzero re... |
sqge0i 14149 | The square of a real is no... |
lt2sqi 14150 | The square function on non... |
le2sqi 14151 | The square function on non... |
sq11i 14152 | The square function is one... |
sq0 14153 | The square of 0 is 0. (Co... |
sq0i 14154 | If a number is zero, then ... |
sq0id 14155 | If a number is zero, then ... |
sq1 14156 | The square of 1 is 1. (Co... |
neg1sqe1 14157 | The square of ` -u 1 ` is ... |
sq2 14158 | The square of 2 is 4. (Co... |
sq3 14159 | The square of 3 is 9. (Co... |
sq4e2t8 14160 | The square of 4 is 2 times... |
cu2 14161 | The cube of 2 is 8. (Cont... |
irec 14162 | The reciprocal of ` _i ` .... |
i2 14163 | ` _i ` squared. (Contribu... |
i3 14164 | ` _i ` cubed. (Contribute... |
i4 14165 | ` _i ` to the fourth power... |
nnlesq 14166 | A positive integer is less... |
zzlesq 14167 | An integer is less than or... |
iexpcyc 14168 | Taking ` _i ` to the ` K `... |
expnass 14169 | A counterexample showing t... |
sqlecan 14170 | Cancel one factor of a squ... |
subsq 14171 | Factor the difference of t... |
subsq2 14172 | Express the difference of ... |
binom2i 14173 | The square of a binomial. ... |
subsqi 14174 | Factor the difference of t... |
sqeqori 14175 | The squares of two complex... |
subsq0i 14176 | The two solutions to the d... |
sqeqor 14177 | The squares of two complex... |
binom2 14178 | The square of a binomial. ... |
binom21 14179 | Special case of ~ binom2 w... |
binom2sub 14180 | Expand the square of a sub... |
binom2sub1 14181 | Special case of ~ binom2su... |
binom2subi 14182 | Expand the square of a sub... |
mulbinom2 14183 | The square of a binomial w... |
binom3 14184 | The cube of a binomial. (... |
sq01 14185 | If a complex number equals... |
zesq 14186 | An integer is even iff its... |
nnesq 14187 | A positive integer is even... |
crreczi 14188 | Reciprocal of a complex nu... |
bernneq 14189 | Bernoulli's inequality, du... |
bernneq2 14190 | Variation of Bernoulli's i... |
bernneq3 14191 | A corollary of ~ bernneq .... |
expnbnd 14192 | Exponentiation with a base... |
expnlbnd 14193 | The reciprocal of exponent... |
expnlbnd2 14194 | The reciprocal of exponent... |
expmulnbnd 14195 | Exponentiation with a base... |
digit2 14196 | Two ways to express the ` ... |
digit1 14197 | Two ways to express the ` ... |
modexp 14198 | Exponentiation property of... |
discr1 14199 | A nonnegative quadratic fo... |
discr 14200 | If a quadratic polynomial ... |
expnngt1 14201 | If an integer power with a... |
expnngt1b 14202 | An integer power with an i... |
sqoddm1div8 14203 | A squared odd number minus... |
nnsqcld 14204 | The naturals are closed un... |
nnexpcld 14205 | Closure of exponentiation ... |
nn0expcld 14206 | Closure of exponentiation ... |
rpexpcld 14207 | Closure law for exponentia... |
ltexp2rd 14208 | The power of a positive nu... |
reexpclzd 14209 | Closure of exponentiation ... |
sqgt0d 14210 | The square of a nonzero re... |
ltexp2d 14211 | Ordering relationship for ... |
leexp2d 14212 | Ordering law for exponenti... |
expcand 14213 | Ordering relationship for ... |
leexp2ad 14214 | Ordering relationship for ... |
leexp2rd 14215 | Ordering relationship for ... |
lt2sqd 14216 | The square function on non... |
le2sqd 14217 | The square function on non... |
sq11d 14218 | The square function is one... |
mulsubdivbinom2 14219 | The square of a binomial w... |
muldivbinom2 14220 | The square of a binomial w... |
sq10 14221 | The square of 10 is 100. ... |
sq10e99m1 14222 | The square of 10 is 99 plu... |
3dec 14223 | A "decimal constructor" wh... |
nn0le2msqi 14224 | The square function on non... |
nn0opthlem1 14225 | A rather pretty lemma for ... |
nn0opthlem2 14226 | Lemma for ~ nn0opthi . (C... |
nn0opthi 14227 | An ordered pair theorem fo... |
nn0opth2i 14228 | An ordered pair theorem fo... |
nn0opth2 14229 | An ordered pair theorem fo... |
facnn 14232 | Value of the factorial fun... |
fac0 14233 | The factorial of 0. (Cont... |
fac1 14234 | The factorial of 1. (Cont... |
facp1 14235 | The factorial of a success... |
fac2 14236 | The factorial of 2. (Cont... |
fac3 14237 | The factorial of 3. (Cont... |
fac4 14238 | The factorial of 4. (Cont... |
facnn2 14239 | Value of the factorial fun... |
faccl 14240 | Closure of the factorial f... |
faccld 14241 | Closure of the factorial f... |
facmapnn 14242 | The factorial function res... |
facne0 14243 | The factorial function is ... |
facdiv 14244 | A positive integer divides... |
facndiv 14245 | No positive integer (great... |
facwordi 14246 | Ordering property of facto... |
faclbnd 14247 | A lower bound for the fact... |
faclbnd2 14248 | A lower bound for the fact... |
faclbnd3 14249 | A lower bound for the fact... |
faclbnd4lem1 14250 | Lemma for ~ faclbnd4 . Pr... |
faclbnd4lem2 14251 | Lemma for ~ faclbnd4 . Us... |
faclbnd4lem3 14252 | Lemma for ~ faclbnd4 . Th... |
faclbnd4lem4 14253 | Lemma for ~ faclbnd4 . Pr... |
faclbnd4 14254 | Variant of ~ faclbnd5 prov... |
faclbnd5 14255 | The factorial function gro... |
faclbnd6 14256 | Geometric lower bound for ... |
facubnd 14257 | An upper bound for the fac... |
facavg 14258 | The product of two factori... |
bcval 14261 | Value of the binomial coef... |
bcval2 14262 | Value of the binomial coef... |
bcval3 14263 | Value of the binomial coef... |
bcval4 14264 | Value of the binomial coef... |
bcrpcl 14265 | Closure of the binomial co... |
bccmpl 14266 | "Complementing" its second... |
bcn0 14267 | ` N ` choose 0 is 1. Rema... |
bc0k 14268 | The binomial coefficient "... |
bcnn 14269 | ` N ` choose ` N ` is 1. ... |
bcn1 14270 | Binomial coefficient: ` N ... |
bcnp1n 14271 | Binomial coefficient: ` N ... |
bcm1k 14272 | The proportion of one bino... |
bcp1n 14273 | The proportion of one bino... |
bcp1nk 14274 | The proportion of one bino... |
bcval5 14275 | Write out the top and bott... |
bcn2 14276 | Binomial coefficient: ` N ... |
bcp1m1 14277 | Compute the binomial coeff... |
bcpasc 14278 | Pascal's rule for the bino... |
bccl 14279 | A binomial coefficient, in... |
bccl2 14280 | A binomial coefficient, in... |
bcn2m1 14281 | Compute the binomial coeff... |
bcn2p1 14282 | Compute the binomial coeff... |
permnn 14283 | The number of permutations... |
bcnm1 14284 | The binomial coefficent of... |
4bc3eq4 14285 | The value of four choose t... |
4bc2eq6 14286 | The value of four choose t... |
hashkf 14289 | The finite part of the siz... |
hashgval 14290 | The value of the ` # ` fun... |
hashginv 14291 | The converse of ` G ` maps... |
hashinf 14292 | The value of the ` # ` fun... |
hashbnd 14293 | If ` A ` has size bounded ... |
hashfxnn0 14294 | The size function is a fun... |
hashf 14295 | The size function maps all... |
hashxnn0 14296 | The value of the hash func... |
hashresfn 14297 | Restriction of the domain ... |
dmhashres 14298 | Restriction of the domain ... |
hashnn0pnf 14299 | The value of the hash func... |
hashnnn0genn0 14300 | If the size of a set is no... |
hashnemnf 14301 | The size of a set is never... |
hashv01gt1 14302 | The size of a set is eithe... |
hashfz1 14303 | The set ` ( 1 ... N ) ` ha... |
hashen 14304 | Two finite sets have the s... |
hasheni 14305 | Equinumerous sets have the... |
hasheqf1o 14306 | The size of two finite set... |
fiinfnf1o 14307 | There is no bijection betw... |
hasheqf1oi 14308 | The size of two sets is eq... |
hashf1rn 14309 | The size of a finite set w... |
hasheqf1od 14310 | The size of two sets is eq... |
fz1eqb 14311 | Two possibly-empty 1-based... |
hashcard 14312 | The size function of the c... |
hashcl 14313 | Closure of the ` # ` funct... |
hashxrcl 14314 | Extended real closure of t... |
hashclb 14315 | Reverse closure of the ` #... |
nfile 14316 | The size of any infinite s... |
hashvnfin 14317 | A set of finite size is a ... |
hashnfinnn0 14318 | The size of an infinite se... |
isfinite4 14319 | A finite set is equinumero... |
hasheq0 14320 | Two ways of saying a set i... |
hashneq0 14321 | Two ways of saying a set i... |
hashgt0n0 14322 | If the size of a set is gr... |
hashnncl 14323 | Positive natural closure o... |
hash0 14324 | The empty set has size zer... |
hashelne0d 14325 | A set with an element has ... |
hashsng 14326 | The size of a singleton. ... |
hashen1 14327 | A set has size 1 if and on... |
hash1elsn 14328 | A set of size 1 with a kno... |
hashrabrsn 14329 | The size of a restricted c... |
hashrabsn01 14330 | The size of a restricted c... |
hashrabsn1 14331 | If the size of a restricte... |
hashfn 14332 | A function is equinumerous... |
fseq1hash 14333 | The value of the size func... |
hashgadd 14334 | ` G ` maps ordinal additio... |
hashgval2 14335 | A short expression for the... |
hashdom 14336 | Dominance relation for the... |
hashdomi 14337 | Non-strict order relation ... |
hashsdom 14338 | Strict dominance relation ... |
hashun 14339 | The size of the union of d... |
hashun2 14340 | The size of the union of f... |
hashun3 14341 | The size of the union of f... |
hashinfxadd 14342 | The extended real addition... |
hashunx 14343 | The size of the union of d... |
hashge0 14344 | The cardinality of a set i... |
hashgt0 14345 | The cardinality of a nonem... |
hashge1 14346 | The cardinality of a nonem... |
1elfz0hash 14347 | 1 is an element of the fin... |
hashnn0n0nn 14348 | If a nonnegative integer i... |
hashunsng 14349 | The size of the union of a... |
hashunsngx 14350 | The size of the union of a... |
hashunsnggt 14351 | The size of a set is great... |
hashprg 14352 | The size of an unordered p... |
elprchashprn2 14353 | If one element of an unord... |
hashprb 14354 | The size of an unordered p... |
hashprdifel 14355 | The elements of an unorder... |
prhash2ex 14356 | There is (at least) one se... |
hashle00 14357 | If the size of a set is le... |
hashgt0elex 14358 | If the size of a set is gr... |
hashgt0elexb 14359 | The size of a set is great... |
hashp1i 14360 | Size of a finite ordinal. ... |
hash1 14361 | Size of a finite ordinal. ... |
hash2 14362 | Size of a finite ordinal. ... |
hash3 14363 | Size of a finite ordinal. ... |
hash4 14364 | Size of a finite ordinal. ... |
pr0hash2ex 14365 | There is (at least) one se... |
hashss 14366 | The size of a subset is le... |
prsshashgt1 14367 | The size of a superset of ... |
hashin 14368 | The size of the intersecti... |
hashssdif 14369 | The size of the difference... |
hashdif 14370 | The size of the difference... |
hashdifsn 14371 | The size of the difference... |
hashdifpr 14372 | The size of the difference... |
hashsn01 14373 | The size of a singleton is... |
hashsnle1 14374 | The size of a singleton is... |
hashsnlei 14375 | Get an upper bound on a co... |
hash1snb 14376 | The size of a set is 1 if ... |
euhash1 14377 | The size of a set is 1 in ... |
hash1n0 14378 | If the size of a set is 1 ... |
hashgt12el 14379 | In a set with more than on... |
hashgt12el2 14380 | In a set with more than on... |
hashgt23el 14381 | A set with more than two e... |
hashunlei 14382 | Get an upper bound on a co... |
hashsslei 14383 | Get an upper bound on a co... |
hashfz 14384 | Value of the numeric cardi... |
fzsdom2 14385 | Condition for finite range... |
hashfzo 14386 | Cardinality of a half-open... |
hashfzo0 14387 | Cardinality of a half-open... |
hashfzp1 14388 | Value of the numeric cardi... |
hashfz0 14389 | Value of the numeric cardi... |
hashxplem 14390 | Lemma for ~ hashxp . (Con... |
hashxp 14391 | The size of the Cartesian ... |
hashmap 14392 | The size of the set expone... |
hashpw 14393 | The size of the power set ... |
hashfun 14394 | A finite set is a function... |
hashres 14395 | The number of elements of ... |
hashreshashfun 14396 | The number of elements of ... |
hashimarn 14397 | The size of the image of a... |
hashimarni 14398 | If the size of the image o... |
hashfundm 14399 | The size of a set function... |
hashf1dmrn 14400 | The size of the domain of ... |
resunimafz0 14401 | TODO-AV: Revise using ` F... |
fnfz0hash 14402 | The size of a function on ... |
ffz0hash 14403 | The size of a function on ... |
fnfz0hashnn0 14404 | The size of a function on ... |
ffzo0hash 14405 | The size of a function on ... |
fnfzo0hash 14406 | The size of a function on ... |
fnfzo0hashnn0 14407 | The value of the size func... |
hashbclem 14408 | Lemma for ~ hashbc : induc... |
hashbc 14409 | The binomial coefficient c... |
hashfacen 14410 | The number of bijections b... |
hashfacenOLD 14411 | Obsolete version of ~ hash... |
hashf1lem1 14412 | Lemma for ~ hashf1 . (Con... |
hashf1lem1OLD 14413 | Obsolete version of ~ hash... |
hashf1lem2 14414 | Lemma for ~ hashf1 . (Con... |
hashf1 14415 | The permutation number ` |... |
hashfac 14416 | A factorial counts the num... |
leiso 14417 | Two ways to write a strict... |
leisorel 14418 | Version of ~ isorel for st... |
fz1isolem 14419 | Lemma for ~ fz1iso . (Con... |
fz1iso 14420 | Any finite ordered set has... |
ishashinf 14421 | Any set that is not finite... |
seqcoll 14422 | The function ` F ` contain... |
seqcoll2 14423 | The function ` F ` contain... |
phphashd 14424 | Corollary of the Pigeonhol... |
phphashrd 14425 | Corollary of the Pigeonhol... |
hashprlei 14426 | An unordered pair has at m... |
hash2pr 14427 | A set of size two is an un... |
hash2prde 14428 | A set of size two is an un... |
hash2exprb 14429 | A set of size two is an un... |
hash2prb 14430 | A set of size two is a pro... |
prprrab 14431 | The set of proper pairs of... |
nehash2 14432 | The cardinality of a set w... |
hash2prd 14433 | A set of size two is an un... |
hash2pwpr 14434 | If the size of a subset of... |
hashle2pr 14435 | A nonempty set of size les... |
hashle2prv 14436 | A nonempty subset of a pow... |
pr2pwpr 14437 | The set of subsets of a pa... |
hashge2el2dif 14438 | A set with size at least 2... |
hashge2el2difr 14439 | A set with at least 2 diff... |
hashge2el2difb 14440 | A set has size at least 2 ... |
hashdmpropge2 14441 | The size of the domain of ... |
hashtplei 14442 | An unordered triple has at... |
hashtpg 14443 | The size of an unordered t... |
hashge3el3dif 14444 | A set with size at least 3... |
elss2prb 14445 | An element of the set of s... |
hash2sspr 14446 | A subset of size two is an... |
exprelprel 14447 | If there is an element of ... |
hash3tr 14448 | A set of size three is an ... |
hash1to3 14449 | If the size of a set is be... |
fundmge2nop0 14450 | A function with a domain c... |
fundmge2nop 14451 | A function with a domain c... |
fun2dmnop0 14452 | A function with a domain c... |
fun2dmnop 14453 | A function with a domain c... |
hashdifsnp1 14454 | If the size of a set is a ... |
fi1uzind 14455 | Properties of an ordered p... |
brfi1uzind 14456 | Properties of a binary rel... |
brfi1ind 14457 | Properties of a binary rel... |
brfi1indALT 14458 | Alternate proof of ~ brfi1... |
opfi1uzind 14459 | Properties of an ordered p... |
opfi1ind 14460 | Properties of an ordered p... |
iswrd 14463 | Property of being a word o... |
wrdval 14464 | Value of the set of words ... |
iswrdi 14465 | A zero-based sequence is a... |
wrdf 14466 | A word is a zero-based seq... |
iswrdb 14467 | A word over an alphabet is... |
wrddm 14468 | The indices of a word (i.e... |
sswrd 14469 | The set of words respects ... |
snopiswrd 14470 | A singleton of an ordered ... |
wrdexg 14471 | The set of words over a se... |
wrdexb 14472 | The set of words over a se... |
wrdexi 14473 | The set of words over a se... |
wrdsymbcl 14474 | A symbol within a word ove... |
wrdfn 14475 | A word is a function with ... |
wrdv 14476 | A word over an alphabet is... |
wrdlndm 14477 | The length of a word is no... |
iswrdsymb 14478 | An arbitrary word is a wor... |
wrdfin 14479 | A word is a finite set. (... |
lencl 14480 | The length of a word is a ... |
lennncl 14481 | The length of a nonempty w... |
wrdffz 14482 | A word is a function from ... |
wrdeq 14483 | Equality theorem for the s... |
wrdeqi 14484 | Equality theorem for the s... |
iswrddm0 14485 | A function with empty doma... |
wrd0 14486 | The empty set is a word (t... |
0wrd0 14487 | The empty word is the only... |
ffz0iswrd 14488 | A sequence with zero-based... |
wrdsymb 14489 | A word is a word over the ... |
nfwrd 14490 | Hypothesis builder for ` W... |
csbwrdg 14491 | Class substitution for the... |
wrdnval 14492 | Words of a fixed length ar... |
wrdmap 14493 | Words as a mapping. (Cont... |
hashwrdn 14494 | If there is only a finite ... |
wrdnfi 14495 | If there is only a finite ... |
wrdsymb0 14496 | A symbol at a position "ou... |
wrdlenge1n0 14497 | A word with length at leas... |
len0nnbi 14498 | The length of a word is a ... |
wrdlenge2n0 14499 | A word with length at leas... |
wrdsymb1 14500 | The first symbol of a none... |
wrdlen1 14501 | A word of length 1 starts ... |
fstwrdne 14502 | The first symbol of a none... |
fstwrdne0 14503 | The first symbol of a none... |
eqwrd 14504 | Two words are equal iff th... |
elovmpowrd 14505 | Implications for the value... |
elovmptnn0wrd 14506 | Implications for the value... |
wrdred1 14507 | A word truncated by a symb... |
wrdred1hash 14508 | The length of a word trunc... |
lsw 14511 | Extract the last symbol of... |
lsw0 14512 | The last symbol of an empt... |
lsw0g 14513 | The last symbol of an empt... |
lsw1 14514 | The last symbol of a word ... |
lswcl 14515 | Closure of the last symbol... |
lswlgt0cl 14516 | The last symbol of a nonem... |
ccatfn 14519 | The concatenation operator... |
ccatfval 14520 | Value of the concatenation... |
ccatcl 14521 | The concatenation of two w... |
ccatlen 14522 | The length of a concatenat... |
ccat0 14523 | The concatenation of two w... |
ccatval1 14524 | Value of a symbol in the l... |
ccatval2 14525 | Value of a symbol in the r... |
ccatval3 14526 | Value of a symbol in the r... |
elfzelfzccat 14527 | An element of a finite set... |
ccatvalfn 14528 | The concatenation of two w... |
ccatsymb 14529 | The symbol at a given posi... |
ccatfv0 14530 | The first symbol of a conc... |
ccatval1lsw 14531 | The last symbol of the lef... |
ccatval21sw 14532 | The first symbol of the ri... |
ccatlid 14533 | Concatenation of a word by... |
ccatrid 14534 | Concatenation of a word by... |
ccatass 14535 | Associative law for concat... |
ccatrn 14536 | The range of a concatenate... |
ccatidid 14537 | Concatenation of the empty... |
lswccatn0lsw 14538 | The last symbol of a word ... |
lswccat0lsw 14539 | The last symbol of a word ... |
ccatalpha 14540 | A concatenation of two arb... |
ccatrcl1 14541 | Reverse closure of a conca... |
ids1 14544 | Identity function protecti... |
s1val 14545 | Value of a singleton word.... |
s1rn 14546 | The range of a singleton w... |
s1eq 14547 | Equality theorem for a sin... |
s1eqd 14548 | Equality theorem for a sin... |
s1cl 14549 | A singleton word is a word... |
s1cld 14550 | A singleton word is a word... |
s1prc 14551 | Value of a singleton word ... |
s1cli 14552 | A singleton word is a word... |
s1len 14553 | Length of a singleton word... |
s1nz 14554 | A singleton word is not th... |
s1dm 14555 | The domain of a singleton ... |
s1dmALT 14556 | Alternate version of ~ s1d... |
s1fv 14557 | Sole symbol of a singleton... |
lsws1 14558 | The last symbol of a singl... |
eqs1 14559 | A word of length 1 is a si... |
wrdl1exs1 14560 | A word of length 1 is a si... |
wrdl1s1 14561 | A word of length 1 is a si... |
s111 14562 | The singleton word functio... |
ccatws1cl 14563 | The concatenation of a wor... |
ccatws1clv 14564 | The concatenation of a wor... |
ccat2s1cl 14565 | The concatenation of two s... |
ccats1alpha 14566 | A concatenation of a word ... |
ccatws1len 14567 | The length of the concaten... |
ccatws1lenp1b 14568 | The length of a word is ` ... |
wrdlenccats1lenm1 14569 | The length of a word is th... |
ccat2s1len 14570 | The length of the concaten... |
ccatw2s1cl 14571 | The concatenation of a wor... |
ccatw2s1len 14572 | The length of the concaten... |
ccats1val1 14573 | Value of a symbol in the l... |
ccats1val2 14574 | Value of the symbol concat... |
ccat1st1st 14575 | The first symbol of a word... |
ccat2s1p1 14576 | Extract the first of two c... |
ccat2s1p2 14577 | Extract the second of two ... |
ccatw2s1ass 14578 | Associative law for a conc... |
ccatws1n0 14579 | The concatenation of a wor... |
ccatws1ls 14580 | The last symbol of the con... |
lswccats1 14581 | The last symbol of a word ... |
lswccats1fst 14582 | The last symbol of a nonem... |
ccatw2s1p1 14583 | Extract the symbol of the ... |
ccatw2s1p2 14584 | Extract the second of two ... |
ccat2s1fvw 14585 | Extract a symbol of a word... |
ccat2s1fst 14586 | The first symbol of the co... |
swrdnznd 14589 | The value of a subword ope... |
swrdval 14590 | Value of a subword. (Cont... |
swrd00 14591 | A zero length substring. ... |
swrdcl 14592 | Closure of the subword ext... |
swrdval2 14593 | Value of the subword extra... |
swrdlen 14594 | Length of an extracted sub... |
swrdfv 14595 | A symbol in an extracted s... |
swrdfv0 14596 | The first symbol in an ext... |
swrdf 14597 | A subword of a word is a f... |
swrdvalfn 14598 | Value of the subword extra... |
swrdrn 14599 | The range of a subword of ... |
swrdlend 14600 | The value of the subword e... |
swrdnd 14601 | The value of the subword e... |
swrdnd2 14602 | Value of the subword extra... |
swrdnnn0nd 14603 | The value of a subword ope... |
swrdnd0 14604 | The value of a subword ope... |
swrd0 14605 | A subword of an empty set ... |
swrdrlen 14606 | Length of a right-anchored... |
swrdlen2 14607 | Length of an extracted sub... |
swrdfv2 14608 | A symbol in an extracted s... |
swrdwrdsymb 14609 | A subword is a word over t... |
swrdsb0eq 14610 | Two subwords with the same... |
swrdsbslen 14611 | Two subwords with the same... |
swrdspsleq 14612 | Two words have a common su... |
swrds1 14613 | Extract a single symbol fr... |
swrdlsw 14614 | Extract the last single sy... |
ccatswrd 14615 | Joining two adjacent subwo... |
swrdccat2 14616 | Recover the right half of ... |
pfxnndmnd 14619 | The value of a prefix oper... |
pfxval 14620 | Value of a prefix operatio... |
pfx00 14621 | The zero length prefix is ... |
pfx0 14622 | A prefix of an empty set i... |
pfxval0 14623 | Value of a prefix operatio... |
pfxcl 14624 | Closure of the prefix extr... |
pfxmpt 14625 | Value of the prefix extrac... |
pfxres 14626 | Value of the subword extra... |
pfxf 14627 | A prefix of a word is a fu... |
pfxfn 14628 | Value of the prefix extrac... |
pfxfv 14629 | A symbol in a prefix of a ... |
pfxlen 14630 | Length of a prefix. (Cont... |
pfxid 14631 | A word is a prefix of itse... |
pfxrn 14632 | The range of a prefix of a... |
pfxn0 14633 | A prefix consisting of at ... |
pfxnd 14634 | The value of a prefix oper... |
pfxnd0 14635 | The value of a prefix oper... |
pfxwrdsymb 14636 | A prefix of a word is a wo... |
addlenrevpfx 14637 | The sum of the lengths of ... |
addlenpfx 14638 | The sum of the lengths of ... |
pfxfv0 14639 | The first symbol of a pref... |
pfxtrcfv 14640 | A symbol in a word truncat... |
pfxtrcfv0 14641 | The first symbol in a word... |
pfxfvlsw 14642 | The last symbol in a nonem... |
pfxeq 14643 | The prefixes of two words ... |
pfxtrcfvl 14644 | The last symbol in a word ... |
pfxsuffeqwrdeq 14645 | Two words are equal if and... |
pfxsuff1eqwrdeq 14646 | Two (nonempty) words are e... |
disjwrdpfx 14647 | Sets of words are disjoint... |
ccatpfx 14648 | Concatenating a prefix wit... |
pfxccat1 14649 | Recover the left half of a... |
pfx1 14650 | The prefix of length one o... |
swrdswrdlem 14651 | Lemma for ~ swrdswrd . (C... |
swrdswrd 14652 | A subword of a subword is ... |
pfxswrd 14653 | A prefix of a subword is a... |
swrdpfx 14654 | A subword of a prefix is a... |
pfxpfx 14655 | A prefix of a prefix is a ... |
pfxpfxid 14656 | A prefix of a prefix with ... |
pfxcctswrd 14657 | The concatenation of the p... |
lenpfxcctswrd 14658 | The length of the concaten... |
lenrevpfxcctswrd 14659 | The length of the concaten... |
pfxlswccat 14660 | Reconstruct a nonempty wor... |
ccats1pfxeq 14661 | The last symbol of a word ... |
ccats1pfxeqrex 14662 | There exists a symbol such... |
ccatopth 14663 | An ~ opth -like theorem fo... |
ccatopth2 14664 | An ~ opth -like theorem fo... |
ccatlcan 14665 | Concatenation of words is ... |
ccatrcan 14666 | Concatenation of words is ... |
wrdeqs1cat 14667 | Decompose a nonempty word ... |
cats1un 14668 | Express a word with an ext... |
wrdind 14669 | Perform induction over the... |
wrd2ind 14670 | Perform induction over the... |
swrdccatfn 14671 | The subword of a concatena... |
swrdccatin1 14672 | The subword of a concatena... |
pfxccatin12lem4 14673 | Lemma 4 for ~ pfxccatin12 ... |
pfxccatin12lem2a 14674 | Lemma for ~ pfxccatin12lem... |
pfxccatin12lem1 14675 | Lemma 1 for ~ pfxccatin12 ... |
swrdccatin2 14676 | The subword of a concatena... |
pfxccatin12lem2c 14677 | Lemma for ~ pfxccatin12lem... |
pfxccatin12lem2 14678 | Lemma 2 for ~ pfxccatin12 ... |
pfxccatin12lem3 14679 | Lemma 3 for ~ pfxccatin12 ... |
pfxccatin12 14680 | The subword of a concatena... |
pfxccat3 14681 | The subword of a concatena... |
swrdccat 14682 | The subword of a concatena... |
pfxccatpfx1 14683 | A prefix of a concatenatio... |
pfxccatpfx2 14684 | A prefix of a concatenatio... |
pfxccat3a 14685 | A prefix of a concatenatio... |
swrdccat3blem 14686 | Lemma for ~ swrdccat3b . ... |
swrdccat3b 14687 | A suffix of a concatenatio... |
pfxccatid 14688 | A prefix of a concatenatio... |
ccats1pfxeqbi 14689 | A word is a prefix of a wo... |
swrdccatin1d 14690 | The subword of a concatena... |
swrdccatin2d 14691 | The subword of a concatena... |
pfxccatin12d 14692 | The subword of a concatena... |
reuccatpfxs1lem 14693 | Lemma for ~ reuccatpfxs1 .... |
reuccatpfxs1 14694 | There is a unique word hav... |
reuccatpfxs1v 14695 | There is a unique word hav... |
splval 14698 | Value of the substring rep... |
splcl 14699 | Closure of the substring r... |
splid 14700 | Splicing a subword for the... |
spllen 14701 | The length of a splice. (... |
splfv1 14702 | Symbols to the left of a s... |
splfv2a 14703 | Symbols within the replace... |
splval2 14704 | Value of a splice, assumin... |
revval 14707 | Value of the word reversin... |
revcl 14708 | The reverse of a word is a... |
revlen 14709 | The reverse of a word has ... |
revfv 14710 | Reverse of a word at a poi... |
rev0 14711 | The empty word is its own ... |
revs1 14712 | Singleton words are their ... |
revccat 14713 | Antiautomorphic property o... |
revrev 14714 | Reversal is an involution ... |
reps 14717 | Construct a function mappi... |
repsundef 14718 | A function mapping a half-... |
repsconst 14719 | Construct a function mappi... |
repsf 14720 | The constructed function m... |
repswsymb 14721 | The symbols of a "repeated... |
repsw 14722 | A function mapping a half-... |
repswlen 14723 | The length of a "repeated ... |
repsw0 14724 | The "repeated symbol word"... |
repsdf2 14725 | Alternative definition of ... |
repswsymball 14726 | All the symbols of a "repe... |
repswsymballbi 14727 | A word is a "repeated symb... |
repswfsts 14728 | The first symbol of a none... |
repswlsw 14729 | The last symbol of a nonem... |
repsw1 14730 | The "repeated symbol word"... |
repswswrd 14731 | A subword of a "repeated s... |
repswpfx 14732 | A prefix of a repeated sym... |
repswccat 14733 | The concatenation of two "... |
repswrevw 14734 | The reverse of a "repeated... |
cshfn 14737 | Perform a cyclical shift f... |
cshword 14738 | Perform a cyclical shift f... |
cshnz 14739 | A cyclical shift is the em... |
0csh0 14740 | Cyclically shifting an emp... |
cshw0 14741 | A word cyclically shifted ... |
cshwmodn 14742 | Cyclically shifting a word... |
cshwsublen 14743 | Cyclically shifting a word... |
cshwn 14744 | A word cyclically shifted ... |
cshwcl 14745 | A cyclically shifted word ... |
cshwlen 14746 | The length of a cyclically... |
cshwf 14747 | A cyclically shifted word ... |
cshwfn 14748 | A cyclically shifted word ... |
cshwrn 14749 | The range of a cyclically ... |
cshwidxmod 14750 | The symbol at a given inde... |
cshwidxmodr 14751 | The symbol at a given inde... |
cshwidx0mod 14752 | The symbol at index 0 of a... |
cshwidx0 14753 | The symbol at index 0 of a... |
cshwidxm1 14754 | The symbol at index ((n-N)... |
cshwidxm 14755 | The symbol at index (n-N) ... |
cshwidxn 14756 | The symbol at index (n-1) ... |
cshf1 14757 | Cyclically shifting a word... |
cshinj 14758 | If a word is injectiv (reg... |
repswcshw 14759 | A cyclically shifted "repe... |
2cshw 14760 | Cyclically shifting a word... |
2cshwid 14761 | Cyclically shifting a word... |
lswcshw 14762 | The last symbol of a word ... |
2cshwcom 14763 | Cyclically shifting a word... |
cshwleneq 14764 | If the results of cyclical... |
3cshw 14765 | Cyclically shifting a word... |
cshweqdif2 14766 | If cyclically shifting two... |
cshweqdifid 14767 | If cyclically shifting a w... |
cshweqrep 14768 | If cyclically shifting a w... |
cshw1 14769 | If cyclically shifting a w... |
cshw1repsw 14770 | If cyclically shifting a w... |
cshwsexa 14771 | The class of (different!) ... |
cshwsexaOLD 14772 | Obsolete version of ~ cshw... |
2cshwcshw 14773 | If a word is a cyclically ... |
scshwfzeqfzo 14774 | For a nonempty word the se... |
cshwcshid 14775 | A cyclically shifted word ... |
cshwcsh2id 14776 | A cyclically shifted word ... |
cshimadifsn 14777 | The image of a cyclically ... |
cshimadifsn0 14778 | The image of a cyclically ... |
wrdco 14779 | Mapping a word by a functi... |
lenco 14780 | Length of a mapped word is... |
s1co 14781 | Mapping of a singleton wor... |
revco 14782 | Mapping of words (i.e., a ... |
ccatco 14783 | Mapping of words commutes ... |
cshco 14784 | Mapping of words commutes ... |
swrdco 14785 | Mapping of words commutes ... |
pfxco 14786 | Mapping of words commutes ... |
lswco 14787 | Mapping of (nonempty) word... |
repsco 14788 | Mapping of words commutes ... |
cats1cld 14803 | Closure of concatenation w... |
cats1co 14804 | Closure of concatenation w... |
cats1cli 14805 | Closure of concatenation w... |
cats1fvn 14806 | The last symbol of a conca... |
cats1fv 14807 | A symbol other than the la... |
cats1len 14808 | The length of concatenatio... |
cats1cat 14809 | Closure of concatenation w... |
cats2cat 14810 | Closure of concatenation o... |
s2eqd 14811 | Equality theorem for a dou... |
s3eqd 14812 | Equality theorem for a len... |
s4eqd 14813 | Equality theorem for a len... |
s5eqd 14814 | Equality theorem for a len... |
s6eqd 14815 | Equality theorem for a len... |
s7eqd 14816 | Equality theorem for a len... |
s8eqd 14817 | Equality theorem for a len... |
s3eq2 14818 | Equality theorem for a len... |
s2cld 14819 | A doubleton word is a word... |
s3cld 14820 | A length 3 string is a wor... |
s4cld 14821 | A length 4 string is a wor... |
s5cld 14822 | A length 5 string is a wor... |
s6cld 14823 | A length 6 string is a wor... |
s7cld 14824 | A length 7 string is a wor... |
s8cld 14825 | A length 7 string is a wor... |
s2cl 14826 | A doubleton word is a word... |
s3cl 14827 | A length 3 string is a wor... |
s2cli 14828 | A doubleton word is a word... |
s3cli 14829 | A length 3 string is a wor... |
s4cli 14830 | A length 4 string is a wor... |
s5cli 14831 | A length 5 string is a wor... |
s6cli 14832 | A length 6 string is a wor... |
s7cli 14833 | A length 7 string is a wor... |
s8cli 14834 | A length 8 string is a wor... |
s2fv0 14835 | Extract the first symbol f... |
s2fv1 14836 | Extract the second symbol ... |
s2len 14837 | The length of a doubleton ... |
s2dm 14838 | The domain of a doubleton ... |
s3fv0 14839 | Extract the first symbol f... |
s3fv1 14840 | Extract the second symbol ... |
s3fv2 14841 | Extract the third symbol f... |
s3len 14842 | The length of a length 3 s... |
s4fv0 14843 | Extract the first symbol f... |
s4fv1 14844 | Extract the second symbol ... |
s4fv2 14845 | Extract the third symbol f... |
s4fv3 14846 | Extract the fourth symbol ... |
s4len 14847 | The length of a length 4 s... |
s5len 14848 | The length of a length 5 s... |
s6len 14849 | The length of a length 6 s... |
s7len 14850 | The length of a length 7 s... |
s8len 14851 | The length of a length 8 s... |
lsws2 14852 | The last symbol of a doubl... |
lsws3 14853 | The last symbol of a 3 let... |
lsws4 14854 | The last symbol of a 4 let... |
s2prop 14855 | A length 2 word is an unor... |
s2dmALT 14856 | Alternate version of ~ s2d... |
s3tpop 14857 | A length 3 word is an unor... |
s4prop 14858 | A length 4 word is a union... |
s3fn 14859 | A length 3 word is a funct... |
funcnvs1 14860 | The converse of a singleto... |
funcnvs2 14861 | The converse of a length 2... |
funcnvs3 14862 | The converse of a length 3... |
funcnvs4 14863 | The converse of a length 4... |
s2f1o 14864 | A length 2 word with mutua... |
f1oun2prg 14865 | A union of unordered pairs... |
s4f1o 14866 | A length 4 word with mutua... |
s4dom 14867 | The domain of a length 4 w... |
s2co 14868 | Mapping a doubleton word b... |
s3co 14869 | Mapping a length 3 string ... |
s0s1 14870 | Concatenation of fixed len... |
s1s2 14871 | Concatenation of fixed len... |
s1s3 14872 | Concatenation of fixed len... |
s1s4 14873 | Concatenation of fixed len... |
s1s5 14874 | Concatenation of fixed len... |
s1s6 14875 | Concatenation of fixed len... |
s1s7 14876 | Concatenation of fixed len... |
s2s2 14877 | Concatenation of fixed len... |
s4s2 14878 | Concatenation of fixed len... |
s4s3 14879 | Concatenation of fixed len... |
s4s4 14880 | Concatenation of fixed len... |
s3s4 14881 | Concatenation of fixed len... |
s2s5 14882 | Concatenation of fixed len... |
s5s2 14883 | Concatenation of fixed len... |
s2eq2s1eq 14884 | Two length 2 words are equ... |
s2eq2seq 14885 | Two length 2 words are equ... |
s3eqs2s1eq 14886 | Two length 3 words are equ... |
s3eq3seq 14887 | Two length 3 words are equ... |
swrds2 14888 | Extract two adjacent symbo... |
swrds2m 14889 | Extract two adjacent symbo... |
wrdlen2i 14890 | Implications of a word of ... |
wrd2pr2op 14891 | A word of length two repre... |
wrdlen2 14892 | A word of length two. (Co... |
wrdlen2s2 14893 | A word of length two as do... |
wrdl2exs2 14894 | A word of length two is a ... |
pfx2 14895 | A prefix of length two. (... |
wrd3tpop 14896 | A word of length three rep... |
wrdlen3s3 14897 | A word of length three as ... |
repsw2 14898 | The "repeated symbol word"... |
repsw3 14899 | The "repeated symbol word"... |
swrd2lsw 14900 | Extract the last two symbo... |
2swrd2eqwrdeq 14901 | Two words of length at lea... |
ccatw2s1ccatws2 14902 | The concatenation of a wor... |
ccat2s1fvwALT 14903 | Alternate proof of ~ ccat2... |
wwlktovf 14904 | Lemma 1 for ~ wrd2f1tovbij... |
wwlktovf1 14905 | Lemma 2 for ~ wrd2f1tovbij... |
wwlktovfo 14906 | Lemma 3 for ~ wrd2f1tovbij... |
wwlktovf1o 14907 | Lemma 4 for ~ wrd2f1tovbij... |
wrd2f1tovbij 14908 | There is a bijection betwe... |
eqwrds3 14909 | A word is equal with a len... |
wrdl3s3 14910 | A word of length 3 is a le... |
s3sndisj 14911 | The singletons consisting ... |
s3iunsndisj 14912 | The union of singletons co... |
ofccat 14913 | Letterwise operations on w... |
ofs1 14914 | Letterwise operations on a... |
ofs2 14915 | Letterwise operations on a... |
coss12d 14916 | Subset deduction for compo... |
trrelssd 14917 | The composition of subclas... |
xpcogend 14918 | The most interesting case ... |
xpcoidgend 14919 | If two classes are not dis... |
cotr2g 14920 | Two ways of saying that th... |
cotr2 14921 | Two ways of saying a relat... |
cotr3 14922 | Two ways of saying a relat... |
coemptyd 14923 | Deduction about compositio... |
xptrrel 14924 | The cross product is alway... |
0trrel 14925 | The empty class is a trans... |
cleq1lem 14926 | Equality implies bijection... |
cleq1 14927 | Equality of relations impl... |
clsslem 14928 | The closure of a subclass ... |
trcleq1 14933 | Equality of relations impl... |
trclsslem 14934 | The transitive closure (as... |
trcleq2lem 14935 | Equality implies bijection... |
cvbtrcl 14936 | Change of bound variable i... |
trcleq12lem 14937 | Equality implies bijection... |
trclexlem 14938 | Existence of relation impl... |
trclublem 14939 | If a relation exists then ... |
trclubi 14940 | The Cartesian product of t... |
trclubgi 14941 | The union with the Cartesi... |
trclub 14942 | The Cartesian product of t... |
trclubg 14943 | The union with the Cartesi... |
trclfv 14944 | The transitive closure of ... |
brintclab 14945 | Two ways to express a bina... |
brtrclfv 14946 | Two ways of expressing the... |
brcnvtrclfv 14947 | Two ways of expressing the... |
brtrclfvcnv 14948 | Two ways of expressing the... |
brcnvtrclfvcnv 14949 | Two ways of expressing the... |
trclfvss 14950 | The transitive closure (as... |
trclfvub 14951 | The transitive closure of ... |
trclfvlb 14952 | The transitive closure of ... |
trclfvcotr 14953 | The transitive closure of ... |
trclfvlb2 14954 | The transitive closure of ... |
trclfvlb3 14955 | The transitive closure of ... |
cotrtrclfv 14956 | The transitive closure of ... |
trclidm 14957 | The transitive closure of ... |
trclun 14958 | Transitive closure of a un... |
trclfvg 14959 | The value of the transitiv... |
trclfvcotrg 14960 | The value of the transitiv... |
reltrclfv 14961 | The transitive closure of ... |
dmtrclfv 14962 | The domain of the transiti... |
reldmrelexp 14965 | The domain of the repeated... |
relexp0g 14966 | A relation composed zero t... |
relexp0 14967 | A relation composed zero t... |
relexp0d 14968 | A relation composed zero t... |
relexpsucnnr 14969 | A reduction for relation e... |
relexp1g 14970 | A relation composed once i... |
dfid5 14971 | Identity relation is equal... |
dfid6 14972 | Identity relation expresse... |
relexp1d 14973 | A relation composed once i... |
relexpsucnnl 14974 | A reduction for relation e... |
relexpsucl 14975 | A reduction for relation e... |
relexpsucr 14976 | A reduction for relation e... |
relexpsucrd 14977 | A reduction for relation e... |
relexpsucld 14978 | A reduction for relation e... |
relexpcnv 14979 | Commutation of converse an... |
relexpcnvd 14980 | Commutation of converse an... |
relexp0rel 14981 | The exponentiation of a cl... |
relexprelg 14982 | The exponentiation of a cl... |
relexprel 14983 | The exponentiation of a re... |
relexpreld 14984 | The exponentiation of a re... |
relexpnndm 14985 | The domain of an exponenti... |
relexpdmg 14986 | The domain of an exponenti... |
relexpdm 14987 | The domain of an exponenti... |
relexpdmd 14988 | The domain of an exponenti... |
relexpnnrn 14989 | The range of an exponentia... |
relexprng 14990 | The range of an exponentia... |
relexprn 14991 | The range of an exponentia... |
relexprnd 14992 | The range of an exponentia... |
relexpfld 14993 | The field of an exponentia... |
relexpfldd 14994 | The field of an exponentia... |
relexpaddnn 14995 | Relation composition becom... |
relexpuzrel 14996 | The exponentiation of a cl... |
relexpaddg 14997 | Relation composition becom... |
relexpaddd 14998 | Relation composition becom... |
rtrclreclem1 15001 | The reflexive, transitive ... |
dfrtrclrec2 15002 | If two elements are connec... |
rtrclreclem2 15003 | The reflexive, transitive ... |
rtrclreclem3 15004 | The reflexive, transitive ... |
rtrclreclem4 15005 | The reflexive, transitive ... |
dfrtrcl2 15006 | The two definitions ` t* `... |
relexpindlem 15007 | Principle of transitive in... |
relexpind 15008 | Principle of transitive in... |
rtrclind 15009 | Principle of transitive in... |
shftlem 15012 | Two ways to write a shifte... |
shftuz 15013 | A shift of the upper integ... |
shftfval 15014 | The value of the sequence ... |
shftdm 15015 | Domain of a relation shift... |
shftfib 15016 | Value of a fiber of the re... |
shftfn 15017 | Functionality and domain o... |
shftval 15018 | Value of a sequence shifte... |
shftval2 15019 | Value of a sequence shifte... |
shftval3 15020 | Value of a sequence shifte... |
shftval4 15021 | Value of a sequence shifte... |
shftval5 15022 | Value of a shifted sequenc... |
shftf 15023 | Functionality of a shifted... |
2shfti 15024 | Composite shift operations... |
shftidt2 15025 | Identity law for the shift... |
shftidt 15026 | Identity law for the shift... |
shftcan1 15027 | Cancellation law for the s... |
shftcan2 15028 | Cancellation law for the s... |
seqshft 15029 | Shifting the index set of ... |
sgnval 15032 | Value of the signum functi... |
sgn0 15033 | The signum of 0 is 0. (Co... |
sgnp 15034 | The signum of a positive e... |
sgnrrp 15035 | The signum of a positive r... |
sgn1 15036 | The signum of 1 is 1. (Co... |
sgnpnf 15037 | The signum of ` +oo ` is 1... |
sgnn 15038 | The signum of a negative e... |
sgnmnf 15039 | The signum of ` -oo ` is -... |
cjval 15046 | The value of the conjugate... |
cjth 15047 | The defining property of t... |
cjf 15048 | Domain and codomain of the... |
cjcl 15049 | The conjugate of a complex... |
reval 15050 | The value of the real part... |
imval 15051 | The value of the imaginary... |
imre 15052 | The imaginary part of a co... |
reim 15053 | The real part of a complex... |
recl 15054 | The real part of a complex... |
imcl 15055 | The imaginary part of a co... |
ref 15056 | Domain and codomain of the... |
imf 15057 | Domain and codomain of the... |
crre 15058 | The real part of a complex... |
crim 15059 | The real part of a complex... |
replim 15060 | Reconstruct a complex numb... |
remim 15061 | Value of the conjugate of ... |
reim0 15062 | The imaginary part of a re... |
reim0b 15063 | A number is real iff its i... |
rereb 15064 | A number is real iff it eq... |
mulre 15065 | A product with a nonzero r... |
rere 15066 | A real number equals its r... |
cjreb 15067 | A number is real iff it eq... |
recj 15068 | Real part of a complex con... |
reneg 15069 | Real part of negative. (C... |
readd 15070 | Real part distributes over... |
resub 15071 | Real part distributes over... |
remullem 15072 | Lemma for ~ remul , ~ immu... |
remul 15073 | Real part of a product. (... |
remul2 15074 | Real part of a product. (... |
rediv 15075 | Real part of a division. ... |
imcj 15076 | Imaginary part of a comple... |
imneg 15077 | The imaginary part of a ne... |
imadd 15078 | Imaginary part distributes... |
imsub 15079 | Imaginary part distributes... |
immul 15080 | Imaginary part of a produc... |
immul2 15081 | Imaginary part of a produc... |
imdiv 15082 | Imaginary part of a divisi... |
cjre 15083 | A real number equals its c... |
cjcj 15084 | The conjugate of the conju... |
cjadd 15085 | Complex conjugate distribu... |
cjmul 15086 | Complex conjugate distribu... |
ipcnval 15087 | Standard inner product on ... |
cjmulrcl 15088 | A complex number times its... |
cjmulval 15089 | A complex number times its... |
cjmulge0 15090 | A complex number times its... |
cjneg 15091 | Complex conjugate of negat... |
addcj 15092 | A number plus its conjugat... |
cjsub 15093 | Complex conjugate distribu... |
cjexp 15094 | Complex conjugate of posit... |
imval2 15095 | The imaginary part of a nu... |
re0 15096 | The real part of zero. (C... |
im0 15097 | The imaginary part of zero... |
re1 15098 | The real part of one. (Co... |
im1 15099 | The imaginary part of one.... |
rei 15100 | The real part of ` _i ` . ... |
imi 15101 | The imaginary part of ` _i... |
cj0 15102 | The conjugate of zero. (C... |
cji 15103 | The complex conjugate of t... |
cjreim 15104 | The conjugate of a represe... |
cjreim2 15105 | The conjugate of the repre... |
cj11 15106 | Complex conjugate is a one... |
cjne0 15107 | A number is nonzero iff it... |
cjdiv 15108 | Complex conjugate distribu... |
cnrecnv 15109 | The inverse to the canonic... |
sqeqd 15110 | A deduction for showing tw... |
recli 15111 | The real part of a complex... |
imcli 15112 | The imaginary part of a co... |
cjcli 15113 | Closure law for complex co... |
replimi 15114 | Construct a complex number... |
cjcji 15115 | The conjugate of the conju... |
reim0bi 15116 | A number is real iff its i... |
rerebi 15117 | A real number equals its r... |
cjrebi 15118 | A number is real iff it eq... |
recji 15119 | Real part of a complex con... |
imcji 15120 | Imaginary part of a comple... |
cjmulrcli 15121 | A complex number times its... |
cjmulvali 15122 | A complex number times its... |
cjmulge0i 15123 | A complex number times its... |
renegi 15124 | Real part of negative. (C... |
imnegi 15125 | Imaginary part of negative... |
cjnegi 15126 | Complex conjugate of negat... |
addcji 15127 | A number plus its conjugat... |
readdi 15128 | Real part distributes over... |
imaddi 15129 | Imaginary part distributes... |
remuli 15130 | Real part of a product. (... |
immuli 15131 | Imaginary part of a produc... |
cjaddi 15132 | Complex conjugate distribu... |
cjmuli 15133 | Complex conjugate distribu... |
ipcni 15134 | Standard inner product on ... |
cjdivi 15135 | Complex conjugate distribu... |
crrei 15136 | The real part of a complex... |
crimi 15137 | The imaginary part of a co... |
recld 15138 | The real part of a complex... |
imcld 15139 | The imaginary part of a co... |
cjcld 15140 | Closure law for complex co... |
replimd 15141 | Construct a complex number... |
remimd 15142 | Value of the conjugate of ... |
cjcjd 15143 | The conjugate of the conju... |
reim0bd 15144 | A number is real iff its i... |
rerebd 15145 | A real number equals its r... |
cjrebd 15146 | A number is real iff it eq... |
cjne0d 15147 | A number is nonzero iff it... |
recjd 15148 | Real part of a complex con... |
imcjd 15149 | Imaginary part of a comple... |
cjmulrcld 15150 | A complex number times its... |
cjmulvald 15151 | A complex number times its... |
cjmulge0d 15152 | A complex number times its... |
renegd 15153 | Real part of negative. (C... |
imnegd 15154 | Imaginary part of negative... |
cjnegd 15155 | Complex conjugate of negat... |
addcjd 15156 | A number plus its conjugat... |
cjexpd 15157 | Complex conjugate of posit... |
readdd 15158 | Real part distributes over... |
imaddd 15159 | Imaginary part distributes... |
resubd 15160 | Real part distributes over... |
imsubd 15161 | Imaginary part distributes... |
remuld 15162 | Real part of a product. (... |
immuld 15163 | Imaginary part of a produc... |
cjaddd 15164 | Complex conjugate distribu... |
cjmuld 15165 | Complex conjugate distribu... |
ipcnd 15166 | Standard inner product on ... |
cjdivd 15167 | Complex conjugate distribu... |
rered 15168 | A real number equals its r... |
reim0d 15169 | The imaginary part of a re... |
cjred 15170 | A real number equals its c... |
remul2d 15171 | Real part of a product. (... |
immul2d 15172 | Imaginary part of a produc... |
redivd 15173 | Real part of a division. ... |
imdivd 15174 | Imaginary part of a divisi... |
crred 15175 | The real part of a complex... |
crimd 15176 | The imaginary part of a co... |
sqrtval 15181 | Value of square root funct... |
absval 15182 | The absolute value (modulu... |
rennim 15183 | A real number does not lie... |
cnpart 15184 | The specification of restr... |
sqrt0 15185 | The square root of zero is... |
01sqrexlem1 15186 | Lemma for ~ 01sqrex . (Co... |
01sqrexlem2 15187 | Lemma for ~ 01sqrex . (Co... |
01sqrexlem3 15188 | Lemma for ~ 01sqrex . (Co... |
01sqrexlem4 15189 | Lemma for ~ 01sqrex . (Co... |
01sqrexlem5 15190 | Lemma for ~ 01sqrex . (Co... |
01sqrexlem6 15191 | Lemma for ~ 01sqrex . (Co... |
01sqrexlem7 15192 | Lemma for ~ 01sqrex . (Co... |
01sqrex 15193 | Existence of a square root... |
resqrex 15194 | Existence of a square root... |
sqrmo 15195 | Uniqueness for the square ... |
resqreu 15196 | Existence and uniqueness f... |
resqrtcl 15197 | Closure of the square root... |
resqrtthlem 15198 | Lemma for ~ resqrtth . (C... |
resqrtth 15199 | Square root theorem over t... |
remsqsqrt 15200 | Square of square root. (C... |
sqrtge0 15201 | The square root function i... |
sqrtgt0 15202 | The square root function i... |
sqrtmul 15203 | Square root distributes ov... |
sqrtle 15204 | Square root is monotonic. ... |
sqrtlt 15205 | Square root is strictly mo... |
sqrt11 15206 | The square root function i... |
sqrt00 15207 | A square root is zero iff ... |
rpsqrtcl 15208 | The square root of a posit... |
sqrtdiv 15209 | Square root distributes ov... |
sqrtneglem 15210 | The square root of a negat... |
sqrtneg 15211 | The square root of a negat... |
sqrtsq2 15212 | Relationship between squar... |
sqrtsq 15213 | Square root of square. (C... |
sqrtmsq 15214 | Square root of square. (C... |
sqrt1 15215 | The square root of 1 is 1.... |
sqrt4 15216 | The square root of 4 is 2.... |
sqrt9 15217 | The square root of 9 is 3.... |
sqrt2gt1lt2 15218 | The square root of 2 is bo... |
sqrtm1 15219 | The imaginary unit is the ... |
nn0sqeq1 15220 | A natural number with squa... |
absneg 15221 | Absolute value of the nega... |
abscl 15222 | Real closure of absolute v... |
abscj 15223 | The absolute value of a nu... |
absvalsq 15224 | Square of value of absolut... |
absvalsq2 15225 | Square of value of absolut... |
sqabsadd 15226 | Square of absolute value o... |
sqabssub 15227 | Square of absolute value o... |
absval2 15228 | Value of absolute value fu... |
abs0 15229 | The absolute value of 0. ... |
absi 15230 | The absolute value of the ... |
absge0 15231 | Absolute value is nonnegat... |
absrpcl 15232 | The absolute value of a no... |
abs00 15233 | The absolute value of a nu... |
abs00ad 15234 | A complex number is zero i... |
abs00bd 15235 | If a complex number is zer... |
absreimsq 15236 | Square of the absolute val... |
absreim 15237 | Absolute value of a number... |
absmul 15238 | Absolute value distributes... |
absdiv 15239 | Absolute value distributes... |
absid 15240 | A nonnegative number is it... |
abs1 15241 | The absolute value of one ... |
absnid 15242 | For a negative number, its... |
leabs 15243 | A real number is less than... |
absor 15244 | The absolute value of a re... |
absre 15245 | Absolute value of a real n... |
absresq 15246 | Square of the absolute val... |
absmod0 15247 | ` A ` is divisible by ` B ... |
absexp 15248 | Absolute value of positive... |
absexpz 15249 | Absolute value of integer ... |
abssq 15250 | Square can be moved in and... |
sqabs 15251 | The squares of two reals a... |
absrele 15252 | The absolute value of a co... |
absimle 15253 | The absolute value of a co... |
max0add 15254 | The sum of the positive an... |
absz 15255 | A real number is an intege... |
nn0abscl 15256 | The absolute value of an i... |
zabscl 15257 | The absolute value of an i... |
abslt 15258 | Absolute value and 'less t... |
absle 15259 | Absolute value and 'less t... |
abssubne0 15260 | If the absolute value of a... |
absdiflt 15261 | The absolute value of a di... |
absdifle 15262 | The absolute value of a di... |
elicc4abs 15263 | Membership in a symmetric ... |
lenegsq 15264 | Comparison to a nonnegativ... |
releabs 15265 | The real part of a number ... |
recval 15266 | Reciprocal expressed with ... |
absidm 15267 | The absolute value functio... |
absgt0 15268 | The absolute value of a no... |
nnabscl 15269 | The absolute value of a no... |
abssub 15270 | Swapping order of subtract... |
abssubge0 15271 | Absolute value of a nonneg... |
abssuble0 15272 | Absolute value of a nonpos... |
absmax 15273 | The maximum of two numbers... |
abstri 15274 | Triangle inequality for ab... |
abs3dif 15275 | Absolute value of differen... |
abs2dif 15276 | Difference of absolute val... |
abs2dif2 15277 | Difference of absolute val... |
abs2difabs 15278 | Absolute value of differen... |
abs1m 15279 | For any complex number, th... |
recan 15280 | Cancellation law involving... |
absf 15281 | Mapping domain and codomai... |
abs3lem 15282 | Lemma involving absolute v... |
abslem2 15283 | Lemma involving absolute v... |
rddif 15284 | The difference between a r... |
absrdbnd 15285 | Bound on the absolute valu... |
fzomaxdiflem 15286 | Lemma for ~ fzomaxdif . (... |
fzomaxdif 15287 | A bound on the separation ... |
uzin2 15288 | The upper integers are clo... |
rexanuz 15289 | Combine two different uppe... |
rexanre 15290 | Combine two different uppe... |
rexfiuz 15291 | Combine finitely many diff... |
rexuz3 15292 | Restrict the base of the u... |
rexanuz2 15293 | Combine two different uppe... |
r19.29uz 15294 | A version of ~ 19.29 for u... |
r19.2uz 15295 | A version of ~ r19.2z for ... |
rexuzre 15296 | Convert an upper real quan... |
rexico 15297 | Restrict the base of an up... |
cau3lem 15298 | Lemma for ~ cau3 . (Contr... |
cau3 15299 | Convert between three-quan... |
cau4 15300 | Change the base of a Cauch... |
caubnd2 15301 | A Cauchy sequence of compl... |
caubnd 15302 | A Cauchy sequence of compl... |
sqreulem 15303 | Lemma for ~ sqreu : write ... |
sqreu 15304 | Existence and uniqueness f... |
sqrtcl 15305 | Closure of the square root... |
sqrtthlem 15306 | Lemma for ~ sqrtth . (Con... |
sqrtf 15307 | Mapping domain and codomai... |
sqrtth 15308 | Square root theorem over t... |
sqrtrege0 15309 | The square root function m... |
eqsqrtor 15310 | Solve an equation containi... |
eqsqrtd 15311 | A deduction for showing th... |
eqsqrt2d 15312 | A deduction for showing th... |
amgm2 15313 | Arithmetic-geometric mean ... |
sqrtthi 15314 | Square root theorem. Theo... |
sqrtcli 15315 | The square root of a nonne... |
sqrtgt0i 15316 | The square root of a posit... |
sqrtmsqi 15317 | Square root of square. (C... |
sqrtsqi 15318 | Square root of square. (C... |
sqsqrti 15319 | Square of square root. (C... |
sqrtge0i 15320 | The square root of a nonne... |
absidi 15321 | A nonnegative number is it... |
absnidi 15322 | A negative number is the n... |
leabsi 15323 | A real number is less than... |
absori 15324 | The absolute value of a re... |
absrei 15325 | Absolute value of a real n... |
sqrtpclii 15326 | The square root of a posit... |
sqrtgt0ii 15327 | The square root of a posit... |
sqrt11i 15328 | The square root function i... |
sqrtmuli 15329 | Square root distributes ov... |
sqrtmulii 15330 | Square root distributes ov... |
sqrtmsq2i 15331 | Relationship between squar... |
sqrtlei 15332 | Square root is monotonic. ... |
sqrtlti 15333 | Square root is strictly mo... |
abslti 15334 | Absolute value and 'less t... |
abslei 15335 | Absolute value and 'less t... |
cnsqrt00 15336 | A square root of a complex... |
absvalsqi 15337 | Square of value of absolut... |
absvalsq2i 15338 | Square of value of absolut... |
abscli 15339 | Real closure of absolute v... |
absge0i 15340 | Absolute value is nonnegat... |
absval2i 15341 | Value of absolute value fu... |
abs00i 15342 | The absolute value of a nu... |
absgt0i 15343 | The absolute value of a no... |
absnegi 15344 | Absolute value of negative... |
abscji 15345 | The absolute value of a nu... |
releabsi 15346 | The real part of a number ... |
abssubi 15347 | Swapping order of subtract... |
absmuli 15348 | Absolute value distributes... |
sqabsaddi 15349 | Square of absolute value o... |
sqabssubi 15350 | Square of absolute value o... |
absdivzi 15351 | Absolute value distributes... |
abstrii 15352 | Triangle inequality for ab... |
abs3difi 15353 | Absolute value of differen... |
abs3lemi 15354 | Lemma involving absolute v... |
rpsqrtcld 15355 | The square root of a posit... |
sqrtgt0d 15356 | The square root of a posit... |
absnidd 15357 | A negative number is the n... |
leabsd 15358 | A real number is less than... |
absord 15359 | The absolute value of a re... |
absred 15360 | Absolute value of a real n... |
resqrtcld 15361 | The square root of a nonne... |
sqrtmsqd 15362 | Square root of square. (C... |
sqrtsqd 15363 | Square root of square. (C... |
sqrtge0d 15364 | The square root of a nonne... |
sqrtnegd 15365 | The square root of a negat... |
absidd 15366 | A nonnegative number is it... |
sqrtdivd 15367 | Square root distributes ov... |
sqrtmuld 15368 | Square root distributes ov... |
sqrtsq2d 15369 | Relationship between squar... |
sqrtled 15370 | Square root is monotonic. ... |
sqrtltd 15371 | Square root is strictly mo... |
sqr11d 15372 | The square root function i... |
absltd 15373 | Absolute value and 'less t... |
absled 15374 | Absolute value and 'less t... |
abssubge0d 15375 | Absolute value of a nonneg... |
abssuble0d 15376 | Absolute value of a nonpos... |
absdifltd 15377 | The absolute value of a di... |
absdifled 15378 | The absolute value of a di... |
icodiamlt 15379 | Two elements in a half-ope... |
abscld 15380 | Real closure of absolute v... |
sqrtcld 15381 | Closure of the square root... |
sqrtrege0d 15382 | The real part of the squar... |
sqsqrtd 15383 | Square root theorem. Theo... |
msqsqrtd 15384 | Square root theorem. Theo... |
sqr00d 15385 | A square root is zero iff ... |
absvalsqd 15386 | Square of value of absolut... |
absvalsq2d 15387 | Square of value of absolut... |
absge0d 15388 | Absolute value is nonnegat... |
absval2d 15389 | Value of absolute value fu... |
abs00d 15390 | The absolute value of a nu... |
absne0d 15391 | The absolute value of a nu... |
absrpcld 15392 | The absolute value of a no... |
absnegd 15393 | Absolute value of negative... |
abscjd 15394 | The absolute value of a nu... |
releabsd 15395 | The real part of a number ... |
absexpd 15396 | Absolute value of positive... |
abssubd 15397 | Swapping order of subtract... |
absmuld 15398 | Absolute value distributes... |
absdivd 15399 | Absolute value distributes... |
abstrid 15400 | Triangle inequality for ab... |
abs2difd 15401 | Difference of absolute val... |
abs2dif2d 15402 | Difference of absolute val... |
abs2difabsd 15403 | Absolute value of differen... |
abs3difd 15404 | Absolute value of differen... |
abs3lemd 15405 | Lemma involving absolute v... |
reusq0 15406 | A complex number is the sq... |
bhmafibid1cn 15407 | The Brahmagupta-Fibonacci ... |
bhmafibid2cn 15408 | The Brahmagupta-Fibonacci ... |
bhmafibid1 15409 | The Brahmagupta-Fibonacci ... |
bhmafibid2 15410 | The Brahmagupta-Fibonacci ... |
limsupgord 15413 | Ordering property of the s... |
limsupcl 15414 | Closure of the superior li... |
limsupval 15415 | The superior limit of an i... |
limsupgf 15416 | Closure of the superior li... |
limsupgval 15417 | Value of the superior limi... |
limsupgle 15418 | The defining property of t... |
limsuple 15419 | The defining property of t... |
limsuplt 15420 | The defining property of t... |
limsupval2 15421 | The superior limit, relati... |
limsupgre 15422 | If a sequence of real numb... |
limsupbnd1 15423 | If a sequence is eventuall... |
limsupbnd2 15424 | If a sequence is eventuall... |
climrel 15433 | The limit relation is a re... |
rlimrel 15434 | The limit relation is a re... |
clim 15435 | Express the predicate: Th... |
rlim 15436 | Express the predicate: Th... |
rlim2 15437 | Rewrite ~ rlim for a mappi... |
rlim2lt 15438 | Use strictly less-than in ... |
rlim3 15439 | Restrict the range of the ... |
climcl 15440 | Closure of the limit of a ... |
rlimpm 15441 | Closure of a function with... |
rlimf 15442 | Closure of a function with... |
rlimss 15443 | Domain closure of a functi... |
rlimcl 15444 | Closure of the limit of a ... |
clim2 15445 | Express the predicate: Th... |
clim2c 15446 | Express the predicate ` F ... |
clim0 15447 | Express the predicate ` F ... |
clim0c 15448 | Express the predicate ` F ... |
rlim0 15449 | Express the predicate ` B ... |
rlim0lt 15450 | Use strictly less-than in ... |
climi 15451 | Convergence of a sequence ... |
climi2 15452 | Convergence of a sequence ... |
climi0 15453 | Convergence of a sequence ... |
rlimi 15454 | Convergence at infinity of... |
rlimi2 15455 | Convergence at infinity of... |
ello1 15456 | Elementhood in the set of ... |
ello12 15457 | Elementhood in the set of ... |
ello12r 15458 | Sufficient condition for e... |
lo1f 15459 | An eventually upper bounde... |
lo1dm 15460 | An eventually upper bounde... |
lo1bdd 15461 | The defining property of a... |
ello1mpt 15462 | Elementhood in the set of ... |
ello1mpt2 15463 | Elementhood in the set of ... |
ello1d 15464 | Sufficient condition for e... |
lo1bdd2 15465 | If an eventually bounded f... |
lo1bddrp 15466 | Refine ~ o1bdd2 to give a ... |
elo1 15467 | Elementhood in the set of ... |
elo12 15468 | Elementhood in the set of ... |
elo12r 15469 | Sufficient condition for e... |
o1f 15470 | An eventually bounded func... |
o1dm 15471 | An eventually bounded func... |
o1bdd 15472 | The defining property of a... |
lo1o1 15473 | A function is eventually b... |
lo1o12 15474 | A function is eventually b... |
elo1mpt 15475 | Elementhood in the set of ... |
elo1mpt2 15476 | Elementhood in the set of ... |
elo1d 15477 | Sufficient condition for e... |
o1lo1 15478 | A real function is eventua... |
o1lo12 15479 | A lower bounded real funct... |
o1lo1d 15480 | A real eventually bounded ... |
icco1 15481 | Derive eventual boundednes... |
o1bdd2 15482 | If an eventually bounded f... |
o1bddrp 15483 | Refine ~ o1bdd2 to give a ... |
climconst 15484 | An (eventually) constant s... |
rlimconst 15485 | A constant sequence conver... |
rlimclim1 15486 | Forward direction of ~ rli... |
rlimclim 15487 | A sequence on an upper int... |
climrlim2 15488 | Produce a real limit from ... |
climconst2 15489 | A constant sequence conver... |
climz 15490 | The zero sequence converge... |
rlimuni 15491 | A real function whose doma... |
rlimdm 15492 | Two ways to express that a... |
climuni 15493 | An infinite sequence of co... |
fclim 15494 | The limit relation is func... |
climdm 15495 | Two ways to express that a... |
climeu 15496 | An infinite sequence of co... |
climreu 15497 | An infinite sequence of co... |
climmo 15498 | An infinite sequence of co... |
rlimres 15499 | The restriction of a funct... |
lo1res 15500 | The restriction of an even... |
o1res 15501 | The restriction of an even... |
rlimres2 15502 | The restriction of a funct... |
lo1res2 15503 | The restriction of a funct... |
o1res2 15504 | The restriction of a funct... |
lo1resb 15505 | The restriction of a funct... |
rlimresb 15506 | The restriction of a funct... |
o1resb 15507 | The restriction of a funct... |
climeq 15508 | Two functions that are eve... |
lo1eq 15509 | Two functions that are eve... |
rlimeq 15510 | Two functions that are eve... |
o1eq 15511 | Two functions that are eve... |
climmpt 15512 | Exhibit a function ` G ` w... |
2clim 15513 | If two sequences converge ... |
climmpt2 15514 | Relate an integer limit on... |
climshftlem 15515 | A shifted function converg... |
climres 15516 | A function restricted to u... |
climshft 15517 | A shifted function converg... |
serclim0 15518 | The zero series converges ... |
rlimcld2 15519 | If ` D ` is a closed set i... |
rlimrege0 15520 | The limit of a sequence of... |
rlimrecl 15521 | The limit of a real sequen... |
rlimge0 15522 | The limit of a sequence of... |
climshft2 15523 | A shifted function converg... |
climrecl 15524 | The limit of a convergent ... |
climge0 15525 | A nonnegative sequence con... |
climabs0 15526 | Convergence to zero of the... |
o1co 15527 | Sufficient condition for t... |
o1compt 15528 | Sufficient condition for t... |
rlimcn1 15529 | Image of a limit under a c... |
rlimcn1b 15530 | Image of a limit under a c... |
rlimcn3 15531 | Image of a limit under a c... |
rlimcn2 15532 | Image of a limit under a c... |
climcn1 15533 | Image of a limit under a c... |
climcn2 15534 | Image of a limit under a c... |
addcn2 15535 | Complex number addition is... |
subcn2 15536 | Complex number subtraction... |
mulcn2 15537 | Complex number multiplicat... |
reccn2 15538 | The reciprocal function is... |
cn1lem 15539 | A sufficient condition for... |
abscn2 15540 | The absolute value functio... |
cjcn2 15541 | The complex conjugate func... |
recn2 15542 | The real part function is ... |
imcn2 15543 | The imaginary part functio... |
climcn1lem 15544 | The limit of a continuous ... |
climabs 15545 | Limit of the absolute valu... |
climcj 15546 | Limit of the complex conju... |
climre 15547 | Limit of the real part of ... |
climim 15548 | Limit of the imaginary par... |
rlimmptrcl 15549 | Reverse closure for a real... |
rlimabs 15550 | Limit of the absolute valu... |
rlimcj 15551 | Limit of the complex conju... |
rlimre 15552 | Limit of the real part of ... |
rlimim 15553 | Limit of the imaginary par... |
o1of2 15554 | Show that a binary operati... |
o1add 15555 | The sum of two eventually ... |
o1mul 15556 | The product of two eventua... |
o1sub 15557 | The difference of two even... |
rlimo1 15558 | Any function with a finite... |
rlimdmo1 15559 | A convergent function is e... |
o1rlimmul 15560 | The product of an eventual... |
o1const 15561 | A constant function is eve... |
lo1const 15562 | A constant function is eve... |
lo1mptrcl 15563 | Reverse closure for an eve... |
o1mptrcl 15564 | Reverse closure for an eve... |
o1add2 15565 | The sum of two eventually ... |
o1mul2 15566 | The product of two eventua... |
o1sub2 15567 | The product of two eventua... |
lo1add 15568 | The sum of two eventually ... |
lo1mul 15569 | The product of an eventual... |
lo1mul2 15570 | The product of an eventual... |
o1dif 15571 | If the difference of two f... |
lo1sub 15572 | The difference of an event... |
climadd 15573 | Limit of the sum of two co... |
climmul 15574 | Limit of the product of tw... |
climsub 15575 | Limit of the difference of... |
climaddc1 15576 | Limit of a constant ` C ` ... |
climaddc2 15577 | Limit of a constant ` C ` ... |
climmulc2 15578 | Limit of a sequence multip... |
climsubc1 15579 | Limit of a constant ` C ` ... |
climsubc2 15580 | Limit of a constant ` C ` ... |
climle 15581 | Comparison of the limits o... |
climsqz 15582 | Convergence of a sequence ... |
climsqz2 15583 | Convergence of a sequence ... |
rlimadd 15584 | Limit of the sum of two co... |
rlimaddOLD 15585 | Obsolete version of ~ rlim... |
rlimsub 15586 | Limit of the difference of... |
rlimmul 15587 | Limit of the product of tw... |
rlimmulOLD 15588 | Obsolete version of ~ rlim... |
rlimdiv 15589 | Limit of the quotient of t... |
rlimneg 15590 | Limit of the negative of a... |
rlimle 15591 | Comparison of the limits o... |
rlimsqzlem 15592 | Lemma for ~ rlimsqz and ~ ... |
rlimsqz 15593 | Convergence of a sequence ... |
rlimsqz2 15594 | Convergence of a sequence ... |
lo1le 15595 | Transfer eventual upper bo... |
o1le 15596 | Transfer eventual boundedn... |
rlimno1 15597 | A function whose inverse c... |
clim2ser 15598 | The limit of an infinite s... |
clim2ser2 15599 | The limit of an infinite s... |
iserex 15600 | An infinite series converg... |
isermulc2 15601 | Multiplication of an infin... |
climlec2 15602 | Comparison of a constant t... |
iserle 15603 | Comparison of the limits o... |
iserge0 15604 | The limit of an infinite s... |
climub 15605 | The limit of a monotonic s... |
climserle 15606 | The partial sums of a conv... |
isershft 15607 | Index shift of the limit o... |
isercolllem1 15608 | Lemma for ~ isercoll . (C... |
isercolllem2 15609 | Lemma for ~ isercoll . (C... |
isercolllem3 15610 | Lemma for ~ isercoll . (C... |
isercoll 15611 | Rearrange an infinite seri... |
isercoll2 15612 | Generalize ~ isercoll so t... |
climsup 15613 | A bounded monotonic sequen... |
climcau 15614 | A converging sequence of c... |
climbdd 15615 | A converging sequence of c... |
caucvgrlem 15616 | Lemma for ~ caurcvgr . (C... |
caurcvgr 15617 | A Cauchy sequence of real ... |
caucvgrlem2 15618 | Lemma for ~ caucvgr . (Co... |
caucvgr 15619 | A Cauchy sequence of compl... |
caurcvg 15620 | A Cauchy sequence of real ... |
caurcvg2 15621 | A Cauchy sequence of real ... |
caucvg 15622 | A Cauchy sequence of compl... |
caucvgb 15623 | A function is convergent i... |
serf0 15624 | If an infinite series conv... |
iseraltlem1 15625 | Lemma for ~ iseralt . A d... |
iseraltlem2 15626 | Lemma for ~ iseralt . The... |
iseraltlem3 15627 | Lemma for ~ iseralt . Fro... |
iseralt 15628 | The alternating series tes... |
sumex 15631 | A sum is a set. (Contribu... |
sumeq1 15632 | Equality theorem for a sum... |
nfsum1 15633 | Bound-variable hypothesis ... |
nfsum 15634 | Bound-variable hypothesis ... |
sumeq2w 15635 | Equality theorem for sum, ... |
sumeq2ii 15636 | Equality theorem for sum, ... |
sumeq2 15637 | Equality theorem for sum. ... |
cbvsum 15638 | Change bound variable in a... |
cbvsumv 15639 | Change bound variable in a... |
cbvsumi 15640 | Change bound variable in a... |
sumeq1i 15641 | Equality inference for sum... |
sumeq2i 15642 | Equality inference for sum... |
sumeq12i 15643 | Equality inference for sum... |
sumeq1d 15644 | Equality deduction for sum... |
sumeq2d 15645 | Equality deduction for sum... |
sumeq2dv 15646 | Equality deduction for sum... |
sumeq2sdv 15647 | Equality deduction for sum... |
2sumeq2dv 15648 | Equality deduction for dou... |
sumeq12dv 15649 | Equality deduction for sum... |
sumeq12rdv 15650 | Equality deduction for sum... |
sum2id 15651 | The second class argument ... |
sumfc 15652 | A lemma to facilitate conv... |
fz1f1o 15653 | A lemma for working with f... |
sumrblem 15654 | Lemma for ~ sumrb . (Cont... |
fsumcvg 15655 | The sequence of partial su... |
sumrb 15656 | Rebase the starting point ... |
summolem3 15657 | Lemma for ~ summo . (Cont... |
summolem2a 15658 | Lemma for ~ summo . (Cont... |
summolem2 15659 | Lemma for ~ summo . (Cont... |
summo 15660 | A sum has at most one limi... |
zsum 15661 | Series sum with index set ... |
isum 15662 | Series sum with an upper i... |
fsum 15663 | The value of a sum over a ... |
sum0 15664 | Any sum over the empty set... |
sumz 15665 | Any sum of zero over a sum... |
fsumf1o 15666 | Re-index a finite sum usin... |
sumss 15667 | Change the index set to a ... |
fsumss 15668 | Change the index set to a ... |
sumss2 15669 | Change the index set of a ... |
fsumcvg2 15670 | The sequence of partial su... |
fsumsers 15671 | Special case of series sum... |
fsumcvg3 15672 | A finite sum is convergent... |
fsumser 15673 | A finite sum expressed in ... |
fsumcl2lem 15674 | - Lemma for finite sum clo... |
fsumcllem 15675 | - Lemma for finite sum clo... |
fsumcl 15676 | Closure of a finite sum of... |
fsumrecl 15677 | Closure of a finite sum of... |
fsumzcl 15678 | Closure of a finite sum of... |
fsumnn0cl 15679 | Closure of a finite sum of... |
fsumrpcl 15680 | Closure of a finite sum of... |
fsumclf 15681 | Closure of a finite sum of... |
fsumzcl2 15682 | A finite sum with integer ... |
fsumadd 15683 | The sum of two finite sums... |
fsumsplit 15684 | Split a sum into two parts... |
fsumsplitf 15685 | Split a sum into two parts... |
sumsnf 15686 | A sum of a singleton is th... |
fsumsplitsn 15687 | Separate out a term in a f... |
fsumsplit1 15688 | Separate out a term in a f... |
sumsn 15689 | A sum of a singleton is th... |
fsum1 15690 | The finite sum of ` A ( k ... |
sumpr 15691 | A sum over a pair is the s... |
sumtp 15692 | A sum over a triple is the... |
sumsns 15693 | A sum of a singleton is th... |
fsumm1 15694 | Separate out the last term... |
fzosump1 15695 | Separate out the last term... |
fsum1p 15696 | Separate out the first ter... |
fsummsnunz 15697 | A finite sum all of whose ... |
fsumsplitsnun 15698 | Separate out a term in a f... |
fsump1 15699 | The addition of the next t... |
isumclim 15700 | An infinite sum equals the... |
isumclim2 15701 | A converging series conver... |
isumclim3 15702 | The sequence of partial fi... |
sumnul 15703 | The sum of a non-convergen... |
isumcl 15704 | The sum of a converging in... |
isummulc2 15705 | An infinite sum multiplied... |
isummulc1 15706 | An infinite sum multiplied... |
isumdivc 15707 | An infinite sum divided by... |
isumrecl 15708 | The sum of a converging in... |
isumge0 15709 | An infinite sum of nonnega... |
isumadd 15710 | Addition of infinite sums.... |
sumsplit 15711 | Split a sum into two parts... |
fsump1i 15712 | Optimized version of ~ fsu... |
fsum2dlem 15713 | Lemma for ~ fsum2d - induc... |
fsum2d 15714 | Write a double sum as a su... |
fsumxp 15715 | Combine two sums into a si... |
fsumcnv 15716 | Transform a region of summ... |
fsumcom2 15717 | Interchange order of summa... |
fsumcom 15718 | Interchange order of summa... |
fsum0diaglem 15719 | Lemma for ~ fsum0diag . (... |
fsum0diag 15720 | Two ways to express "the s... |
mptfzshft 15721 | 1-1 onto function in maps-... |
fsumrev 15722 | Reversal of a finite sum. ... |
fsumshft 15723 | Index shift of a finite su... |
fsumshftm 15724 | Negative index shift of a ... |
fsumrev2 15725 | Reversal of a finite sum. ... |
fsum0diag2 15726 | Two ways to express "the s... |
fsummulc2 15727 | A finite sum multiplied by... |
fsummulc1 15728 | A finite sum multiplied by... |
fsumdivc 15729 | A finite sum divided by a ... |
fsumneg 15730 | Negation of a finite sum. ... |
fsumsub 15731 | Split a finite sum over a ... |
fsum2mul 15732 | Separate the nested sum of... |
fsumconst 15733 | The sum of constant terms ... |
fsumdifsnconst 15734 | The sum of constant terms ... |
modfsummodslem1 15735 | Lemma 1 for ~ modfsummods ... |
modfsummods 15736 | Induction step for ~ modfs... |
modfsummod 15737 | A finite sum modulo a posi... |
fsumge0 15738 | If all of the terms of a f... |
fsumless 15739 | A shorter sum of nonnegati... |
fsumge1 15740 | A sum of nonnegative numbe... |
fsum00 15741 | A sum of nonnegative numbe... |
fsumle 15742 | If all of the terms of fin... |
fsumlt 15743 | If every term in one finit... |
fsumabs 15744 | Generalized triangle inequ... |
telfsumo 15745 | Sum of a telescoping serie... |
telfsumo2 15746 | Sum of a telescoping serie... |
telfsum 15747 | Sum of a telescoping serie... |
telfsum2 15748 | Sum of a telescoping serie... |
fsumparts 15749 | Summation by parts. (Cont... |
fsumrelem 15750 | Lemma for ~ fsumre , ~ fsu... |
fsumre 15751 | The real part of a sum. (... |
fsumim 15752 | The imaginary part of a su... |
fsumcj 15753 | The complex conjugate of a... |
fsumrlim 15754 | Limit of a finite sum of c... |
fsumo1 15755 | The finite sum of eventual... |
o1fsum 15756 | If ` A ( k ) ` is O(1), th... |
seqabs 15757 | Generalized triangle inequ... |
iserabs 15758 | Generalized triangle inequ... |
cvgcmp 15759 | A comparison test for conv... |
cvgcmpub 15760 | An upper bound for the lim... |
cvgcmpce 15761 | A comparison test for conv... |
abscvgcvg 15762 | An absolutely convergent s... |
climfsum 15763 | Limit of a finite sum of c... |
fsumiun 15764 | Sum over a disjoint indexe... |
hashiun 15765 | The cardinality of a disjo... |
hash2iun 15766 | The cardinality of a neste... |
hash2iun1dif1 15767 | The cardinality of a neste... |
hashrabrex 15768 | The number of elements in ... |
hashuni 15769 | The cardinality of a disjo... |
qshash 15770 | The cardinality of a set w... |
ackbijnn 15771 | Translate the Ackermann bi... |
binomlem 15772 | Lemma for ~ binom (binomia... |
binom 15773 | The binomial theorem: ` ( ... |
binom1p 15774 | Special case of the binomi... |
binom11 15775 | Special case of the binomi... |
binom1dif 15776 | A summation for the differ... |
bcxmaslem1 15777 | Lemma for ~ bcxmas . (Con... |
bcxmas 15778 | Parallel summation (Christ... |
incexclem 15779 | Lemma for ~ incexc . (Con... |
incexc 15780 | The inclusion/exclusion pr... |
incexc2 15781 | The inclusion/exclusion pr... |
isumshft 15782 | Index shift of an infinite... |
isumsplit 15783 | Split off the first ` N ` ... |
isum1p 15784 | The infinite sum of a conv... |
isumnn0nn 15785 | Sum from 0 to infinity in ... |
isumrpcl 15786 | The infinite sum of positi... |
isumle 15787 | Comparison of two infinite... |
isumless 15788 | A finite sum of nonnegativ... |
isumsup2 15789 | An infinite sum of nonnega... |
isumsup 15790 | An infinite sum of nonnega... |
isumltss 15791 | A partial sum of a series ... |
climcndslem1 15792 | Lemma for ~ climcnds : bou... |
climcndslem2 15793 | Lemma for ~ climcnds : bou... |
climcnds 15794 | The Cauchy condensation te... |
divrcnv 15795 | The sequence of reciprocal... |
divcnv 15796 | The sequence of reciprocal... |
flo1 15797 | The floor function satisfi... |
divcnvshft 15798 | Limit of a ratio function.... |
supcvg 15799 | Extract a sequence ` f ` i... |
infcvgaux1i 15800 | Auxiliary theorem for appl... |
infcvgaux2i 15801 | Auxiliary theorem for appl... |
harmonic 15802 | The harmonic series ` H ` ... |
arisum 15803 | Arithmetic series sum of t... |
arisum2 15804 | Arithmetic series sum of t... |
trireciplem 15805 | Lemma for ~ trirecip . Sh... |
trirecip 15806 | The sum of the reciprocals... |
expcnv 15807 | A sequence of powers of a ... |
explecnv 15808 | A sequence of terms conver... |
geoserg 15809 | The value of the finite ge... |
geoser 15810 | The value of the finite ge... |
pwdif 15811 | The difference of two numb... |
pwm1geoser 15812 | The n-th power of a number... |
geolim 15813 | The partial sums in the in... |
geolim2 15814 | The partial sums in the ge... |
georeclim 15815 | The limit of a geometric s... |
geo2sum 15816 | The value of the finite ge... |
geo2sum2 15817 | The value of the finite ge... |
geo2lim 15818 | The value of the infinite ... |
geomulcvg 15819 | The geometric series conve... |
geoisum 15820 | The infinite sum of ` 1 + ... |
geoisumr 15821 | The infinite sum of recipr... |
geoisum1 15822 | The infinite sum of ` A ^ ... |
geoisum1c 15823 | The infinite sum of ` A x.... |
0.999... 15824 | The recurring decimal 0.99... |
geoihalfsum 15825 | Prove that the infinite ge... |
cvgrat 15826 | Ratio test for convergence... |
mertenslem1 15827 | Lemma for ~ mertens . (Co... |
mertenslem2 15828 | Lemma for ~ mertens . (Co... |
mertens 15829 | Mertens' theorem. If ` A ... |
prodf 15830 | An infinite product of com... |
clim2prod 15831 | The limit of an infinite p... |
clim2div 15832 | The limit of an infinite p... |
prodfmul 15833 | The product of two infinit... |
prodf1 15834 | The value of the partial p... |
prodf1f 15835 | A one-valued infinite prod... |
prodfclim1 15836 | The constant one product c... |
prodfn0 15837 | No term of a nonzero infin... |
prodfrec 15838 | The reciprocal of an infin... |
prodfdiv 15839 | The quotient of two infini... |
ntrivcvg 15840 | A non-trivially converging... |
ntrivcvgn0 15841 | A product that converges t... |
ntrivcvgfvn0 15842 | Any value of a product seq... |
ntrivcvgtail 15843 | A tail of a non-trivially ... |
ntrivcvgmullem 15844 | Lemma for ~ ntrivcvgmul . ... |
ntrivcvgmul 15845 | The product of two non-tri... |
prodex 15848 | A product is a set. (Cont... |
prodeq1f 15849 | Equality theorem for a pro... |
prodeq1 15850 | Equality theorem for a pro... |
nfcprod1 15851 | Bound-variable hypothesis ... |
nfcprod 15852 | Bound-variable hypothesis ... |
prodeq2w 15853 | Equality theorem for produ... |
prodeq2ii 15854 | Equality theorem for produ... |
prodeq2 15855 | Equality theorem for produ... |
cbvprod 15856 | Change bound variable in a... |
cbvprodv 15857 | Change bound variable in a... |
cbvprodi 15858 | Change bound variable in a... |
prodeq1i 15859 | Equality inference for pro... |
prodeq2i 15860 | Equality inference for pro... |
prodeq12i 15861 | Equality inference for pro... |
prodeq1d 15862 | Equality deduction for pro... |
prodeq2d 15863 | Equality deduction for pro... |
prodeq2dv 15864 | Equality deduction for pro... |
prodeq2sdv 15865 | Equality deduction for pro... |
2cprodeq2dv 15866 | Equality deduction for dou... |
prodeq12dv 15867 | Equality deduction for pro... |
prodeq12rdv 15868 | Equality deduction for pro... |
prod2id 15869 | The second class argument ... |
prodrblem 15870 | Lemma for ~ prodrb . (Con... |
fprodcvg 15871 | The sequence of partial pr... |
prodrblem2 15872 | Lemma for ~ prodrb . (Con... |
prodrb 15873 | Rebase the starting point ... |
prodmolem3 15874 | Lemma for ~ prodmo . (Con... |
prodmolem2a 15875 | Lemma for ~ prodmo . (Con... |
prodmolem2 15876 | Lemma for ~ prodmo . (Con... |
prodmo 15877 | A product has at most one ... |
zprod 15878 | Series product with index ... |
iprod 15879 | Series product with an upp... |
zprodn0 15880 | Nonzero series product wit... |
iprodn0 15881 | Nonzero series product wit... |
fprod 15882 | The value of a product ove... |
fprodntriv 15883 | A non-triviality lemma for... |
prod0 15884 | A product over the empty s... |
prod1 15885 | Any product of one over a ... |
prodfc 15886 | A lemma to facilitate conv... |
fprodf1o 15887 | Re-index a finite product ... |
prodss 15888 | Change the index set to a ... |
fprodss 15889 | Change the index set to a ... |
fprodser 15890 | A finite product expressed... |
fprodcl2lem 15891 | Finite product closure lem... |
fprodcllem 15892 | Finite product closure lem... |
fprodcl 15893 | Closure of a finite produc... |
fprodrecl 15894 | Closure of a finite produc... |
fprodzcl 15895 | Closure of a finite produc... |
fprodnncl 15896 | Closure of a finite produc... |
fprodrpcl 15897 | Closure of a finite produc... |
fprodnn0cl 15898 | Closure of a finite produc... |
fprodcllemf 15899 | Finite product closure lem... |
fprodreclf 15900 | Closure of a finite produc... |
fprodmul 15901 | The product of two finite ... |
fproddiv 15902 | The quotient of two finite... |
prodsn 15903 | A product of a singleton i... |
fprod1 15904 | A finite product of only o... |
prodsnf 15905 | A product of a singleton i... |
climprod1 15906 | The limit of a product ove... |
fprodsplit 15907 | Split a finite product int... |
fprodm1 15908 | Separate out the last term... |
fprod1p 15909 | Separate out the first ter... |
fprodp1 15910 | Multiply in the last term ... |
fprodm1s 15911 | Separate out the last term... |
fprodp1s 15912 | Multiply in the last term ... |
prodsns 15913 | A product of the singleton... |
fprodfac 15914 | Factorial using product no... |
fprodabs 15915 | The absolute value of a fi... |
fprodeq0 15916 | Any finite product contain... |
fprodshft 15917 | Shift the index of a finit... |
fprodrev 15918 | Reversal of a finite produ... |
fprodconst 15919 | The product of constant te... |
fprodn0 15920 | A finite product of nonzer... |
fprod2dlem 15921 | Lemma for ~ fprod2d - indu... |
fprod2d 15922 | Write a double product as ... |
fprodxp 15923 | Combine two products into ... |
fprodcnv 15924 | Transform a product region... |
fprodcom2 15925 | Interchange order of multi... |
fprodcom 15926 | Interchange product order.... |
fprod0diag 15927 | Two ways to express "the p... |
fproddivf 15928 | The quotient of two finite... |
fprodsplitf 15929 | Split a finite product int... |
fprodsplitsn 15930 | Separate out a term in a f... |
fprodsplit1f 15931 | Separate out a term in a f... |
fprodn0f 15932 | A finite product of nonzer... |
fprodclf 15933 | Closure of a finite produc... |
fprodge0 15934 | If all the terms of a fini... |
fprodeq0g 15935 | Any finite product contain... |
fprodge1 15936 | If all of the terms of a f... |
fprodle 15937 | If all the terms of two fi... |
fprodmodd 15938 | If all factors of two fini... |
iprodclim 15939 | An infinite product equals... |
iprodclim2 15940 | A converging product conve... |
iprodclim3 15941 | The sequence of partial fi... |
iprodcl 15942 | The product of a non-trivi... |
iprodrecl 15943 | The product of a non-trivi... |
iprodmul 15944 | Multiplication of infinite... |
risefacval 15949 | The value of the rising fa... |
fallfacval 15950 | The value of the falling f... |
risefacval2 15951 | One-based value of rising ... |
fallfacval2 15952 | One-based value of falling... |
fallfacval3 15953 | A product representation o... |
risefaccllem 15954 | Lemma for rising factorial... |
fallfaccllem 15955 | Lemma for falling factoria... |
risefaccl 15956 | Closure law for rising fac... |
fallfaccl 15957 | Closure law for falling fa... |
rerisefaccl 15958 | Closure law for rising fac... |
refallfaccl 15959 | Closure law for falling fa... |
nnrisefaccl 15960 | Closure law for rising fac... |
zrisefaccl 15961 | Closure law for rising fac... |
zfallfaccl 15962 | Closure law for falling fa... |
nn0risefaccl 15963 | Closure law for rising fac... |
rprisefaccl 15964 | Closure law for rising fac... |
risefallfac 15965 | A relationship between ris... |
fallrisefac 15966 | A relationship between fal... |
risefall0lem 15967 | Lemma for ~ risefac0 and ~... |
risefac0 15968 | The value of the rising fa... |
fallfac0 15969 | The value of the falling f... |
risefacp1 15970 | The value of the rising fa... |
fallfacp1 15971 | The value of the falling f... |
risefacp1d 15972 | The value of the rising fa... |
fallfacp1d 15973 | The value of the falling f... |
risefac1 15974 | The value of rising factor... |
fallfac1 15975 | The value of falling facto... |
risefacfac 15976 | Relate rising factorial to... |
fallfacfwd 15977 | The forward difference of ... |
0fallfac 15978 | The value of the zero fall... |
0risefac 15979 | The value of the zero risi... |
binomfallfaclem1 15980 | Lemma for ~ binomfallfac .... |
binomfallfaclem2 15981 | Lemma for ~ binomfallfac .... |
binomfallfac 15982 | A version of the binomial ... |
binomrisefac 15983 | A version of the binomial ... |
fallfacval4 15984 | Represent the falling fact... |
bcfallfac 15985 | Binomial coefficient in te... |
fallfacfac 15986 | Relate falling factorial t... |
bpolylem 15989 | Lemma for ~ bpolyval . (C... |
bpolyval 15990 | The value of the Bernoulli... |
bpoly0 15991 | The value of the Bernoulli... |
bpoly1 15992 | The value of the Bernoulli... |
bpolycl 15993 | Closure law for Bernoulli ... |
bpolysum 15994 | A sum for Bernoulli polyno... |
bpolydiflem 15995 | Lemma for ~ bpolydif . (C... |
bpolydif 15996 | Calculate the difference b... |
fsumkthpow 15997 | A closed-form expression f... |
bpoly2 15998 | The Bernoulli polynomials ... |
bpoly3 15999 | The Bernoulli polynomials ... |
bpoly4 16000 | The Bernoulli polynomials ... |
fsumcube 16001 | Express the sum of cubes i... |
eftcl 16014 | Closure of a term in the s... |
reeftcl 16015 | The terms of the series ex... |
eftabs 16016 | The absolute value of a te... |
eftval 16017 | The value of a term in the... |
efcllem 16018 | Lemma for ~ efcl . The se... |
ef0lem 16019 | The series defining the ex... |
efval 16020 | Value of the exponential f... |
esum 16021 | Value of Euler's constant ... |
eff 16022 | Domain and codomain of the... |
efcl 16023 | Closure law for the expone... |
efval2 16024 | Value of the exponential f... |
efcvg 16025 | The series that defines th... |
efcvgfsum 16026 | Exponential function conve... |
reefcl 16027 | The exponential function i... |
reefcld 16028 | The exponential function i... |
ere 16029 | Euler's constant ` _e ` = ... |
ege2le3 16030 | Lemma for ~ egt2lt3 . (Co... |
ef0 16031 | Value of the exponential f... |
efcj 16032 | The exponential of a compl... |
efaddlem 16033 | Lemma for ~ efadd (exponen... |
efadd 16034 | Sum of exponents law for e... |
fprodefsum 16035 | Move the exponential funct... |
efcan 16036 | Cancellation law for expon... |
efne0 16037 | The exponential of a compl... |
efneg 16038 | The exponential of the opp... |
eff2 16039 | The exponential function m... |
efsub 16040 | Difference of exponents la... |
efexp 16041 | The exponential of an inte... |
efzval 16042 | Value of the exponential f... |
efgt0 16043 | The exponential of a real ... |
rpefcl 16044 | The exponential of a real ... |
rpefcld 16045 | The exponential of a real ... |
eftlcvg 16046 | The tail series of the exp... |
eftlcl 16047 | Closure of the sum of an i... |
reeftlcl 16048 | Closure of the sum of an i... |
eftlub 16049 | An upper bound on the abso... |
efsep 16050 | Separate out the next term... |
effsumlt 16051 | The partial sums of the se... |
eft0val 16052 | The value of the first ter... |
ef4p 16053 | Separate out the first fou... |
efgt1p2 16054 | The exponential of a posit... |
efgt1p 16055 | The exponential of a posit... |
efgt1 16056 | The exponential of a posit... |
eflt 16057 | The exponential function o... |
efle 16058 | The exponential function o... |
reef11 16059 | The exponential function o... |
reeff1 16060 | The exponential function m... |
eflegeo 16061 | The exponential function o... |
sinval 16062 | Value of the sine function... |
cosval 16063 | Value of the cosine functi... |
sinf 16064 | Domain and codomain of the... |
cosf 16065 | Domain and codomain of the... |
sincl 16066 | Closure of the sine functi... |
coscl 16067 | Closure of the cosine func... |
tanval 16068 | Value of the tangent funct... |
tancl 16069 | The closure of the tangent... |
sincld 16070 | Closure of the sine functi... |
coscld 16071 | Closure of the cosine func... |
tancld 16072 | Closure of the tangent fun... |
tanval2 16073 | Express the tangent functi... |
tanval3 16074 | Express the tangent functi... |
resinval 16075 | The sine of a real number ... |
recosval 16076 | The cosine of a real numbe... |
efi4p 16077 | Separate out the first fou... |
resin4p 16078 | Separate out the first fou... |
recos4p 16079 | Separate out the first fou... |
resincl 16080 | The sine of a real number ... |
recoscl 16081 | The cosine of a real numbe... |
retancl 16082 | The closure of the tangent... |
resincld 16083 | Closure of the sine functi... |
recoscld 16084 | Closure of the cosine func... |
retancld 16085 | Closure of the tangent fun... |
sinneg 16086 | The sine of a negative is ... |
cosneg 16087 | The cosines of a number an... |
tanneg 16088 | The tangent of a negative ... |
sin0 16089 | Value of the sine function... |
cos0 16090 | Value of the cosine functi... |
tan0 16091 | The value of the tangent f... |
efival 16092 | The exponential function i... |
efmival 16093 | The exponential function i... |
sinhval 16094 | Value of the hyperbolic si... |
coshval 16095 | Value of the hyperbolic co... |
resinhcl 16096 | The hyperbolic sine of a r... |
rpcoshcl 16097 | The hyperbolic cosine of a... |
recoshcl 16098 | The hyperbolic cosine of a... |
retanhcl 16099 | The hyperbolic tangent of ... |
tanhlt1 16100 | The hyperbolic tangent of ... |
tanhbnd 16101 | The hyperbolic tangent of ... |
efeul 16102 | Eulerian representation of... |
efieq 16103 | The exponentials of two im... |
sinadd 16104 | Addition formula for sine.... |
cosadd 16105 | Addition formula for cosin... |
tanaddlem 16106 | A useful intermediate step... |
tanadd 16107 | Addition formula for tange... |
sinsub 16108 | Sine of difference. (Cont... |
cossub 16109 | Cosine of difference. (Co... |
addsin 16110 | Sum of sines. (Contribute... |
subsin 16111 | Difference of sines. (Con... |
sinmul 16112 | Product of sines can be re... |
cosmul 16113 | Product of cosines can be ... |
addcos 16114 | Sum of cosines. (Contribu... |
subcos 16115 | Difference of cosines. (C... |
sincossq 16116 | Sine squared plus cosine s... |
sin2t 16117 | Double-angle formula for s... |
cos2t 16118 | Double-angle formula for c... |
cos2tsin 16119 | Double-angle formula for c... |
sinbnd 16120 | The sine of a real number ... |
cosbnd 16121 | The cosine of a real numbe... |
sinbnd2 16122 | The sine of a real number ... |
cosbnd2 16123 | The cosine of a real numbe... |
ef01bndlem 16124 | Lemma for ~ sin01bnd and ~... |
sin01bnd 16125 | Bounds on the sine of a po... |
cos01bnd 16126 | Bounds on the cosine of a ... |
cos1bnd 16127 | Bounds on the cosine of 1.... |
cos2bnd 16128 | Bounds on the cosine of 2.... |
sinltx 16129 | The sine of a positive rea... |
sin01gt0 16130 | The sine of a positive rea... |
cos01gt0 16131 | The cosine of a positive r... |
sin02gt0 16132 | The sine of a positive rea... |
sincos1sgn 16133 | The signs of the sine and ... |
sincos2sgn 16134 | The signs of the sine and ... |
sin4lt0 16135 | The sine of 4 is negative.... |
absefi 16136 | The absolute value of the ... |
absef 16137 | The absolute value of the ... |
absefib 16138 | A complex number is real i... |
efieq1re 16139 | A number whose imaginary e... |
demoivre 16140 | De Moivre's Formula. Proo... |
demoivreALT 16141 | Alternate proof of ~ demoi... |
eirrlem 16144 | Lemma for ~ eirr . (Contr... |
eirr 16145 | ` _e ` is irrational. (Co... |
egt2lt3 16146 | Euler's constant ` _e ` = ... |
epos 16147 | Euler's constant ` _e ` is... |
epr 16148 | Euler's constant ` _e ` is... |
ene0 16149 | ` _e ` is not 0. (Contrib... |
ene1 16150 | ` _e ` is not 1. (Contrib... |
xpnnen 16151 | The Cartesian product of t... |
znnen 16152 | The set of integers and th... |
qnnen 16153 | The rational numbers are c... |
rpnnen2lem1 16154 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem2 16155 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem3 16156 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem4 16157 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem5 16158 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem6 16159 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem7 16160 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem8 16161 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem9 16162 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem10 16163 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem11 16164 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem12 16165 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2 16166 | The other half of ~ rpnnen... |
rpnnen 16167 | The cardinality of the con... |
rexpen 16168 | The real numbers are equin... |
cpnnen 16169 | The complex numbers are eq... |
rucALT 16170 | Alternate proof of ~ ruc .... |
ruclem1 16171 | Lemma for ~ ruc (the reals... |
ruclem2 16172 | Lemma for ~ ruc . Orderin... |
ruclem3 16173 | Lemma for ~ ruc . The con... |
ruclem4 16174 | Lemma for ~ ruc . Initial... |
ruclem6 16175 | Lemma for ~ ruc . Domain ... |
ruclem7 16176 | Lemma for ~ ruc . Success... |
ruclem8 16177 | Lemma for ~ ruc . The int... |
ruclem9 16178 | Lemma for ~ ruc . The fir... |
ruclem10 16179 | Lemma for ~ ruc . Every f... |
ruclem11 16180 | Lemma for ~ ruc . Closure... |
ruclem12 16181 | Lemma for ~ ruc . The sup... |
ruclem13 16182 | Lemma for ~ ruc . There i... |
ruc 16183 | The set of positive intege... |
resdomq 16184 | The set of rationals is st... |
aleph1re 16185 | There are at least aleph-o... |
aleph1irr 16186 | There are at least aleph-o... |
cnso 16187 | The complex numbers can be... |
sqrt2irrlem 16188 | Lemma for ~ sqrt2irr . Th... |
sqrt2irr 16189 | The square root of 2 is ir... |
sqrt2re 16190 | The square root of 2 exist... |
sqrt2irr0 16191 | The square root of 2 is an... |
nthruc 16192 | The sequence ` NN ` , ` ZZ... |
nthruz 16193 | The sequence ` NN ` , ` NN... |
divides 16196 | Define the divides relatio... |
dvdsval2 16197 | One nonzero integer divide... |
dvdsval3 16198 | One nonzero integer divide... |
dvdszrcl 16199 | Reverse closure for the di... |
dvdsmod0 16200 | If a positive integer divi... |
p1modz1 16201 | If a number greater than 1... |
dvdsmodexp 16202 | If a positive integer divi... |
nndivdvds 16203 | Strong form of ~ dvdsval2 ... |
nndivides 16204 | Definition of the divides ... |
moddvds 16205 | Two ways to say ` A == B `... |
modm1div 16206 | An integer greater than on... |
dvds0lem 16207 | A lemma to assist theorems... |
dvds1lem 16208 | A lemma to assist theorems... |
dvds2lem 16209 | A lemma to assist theorems... |
iddvds 16210 | An integer divides itself.... |
1dvds 16211 | 1 divides any integer. Th... |
dvds0 16212 | Any integer divides 0. Th... |
negdvdsb 16213 | An integer divides another... |
dvdsnegb 16214 | An integer divides another... |
absdvdsb 16215 | An integer divides another... |
dvdsabsb 16216 | An integer divides another... |
0dvds 16217 | Only 0 is divisible by 0. ... |
dvdsmul1 16218 | An integer divides a multi... |
dvdsmul2 16219 | An integer divides a multi... |
iddvdsexp 16220 | An integer divides a posit... |
muldvds1 16221 | If a product divides an in... |
muldvds2 16222 | If a product divides an in... |
dvdscmul 16223 | Multiplication by a consta... |
dvdsmulc 16224 | Multiplication by a consta... |
dvdscmulr 16225 | Cancellation law for the d... |
dvdsmulcr 16226 | Cancellation law for the d... |
summodnegmod 16227 | The sum of two integers mo... |
modmulconst 16228 | Constant multiplication in... |
dvds2ln 16229 | If an integer divides each... |
dvds2add 16230 | If an integer divides each... |
dvds2sub 16231 | If an integer divides each... |
dvds2addd 16232 | Deduction form of ~ dvds2a... |
dvds2subd 16233 | Deduction form of ~ dvds2s... |
dvdstr 16234 | The divides relation is tr... |
dvdstrd 16235 | The divides relation is tr... |
dvdsmultr1 16236 | If an integer divides anot... |
dvdsmultr1d 16237 | Deduction form of ~ dvdsmu... |
dvdsmultr2 16238 | If an integer divides anot... |
dvdsmultr2d 16239 | Deduction form of ~ dvdsmu... |
ordvdsmul 16240 | If an integer divides eith... |
dvdssub2 16241 | If an integer divides a di... |
dvdsadd 16242 | An integer divides another... |
dvdsaddr 16243 | An integer divides another... |
dvdssub 16244 | An integer divides another... |
dvdssubr 16245 | An integer divides another... |
dvdsadd2b 16246 | Adding a multiple of the b... |
dvdsaddre2b 16247 | Adding a multiple of the b... |
fsumdvds 16248 | If every term in a sum is ... |
dvdslelem 16249 | Lemma for ~ dvdsle . (Con... |
dvdsle 16250 | The divisors of a positive... |
dvdsleabs 16251 | The divisors of a nonzero ... |
dvdsleabs2 16252 | Transfer divisibility to a... |
dvdsabseq 16253 | If two integers divide eac... |
dvdseq 16254 | If two nonnegative integer... |
divconjdvds 16255 | If a nonzero integer ` M `... |
dvdsdivcl 16256 | The complement of a diviso... |
dvdsflip 16257 | An involution of the divis... |
dvdsssfz1 16258 | The set of divisors of a n... |
dvds1 16259 | The only nonnegative integ... |
alzdvds 16260 | Only 0 is divisible by all... |
dvdsext 16261 | Poset extensionality for d... |
fzm1ndvds 16262 | No number between ` 1 ` an... |
fzo0dvdseq 16263 | Zero is the only one of th... |
fzocongeq 16264 | Two different elements of ... |
addmodlteqALT 16265 | Two nonnegative integers l... |
dvdsfac 16266 | A positive integer divides... |
dvdsexp2im 16267 | If an integer divides anot... |
dvdsexp 16268 | A power divides a power wi... |
dvdsmod 16269 | Any number ` K ` whose mod... |
mulmoddvds 16270 | If an integer is divisible... |
3dvds 16271 | A rule for divisibility by... |
3dvdsdec 16272 | A decimal number is divisi... |
3dvds2dec 16273 | A decimal number is divisi... |
fprodfvdvdsd 16274 | A finite product of intege... |
fproddvdsd 16275 | A finite product of intege... |
evenelz 16276 | An even number is an integ... |
zeo3 16277 | An integer is even or odd.... |
zeo4 16278 | An integer is even or odd ... |
zeneo 16279 | No even integer equals an ... |
odd2np1lem 16280 | Lemma for ~ odd2np1 . (Co... |
odd2np1 16281 | An integer is odd iff it i... |
even2n 16282 | An integer is even iff it ... |
oddm1even 16283 | An integer is odd iff its ... |
oddp1even 16284 | An integer is odd iff its ... |
oexpneg 16285 | The exponential of the neg... |
mod2eq0even 16286 | An integer is 0 modulo 2 i... |
mod2eq1n2dvds 16287 | An integer is 1 modulo 2 i... |
oddnn02np1 16288 | A nonnegative integer is o... |
oddge22np1 16289 | An integer greater than on... |
evennn02n 16290 | A nonnegative integer is e... |
evennn2n 16291 | A positive integer is even... |
2tp1odd 16292 | A number which is twice an... |
mulsucdiv2z 16293 | An integer multiplied with... |
sqoddm1div8z 16294 | A squared odd number minus... |
2teven 16295 | A number which is twice an... |
zeo5 16296 | An integer is either even ... |
evend2 16297 | An integer is even iff its... |
oddp1d2 16298 | An integer is odd iff its ... |
zob 16299 | Alternate characterization... |
oddm1d2 16300 | An integer is odd iff its ... |
ltoddhalfle 16301 | An integer is less than ha... |
halfleoddlt 16302 | An integer is greater than... |
opoe 16303 | The sum of two odds is eve... |
omoe 16304 | The difference of two odds... |
opeo 16305 | The sum of an odd and an e... |
omeo 16306 | The difference of an odd a... |
z0even 16307 | 2 divides 0. That means 0... |
n2dvds1 16308 | 2 does not divide 1. That... |
n2dvdsm1 16309 | 2 does not divide -1. Tha... |
z2even 16310 | 2 divides 2. That means 2... |
n2dvds3 16311 | 2 does not divide 3. That... |
z4even 16312 | 2 divides 4. That means 4... |
4dvdseven 16313 | An integer which is divisi... |
m1expe 16314 | Exponentiation of -1 by an... |
m1expo 16315 | Exponentiation of -1 by an... |
m1exp1 16316 | Exponentiation of negative... |
nn0enne 16317 | A positive integer is an e... |
nn0ehalf 16318 | The half of an even nonneg... |
nnehalf 16319 | The half of an even positi... |
nn0onn 16320 | An odd nonnegative integer... |
nn0o1gt2 16321 | An odd nonnegative integer... |
nno 16322 | An alternate characterizat... |
nn0o 16323 | An alternate characterizat... |
nn0ob 16324 | Alternate characterization... |
nn0oddm1d2 16325 | A positive integer is odd ... |
nnoddm1d2 16326 | A positive integer is odd ... |
sumeven 16327 | If every term in a sum is ... |
sumodd 16328 | If every term in a sum is ... |
evensumodd 16329 | If every term in a sum wit... |
oddsumodd 16330 | If every term in a sum wit... |
pwp1fsum 16331 | The n-th power of a number... |
oddpwp1fsum 16332 | An odd power of a number i... |
divalglem0 16333 | Lemma for ~ divalg . (Con... |
divalglem1 16334 | Lemma for ~ divalg . (Con... |
divalglem2 16335 | Lemma for ~ divalg . (Con... |
divalglem4 16336 | Lemma for ~ divalg . (Con... |
divalglem5 16337 | Lemma for ~ divalg . (Con... |
divalglem6 16338 | Lemma for ~ divalg . (Con... |
divalglem7 16339 | Lemma for ~ divalg . (Con... |
divalglem8 16340 | Lemma for ~ divalg . (Con... |
divalglem9 16341 | Lemma for ~ divalg . (Con... |
divalglem10 16342 | Lemma for ~ divalg . (Con... |
divalg 16343 | The division algorithm (th... |
divalgb 16344 | Express the division algor... |
divalg2 16345 | The division algorithm (th... |
divalgmod 16346 | The result of the ` mod ` ... |
divalgmodcl 16347 | The result of the ` mod ` ... |
modremain 16348 | The result of the modulo o... |
ndvdssub 16349 | Corollary of the division ... |
ndvdsadd 16350 | Corollary of the division ... |
ndvdsp1 16351 | Special case of ~ ndvdsadd... |
ndvdsi 16352 | A quick test for non-divis... |
flodddiv4 16353 | The floor of an odd intege... |
fldivndvdslt 16354 | The floor of an integer di... |
flodddiv4lt 16355 | The floor of an odd number... |
flodddiv4t2lthalf 16356 | The floor of an odd number... |
bitsfval 16361 | Expand the definition of t... |
bitsval 16362 | Expand the definition of t... |
bitsval2 16363 | Expand the definition of t... |
bitsss 16364 | The set of bits of an inte... |
bitsf 16365 | The ` bits ` function is a... |
bits0 16366 | Value of the zeroth bit. ... |
bits0e 16367 | The zeroth bit of an even ... |
bits0o 16368 | The zeroth bit of an odd n... |
bitsp1 16369 | The ` M + 1 ` -th bit of `... |
bitsp1e 16370 | The ` M + 1 ` -th bit of `... |
bitsp1o 16371 | The ` M + 1 ` -th bit of `... |
bitsfzolem 16372 | Lemma for ~ bitsfzo . (Co... |
bitsfzo 16373 | The bits of a number are a... |
bitsmod 16374 | Truncating the bit sequenc... |
bitsfi 16375 | Every number is associated... |
bitscmp 16376 | The bit complement of ` N ... |
0bits 16377 | The bits of zero. (Contri... |
m1bits 16378 | The bits of negative one. ... |
bitsinv1lem 16379 | Lemma for ~ bitsinv1 . (C... |
bitsinv1 16380 | There is an explicit inver... |
bitsinv2 16381 | There is an explicit inver... |
bitsf1ocnv 16382 | The ` bits ` function rest... |
bitsf1o 16383 | The ` bits ` function rest... |
bitsf1 16384 | The ` bits ` function is a... |
2ebits 16385 | The bits of a power of two... |
bitsinv 16386 | The inverse of the ` bits ... |
bitsinvp1 16387 | Recursive definition of th... |
sadadd2lem2 16388 | The core of the proof of ~... |
sadfval 16390 | Define the addition of two... |
sadcf 16391 | The carry sequence is a se... |
sadc0 16392 | The initial element of the... |
sadcp1 16393 | The carry sequence (which ... |
sadval 16394 | The full adder sequence is... |
sadcaddlem 16395 | Lemma for ~ sadcadd . (Co... |
sadcadd 16396 | Non-recursive definition o... |
sadadd2lem 16397 | Lemma for ~ sadadd2 . (Co... |
sadadd2 16398 | Sum of initial segments of... |
sadadd3 16399 | Sum of initial segments of... |
sadcl 16400 | The sum of two sequences i... |
sadcom 16401 | The adder sequence functio... |
saddisjlem 16402 | Lemma for ~ sadadd . (Con... |
saddisj 16403 | The sum of disjoint sequen... |
sadaddlem 16404 | Lemma for ~ sadadd . (Con... |
sadadd 16405 | For sequences that corresp... |
sadid1 16406 | The adder sequence functio... |
sadid2 16407 | The adder sequence functio... |
sadasslem 16408 | Lemma for ~ sadass . (Con... |
sadass 16409 | Sequence addition is assoc... |
sadeq 16410 | Any element of a sequence ... |
bitsres 16411 | Restrict the bits of a num... |
bitsuz 16412 | The bits of a number are a... |
bitsshft 16413 | Shifting a bit sequence to... |
smufval 16415 | The multiplication of two ... |
smupf 16416 | The sequence of partial su... |
smup0 16417 | The initial element of the... |
smupp1 16418 | The initial element of the... |
smuval 16419 | Define the addition of two... |
smuval2 16420 | The partial sum sequence s... |
smupvallem 16421 | If ` A ` only has elements... |
smucl 16422 | The product of two sequenc... |
smu01lem 16423 | Lemma for ~ smu01 and ~ sm... |
smu01 16424 | Multiplication of a sequen... |
smu02 16425 | Multiplication of a sequen... |
smupval 16426 | Rewrite the elements of th... |
smup1 16427 | Rewrite ~ smupp1 using onl... |
smueqlem 16428 | Any element of a sequence ... |
smueq 16429 | Any element of a sequence ... |
smumullem 16430 | Lemma for ~ smumul . (Con... |
smumul 16431 | For sequences that corresp... |
gcdval 16434 | The value of the ` gcd ` o... |
gcd0val 16435 | The value, by convention, ... |
gcdn0val 16436 | The value of the ` gcd ` o... |
gcdcllem1 16437 | Lemma for ~ gcdn0cl , ~ gc... |
gcdcllem2 16438 | Lemma for ~ gcdn0cl , ~ gc... |
gcdcllem3 16439 | Lemma for ~ gcdn0cl , ~ gc... |
gcdn0cl 16440 | Closure of the ` gcd ` ope... |
gcddvds 16441 | The gcd of two integers di... |
dvdslegcd 16442 | An integer which divides b... |
nndvdslegcd 16443 | A positive integer which d... |
gcdcl 16444 | Closure of the ` gcd ` ope... |
gcdnncl 16445 | Closure of the ` gcd ` ope... |
gcdcld 16446 | Closure of the ` gcd ` ope... |
gcd2n0cl 16447 | Closure of the ` gcd ` ope... |
zeqzmulgcd 16448 | An integer is the product ... |
divgcdz 16449 | An integer divided by the ... |
gcdf 16450 | Domain and codomain of the... |
gcdcom 16451 | The ` gcd ` operator is co... |
gcdcomd 16452 | The ` gcd ` operator is co... |
divgcdnn 16453 | A positive integer divided... |
divgcdnnr 16454 | A positive integer divided... |
gcdeq0 16455 | The gcd of two integers is... |
gcdn0gt0 16456 | The gcd of two integers is... |
gcd0id 16457 | The gcd of 0 and an intege... |
gcdid0 16458 | The gcd of an integer and ... |
nn0gcdid0 16459 | The gcd of a nonnegative i... |
gcdneg 16460 | Negating one operand of th... |
neggcd 16461 | Negating one operand of th... |
gcdaddmlem 16462 | Lemma for ~ gcdaddm . (Co... |
gcdaddm 16463 | Adding a multiple of one o... |
gcdadd 16464 | The GCD of two numbers is ... |
gcdid 16465 | The gcd of a number and it... |
gcd1 16466 | The gcd of a number with 1... |
gcdabs1 16467 | ` gcd ` of the absolute va... |
gcdabs2 16468 | ` gcd ` of the absolute va... |
gcdabs 16469 | The gcd of two integers is... |
gcdabsOLD 16470 | Obsolete version of ~ gcda... |
modgcd 16471 | The gcd remains unchanged ... |
1gcd 16472 | The GCD of one and an inte... |
gcdmultipled 16473 | The greatest common diviso... |
gcdmultiplez 16474 | The GCD of a multiple of a... |
gcdmultiple 16475 | The GCD of a multiple of a... |
dvdsgcdidd 16476 | The greatest common diviso... |
6gcd4e2 16477 | The greatest common diviso... |
bezoutlem1 16478 | Lemma for ~ bezout . (Con... |
bezoutlem2 16479 | Lemma for ~ bezout . (Con... |
bezoutlem3 16480 | Lemma for ~ bezout . (Con... |
bezoutlem4 16481 | Lemma for ~ bezout . (Con... |
bezout 16482 | Bézout's identity: ... |
dvdsgcd 16483 | An integer which divides e... |
dvdsgcdb 16484 | Biconditional form of ~ dv... |
dfgcd2 16485 | Alternate definition of th... |
gcdass 16486 | Associative law for ` gcd ... |
mulgcd 16487 | Distribute multiplication ... |
absmulgcd 16488 | Distribute absolute value ... |
mulgcdr 16489 | Reverse distribution law f... |
gcddiv 16490 | Division law for GCD. (Con... |
gcdzeq 16491 | A positive integer ` A ` i... |
gcdeq 16492 | ` A ` is equal to its gcd ... |
dvdssqim 16493 | Unidirectional form of ~ d... |
dvdsmulgcd 16494 | A divisibility equivalent ... |
rpmulgcd 16495 | If ` K ` and ` M ` are rel... |
rplpwr 16496 | If ` A ` and ` B ` are rel... |
rprpwr 16497 | If ` A ` and ` B ` are rel... |
rppwr 16498 | If ` A ` and ` B ` are rel... |
sqgcd 16499 | Square distributes over gc... |
dvdssqlem 16500 | Lemma for ~ dvdssq . (Con... |
dvdssq 16501 | Two numbers are divisible ... |
bezoutr 16502 | Partial converse to ~ bezo... |
bezoutr1 16503 | Converse of ~ bezout for w... |
nn0seqcvgd 16504 | A strictly-decreasing nonn... |
seq1st 16505 | A sequence whose iteration... |
algr0 16506 | The value of the algorithm... |
algrf 16507 | An algorithm is a step fun... |
algrp1 16508 | The value of the algorithm... |
alginv 16509 | If ` I ` is an invariant o... |
algcvg 16510 | One way to prove that an a... |
algcvgblem 16511 | Lemma for ~ algcvgb . (Co... |
algcvgb 16512 | Two ways of expressing tha... |
algcvga 16513 | The countdown function ` C... |
algfx 16514 | If ` F ` reaches a fixed p... |
eucalgval2 16515 | The value of the step func... |
eucalgval 16516 | Euclid's Algorithm ~ eucal... |
eucalgf 16517 | Domain and codomain of the... |
eucalginv 16518 | The invariant of the step ... |
eucalglt 16519 | The second member of the s... |
eucalgcvga 16520 | Once Euclid's Algorithm ha... |
eucalg 16521 | Euclid's Algorithm compute... |
lcmval 16526 | Value of the ` lcm ` opera... |
lcmcom 16527 | The ` lcm ` operator is co... |
lcm0val 16528 | The value, by convention, ... |
lcmn0val 16529 | The value of the ` lcm ` o... |
lcmcllem 16530 | Lemma for ~ lcmn0cl and ~ ... |
lcmn0cl 16531 | Closure of the ` lcm ` ope... |
dvdslcm 16532 | The lcm of two integers is... |
lcmledvds 16533 | A positive integer which b... |
lcmeq0 16534 | The lcm of two integers is... |
lcmcl 16535 | Closure of the ` lcm ` ope... |
gcddvdslcm 16536 | The greatest common diviso... |
lcmneg 16537 | Negating one operand of th... |
neglcm 16538 | Negating one operand of th... |
lcmabs 16539 | The lcm of two integers is... |
lcmgcdlem 16540 | Lemma for ~ lcmgcd and ~ l... |
lcmgcd 16541 | The product of two numbers... |
lcmdvds 16542 | The lcm of two integers di... |
lcmid 16543 | The lcm of an integer and ... |
lcm1 16544 | The lcm of an integer and ... |
lcmgcdnn 16545 | The product of two positiv... |
lcmgcdeq 16546 | Two integers' absolute val... |
lcmdvdsb 16547 | Biconditional form of ~ lc... |
lcmass 16548 | Associative law for ` lcm ... |
3lcm2e6woprm 16549 | The least common multiple ... |
6lcm4e12 16550 | The least common multiple ... |
absproddvds 16551 | The absolute value of the ... |
absprodnn 16552 | The absolute value of the ... |
fissn0dvds 16553 | For each finite subset of ... |
fissn0dvdsn0 16554 | For each finite subset of ... |
lcmfval 16555 | Value of the ` _lcm ` func... |
lcmf0val 16556 | The value, by convention, ... |
lcmfn0val 16557 | The value of the ` _lcm ` ... |
lcmfnnval 16558 | The value of the ` _lcm ` ... |
lcmfcllem 16559 | Lemma for ~ lcmfn0cl and ~... |
lcmfn0cl 16560 | Closure of the ` _lcm ` fu... |
lcmfpr 16561 | The value of the ` _lcm ` ... |
lcmfcl 16562 | Closure of the ` _lcm ` fu... |
lcmfnncl 16563 | Closure of the ` _lcm ` fu... |
lcmfeq0b 16564 | The least common multiple ... |
dvdslcmf 16565 | The least common multiple ... |
lcmfledvds 16566 | A positive integer which i... |
lcmf 16567 | Characterization of the le... |
lcmf0 16568 | The least common multiple ... |
lcmfsn 16569 | The least common multiple ... |
lcmftp 16570 | The least common multiple ... |
lcmfunsnlem1 16571 | Lemma for ~ lcmfdvds and ~... |
lcmfunsnlem2lem1 16572 | Lemma 1 for ~ lcmfunsnlem2... |
lcmfunsnlem2lem2 16573 | Lemma 2 for ~ lcmfunsnlem2... |
lcmfunsnlem2 16574 | Lemma for ~ lcmfunsn and ~... |
lcmfunsnlem 16575 | Lemma for ~ lcmfdvds and ~... |
lcmfdvds 16576 | The least common multiple ... |
lcmfdvdsb 16577 | Biconditional form of ~ lc... |
lcmfunsn 16578 | The ` _lcm ` function for ... |
lcmfun 16579 | The ` _lcm ` function for ... |
lcmfass 16580 | Associative law for the ` ... |
lcmf2a3a4e12 16581 | The least common multiple ... |
lcmflefac 16582 | The least common multiple ... |
coprmgcdb 16583 | Two positive integers are ... |
ncoprmgcdne1b 16584 | Two positive integers are ... |
ncoprmgcdgt1b 16585 | Two positive integers are ... |
coprmdvds1 16586 | If two positive integers a... |
coprmdvds 16587 | Euclid's Lemma (see ProofW... |
coprmdvds2 16588 | If an integer is divisible... |
mulgcddvds 16589 | One half of ~ rpmulgcd2 , ... |
rpmulgcd2 16590 | If ` M ` is relatively pri... |
qredeq 16591 | Two equal reduced fraction... |
qredeu 16592 | Every rational number has ... |
rpmul 16593 | If ` K ` is relatively pri... |
rpdvds 16594 | If ` K ` is relatively pri... |
coprmprod 16595 | The product of the element... |
coprmproddvdslem 16596 | Lemma for ~ coprmproddvds ... |
coprmproddvds 16597 | If a positive integer is d... |
congr 16598 | Definition of congruence b... |
divgcdcoprm0 16599 | Integers divided by gcd ar... |
divgcdcoprmex 16600 | Integers divided by gcd ar... |
cncongr1 16601 | One direction of the bicon... |
cncongr2 16602 | The other direction of the... |
cncongr 16603 | Cancellability of Congruen... |
cncongrcoprm 16604 | Corollary 1 of Cancellabil... |
isprm 16607 | The predicate "is a prime ... |
prmnn 16608 | A prime number is a positi... |
prmz 16609 | A prime number is an integ... |
prmssnn 16610 | The prime numbers are a su... |
prmex 16611 | The set of prime numbers e... |
0nprm 16612 | 0 is not a prime number. ... |
1nprm 16613 | 1 is not a prime number. ... |
1idssfct 16614 | The positive divisors of a... |
isprm2lem 16615 | Lemma for ~ isprm2 . (Con... |
isprm2 16616 | The predicate "is a prime ... |
isprm3 16617 | The predicate "is a prime ... |
isprm4 16618 | The predicate "is a prime ... |
prmind2 16619 | A variation on ~ prmind as... |
prmind 16620 | Perform induction over the... |
dvdsprime 16621 | If ` M ` divides a prime, ... |
nprm 16622 | A product of two integers ... |
nprmi 16623 | An inference for composite... |
dvdsnprmd 16624 | If a number is divisible b... |
prm2orodd 16625 | A prime number is either 2... |
2prm 16626 | 2 is a prime number. (Con... |
2mulprm 16627 | A multiple of two is prime... |
3prm 16628 | 3 is a prime number. (Con... |
4nprm 16629 | 4 is not a prime number. ... |
prmuz2 16630 | A prime number is an integ... |
prmgt1 16631 | A prime number is an integ... |
prmm2nn0 16632 | Subtracting 2 from a prime... |
oddprmgt2 16633 | An odd prime is greater th... |
oddprmge3 16634 | An odd prime is greater th... |
ge2nprmge4 16635 | A composite integer greate... |
sqnprm 16636 | A square is never prime. ... |
dvdsprm 16637 | An integer greater than or... |
exprmfct 16638 | Every integer greater than... |
prmdvdsfz 16639 | Each integer greater than ... |
nprmdvds1 16640 | No prime number divides 1.... |
isprm5 16641 | One need only check prime ... |
isprm7 16642 | One need only check prime ... |
maxprmfct 16643 | The set of prime factors o... |
divgcdodd 16644 | Either ` A / ( A gcd B ) `... |
coprm 16645 | A prime number either divi... |
prmrp 16646 | Unequal prime numbers are ... |
euclemma 16647 | Euclid's lemma. A prime n... |
isprm6 16648 | A number is prime iff it s... |
prmdvdsexp 16649 | A prime divides a positive... |
prmdvdsexpb 16650 | A prime divides a positive... |
prmdvdsexpr 16651 | If a prime divides a nonne... |
prmdvdssq 16652 | Condition for a prime divi... |
prmdvdssqOLD 16653 | Obsolete version of ~ prmd... |
prmexpb 16654 | Two positive prime powers ... |
prmfac1 16655 | The factorial of a number ... |
dvdszzq 16656 | Divisibility for an intege... |
rpexp 16657 | If two numbers ` A ` and `... |
rpexp1i 16658 | Relative primality passes ... |
rpexp12i 16659 | Relative primality passes ... |
prmndvdsfaclt 16660 | A prime number does not di... |
prmdvdsbc 16661 | Condition for a prime numb... |
prmdvdsncoprmbd 16662 | Two positive integers are ... |
ncoprmlnprm 16663 | If two positive integers a... |
cncongrprm 16664 | Corollary 2 of Cancellabil... |
isevengcd2 16665 | The predicate "is an even ... |
isoddgcd1 16666 | The predicate "is an odd n... |
3lcm2e6 16667 | The least common multiple ... |
qnumval 16672 | Value of the canonical num... |
qdenval 16673 | Value of the canonical den... |
qnumdencl 16674 | Lemma for ~ qnumcl and ~ q... |
qnumcl 16675 | The canonical numerator of... |
qdencl 16676 | The canonical denominator ... |
fnum 16677 | Canonical numerator define... |
fden 16678 | Canonical denominator defi... |
qnumdenbi 16679 | Two numbers are the canoni... |
qnumdencoprm 16680 | The canonical representati... |
qeqnumdivden 16681 | Recover a rational number ... |
qmuldeneqnum 16682 | Multiplying a rational by ... |
divnumden 16683 | Calculate the reduced form... |
divdenle 16684 | Reducing a quotient never ... |
qnumgt0 16685 | A rational is positive iff... |
qgt0numnn 16686 | A rational is positive iff... |
nn0gcdsq 16687 | Squaring commutes with GCD... |
zgcdsq 16688 | ~ nn0gcdsq extended to int... |
numdensq 16689 | Squaring a rational square... |
numsq 16690 | Square commutes with canon... |
densq 16691 | Square commutes with canon... |
qden1elz 16692 | A rational is an integer i... |
zsqrtelqelz 16693 | If an integer has a ration... |
nonsq 16694 | Any integer strictly betwe... |
phival 16699 | Value of the Euler ` phi `... |
phicl2 16700 | Bounds and closure for the... |
phicl 16701 | Closure for the value of t... |
phibndlem 16702 | Lemma for ~ phibnd . (Con... |
phibnd 16703 | A slightly tighter bound o... |
phicld 16704 | Closure for the value of t... |
phi1 16705 | Value of the Euler ` phi `... |
dfphi2 16706 | Alternate definition of th... |
hashdvds 16707 | The number of numbers in a... |
phiprmpw 16708 | Value of the Euler ` phi `... |
phiprm 16709 | Value of the Euler ` phi `... |
crth 16710 | The Chinese Remainder Theo... |
phimullem 16711 | Lemma for ~ phimul . (Con... |
phimul 16712 | The Euler ` phi ` function... |
eulerthlem1 16713 | Lemma for ~ eulerth . (Co... |
eulerthlem2 16714 | Lemma for ~ eulerth . (Co... |
eulerth 16715 | Euler's theorem, a general... |
fermltl 16716 | Fermat's little theorem. ... |
prmdiv 16717 | Show an explicit expressio... |
prmdiveq 16718 | The modular inverse of ` A... |
prmdivdiv 16719 | The (modular) inverse of t... |
hashgcdlem 16720 | A correspondence between e... |
hashgcdeq 16721 | Number of initial positive... |
phisum 16722 | The divisor sum identity o... |
odzval 16723 | Value of the order functio... |
odzcllem 16724 | - Lemma for ~ odzcl , show... |
odzcl 16725 | The order of a group eleme... |
odzid 16726 | Any element raised to the ... |
odzdvds 16727 | The only powers of ` A ` t... |
odzphi 16728 | The order of any group ele... |
modprm1div 16729 | A prime number divides an ... |
m1dvdsndvds 16730 | If an integer minus 1 is d... |
modprminv 16731 | Show an explicit expressio... |
modprminveq 16732 | The modular inverse of ` A... |
vfermltl 16733 | Variant of Fermat's little... |
vfermltlALT 16734 | Alternate proof of ~ vferm... |
powm2modprm 16735 | If an integer minus 1 is d... |
reumodprminv 16736 | For any prime number and f... |
modprm0 16737 | For two positive integers ... |
nnnn0modprm0 16738 | For a positive integer and... |
modprmn0modprm0 16739 | For an integer not being 0... |
coprimeprodsq 16740 | If three numbers are copri... |
coprimeprodsq2 16741 | If three numbers are copri... |
oddprm 16742 | A prime not equal to ` 2 `... |
nnoddn2prm 16743 | A prime not equal to ` 2 `... |
oddn2prm 16744 | A prime not equal to ` 2 `... |
nnoddn2prmb 16745 | A number is a prime number... |
prm23lt5 16746 | A prime less than 5 is eit... |
prm23ge5 16747 | A prime is either 2 or 3 o... |
pythagtriplem1 16748 | Lemma for ~ pythagtrip . ... |
pythagtriplem2 16749 | Lemma for ~ pythagtrip . ... |
pythagtriplem3 16750 | Lemma for ~ pythagtrip . ... |
pythagtriplem4 16751 | Lemma for ~ pythagtrip . ... |
pythagtriplem10 16752 | Lemma for ~ pythagtrip . ... |
pythagtriplem6 16753 | Lemma for ~ pythagtrip . ... |
pythagtriplem7 16754 | Lemma for ~ pythagtrip . ... |
pythagtriplem8 16755 | Lemma for ~ pythagtrip . ... |
pythagtriplem9 16756 | Lemma for ~ pythagtrip . ... |
pythagtriplem11 16757 | Lemma for ~ pythagtrip . ... |
pythagtriplem12 16758 | Lemma for ~ pythagtrip . ... |
pythagtriplem13 16759 | Lemma for ~ pythagtrip . ... |
pythagtriplem14 16760 | Lemma for ~ pythagtrip . ... |
pythagtriplem15 16761 | Lemma for ~ pythagtrip . ... |
pythagtriplem16 16762 | Lemma for ~ pythagtrip . ... |
pythagtriplem17 16763 | Lemma for ~ pythagtrip . ... |
pythagtriplem18 16764 | Lemma for ~ pythagtrip . ... |
pythagtriplem19 16765 | Lemma for ~ pythagtrip . ... |
pythagtrip 16766 | Parameterize the Pythagore... |
iserodd 16767 | Collect the odd terms in a... |
pclem 16770 | - Lemma for the prime powe... |
pcprecl 16771 | Closure of the prime power... |
pcprendvds 16772 | Non-divisibility property ... |
pcprendvds2 16773 | Non-divisibility property ... |
pcpre1 16774 | Value of the prime power p... |
pcpremul 16775 | Multiplicative property of... |
pcval 16776 | The value of the prime pow... |
pceulem 16777 | Lemma for ~ pceu . (Contr... |
pceu 16778 | Uniqueness for the prime p... |
pczpre 16779 | Connect the prime count pr... |
pczcl 16780 | Closure of the prime power... |
pccl 16781 | Closure of the prime power... |
pccld 16782 | Closure of the prime power... |
pcmul 16783 | Multiplication property of... |
pcdiv 16784 | Division property of the p... |
pcqmul 16785 | Multiplication property of... |
pc0 16786 | The value of the prime pow... |
pc1 16787 | Value of the prime count f... |
pcqcl 16788 | Closure of the general pri... |
pcqdiv 16789 | Division property of the p... |
pcrec 16790 | Prime power of a reciproca... |
pcexp 16791 | Prime power of an exponent... |
pcxnn0cl 16792 | Extended nonnegative integ... |
pcxcl 16793 | Extended real closure of t... |
pcge0 16794 | The prime count of an inte... |
pczdvds 16795 | Defining property of the p... |
pcdvds 16796 | Defining property of the p... |
pczndvds 16797 | Defining property of the p... |
pcndvds 16798 | Defining property of the p... |
pczndvds2 16799 | The remainder after dividi... |
pcndvds2 16800 | The remainder after dividi... |
pcdvdsb 16801 | ` P ^ A ` divides ` N ` if... |
pcelnn 16802 | There are a positive numbe... |
pceq0 16803 | There are zero powers of a... |
pcidlem 16804 | The prime count of a prime... |
pcid 16805 | The prime count of a prime... |
pcneg 16806 | The prime count of a negat... |
pcabs 16807 | The prime count of an abso... |
pcdvdstr 16808 | The prime count increases ... |
pcgcd1 16809 | The prime count of a GCD i... |
pcgcd 16810 | The prime count of a GCD i... |
pc2dvds 16811 | A characterization of divi... |
pc11 16812 | The prime count function, ... |
pcz 16813 | The prime count function c... |
pcprmpw2 16814 | Self-referential expressio... |
pcprmpw 16815 | Self-referential expressio... |
dvdsprmpweq 16816 | If a positive integer divi... |
dvdsprmpweqnn 16817 | If an integer greater than... |
dvdsprmpweqle 16818 | If a positive integer divi... |
difsqpwdvds 16819 | If the difference of two s... |
pcaddlem 16820 | Lemma for ~ pcadd . The o... |
pcadd 16821 | An inequality for the prim... |
pcadd2 16822 | The inequality of ~ pcadd ... |
pcmptcl 16823 | Closure for the prime powe... |
pcmpt 16824 | Construct a function with ... |
pcmpt2 16825 | Dividing two prime count m... |
pcmptdvds 16826 | The partial products of th... |
pcprod 16827 | The product of the primes ... |
sumhash 16828 | The sum of 1 over a set is... |
fldivp1 16829 | The difference between the... |
pcfaclem 16830 | Lemma for ~ pcfac . (Cont... |
pcfac 16831 | Calculate the prime count ... |
pcbc 16832 | Calculate the prime count ... |
qexpz 16833 | If a power of a rational n... |
expnprm 16834 | A second or higher power o... |
oddprmdvds 16835 | Every positive integer whi... |
prmpwdvds 16836 | A relation involving divis... |
pockthlem 16837 | Lemma for ~ pockthg . (Co... |
pockthg 16838 | The generalized Pocklingto... |
pockthi 16839 | Pocklington's theorem, whi... |
unbenlem 16840 | Lemma for ~ unben . (Cont... |
unben 16841 | An unbounded set of positi... |
infpnlem1 16842 | Lemma for ~ infpn . The s... |
infpnlem2 16843 | Lemma for ~ infpn . For a... |
infpn 16844 | There exist infinitely man... |
infpn2 16845 | There exist infinitely man... |
prmunb 16846 | The primes are unbounded. ... |
prminf 16847 | There are an infinite numb... |
prmreclem1 16848 | Lemma for ~ prmrec . Prop... |
prmreclem2 16849 | Lemma for ~ prmrec . Ther... |
prmreclem3 16850 | Lemma for ~ prmrec . The ... |
prmreclem4 16851 | Lemma for ~ prmrec . Show... |
prmreclem5 16852 | Lemma for ~ prmrec . Here... |
prmreclem6 16853 | Lemma for ~ prmrec . If t... |
prmrec 16854 | The sum of the reciprocals... |
1arithlem1 16855 | Lemma for ~ 1arith . (Con... |
1arithlem2 16856 | Lemma for ~ 1arith . (Con... |
1arithlem3 16857 | Lemma for ~ 1arith . (Con... |
1arithlem4 16858 | Lemma for ~ 1arith . (Con... |
1arith 16859 | Fundamental theorem of ari... |
1arith2 16860 | Fundamental theorem of ari... |
elgz 16863 | Elementhood in the gaussia... |
gzcn 16864 | A gaussian integer is a co... |
zgz 16865 | An integer is a gaussian i... |
igz 16866 | ` _i ` is a gaussian integ... |
gznegcl 16867 | The gaussian integers are ... |
gzcjcl 16868 | The gaussian integers are ... |
gzaddcl 16869 | The gaussian integers are ... |
gzmulcl 16870 | The gaussian integers are ... |
gzreim 16871 | Construct a gaussian integ... |
gzsubcl 16872 | The gaussian integers are ... |
gzabssqcl 16873 | The squared norm of a gaus... |
4sqlem5 16874 | Lemma for ~ 4sq . (Contri... |
4sqlem6 16875 | Lemma for ~ 4sq . (Contri... |
4sqlem7 16876 | Lemma for ~ 4sq . (Contri... |
4sqlem8 16877 | Lemma for ~ 4sq . (Contri... |
4sqlem9 16878 | Lemma for ~ 4sq . (Contri... |
4sqlem10 16879 | Lemma for ~ 4sq . (Contri... |
4sqlem1 16880 | Lemma for ~ 4sq . The set... |
4sqlem2 16881 | Lemma for ~ 4sq . Change ... |
4sqlem3 16882 | Lemma for ~ 4sq . Suffici... |
4sqlem4a 16883 | Lemma for ~ 4sqlem4 . (Co... |
4sqlem4 16884 | Lemma for ~ 4sq . We can ... |
mul4sqlem 16885 | Lemma for ~ mul4sq : algeb... |
mul4sq 16886 | Euler's four-square identi... |
4sqlem11 16887 | Lemma for ~ 4sq . Use the... |
4sqlem12 16888 | Lemma for ~ 4sq . For any... |
4sqlem13 16889 | Lemma for ~ 4sq . (Contri... |
4sqlem14 16890 | Lemma for ~ 4sq . (Contri... |
4sqlem15 16891 | Lemma for ~ 4sq . (Contri... |
4sqlem16 16892 | Lemma for ~ 4sq . (Contri... |
4sqlem17 16893 | Lemma for ~ 4sq . (Contri... |
4sqlem18 16894 | Lemma for ~ 4sq . Inducti... |
4sqlem19 16895 | Lemma for ~ 4sq . The pro... |
4sq 16896 | Lagrange's four-square the... |
vdwapfval 16903 | Define the arithmetic prog... |
vdwapf 16904 | The arithmetic progression... |
vdwapval 16905 | Value of the arithmetic pr... |
vdwapun 16906 | Remove the first element o... |
vdwapid1 16907 | The first element of an ar... |
vdwap0 16908 | Value of a length-1 arithm... |
vdwap1 16909 | Value of a length-1 arithm... |
vdwmc 16910 | The predicate " The ` <. R... |
vdwmc2 16911 | Expand out the definition ... |
vdwpc 16912 | The predicate " The colori... |
vdwlem1 16913 | Lemma for ~ vdw . (Contri... |
vdwlem2 16914 | Lemma for ~ vdw . (Contri... |
vdwlem3 16915 | Lemma for ~ vdw . (Contri... |
vdwlem4 16916 | Lemma for ~ vdw . (Contri... |
vdwlem5 16917 | Lemma for ~ vdw . (Contri... |
vdwlem6 16918 | Lemma for ~ vdw . (Contri... |
vdwlem7 16919 | Lemma for ~ vdw . (Contri... |
vdwlem8 16920 | Lemma for ~ vdw . (Contri... |
vdwlem9 16921 | Lemma for ~ vdw . (Contri... |
vdwlem10 16922 | Lemma for ~ vdw . Set up ... |
vdwlem11 16923 | Lemma for ~ vdw . (Contri... |
vdwlem12 16924 | Lemma for ~ vdw . ` K = 2 ... |
vdwlem13 16925 | Lemma for ~ vdw . Main in... |
vdw 16926 | Van der Waerden's theorem.... |
vdwnnlem1 16927 | Corollary of ~ vdw , and l... |
vdwnnlem2 16928 | Lemma for ~ vdwnn . The s... |
vdwnnlem3 16929 | Lemma for ~ vdwnn . (Cont... |
vdwnn 16930 | Van der Waerden's theorem,... |
ramtlecl 16932 | The set ` T ` of numbers w... |
hashbcval 16934 | Value of the "binomial set... |
hashbccl 16935 | The binomial set is a fini... |
hashbcss 16936 | Subset relation for the bi... |
hashbc0 16937 | The set of subsets of size... |
hashbc2 16938 | The size of the binomial s... |
0hashbc 16939 | There are no subsets of th... |
ramval 16940 | The value of the Ramsey nu... |
ramcl2lem 16941 | Lemma for extended real cl... |
ramtcl 16942 | The Ramsey number has the ... |
ramtcl2 16943 | The Ramsey number is an in... |
ramtub 16944 | The Ramsey number is a low... |
ramub 16945 | The Ramsey number is a low... |
ramub2 16946 | It is sufficient to check ... |
rami 16947 | The defining property of a... |
ramcl2 16948 | The Ramsey number is eithe... |
ramxrcl 16949 | The Ramsey number is an ex... |
ramubcl 16950 | If the Ramsey number is up... |
ramlb 16951 | Establish a lower bound on... |
0ram 16952 | The Ramsey number when ` M... |
0ram2 16953 | The Ramsey number when ` M... |
ram0 16954 | The Ramsey number when ` R... |
0ramcl 16955 | Lemma for ~ ramcl : Exist... |
ramz2 16956 | The Ramsey number when ` F... |
ramz 16957 | The Ramsey number when ` F... |
ramub1lem1 16958 | Lemma for ~ ramub1 . (Con... |
ramub1lem2 16959 | Lemma for ~ ramub1 . (Con... |
ramub1 16960 | Inductive step for Ramsey'... |
ramcl 16961 | Ramsey's theorem: the Rams... |
ramsey 16962 | Ramsey's theorem with the ... |
prmoval 16965 | Value of the primorial fun... |
prmocl 16966 | Closure of the primorial f... |
prmone0 16967 | The primorial function is ... |
prmo0 16968 | The primorial of 0. (Cont... |
prmo1 16969 | The primorial of 1. (Cont... |
prmop1 16970 | The primorial of a success... |
prmonn2 16971 | Value of the primorial fun... |
prmo2 16972 | The primorial of 2. (Cont... |
prmo3 16973 | The primorial of 3. (Cont... |
prmdvdsprmo 16974 | The primorial of a number ... |
prmdvdsprmop 16975 | The primorial of a number ... |
fvprmselelfz 16976 | The value of the prime sel... |
fvprmselgcd1 16977 | The greatest common diviso... |
prmolefac 16978 | The primorial of a positiv... |
prmodvdslcmf 16979 | The primorial of a nonnega... |
prmolelcmf 16980 | The primorial of a positiv... |
prmgaplem1 16981 | Lemma for ~ prmgap : The ... |
prmgaplem2 16982 | Lemma for ~ prmgap : The ... |
prmgaplcmlem1 16983 | Lemma for ~ prmgaplcm : T... |
prmgaplcmlem2 16984 | Lemma for ~ prmgaplcm : T... |
prmgaplem3 16985 | Lemma for ~ prmgap . (Con... |
prmgaplem4 16986 | Lemma for ~ prmgap . (Con... |
prmgaplem5 16987 | Lemma for ~ prmgap : for e... |
prmgaplem6 16988 | Lemma for ~ prmgap : for e... |
prmgaplem7 16989 | Lemma for ~ prmgap . (Con... |
prmgaplem8 16990 | Lemma for ~ prmgap . (Con... |
prmgap 16991 | The prime gap theorem: for... |
prmgaplcm 16992 | Alternate proof of ~ prmga... |
prmgapprmolem 16993 | Lemma for ~ prmgapprmo : ... |
prmgapprmo 16994 | Alternate proof of ~ prmga... |
dec2dvds 16995 | Divisibility by two is obv... |
dec5dvds 16996 | Divisibility by five is ob... |
dec5dvds2 16997 | Divisibility by five is ob... |
dec5nprm 16998 | Divisibility by five is ob... |
dec2nprm 16999 | Divisibility by two is obv... |
modxai 17000 | Add exponents in a power m... |
mod2xi 17001 | Double exponents in a powe... |
modxp1i 17002 | Add one to an exponent in ... |
mod2xnegi 17003 | Version of ~ mod2xi with a... |
modsubi 17004 | Subtract from within a mod... |
gcdi 17005 | Calculate a GCD via Euclid... |
gcdmodi 17006 | Calculate a GCD via Euclid... |
decexp2 17007 | Calculate a power of two. ... |
numexp0 17008 | Calculate an integer power... |
numexp1 17009 | Calculate an integer power... |
numexpp1 17010 | Calculate an integer power... |
numexp2x 17011 | Double an integer power. ... |
decsplit0b 17012 | Split a decimal number int... |
decsplit0 17013 | Split a decimal number int... |
decsplit1 17014 | Split a decimal number int... |
decsplit 17015 | Split a decimal number int... |
karatsuba 17016 | The Karatsuba multiplicati... |
2exp4 17017 | Two to the fourth power is... |
2exp5 17018 | Two to the fifth power is ... |
2exp6 17019 | Two to the sixth power is ... |
2exp7 17020 | Two to the seventh power i... |
2exp8 17021 | Two to the eighth power is... |
2exp11 17022 | Two to the eleventh power ... |
2exp16 17023 | Two to the sixteenth power... |
3exp3 17024 | Three to the third power i... |
2expltfac 17025 | The factorial grows faster... |
cshwsidrepsw 17026 | If cyclically shifting a w... |
cshwsidrepswmod0 17027 | If cyclically shifting a w... |
cshwshashlem1 17028 | If cyclically shifting a w... |
cshwshashlem2 17029 | If cyclically shifting a w... |
cshwshashlem3 17030 | If cyclically shifting a w... |
cshwsdisj 17031 | The singletons resulting b... |
cshwsiun 17032 | The set of (different!) wo... |
cshwsex 17033 | The class of (different!) ... |
cshws0 17034 | The size of the set of (di... |
cshwrepswhash1 17035 | The size of the set of (di... |
cshwshashnsame 17036 | If a word (not consisting ... |
cshwshash 17037 | If a word has a length bei... |
prmlem0 17038 | Lemma for ~ prmlem1 and ~ ... |
prmlem1a 17039 | A quick proof skeleton to ... |
prmlem1 17040 | A quick proof skeleton to ... |
5prm 17041 | 5 is a prime number. (Con... |
6nprm 17042 | 6 is not a prime number. ... |
7prm 17043 | 7 is a prime number. (Con... |
8nprm 17044 | 8 is not a prime number. ... |
9nprm 17045 | 9 is not a prime number. ... |
10nprm 17046 | 10 is not a prime number. ... |
11prm 17047 | 11 is a prime number. (Co... |
13prm 17048 | 13 is a prime number. (Co... |
17prm 17049 | 17 is a prime number. (Co... |
19prm 17050 | 19 is a prime number. (Co... |
23prm 17051 | 23 is a prime number. (Co... |
prmlem2 17052 | Our last proving session g... |
37prm 17053 | 37 is a prime number. (Co... |
43prm 17054 | 43 is a prime number. (Co... |
83prm 17055 | 83 is a prime number. (Co... |
139prm 17056 | 139 is a prime number. (C... |
163prm 17057 | 163 is a prime number. (C... |
317prm 17058 | 317 is a prime number. (C... |
631prm 17059 | 631 is a prime number. (C... |
prmo4 17060 | The primorial of 4. (Cont... |
prmo5 17061 | The primorial of 5. (Cont... |
prmo6 17062 | The primorial of 6. (Cont... |
1259lem1 17063 | Lemma for ~ 1259prm . Cal... |
1259lem2 17064 | Lemma for ~ 1259prm . Cal... |
1259lem3 17065 | Lemma for ~ 1259prm . Cal... |
1259lem4 17066 | Lemma for ~ 1259prm . Cal... |
1259lem5 17067 | Lemma for ~ 1259prm . Cal... |
1259prm 17068 | 1259 is a prime number. (... |
2503lem1 17069 | Lemma for ~ 2503prm . Cal... |
2503lem2 17070 | Lemma for ~ 2503prm . Cal... |
2503lem3 17071 | Lemma for ~ 2503prm . Cal... |
2503prm 17072 | 2503 is a prime number. (... |
4001lem1 17073 | Lemma for ~ 4001prm . Cal... |
4001lem2 17074 | Lemma for ~ 4001prm . Cal... |
4001lem3 17075 | Lemma for ~ 4001prm . Cal... |
4001lem4 17076 | Lemma for ~ 4001prm . Cal... |
4001prm 17077 | 4001 is a prime number. (... |
brstruct 17080 | The structure relation is ... |
isstruct2 17081 | The property of being a st... |
structex 17082 | A structure is a set. (Co... |
structn0fun 17083 | A structure without the em... |
isstruct 17084 | The property of being a st... |
structcnvcnv 17085 | Two ways to express the re... |
structfung 17086 | The converse of the conver... |
structfun 17087 | Convert between two kinds ... |
structfn 17088 | Convert between two kinds ... |
strleun 17089 | Combine two structures int... |
strle1 17090 | Make a structure from a si... |
strle2 17091 | Make a structure from a pa... |
strle3 17092 | Make a structure from a tr... |
sbcie2s 17093 | A special version of class... |
sbcie3s 17094 | A special version of class... |
reldmsets 17097 | The structure override ope... |
setsvalg 17098 | Value of the structure rep... |
setsval 17099 | Value of the structure rep... |
fvsetsid 17100 | The value of the structure... |
fsets 17101 | The structure replacement ... |
setsdm 17102 | The domain of a structure ... |
setsfun 17103 | A structure with replaceme... |
setsfun0 17104 | A structure with replaceme... |
setsn0fun 17105 | The value of the structure... |
setsstruct2 17106 | An extensible structure wi... |
setsexstruct2 17107 | An extensible structure wi... |
setsstruct 17108 | An extensible structure wi... |
wunsets 17109 | Closure of structure repla... |
setsres 17110 | The structure replacement ... |
setsabs 17111 | Replacing the same compone... |
setscom 17112 | Different components can b... |
sloteq 17115 | Equality theorem for the `... |
slotfn 17116 | A slot is a function on se... |
strfvnd 17117 | Deduction version of ~ str... |
strfvn 17118 | Value of a structure compo... |
strfvss 17119 | A structure component extr... |
wunstr 17120 | Closure of a structure ind... |
str0 17121 | All components of the empt... |
strfvi 17122 | Structure slot extractors ... |
fveqprc 17123 | Lemma for showing the equa... |
oveqprc 17124 | Lemma for showing the equa... |
wunndx 17127 | Closure of the index extra... |
ndxarg 17128 | Get the numeric argument f... |
ndxid 17129 | A structure component extr... |
strndxid 17130 | The value of a structure c... |
setsidvald 17131 | Value of the structure rep... |
setsidvaldOLD 17132 | Obsolete version of ~ sets... |
strfvd 17133 | Deduction version of ~ str... |
strfv2d 17134 | Deduction version of ~ str... |
strfv2 17135 | A variation on ~ strfv to ... |
strfv 17136 | Extract a structure compon... |
strfv3 17137 | Variant on ~ strfv for lar... |
strssd 17138 | Deduction version of ~ str... |
strss 17139 | Propagate component extrac... |
setsid 17140 | Value of the structure rep... |
setsnid 17141 | Value of the structure rep... |
setsnidOLD 17142 | Obsolete proof of ~ setsni... |
baseval 17145 | Value of the base set extr... |
baseid 17146 | Utility theorem: index-ind... |
basfn 17147 | The base set extractor is ... |
base0 17148 | The base set of the empty ... |
elbasfv 17149 | Utility theorem: reverse c... |
elbasov 17150 | Utility theorem: reverse c... |
strov2rcl 17151 | Partial reverse closure fo... |
basendx 17152 | Index value of the base se... |
basendxnn 17153 | The index value of the bas... |
basendxnnOLD 17154 | Obsolete proof of ~ basend... |
basndxelwund 17155 | The index of the base set ... |
basprssdmsets 17156 | The pair of the base index... |
opelstrbas 17157 | The base set of a structur... |
1strstr 17158 | A constructed one-slot str... |
1strstr1 17159 | A constructed one-slot str... |
1strbas 17160 | The base set of a construc... |
1strbasOLD 17161 | Obsolete proof of ~ 1strba... |
1strwunbndx 17162 | A constructed one-slot str... |
1strwun 17163 | A constructed one-slot str... |
1strwunOLD 17164 | Obsolete version of ~ 1str... |
2strstr 17165 | A constructed two-slot str... |
2strbas 17166 | The base set of a construc... |
2strop 17167 | The other slot of a constr... |
2strstr1 17168 | A constructed two-slot str... |
2strstr1OLD 17169 | Obsolete version of ~ 2str... |
2strbas1 17170 | The base set of a construc... |
2strop1 17171 | The other slot of a constr... |
reldmress 17174 | The structure restriction ... |
ressval 17175 | Value of structure restric... |
ressid2 17176 | General behavior of trivia... |
ressval2 17177 | Value of nontrivial struct... |
ressbas 17178 | Base set of a structure re... |
ressbasOLD 17179 | Obsolete proof of ~ ressba... |
ressbasssg 17180 | The base set of a restrict... |
ressbas2 17181 | Base set of a structure re... |
ressbasss 17182 | The base set of a restrict... |
ressbasssOLD 17183 | Obsolete proof of ~ ressba... |
ressbasss2 17184 | The base set of a restrict... |
resseqnbas 17185 | The components of an exten... |
resslemOLD 17186 | Obsolete version of ~ ress... |
ress0 17187 | All restrictions of the nu... |
ressid 17188 | Behavior of trivial restri... |
ressinbas 17189 | Restriction only cares abo... |
ressval3d 17190 | Value of structure restric... |
ressval3dOLD 17191 | Obsolete version of ~ ress... |
ressress 17192 | Restriction composition la... |
ressabs 17193 | Restriction absorption law... |
wunress 17194 | Closure of structure restr... |
wunressOLD 17195 | Obsolete proof of ~ wunres... |
plusgndx 17222 | Index value of the ~ df-pl... |
plusgid 17223 | Utility theorem: index-ind... |
plusgndxnn 17224 | The index of the slot for ... |
basendxltplusgndx 17225 | The index of the slot for ... |
basendxnplusgndx 17226 | The slot for the base set ... |
basendxnplusgndxOLD 17227 | Obsolete version of ~ base... |
grpstr 17228 | A constructed group is a s... |
grpstrndx 17229 | A constructed group is a s... |
grpbase 17230 | The base set of a construc... |
grpbaseOLD 17231 | Obsolete version of ~ grpb... |
grpplusg 17232 | The operation of a constru... |
grpplusgOLD 17233 | Obsolete version of ~ grpp... |
ressplusg 17234 | ` +g ` is unaffected by re... |
grpbasex 17235 | The base of an explicitly ... |
grpplusgx 17236 | The operation of an explic... |
mulrndx 17237 | Index value of the ~ df-mu... |
mulridx 17238 | Utility theorem: index-ind... |
basendxnmulrndx 17239 | The slot for the base set ... |
basendxnmulrndxOLD 17240 | Obsolete proof of ~ basend... |
plusgndxnmulrndx 17241 | The slot for the group (ad... |
rngstr 17242 | A constructed ring is a st... |
rngbase 17243 | The base set of a construc... |
rngplusg 17244 | The additive operation of ... |
rngmulr 17245 | The multiplicative operati... |
starvndx 17246 | Index value of the ~ df-st... |
starvid 17247 | Utility theorem: index-ind... |
starvndxnbasendx 17248 | The slot for the involutio... |
starvndxnplusgndx 17249 | The slot for the involutio... |
starvndxnmulrndx 17250 | The slot for the involutio... |
ressmulr 17251 | ` .r ` is unaffected by re... |
ressstarv 17252 | ` *r ` is unaffected by re... |
srngstr 17253 | A constructed star ring is... |
srngbase 17254 | The base set of a construc... |
srngplusg 17255 | The addition operation of ... |
srngmulr 17256 | The multiplication operati... |
srnginvl 17257 | The involution function of... |
scandx 17258 | Index value of the ~ df-sc... |
scaid 17259 | Utility theorem: index-ind... |
scandxnbasendx 17260 | The slot for the scalar is... |
scandxnplusgndx 17261 | The slot for the scalar fi... |
scandxnmulrndx 17262 | The slot for the scalar fi... |
vscandx 17263 | Index value of the ~ df-vs... |
vscaid 17264 | Utility theorem: index-ind... |
vscandxnbasendx 17265 | The slot for the scalar pr... |
vscandxnplusgndx 17266 | The slot for the scalar pr... |
vscandxnmulrndx 17267 | The slot for the scalar pr... |
vscandxnscandx 17268 | The slot for the scalar pr... |
lmodstr 17269 | A constructed left module ... |
lmodbase 17270 | The base set of a construc... |
lmodplusg 17271 | The additive operation of ... |
lmodsca 17272 | The set of scalars of a co... |
lmodvsca 17273 | The scalar product operati... |
ipndx 17274 | Index value of the ~ df-ip... |
ipid 17275 | Utility theorem: index-ind... |
ipndxnbasendx 17276 | The slot for the inner pro... |
ipndxnplusgndx 17277 | The slot for the inner pro... |
ipndxnmulrndx 17278 | The slot for the inner pro... |
slotsdifipndx 17279 | The slot for the scalar is... |
ipsstr 17280 | Lemma to shorten proofs of... |
ipsbase 17281 | The base set of a construc... |
ipsaddg 17282 | The additive operation of ... |
ipsmulr 17283 | The multiplicative operati... |
ipssca 17284 | The set of scalars of a co... |
ipsvsca 17285 | The scalar product operati... |
ipsip 17286 | The multiplicative operati... |
resssca 17287 | ` Scalar ` is unaffected b... |
ressvsca 17288 | ` .s ` is unaffected by re... |
ressip 17289 | The inner product is unaff... |
phlstr 17290 | A constructed pre-Hilbert ... |
phlbase 17291 | The base set of a construc... |
phlplusg 17292 | The additive operation of ... |
phlsca 17293 | The ring of scalars of a c... |
phlvsca 17294 | The scalar product operati... |
phlip 17295 | The inner product (Hermiti... |
tsetndx 17296 | Index value of the ~ df-ts... |
tsetid 17297 | Utility theorem: index-ind... |
tsetndxnn 17298 | The index of the slot for ... |
basendxlttsetndx 17299 | The index of the slot for ... |
tsetndxnbasendx 17300 | The slot for the topology ... |
tsetndxnplusgndx 17301 | The slot for the topology ... |
tsetndxnmulrndx 17302 | The slot for the topology ... |
tsetndxnstarvndx 17303 | The slot for the topology ... |
slotstnscsi 17304 | The slots ` Scalar ` , ` .... |
topgrpstr 17305 | A constructed topological ... |
topgrpbas 17306 | The base set of a construc... |
topgrpplusg 17307 | The additive operation of ... |
topgrptset 17308 | The topology of a construc... |
resstset 17309 | ` TopSet ` is unaffected b... |
plendx 17310 | Index value of the ~ df-pl... |
pleid 17311 | Utility theorem: self-refe... |
plendxnn 17312 | The index value of the ord... |
basendxltplendx 17313 | The index value of the ` B... |
plendxnbasendx 17314 | The slot for the order is ... |
plendxnplusgndx 17315 | The slot for the "less tha... |
plendxnmulrndx 17316 | The slot for the "less tha... |
plendxnscandx 17317 | The slot for the "less tha... |
plendxnvscandx 17318 | The slot for the "less tha... |
slotsdifplendx 17319 | The index of the slot for ... |
otpsstr 17320 | Functionality of a topolog... |
otpsbas 17321 | The base set of a topologi... |
otpstset 17322 | The open sets of a topolog... |
otpsle 17323 | The order of a topological... |
ressle 17324 | ` le ` is unaffected by re... |
ocndx 17325 | Index value of the ~ df-oc... |
ocid 17326 | Utility theorem: index-ind... |
basendxnocndx 17327 | The slot for the orthocomp... |
plendxnocndx 17328 | The slot for the orthocomp... |
dsndx 17329 | Index value of the ~ df-ds... |
dsid 17330 | Utility theorem: index-ind... |
dsndxnn 17331 | The index of the slot for ... |
basendxltdsndx 17332 | The index of the slot for ... |
dsndxnbasendx 17333 | The slot for the distance ... |
dsndxnplusgndx 17334 | The slot for the distance ... |
dsndxnmulrndx 17335 | The slot for the distance ... |
slotsdnscsi 17336 | The slots ` Scalar ` , ` .... |
dsndxntsetndx 17337 | The slot for the distance ... |
slotsdifdsndx 17338 | The index of the slot for ... |
unifndx 17339 | Index value of the ~ df-un... |
unifid 17340 | Utility theorem: index-ind... |
unifndxnn 17341 | The index of the slot for ... |
basendxltunifndx 17342 | The index of the slot for ... |
unifndxnbasendx 17343 | The slot for the uniform s... |
unifndxntsetndx 17344 | The slot for the uniform s... |
slotsdifunifndx 17345 | The index of the slot for ... |
ressunif 17346 | ` UnifSet ` is unaffected ... |
odrngstr 17347 | Functionality of an ordere... |
odrngbas 17348 | The base set of an ordered... |
odrngplusg 17349 | The addition operation of ... |
odrngmulr 17350 | The multiplication operati... |
odrngtset 17351 | The open sets of an ordere... |
odrngle 17352 | The order of an ordered me... |
odrngds 17353 | The metric of an ordered m... |
ressds 17354 | ` dist ` is unaffected by ... |
homndx 17355 | Index value of the ~ df-ho... |
homid 17356 | Utility theorem: index-ind... |
ccondx 17357 | Index value of the ~ df-cc... |
ccoid 17358 | Utility theorem: index-ind... |
slotsbhcdif 17359 | The slots ` Base ` , ` Hom... |
slotsbhcdifOLD 17360 | Obsolete proof of ~ slotsb... |
slotsdifplendx2 17361 | The index of the slot for ... |
slotsdifocndx 17362 | The index of the slot for ... |
resshom 17363 | ` Hom ` is unaffected by r... |
ressco 17364 | ` comp ` is unaffected by ... |
restfn 17369 | The subspace topology oper... |
topnfn 17370 | The topology extractor fun... |
restval 17371 | The subspace topology indu... |
elrest 17372 | The predicate "is an open ... |
elrestr 17373 | Sufficient condition for b... |
0rest 17374 | Value of the structure res... |
restid2 17375 | The subspace topology over... |
restsspw 17376 | The subspace topology is a... |
firest 17377 | The finite intersections o... |
restid 17378 | The subspace topology of t... |
topnval 17379 | Value of the topology extr... |
topnid 17380 | Value of the topology extr... |
topnpropd 17381 | The topology extractor fun... |
reldmprds 17393 | The structure product is a... |
prdsbasex 17395 | Lemma for structure produc... |
imasvalstr 17396 | An image structure value i... |
prdsvalstr 17397 | Structure product value is... |
prdsbaslem 17398 | Lemma for ~ prdsbas and si... |
prdsvallem 17399 | Lemma for ~ prdsval . (Co... |
prdsval 17400 | Value of the structure pro... |
prdssca 17401 | Scalar ring of a structure... |
prdsbas 17402 | Base set of a structure pr... |
prdsplusg 17403 | Addition in a structure pr... |
prdsmulr 17404 | Multiplication in a struct... |
prdsvsca 17405 | Scalar multiplication in a... |
prdsip 17406 | Inner product in a structu... |
prdsle 17407 | Structure product weak ord... |
prdsless 17408 | Closure of the order relat... |
prdsds 17409 | Structure product distance... |
prdsdsfn 17410 | Structure product distance... |
prdstset 17411 | Structure product topology... |
prdshom 17412 | Structure product hom-sets... |
prdsco 17413 | Structure product composit... |
prdsbas2 17414 | The base set of a structur... |
prdsbasmpt 17415 | A constructed tuple is a p... |
prdsbasfn 17416 | Points in the structure pr... |
prdsbasprj 17417 | Each point in a structure ... |
prdsplusgval 17418 | Value of a componentwise s... |
prdsplusgfval 17419 | Value of a structure produ... |
prdsmulrval 17420 | Value of a componentwise r... |
prdsmulrfval 17421 | Value of a structure produ... |
prdsleval 17422 | Value of the product order... |
prdsdsval 17423 | Value of the metric in a s... |
prdsvscaval 17424 | Scalar multiplication in a... |
prdsvscafval 17425 | Scalar multiplication of a... |
prdsbas3 17426 | The base set of an indexed... |
prdsbasmpt2 17427 | A constructed tuple is a p... |
prdsbascl 17428 | An element of the base has... |
prdsdsval2 17429 | Value of the metric in a s... |
prdsdsval3 17430 | Value of the metric in a s... |
pwsval 17431 | Value of a structure power... |
pwsbas 17432 | Base set of a structure po... |
pwselbasb 17433 | Membership in the base set... |
pwselbas 17434 | An element of a structure ... |
pwsplusgval 17435 | Value of addition in a str... |
pwsmulrval 17436 | Value of multiplication in... |
pwsle 17437 | Ordering in a structure po... |
pwsleval 17438 | Ordering in a structure po... |
pwsvscafval 17439 | Scalar multiplication in a... |
pwsvscaval 17440 | Scalar multiplication of a... |
pwssca 17441 | The ring of scalars of a s... |
pwsdiagel 17442 | Membership of diagonal ele... |
pwssnf1o 17443 | Triviality of singleton po... |
imasval 17456 | Value of an image structur... |
imasbas 17457 | The base set of an image s... |
imasds 17458 | The distance function of a... |
imasdsfn 17459 | The distance function is a... |
imasdsval 17460 | The distance function of a... |
imasdsval2 17461 | The distance function of a... |
imasplusg 17462 | The group operation in an ... |
imasmulr 17463 | The ring multiplication in... |
imassca 17464 | The scalar field of an ima... |
imasvsca 17465 | The scalar multiplication ... |
imasip 17466 | The inner product of an im... |
imastset 17467 | The topology of an image s... |
imasle 17468 | The ordering of an image s... |
f1ocpbllem 17469 | Lemma for ~ f1ocpbl . (Co... |
f1ocpbl 17470 | An injection is compatible... |
f1ovscpbl 17471 | An injection is compatible... |
f1olecpbl 17472 | An injection is compatible... |
imasaddfnlem 17473 | The image structure operat... |
imasaddvallem 17474 | The operation of an image ... |
imasaddflem 17475 | The image set operations a... |
imasaddfn 17476 | The image structure's grou... |
imasaddval 17477 | The value of an image stru... |
imasaddf 17478 | The image structure's grou... |
imasmulfn 17479 | The image structure's ring... |
imasmulval 17480 | The value of an image stru... |
imasmulf 17481 | The image structure's ring... |
imasvscafn 17482 | The image structure's scal... |
imasvscaval 17483 | The value of an image stru... |
imasvscaf 17484 | The image structure's scal... |
imasless 17485 | The order relation defined... |
imasleval 17486 | The value of the image str... |
qusval 17487 | Value of a quotient struct... |
quslem 17488 | The function in ~ qusval i... |
qusin 17489 | Restrict the equivalence r... |
qusbas 17490 | Base set of a quotient str... |
quss 17491 | The scalar field of a quot... |
divsfval 17492 | Value of the function in ~... |
ercpbllem 17493 | Lemma for ~ ercpbl . (Con... |
ercpbl 17494 | Translate the function com... |
erlecpbl 17495 | Translate the relation com... |
qusaddvallem 17496 | Value of an operation defi... |
qusaddflem 17497 | The operation of a quotien... |
qusaddval 17498 | The addition in a quotient... |
qusaddf 17499 | The addition in a quotient... |
qusmulval 17500 | The multiplication in a qu... |
qusmulf 17501 | The multiplication in a qu... |
fnpr2o 17502 | Function with a domain of ... |
fnpr2ob 17503 | Biconditional version of ~... |
fvpr0o 17504 | The value of a function wi... |
fvpr1o 17505 | The value of a function wi... |
fvprif 17506 | The value of the pair func... |
xpsfrnel 17507 | Elementhood in the target ... |
xpsfeq 17508 | A function on ` 2o ` is de... |
xpsfrnel2 17509 | Elementhood in the target ... |
xpscf 17510 | Equivalent condition for t... |
xpsfval 17511 | The value of the function ... |
xpsff1o 17512 | The function appearing in ... |
xpsfrn 17513 | A short expression for the... |
xpsff1o2 17514 | The function appearing in ... |
xpsval 17515 | Value of the binary struct... |
xpsrnbas 17516 | The indexed structure prod... |
xpsbas 17517 | The base set of the binary... |
xpsaddlem 17518 | Lemma for ~ xpsadd and ~ x... |
xpsadd 17519 | Value of the addition oper... |
xpsmul 17520 | Value of the multiplicatio... |
xpssca 17521 | Value of the scalar field ... |
xpsvsca 17522 | Value of the scalar multip... |
xpsless 17523 | Closure of the ordering in... |
xpsle 17524 | Value of the ordering in a... |
ismre 17533 | Property of being a Moore ... |
fnmre 17534 | The Moore collection gener... |
mresspw 17535 | A Moore collection is a su... |
mress 17536 | A Moore-closed subset is a... |
mre1cl 17537 | In any Moore collection th... |
mreintcl 17538 | A nonempty collection of c... |
mreiincl 17539 | A nonempty indexed interse... |
mrerintcl 17540 | The relative intersection ... |
mreriincl 17541 | The relative intersection ... |
mreincl 17542 | Two closed sets have a clo... |
mreuni 17543 | Since the entire base set ... |
mreunirn 17544 | Two ways to express the no... |
ismred 17545 | Properties that determine ... |
ismred2 17546 | Properties that determine ... |
mremre 17547 | The Moore collections of s... |
submre 17548 | The subcollection of a clo... |
mrcflem 17549 | The domain and codomain of... |
fnmrc 17550 | Moore-closure is a well-be... |
mrcfval 17551 | Value of the function expr... |
mrcf 17552 | The Moore closure is a fun... |
mrcval 17553 | Evaluation of the Moore cl... |
mrccl 17554 | The Moore closure of a set... |
mrcsncl 17555 | The Moore closure of a sin... |
mrcid 17556 | The closure of a closed se... |
mrcssv 17557 | The closure of a set is a ... |
mrcidb 17558 | A set is closed iff it is ... |
mrcss 17559 | Closure preserves subset o... |
mrcssid 17560 | The closure of a set is a ... |
mrcidb2 17561 | A set is closed iff it con... |
mrcidm 17562 | The closure operation is i... |
mrcsscl 17563 | The closure is the minimal... |
mrcuni 17564 | Idempotence of closure und... |
mrcun 17565 | Idempotence of closure und... |
mrcssvd 17566 | The Moore closure of a set... |
mrcssd 17567 | Moore closure preserves su... |
mrcssidd 17568 | A set is contained in its ... |
mrcidmd 17569 | Moore closure is idempoten... |
mressmrcd 17570 | In a Moore system, if a se... |
submrc 17571 | In a closure system which ... |
mrieqvlemd 17572 | In a Moore system, if ` Y ... |
mrisval 17573 | Value of the set of indepe... |
ismri 17574 | Criterion for a set to be ... |
ismri2 17575 | Criterion for a subset of ... |
ismri2d 17576 | Criterion for a subset of ... |
ismri2dd 17577 | Definition of independence... |
mriss 17578 | An independent set of a Mo... |
mrissd 17579 | An independent set of a Mo... |
ismri2dad 17580 | Consequence of a set in a ... |
mrieqvd 17581 | In a Moore system, a set i... |
mrieqv2d 17582 | In a Moore system, a set i... |
mrissmrcd 17583 | In a Moore system, if an i... |
mrissmrid 17584 | In a Moore system, subsets... |
mreexd 17585 | In a Moore system, the clo... |
mreexmrid 17586 | In a Moore system whose cl... |
mreexexlemd 17587 | This lemma is used to gene... |
mreexexlem2d 17588 | Used in ~ mreexexlem4d to ... |
mreexexlem3d 17589 | Base case of the induction... |
mreexexlem4d 17590 | Induction step of the indu... |
mreexexd 17591 | Exchange-type theorem. In... |
mreexdomd 17592 | In a Moore system whose cl... |
mreexfidimd 17593 | In a Moore system whose cl... |
isacs 17594 | A set is an algebraic clos... |
acsmre 17595 | Algebraic closure systems ... |
isacs2 17596 | In the definition of an al... |
acsfiel 17597 | A set is closed in an alge... |
acsfiel2 17598 | A set is closed in an alge... |
acsmred 17599 | An algebraic closure syste... |
isacs1i 17600 | A closure system determine... |
mreacs 17601 | Algebraicity is a composab... |
acsfn 17602 | Algebraicity of a conditio... |
acsfn0 17603 | Algebraicity of a point cl... |
acsfn1 17604 | Algebraicity of a one-argu... |
acsfn1c 17605 | Algebraicity of a one-argu... |
acsfn2 17606 | Algebraicity of a two-argu... |
iscat 17615 | The predicate "is a catego... |
iscatd 17616 | Properties that determine ... |
catidex 17617 | Each object in a category ... |
catideu 17618 | Each object in a category ... |
cidfval 17619 | Each object in a category ... |
cidval 17620 | Each object in a category ... |
cidffn 17621 | The identity arrow constru... |
cidfn 17622 | The identity arrow operato... |
catidd 17623 | Deduce the identity arrow ... |
iscatd2 17624 | Version of ~ iscatd with a... |
catidcl 17625 | Each object in a category ... |
catlid 17626 | Left identity property of ... |
catrid 17627 | Right identity property of... |
catcocl 17628 | Closure of a composition a... |
catass 17629 | Associativity of compositi... |
catcone0 17630 | Composition of non-empty h... |
0catg 17631 | Any structure with an empt... |
0cat 17632 | The empty set is a categor... |
homffval 17633 | Value of the functionalize... |
fnhomeqhomf 17634 | If the Hom-set operation i... |
homfval 17635 | Value of the functionalize... |
homffn 17636 | The functionalized Hom-set... |
homfeq 17637 | Condition for two categori... |
homfeqd 17638 | If two structures have the... |
homfeqbas 17639 | Deduce equality of base se... |
homfeqval 17640 | Value of the functionalize... |
comfffval 17641 | Value of the functionalize... |
comffval 17642 | Value of the functionalize... |
comfval 17643 | Value of the functionalize... |
comfffval2 17644 | Value of the functionalize... |
comffval2 17645 | Value of the functionalize... |
comfval2 17646 | Value of the functionalize... |
comfffn 17647 | The functionalized composi... |
comffn 17648 | The functionalized composi... |
comfeq 17649 | Condition for two categori... |
comfeqd 17650 | Condition for two categori... |
comfeqval 17651 | Equality of two compositio... |
catpropd 17652 | Two structures with the sa... |
cidpropd 17653 | Two structures with the sa... |
oppcval 17656 | Value of the opposite cate... |
oppchomfval 17657 | Hom-sets of the opposite c... |
oppchomfvalOLD 17658 | Obsolete proof of ~ oppcho... |
oppchom 17659 | Hom-sets of the opposite c... |
oppccofval 17660 | Composition in the opposit... |
oppcco 17661 | Composition in the opposit... |
oppcbas 17662 | Base set of an opposite ca... |
oppcbasOLD 17663 | Obsolete version of ~ oppc... |
oppccatid 17664 | Lemma for ~ oppccat . (Co... |
oppchomf 17665 | Hom-sets of the opposite c... |
oppcid 17666 | Identity function of an op... |
oppccat 17667 | An opposite category is a ... |
2oppcbas 17668 | The double opposite catego... |
2oppchomf 17669 | The double opposite catego... |
2oppccomf 17670 | The double opposite catego... |
oppchomfpropd 17671 | If two categories have the... |
oppccomfpropd 17672 | If two categories have the... |
oppccatf 17673 | ` oppCat ` restricted to `... |
monfval 17678 | Definition of a monomorphi... |
ismon 17679 | Definition of a monomorphi... |
ismon2 17680 | Write out the monomorphism... |
monhom 17681 | A monomorphism is a morphi... |
moni 17682 | Property of a monomorphism... |
monpropd 17683 | If two categories have the... |
oppcmon 17684 | A monomorphism in the oppo... |
oppcepi 17685 | An epimorphism in the oppo... |
isepi 17686 | Definition of an epimorphi... |
isepi2 17687 | Write out the epimorphism ... |
epihom 17688 | An epimorphism is a morphi... |
epii 17689 | Property of an epimorphism... |
sectffval 17696 | Value of the section opera... |
sectfval 17697 | Value of the section relat... |
sectss 17698 | The section relation is a ... |
issect 17699 | The property " ` F ` is a ... |
issect2 17700 | Property of being a sectio... |
sectcan 17701 | If ` G ` is a section of `... |
sectco 17702 | Composition of two section... |
isofval 17703 | Function value of the func... |
invffval 17704 | Value of the inverse relat... |
invfval 17705 | Value of the inverse relat... |
isinv 17706 | Value of the inverse relat... |
invss 17707 | The inverse relation is a ... |
invsym 17708 | The inverse relation is sy... |
invsym2 17709 | The inverse relation is sy... |
invfun 17710 | The inverse relation is a ... |
isoval 17711 | The isomorphisms are the d... |
inviso1 17712 | If ` G ` is an inverse to ... |
inviso2 17713 | If ` G ` is an inverse to ... |
invf 17714 | The inverse relation is a ... |
invf1o 17715 | The inverse relation is a ... |
invinv 17716 | The inverse of the inverse... |
invco 17717 | The composition of two iso... |
dfiso2 17718 | Alternate definition of an... |
dfiso3 17719 | Alternate definition of an... |
inveq 17720 | If there are two inverses ... |
isofn 17721 | The function value of the ... |
isohom 17722 | An isomorphism is a homomo... |
isoco 17723 | The composition of two iso... |
oppcsect 17724 | A section in the opposite ... |
oppcsect2 17725 | A section in the opposite ... |
oppcinv 17726 | An inverse in the opposite... |
oppciso 17727 | An isomorphism in the oppo... |
sectmon 17728 | If ` F ` is a section of `... |
monsect 17729 | If ` F ` is a monomorphism... |
sectepi 17730 | If ` F ` is a section of `... |
episect 17731 | If ` F ` is an epimorphism... |
sectid 17732 | The identity is a section ... |
invid 17733 | The inverse of the identit... |
idiso 17734 | The identity is an isomorp... |
idinv 17735 | The inverse of the identit... |
invisoinvl 17736 | The inverse of an isomorph... |
invisoinvr 17737 | The inverse of an isomorph... |
invcoisoid 17738 | The inverse of an isomorph... |
isocoinvid 17739 | The inverse of an isomorph... |
rcaninv 17740 | Right cancellation of an i... |
cicfval 17743 | The set of isomorphic obje... |
brcic 17744 | The relation "is isomorphi... |
cic 17745 | Objects ` X ` and ` Y ` in... |
brcici 17746 | Prove that two objects are... |
cicref 17747 | Isomorphism is reflexive. ... |
ciclcl 17748 | Isomorphism implies the le... |
cicrcl 17749 | Isomorphism implies the ri... |
cicsym 17750 | Isomorphism is symmetric. ... |
cictr 17751 | Isomorphism is transitive.... |
cicer 17752 | Isomorphism is an equivale... |
sscrel 17759 | The subcategory subset rel... |
brssc 17760 | The subcategory subset rel... |
sscpwex 17761 | An analogue of ~ pwex for ... |
subcrcl 17762 | Reverse closure for the su... |
sscfn1 17763 | The subcategory subset rel... |
sscfn2 17764 | The subcategory subset rel... |
ssclem 17765 | Lemma for ~ ssc1 and simil... |
isssc 17766 | Value of the subcategory s... |
ssc1 17767 | Infer subset relation on o... |
ssc2 17768 | Infer subset relation on m... |
sscres 17769 | Any function restricted to... |
sscid 17770 | The subcategory subset rel... |
ssctr 17771 | The subcategory subset rel... |
ssceq 17772 | The subcategory subset rel... |
rescval 17773 | Value of the category rest... |
rescval2 17774 | Value of the category rest... |
rescbas 17775 | Base set of the category r... |
rescbasOLD 17776 | Obsolete version of ~ resc... |
reschom 17777 | Hom-sets of the category r... |
reschomf 17778 | Hom-sets of the category r... |
rescco 17779 | Composition in the categor... |
resccoOLD 17780 | Obsolete proof of ~ rescco... |
rescabs 17781 | Restriction absorption law... |
rescabsOLD 17782 | Obsolete proof of ~ seqp1d... |
rescabs2 17783 | Restriction absorption law... |
issubc 17784 | Elementhood in the set of ... |
issubc2 17785 | Elementhood in the set of ... |
0ssc 17786 | For any category ` C ` , t... |
0subcat 17787 | For any category ` C ` , t... |
catsubcat 17788 | For any category ` C ` , `... |
subcssc 17789 | An element in the set of s... |
subcfn 17790 | An element in the set of s... |
subcss1 17791 | The objects of a subcatego... |
subcss2 17792 | The morphisms of a subcate... |
subcidcl 17793 | The identity of the origin... |
subccocl 17794 | A subcategory is closed un... |
subccatid 17795 | A subcategory is a categor... |
subcid 17796 | The identity in a subcateg... |
subccat 17797 | A subcategory is a categor... |
issubc3 17798 | Alternate definition of a ... |
fullsubc 17799 | The full subcategory gener... |
fullresc 17800 | The category formed by str... |
resscat 17801 | A category restricted to a... |
subsubc 17802 | A subcategory of a subcate... |
relfunc 17811 | The set of functors is a r... |
funcrcl 17812 | Reverse closure for a func... |
isfunc 17813 | Value of the set of functo... |
isfuncd 17814 | Deduce that an operation i... |
funcf1 17815 | The object part of a funct... |
funcixp 17816 | The morphism part of a fun... |
funcf2 17817 | The morphism part of a fun... |
funcfn2 17818 | The morphism part of a fun... |
funcid 17819 | A functor maps each identi... |
funcco 17820 | A functor maps composition... |
funcsect 17821 | The image of a section und... |
funcinv 17822 | The image of an inverse un... |
funciso 17823 | The image of an isomorphis... |
funcoppc 17824 | A functor on categories yi... |
idfuval 17825 | Value of the identity func... |
idfu2nd 17826 | Value of the morphism part... |
idfu2 17827 | Value of the morphism part... |
idfu1st 17828 | Value of the object part o... |
idfu1 17829 | Value of the object part o... |
idfucl 17830 | The identity functor is a ... |
cofuval 17831 | Value of the composition o... |
cofu1st 17832 | Value of the object part o... |
cofu1 17833 | Value of the object part o... |
cofu2nd 17834 | Value of the morphism part... |
cofu2 17835 | Value of the morphism part... |
cofuval2 17836 | Value of the composition o... |
cofucl 17837 | The composition of two fun... |
cofuass 17838 | Functor composition is ass... |
cofulid 17839 | The identity functor is a ... |
cofurid 17840 | The identity functor is a ... |
resfval 17841 | Value of the functor restr... |
resfval2 17842 | Value of the functor restr... |
resf1st 17843 | Value of the functor restr... |
resf2nd 17844 | Value of the functor restr... |
funcres 17845 | A functor restricted to a ... |
funcres2b 17846 | Condition for a functor to... |
funcres2 17847 | A functor into a restricte... |
idfusubc0 17848 | The identity functor for a... |
idfusubc 17849 | The identity functor for a... |
wunfunc 17850 | A weak universe is closed ... |
wunfuncOLD 17851 | Obsolete proof of ~ wunfun... |
funcpropd 17852 | If two categories have the... |
funcres2c 17853 | Condition for a functor to... |
fullfunc 17858 | A full functor is a functo... |
fthfunc 17859 | A faithful functor is a fu... |
relfull 17860 | The set of full functors i... |
relfth 17861 | The set of faithful functo... |
isfull 17862 | Value of the set of full f... |
isfull2 17863 | Equivalent condition for a... |
fullfo 17864 | The morphism map of a full... |
fulli 17865 | The morphism map of a full... |
isfth 17866 | Value of the set of faithf... |
isfth2 17867 | Equivalent condition for a... |
isffth2 17868 | A fully faithful functor i... |
fthf1 17869 | The morphism map of a fait... |
fthi 17870 | The morphism map of a fait... |
ffthf1o 17871 | The morphism map of a full... |
fullpropd 17872 | If two categories have the... |
fthpropd 17873 | If two categories have the... |
fulloppc 17874 | The opposite functor of a ... |
fthoppc 17875 | The opposite functor of a ... |
ffthoppc 17876 | The opposite functor of a ... |
fthsect 17877 | A faithful functor reflect... |
fthinv 17878 | A faithful functor reflect... |
fthmon 17879 | A faithful functor reflect... |
fthepi 17880 | A faithful functor reflect... |
ffthiso 17881 | A fully faithful functor r... |
fthres2b 17882 | Condition for a faithful f... |
fthres2c 17883 | Condition for a faithful f... |
fthres2 17884 | A faithful functor into a ... |
idffth 17885 | The identity functor is a ... |
cofull 17886 | The composition of two ful... |
cofth 17887 | The composition of two fai... |
coffth 17888 | The composition of two ful... |
rescfth 17889 | The inclusion functor from... |
ressffth 17890 | The inclusion functor from... |
fullres2c 17891 | Condition for a full funct... |
ffthres2c 17892 | Condition for a fully fait... |
inclfusubc 17893 | The "inclusion functor" fr... |
fnfuc 17898 | The ` FuncCat ` operation ... |
natfval 17899 | Value of the function givi... |
isnat 17900 | Property of being a natura... |
isnat2 17901 | Property of being a natura... |
natffn 17902 | The natural transformation... |
natrcl 17903 | Reverse closure for a natu... |
nat1st2nd 17904 | Rewrite the natural transf... |
natixp 17905 | A natural transformation i... |
natcl 17906 | A component of a natural t... |
natfn 17907 | A natural transformation i... |
nati 17908 | Naturality property of a n... |
wunnat 17909 | A weak universe is closed ... |
wunnatOLD 17910 | Obsolete proof of ~ wunnat... |
catstr 17911 | A category structure is a ... |
fucval 17912 | Value of the functor categ... |
fuccofval 17913 | Value of the functor categ... |
fucbas 17914 | The objects of the functor... |
fuchom 17915 | The morphisms in the funct... |
fuchomOLD 17916 | Obsolete proof of ~ fuchom... |
fucco 17917 | Value of the composition o... |
fuccoval 17918 | Value of the functor categ... |
fuccocl 17919 | The composition of two nat... |
fucidcl 17920 | The identity natural trans... |
fuclid 17921 | Left identity of natural t... |
fucrid 17922 | Right identity of natural ... |
fucass 17923 | Associativity of natural t... |
fuccatid 17924 | The functor category is a ... |
fuccat 17925 | The functor category is a ... |
fucid 17926 | The identity morphism in t... |
fucsect 17927 | Two natural transformation... |
fucinv 17928 | Two natural transformation... |
invfuc 17929 | If ` V ( x ) ` is an inver... |
fuciso 17930 | A natural transformation i... |
natpropd 17931 | If two categories have the... |
fucpropd 17932 | If two categories have the... |
initofn 17939 | ` InitO ` is a function on... |
termofn 17940 | ` TermO ` is a function on... |
zeroofn 17941 | ` ZeroO ` is a function on... |
initorcl 17942 | Reverse closure for an ini... |
termorcl 17943 | Reverse closure for a term... |
zeroorcl 17944 | Reverse closure for a zero... |
initoval 17945 | The value of the initial o... |
termoval 17946 | The value of the terminal ... |
zerooval 17947 | The value of the zero obje... |
isinito 17948 | The predicate "is an initi... |
istermo 17949 | The predicate "is a termin... |
iszeroo 17950 | The predicate "is a zero o... |
isinitoi 17951 | Implication of a class bei... |
istermoi 17952 | Implication of a class bei... |
initoid 17953 | For an initial object, the... |
termoid 17954 | For a terminal object, the... |
dfinito2 17955 | An initial object is a ter... |
dftermo2 17956 | A terminal object is an in... |
dfinito3 17957 | An alternate definition of... |
dftermo3 17958 | An alternate definition of... |
initoo 17959 | An initial object is an ob... |
termoo 17960 | A terminal object is an ob... |
iszeroi 17961 | Implication of a class bei... |
2initoinv 17962 | Morphisms between two init... |
initoeu1 17963 | Initial objects are essent... |
initoeu1w 17964 | Initial objects are essent... |
initoeu2lem0 17965 | Lemma 0 for ~ initoeu2 . ... |
initoeu2lem1 17966 | Lemma 1 for ~ initoeu2 . ... |
initoeu2lem2 17967 | Lemma 2 for ~ initoeu2 . ... |
initoeu2 17968 | Initial objects are essent... |
2termoinv 17969 | Morphisms between two term... |
termoeu1 17970 | Terminal objects are essen... |
termoeu1w 17971 | Terminal objects are essen... |
homarcl 17980 | Reverse closure for an arr... |
homafval 17981 | Value of the disjointified... |
homaf 17982 | Functionality of the disjo... |
homaval 17983 | Value of the disjointified... |
elhoma 17984 | Value of the disjointified... |
elhomai 17985 | Produce an arrow from a mo... |
elhomai2 17986 | Produce an arrow from a mo... |
homarcl2 17987 | Reverse closure for the do... |
homarel 17988 | An arrow is an ordered pai... |
homa1 17989 | The first component of an ... |
homahom2 17990 | The second component of an... |
homahom 17991 | The second component of an... |
homadm 17992 | The domain of an arrow wit... |
homacd 17993 | The codomain of an arrow w... |
homadmcd 17994 | Decompose an arrow into do... |
arwval 17995 | The set of arrows is the u... |
arwrcl 17996 | The first component of an ... |
arwhoma 17997 | An arrow is contained in t... |
homarw 17998 | A hom-set is a subset of t... |
arwdm 17999 | The domain of an arrow is ... |
arwcd 18000 | The codomain of an arrow i... |
dmaf 18001 | The domain function is a f... |
cdaf 18002 | The codomain function is a... |
arwhom 18003 | The second component of an... |
arwdmcd 18004 | Decompose an arrow into do... |
idafval 18009 | Value of the identity arro... |
idaval 18010 | Value of the identity arro... |
ida2 18011 | Morphism part of the ident... |
idahom 18012 | Domain and codomain of the... |
idadm 18013 | Domain of the identity arr... |
idacd 18014 | Codomain of the identity a... |
idaf 18015 | The identity arrow functio... |
coafval 18016 | The value of the compositi... |
eldmcoa 18017 | A pair ` <. G , F >. ` is ... |
dmcoass 18018 | The domain of composition ... |
homdmcoa 18019 | If ` F : X --> Y ` and ` G... |
coaval 18020 | Value of composition for c... |
coa2 18021 | The morphism part of arrow... |
coahom 18022 | The composition of two com... |
coapm 18023 | Composition of arrows is a... |
arwlid 18024 | Left identity of a categor... |
arwrid 18025 | Right identity of a catego... |
arwass 18026 | Associativity of compositi... |
setcval 18029 | Value of the category of s... |
setcbas 18030 | Set of objects of the cate... |
setchomfval 18031 | Set of arrows of the categ... |
setchom 18032 | Set of arrows of the categ... |
elsetchom 18033 | A morphism of sets is a fu... |
setccofval 18034 | Composition in the categor... |
setcco 18035 | Composition in the categor... |
setccatid 18036 | Lemma for ~ setccat . (Co... |
setccat 18037 | The category of sets is a ... |
setcid 18038 | The identity arrow in the ... |
setcmon 18039 | A monomorphism of sets is ... |
setcepi 18040 | An epimorphism of sets is ... |
setcsect 18041 | A section in the category ... |
setcinv 18042 | An inverse in the category... |
setciso 18043 | An isomorphism in the cate... |
resssetc 18044 | The restriction of the cat... |
funcsetcres2 18045 | A functor into a smaller c... |
setc2obas 18046 | ` (/) ` and ` 1o ` are dis... |
setc2ohom 18047 | ` ( SetCat `` 2o ) ` is a ... |
cat1lem 18048 | The category of sets in a ... |
cat1 18049 | The definition of category... |
catcval 18052 | Value of the category of c... |
catcbas 18053 | Set of objects of the cate... |
catchomfval 18054 | Set of arrows of the categ... |
catchom 18055 | Set of arrows of the categ... |
catccofval 18056 | Composition in the categor... |
catcco 18057 | Composition in the categor... |
catccatid 18058 | Lemma for ~ catccat . (Co... |
catcid 18059 | The identity arrow in the ... |
catccat 18060 | The category of categories... |
resscatc 18061 | The restriction of the cat... |
catcisolem 18062 | Lemma for ~ catciso . (Co... |
catciso 18063 | A functor is an isomorphis... |
catcbascl 18064 | An element of the base set... |
catcslotelcl 18065 | A slot entry of an element... |
catcbaselcl 18066 | The base set of an element... |
catchomcl 18067 | The Hom-set of an element ... |
catcccocl 18068 | The composition operation ... |
catcoppccl 18069 | The category of categories... |
catcoppcclOLD 18070 | Obsolete proof of ~ catcop... |
catcfuccl 18071 | The category of categories... |
catcfucclOLD 18072 | Obsolete proof of ~ catcfu... |
fncnvimaeqv 18073 | The inverse images of the ... |
bascnvimaeqv 18074 | The inverse image of the u... |
estrcval 18077 | Value of the category of e... |
estrcbas 18078 | Set of objects of the cate... |
estrchomfval 18079 | Set of morphisms ("arrows"... |
estrchom 18080 | The morphisms between exte... |
elestrchom 18081 | A morphism between extensi... |
estrccofval 18082 | Composition in the categor... |
estrcco 18083 | Composition in the categor... |
estrcbasbas 18084 | An element of the base set... |
estrccatid 18085 | Lemma for ~ estrccat . (C... |
estrccat 18086 | The category of extensible... |
estrcid 18087 | The identity arrow in the ... |
estrchomfn 18088 | The Hom-set operation in t... |
estrchomfeqhom 18089 | The functionalized Hom-set... |
estrreslem1 18090 | Lemma 1 for ~ estrres . (... |
estrreslem1OLD 18091 | Obsolete version of ~ estr... |
estrreslem2 18092 | Lemma 2 for ~ estrres . (... |
estrres 18093 | Any restriction of a categ... |
funcestrcsetclem1 18094 | Lemma 1 for ~ funcestrcset... |
funcestrcsetclem2 18095 | Lemma 2 for ~ funcestrcset... |
funcestrcsetclem3 18096 | Lemma 3 for ~ funcestrcset... |
funcestrcsetclem4 18097 | Lemma 4 for ~ funcestrcset... |
funcestrcsetclem5 18098 | Lemma 5 for ~ funcestrcset... |
funcestrcsetclem6 18099 | Lemma 6 for ~ funcestrcset... |
funcestrcsetclem7 18100 | Lemma 7 for ~ funcestrcset... |
funcestrcsetclem8 18101 | Lemma 8 for ~ funcestrcset... |
funcestrcsetclem9 18102 | Lemma 9 for ~ funcestrcset... |
funcestrcsetc 18103 | The "natural forgetful fun... |
fthestrcsetc 18104 | The "natural forgetful fun... |
fullestrcsetc 18105 | The "natural forgetful fun... |
equivestrcsetc 18106 | The "natural forgetful fun... |
setc1strwun 18107 | A constructed one-slot str... |
funcsetcestrclem1 18108 | Lemma 1 for ~ funcsetcestr... |
funcsetcestrclem2 18109 | Lemma 2 for ~ funcsetcestr... |
funcsetcestrclem3 18110 | Lemma 3 for ~ funcsetcestr... |
embedsetcestrclem 18111 | Lemma for ~ embedsetcestrc... |
funcsetcestrclem4 18112 | Lemma 4 for ~ funcsetcestr... |
funcsetcestrclem5 18113 | Lemma 5 for ~ funcsetcestr... |
funcsetcestrclem6 18114 | Lemma 6 for ~ funcsetcestr... |
funcsetcestrclem7 18115 | Lemma 7 for ~ funcsetcestr... |
funcsetcestrclem8 18116 | Lemma 8 for ~ funcsetcestr... |
funcsetcestrclem9 18117 | Lemma 9 for ~ funcsetcestr... |
funcsetcestrc 18118 | The "embedding functor" fr... |
fthsetcestrc 18119 | The "embedding functor" fr... |
fullsetcestrc 18120 | The "embedding functor" fr... |
embedsetcestrc 18121 | The "embedding functor" fr... |
fnxpc 18130 | The binary product of cate... |
xpcval 18131 | Value of the binary produc... |
xpcbas 18132 | Set of objects of the bina... |
xpchomfval 18133 | Set of morphisms of the bi... |
xpchom 18134 | Set of morphisms of the bi... |
relxpchom 18135 | A hom-set in the binary pr... |
xpccofval 18136 | Value of composition in th... |
xpcco 18137 | Value of composition in th... |
xpcco1st 18138 | Value of composition in th... |
xpcco2nd 18139 | Value of composition in th... |
xpchom2 18140 | Value of the set of morphi... |
xpcco2 18141 | Value of composition in th... |
xpccatid 18142 | The product of two categor... |
xpcid 18143 | The identity morphism in t... |
xpccat 18144 | The product of two categor... |
1stfval 18145 | Value of the first project... |
1stf1 18146 | Value of the first project... |
1stf2 18147 | Value of the first project... |
2ndfval 18148 | Value of the first project... |
2ndf1 18149 | Value of the first project... |
2ndf2 18150 | Value of the first project... |
1stfcl 18151 | The first projection funct... |
2ndfcl 18152 | The second projection func... |
prfval 18153 | Value of the pairing funct... |
prf1 18154 | Value of the pairing funct... |
prf2fval 18155 | Value of the pairing funct... |
prf2 18156 | Value of the pairing funct... |
prfcl 18157 | The pairing of functors ` ... |
prf1st 18158 | Cancellation of pairing wi... |
prf2nd 18159 | Cancellation of pairing wi... |
1st2ndprf 18160 | Break a functor into a pro... |
catcxpccl 18161 | The category of categories... |
catcxpcclOLD 18162 | Obsolete proof of ~ catcxp... |
xpcpropd 18163 | If two categories have the... |
evlfval 18172 | Value of the evaluation fu... |
evlf2 18173 | Value of the evaluation fu... |
evlf2val 18174 | Value of the evaluation na... |
evlf1 18175 | Value of the evaluation fu... |
evlfcllem 18176 | Lemma for ~ evlfcl . (Con... |
evlfcl 18177 | The evaluation functor is ... |
curfval 18178 | Value of the curry functor... |
curf1fval 18179 | Value of the object part o... |
curf1 18180 | Value of the object part o... |
curf11 18181 | Value of the double evalua... |
curf12 18182 | The partially evaluated cu... |
curf1cl 18183 | The partially evaluated cu... |
curf2 18184 | Value of the curry functor... |
curf2val 18185 | Value of a component of th... |
curf2cl 18186 | The curry functor at a mor... |
curfcl 18187 | The curry functor of a fun... |
curfpropd 18188 | If two categories have the... |
uncfval 18189 | Value of the uncurry funct... |
uncfcl 18190 | The uncurry operation take... |
uncf1 18191 | Value of the uncurry funct... |
uncf2 18192 | Value of the uncurry funct... |
curfuncf 18193 | Cancellation of curry with... |
uncfcurf 18194 | Cancellation of uncurry wi... |
diagval 18195 | Define the diagonal functo... |
diagcl 18196 | The diagonal functor is a ... |
diag1cl 18197 | The constant functor of ` ... |
diag11 18198 | Value of the constant func... |
diag12 18199 | Value of the constant func... |
diag2 18200 | Value of the diagonal func... |
diag2cl 18201 | The diagonal functor at a ... |
curf2ndf 18202 | As shown in ~ diagval , th... |
hofval 18207 | Value of the Hom functor, ... |
hof1fval 18208 | The object part of the Hom... |
hof1 18209 | The object part of the Hom... |
hof2fval 18210 | The morphism part of the H... |
hof2val 18211 | The morphism part of the H... |
hof2 18212 | The morphism part of the H... |
hofcllem 18213 | Lemma for ~ hofcl . (Cont... |
hofcl 18214 | Closure of the Hom functor... |
oppchofcl 18215 | Closure of the opposite Ho... |
yonval 18216 | Value of the Yoneda embedd... |
yoncl 18217 | The Yoneda embedding is a ... |
yon1cl 18218 | The Yoneda embedding at an... |
yon11 18219 | Value of the Yoneda embedd... |
yon12 18220 | Value of the Yoneda embedd... |
yon2 18221 | Value of the Yoneda embedd... |
hofpropd 18222 | If two categories have the... |
yonpropd 18223 | If two categories have the... |
oppcyon 18224 | Value of the opposite Yone... |
oyoncl 18225 | The opposite Yoneda embedd... |
oyon1cl 18226 | The opposite Yoneda embedd... |
yonedalem1 18227 | Lemma for ~ yoneda . (Con... |
yonedalem21 18228 | Lemma for ~ yoneda . (Con... |
yonedalem3a 18229 | Lemma for ~ yoneda . (Con... |
yonedalem4a 18230 | Lemma for ~ yoneda . (Con... |
yonedalem4b 18231 | Lemma for ~ yoneda . (Con... |
yonedalem4c 18232 | Lemma for ~ yoneda . (Con... |
yonedalem22 18233 | Lemma for ~ yoneda . (Con... |
yonedalem3b 18234 | Lemma for ~ yoneda . (Con... |
yonedalem3 18235 | Lemma for ~ yoneda . (Con... |
yonedainv 18236 | The Yoneda Lemma with expl... |
yonffthlem 18237 | Lemma for ~ yonffth . (Co... |
yoneda 18238 | The Yoneda Lemma. There i... |
yonffth 18239 | The Yoneda Lemma. The Yon... |
yoniso 18240 | If the codomain is recover... |
oduval 18243 | Value of an order dual str... |
oduleval 18244 | Value of the less-equal re... |
oduleg 18245 | Truth of the less-equal re... |
odubas 18246 | Base set of an order dual ... |
odubasOLD 18247 | Obsolete proof of ~ odubas... |
isprs 18252 | Property of being a preord... |
prslem 18253 | Lemma for ~ prsref and ~ p... |
prsref 18254 | "Less than or equal to" is... |
prstr 18255 | "Less than or equal to" is... |
isdrs 18256 | Property of being a direct... |
drsdir 18257 | Direction of a directed se... |
drsprs 18258 | A directed set is a proset... |
drsbn0 18259 | The base of a directed set... |
drsdirfi 18260 | Any _finite_ number of ele... |
isdrs2 18261 | Directed sets may be defin... |
ispos 18269 | The predicate "is a poset"... |
ispos2 18270 | A poset is an antisymmetri... |
posprs 18271 | A poset is a proset. (Con... |
posi 18272 | Lemma for poset properties... |
posref 18273 | A poset ordering is reflex... |
posasymb 18274 | A poset ordering is asymme... |
postr 18275 | A poset ordering is transi... |
0pos 18276 | Technical lemma to simplif... |
0posOLD 18277 | Obsolete proof of ~ 0pos a... |
isposd 18278 | Properties that determine ... |
isposi 18279 | Properties that determine ... |
isposix 18280 | Properties that determine ... |
isposixOLD 18281 | Obsolete proof of ~ isposi... |
pospropd 18282 | Posethood is determined on... |
odupos 18283 | Being a poset is a self-du... |
oduposb 18284 | Being a poset is a self-du... |
pltfval 18286 | Value of the less-than rel... |
pltval 18287 | Less-than relation. ( ~ d... |
pltle 18288 | "Less than" implies "less ... |
pltne 18289 | The "less than" relation i... |
pltirr 18290 | The "less than" relation i... |
pleval2i 18291 | One direction of ~ pleval2... |
pleval2 18292 | "Less than or equal to" in... |
pltnle 18293 | "Less than" implies not co... |
pltval3 18294 | Alternate expression for t... |
pltnlt 18295 | The less-than relation imp... |
pltn2lp 18296 | The less-than relation has... |
plttr 18297 | The less-than relation is ... |
pltletr 18298 | Transitive law for chained... |
plelttr 18299 | Transitive law for chained... |
pospo 18300 | Write a poset structure in... |
lubfval 18305 | Value of the least upper b... |
lubdm 18306 | Domain of the least upper ... |
lubfun 18307 | The LUB is a function. (C... |
lubeldm 18308 | Member of the domain of th... |
lubelss 18309 | A member of the domain of ... |
lubeu 18310 | Unique existence proper of... |
lubval 18311 | Value of the least upper b... |
lubcl 18312 | The least upper bound func... |
lubprop 18313 | Properties of greatest low... |
luble 18314 | The greatest lower bound i... |
lublecllem 18315 | Lemma for ~ lublecl and ~ ... |
lublecl 18316 | The set of all elements le... |
lubid 18317 | The LUB of elements less t... |
glbfval 18318 | Value of the greatest lowe... |
glbdm 18319 | Domain of the greatest low... |
glbfun 18320 | The GLB is a function. (C... |
glbeldm 18321 | Member of the domain of th... |
glbelss 18322 | A member of the domain of ... |
glbeu 18323 | Unique existence proper of... |
glbval 18324 | Value of the greatest lowe... |
glbcl 18325 | The least upper bound func... |
glbprop 18326 | Properties of greatest low... |
glble 18327 | The greatest lower bound i... |
joinfval 18328 | Value of join function for... |
joinfval2 18329 | Value of join function for... |
joindm 18330 | Domain of join function fo... |
joindef 18331 | Two ways to say that a joi... |
joinval 18332 | Join value. Since both si... |
joincl 18333 | Closure of join of element... |
joindmss 18334 | Subset property of domain ... |
joinval2lem 18335 | Lemma for ~ joinval2 and ~... |
joinval2 18336 | Value of join for a poset ... |
joineu 18337 | Uniqueness of join of elem... |
joinlem 18338 | Lemma for join properties.... |
lejoin1 18339 | A join's first argument is... |
lejoin2 18340 | A join's second argument i... |
joinle 18341 | A join is less than or equ... |
meetfval 18342 | Value of meet function for... |
meetfval2 18343 | Value of meet function for... |
meetdm 18344 | Domain of meet function fo... |
meetdef 18345 | Two ways to say that a mee... |
meetval 18346 | Meet value. Since both si... |
meetcl 18347 | Closure of meet of element... |
meetdmss 18348 | Subset property of domain ... |
meetval2lem 18349 | Lemma for ~ meetval2 and ~... |
meetval2 18350 | Value of meet for a poset ... |
meeteu 18351 | Uniqueness of meet of elem... |
meetlem 18352 | Lemma for meet properties.... |
lemeet1 18353 | A meet's first argument is... |
lemeet2 18354 | A meet's second argument i... |
meetle 18355 | A meet is less than or equ... |
joincomALT 18356 | The join of a poset is com... |
joincom 18357 | The join of a poset is com... |
meetcomALT 18358 | The meet of a poset is com... |
meetcom 18359 | The meet of a poset is com... |
join0 18360 | Lemma for ~ odumeet . (Co... |
meet0 18361 | Lemma for ~ odujoin . (Co... |
odulub 18362 | Least upper bounds in a du... |
odujoin 18363 | Joins in a dual order are ... |
oduglb 18364 | Greatest lower bounds in a... |
odumeet 18365 | Meets in a dual order are ... |
poslubmo 18366 | Least upper bounds in a po... |
posglbmo 18367 | Greatest lower bounds in a... |
poslubd 18368 | Properties which determine... |
poslubdg 18369 | Properties which determine... |
posglbdg 18370 | Properties which determine... |
istos 18373 | The predicate "is a toset"... |
tosso 18374 | Write the totally ordered ... |
tospos 18375 | A Toset is a Poset. (Cont... |
tleile 18376 | In a Toset, any two elemen... |
tltnle 18377 | In a Toset, "less than" is... |
p0val 18382 | Value of poset zero. (Con... |
p1val 18383 | Value of poset zero. (Con... |
p0le 18384 | Any element is less than o... |
ple1 18385 | Any element is less than o... |
islat 18388 | The predicate "is a lattic... |
odulatb 18389 | Being a lattice is self-du... |
odulat 18390 | Being a lattice is self-du... |
latcl2 18391 | The join and meet of any t... |
latlem 18392 | Lemma for lattice properti... |
latpos 18393 | A lattice is a poset. (Co... |
latjcl 18394 | Closure of join operation ... |
latmcl 18395 | Closure of meet operation ... |
latref 18396 | A lattice ordering is refl... |
latasymb 18397 | A lattice ordering is asym... |
latasym 18398 | A lattice ordering is asym... |
lattr 18399 | A lattice ordering is tran... |
latasymd 18400 | Deduce equality from latti... |
lattrd 18401 | A lattice ordering is tran... |
latjcom 18402 | The join of a lattice comm... |
latlej1 18403 | A join's first argument is... |
latlej2 18404 | A join's second argument i... |
latjle12 18405 | A join is less than or equ... |
latleeqj1 18406 | "Less than or equal to" in... |
latleeqj2 18407 | "Less than or equal to" in... |
latjlej1 18408 | Add join to both sides of ... |
latjlej2 18409 | Add join to both sides of ... |
latjlej12 18410 | Add join to both sides of ... |
latnlej 18411 | An idiom to express that a... |
latnlej1l 18412 | An idiom to express that a... |
latnlej1r 18413 | An idiom to express that a... |
latnlej2 18414 | An idiom to express that a... |
latnlej2l 18415 | An idiom to express that a... |
latnlej2r 18416 | An idiom to express that a... |
latjidm 18417 | Lattice join is idempotent... |
latmcom 18418 | The join of a lattice comm... |
latmle1 18419 | A meet is less than or equ... |
latmle2 18420 | A meet is less than or equ... |
latlem12 18421 | An element is less than or... |
latleeqm1 18422 | "Less than or equal to" in... |
latleeqm2 18423 | "Less than or equal to" in... |
latmlem1 18424 | Add meet to both sides of ... |
latmlem2 18425 | Add meet to both sides of ... |
latmlem12 18426 | Add join to both sides of ... |
latnlemlt 18427 | Negation of "less than or ... |
latnle 18428 | Equivalent expressions for... |
latmidm 18429 | Lattice meet is idempotent... |
latabs1 18430 | Lattice absorption law. F... |
latabs2 18431 | Lattice absorption law. F... |
latledi 18432 | An ortholattice is distrib... |
latmlej11 18433 | Ordering of a meet and joi... |
latmlej12 18434 | Ordering of a meet and joi... |
latmlej21 18435 | Ordering of a meet and joi... |
latmlej22 18436 | Ordering of a meet and joi... |
lubsn 18437 | The least upper bound of a... |
latjass 18438 | Lattice join is associativ... |
latj12 18439 | Swap 1st and 2nd members o... |
latj32 18440 | Swap 2nd and 3rd members o... |
latj13 18441 | Swap 1st and 3rd members o... |
latj31 18442 | Swap 2nd and 3rd members o... |
latjrot 18443 | Rotate lattice join of 3 c... |
latj4 18444 | Rearrangement of lattice j... |
latj4rot 18445 | Rotate lattice join of 4 c... |
latjjdi 18446 | Lattice join distributes o... |
latjjdir 18447 | Lattice join distributes o... |
mod1ile 18448 | The weak direction of the ... |
mod2ile 18449 | The weak direction of the ... |
latmass 18450 | Lattice meet is associativ... |
latdisdlem 18451 | Lemma for ~ latdisd . (Co... |
latdisd 18452 | In a lattice, joins distri... |
isclat 18455 | The predicate "is a comple... |
clatpos 18456 | A complete lattice is a po... |
clatlem 18457 | Lemma for properties of a ... |
clatlubcl 18458 | Any subset of the base set... |
clatlubcl2 18459 | Any subset of the base set... |
clatglbcl 18460 | Any subset of the base set... |
clatglbcl2 18461 | Any subset of the base set... |
oduclatb 18462 | Being a complete lattice i... |
clatl 18463 | A complete lattice is a la... |
isglbd 18464 | Properties that determine ... |
lublem 18465 | Lemma for the least upper ... |
lubub 18466 | The LUB of a complete latt... |
lubl 18467 | The LUB of a complete latt... |
lubss 18468 | Subset law for least upper... |
lubel 18469 | An element of a set is les... |
lubun 18470 | The LUB of a union. (Cont... |
clatglb 18471 | Properties of greatest low... |
clatglble 18472 | The greatest lower bound i... |
clatleglb 18473 | Two ways of expressing "le... |
clatglbss 18474 | Subset law for greatest lo... |
isdlat 18477 | Property of being a distri... |
dlatmjdi 18478 | In a distributive lattice,... |
dlatl 18479 | A distributive lattice is ... |
odudlatb 18480 | The dual of a distributive... |
dlatjmdi 18481 | In a distributive lattice,... |
ipostr 18484 | The structure of ~ df-ipo ... |
ipoval 18485 | Value of the inclusion pos... |
ipobas 18486 | Base set of the inclusion ... |
ipolerval 18487 | Relation of the inclusion ... |
ipotset 18488 | Topology of the inclusion ... |
ipole 18489 | Weak order condition of th... |
ipolt 18490 | Strict order condition of ... |
ipopos 18491 | The inclusion poset on a f... |
isipodrs 18492 | Condition for a family of ... |
ipodrscl 18493 | Direction by inclusion as ... |
ipodrsfi 18494 | Finite upper bound propert... |
fpwipodrs 18495 | The finite subsets of any ... |
ipodrsima 18496 | The monotone image of a di... |
isacs3lem 18497 | An algebraic closure syste... |
acsdrsel 18498 | An algebraic closure syste... |
isacs4lem 18499 | In a closure system in whi... |
isacs5lem 18500 | If closure commutes with d... |
acsdrscl 18501 | In an algebraic closure sy... |
acsficl 18502 | A closure in an algebraic ... |
isacs5 18503 | A closure system is algebr... |
isacs4 18504 | A closure system is algebr... |
isacs3 18505 | A closure system is algebr... |
acsficld 18506 | In an algebraic closure sy... |
acsficl2d 18507 | In an algebraic closure sy... |
acsfiindd 18508 | In an algebraic closure sy... |
acsmapd 18509 | In an algebraic closure sy... |
acsmap2d 18510 | In an algebraic closure sy... |
acsinfd 18511 | In an algebraic closure sy... |
acsdomd 18512 | In an algebraic closure sy... |
acsinfdimd 18513 | In an algebraic closure sy... |
acsexdimd 18514 | In an algebraic closure sy... |
mrelatglb 18515 | Greatest lower bounds in a... |
mrelatglb0 18516 | The empty intersection in ... |
mrelatlub 18517 | Least upper bounds in a Mo... |
mreclatBAD 18518 | A Moore space is a complet... |
isps 18523 | The predicate "is a poset"... |
psrel 18524 | A poset is a relation. (C... |
psref2 18525 | A poset is antisymmetric a... |
pstr2 18526 | A poset is transitive. (C... |
pslem 18527 | Lemma for ~ psref and othe... |
psdmrn 18528 | The domain and range of a ... |
psref 18529 | A poset is reflexive. (Co... |
psrn 18530 | The range of a poset equal... |
psasym 18531 | A poset is antisymmetric. ... |
pstr 18532 | A poset is transitive. (C... |
cnvps 18533 | The converse of a poset is... |
cnvpsb 18534 | The converse of a poset is... |
psss 18535 | Any subset of a partially ... |
psssdm2 18536 | Field of a subposet. (Con... |
psssdm 18537 | Field of a subposet. (Con... |
istsr 18538 | The predicate is a toset. ... |
istsr2 18539 | The predicate is a toset. ... |
tsrlin 18540 | A toset is a linear order.... |
tsrlemax 18541 | Two ways of saying a numbe... |
tsrps 18542 | A toset is a poset. (Cont... |
cnvtsr 18543 | The converse of a toset is... |
tsrss 18544 | Any subset of a totally or... |
ledm 18545 | The domain of ` <_ ` is ` ... |
lern 18546 | The range of ` <_ ` is ` R... |
lefld 18547 | The field of the 'less or ... |
letsr 18548 | The "less than or equal to... |
isdir 18553 | A condition for a relation... |
reldir 18554 | A direction is a relation.... |
dirdm 18555 | A direction's domain is eq... |
dirref 18556 | A direction is reflexive. ... |
dirtr 18557 | A direction is transitive.... |
dirge 18558 | For any two elements of a ... |
tsrdir 18559 | A totally ordered set is a... |
ismgm 18564 | The predicate "is a magma"... |
ismgmn0 18565 | The predicate "is a magma"... |
mgmcl 18566 | Closure of the operation o... |
isnmgm 18567 | A condition for a structur... |
mgmsscl 18568 | If the base set of a magma... |
plusffval 18569 | The group addition operati... |
plusfval 18570 | The group addition operati... |
plusfeq 18571 | If the addition operation ... |
plusffn 18572 | The group addition operati... |
mgmplusf 18573 | The group addition functio... |
mgmpropd 18574 | If two structures have the... |
ismgmd 18575 | Deduce a magma from its pr... |
issstrmgm 18576 | Characterize a substructur... |
intopsn 18577 | The internal operation for... |
mgmb1mgm1 18578 | The only magma with a base... |
mgm0 18579 | Any set with an empty base... |
mgm0b 18580 | The structure with an empt... |
mgm1 18581 | The structure with one ele... |
opifismgm 18582 | A structure with a group a... |
mgmidmo 18583 | A two-sided identity eleme... |
grpidval 18584 | The value of the identity ... |
grpidpropd 18585 | If two structures have the... |
fn0g 18586 | The group zero extractor i... |
0g0 18587 | The identity element funct... |
ismgmid 18588 | The identity element of a ... |
mgmidcl 18589 | The identity element of a ... |
mgmlrid 18590 | The identity element of a ... |
ismgmid2 18591 | Show that a given element ... |
lidrideqd 18592 | If there is a left and rig... |
lidrididd 18593 | If there is a left and rig... |
grpidd 18594 | Deduce the identity elemen... |
mgmidsssn0 18595 | Property of the set of ide... |
grprinvlem 18596 | Lemma for ~ grpinva . (Co... |
grpinva 18597 | Deduce right inverse from ... |
grprida 18598 | Deduce right identity from... |
gsumvalx 18599 | Expand out the substitutio... |
gsumval 18600 | Expand out the substitutio... |
gsumpropd 18601 | The group sum depends only... |
gsumpropd2lem 18602 | Lemma for ~ gsumpropd2 . ... |
gsumpropd2 18603 | A stronger version of ~ gs... |
gsummgmpropd 18604 | A stronger version of ~ gs... |
gsumress 18605 | The group sum in a substru... |
gsumval1 18606 | Value of the group sum ope... |
gsum0 18607 | Value of the empty group s... |
gsumval2a 18608 | Value of the group sum ope... |
gsumval2 18609 | Value of the group sum ope... |
gsumsplit1r 18610 | Splitting off the rightmos... |
gsumprval 18611 | Value of the group sum ope... |
gsumpr12val 18612 | Value of the group sum ope... |
mgmhmrcl 18617 | Reverse closure of a magma... |
submgmrcl 18618 | Reverse closure for submag... |
ismgmhm 18619 | Property of a magma homomo... |
mgmhmf 18620 | A magma homomorphism is a ... |
mgmhmpropd 18621 | Magma homomorphism depends... |
mgmhmlin 18622 | A magma homomorphism prese... |
mgmhmf1o 18623 | A magma homomorphism is bi... |
idmgmhm 18624 | The identity homomorphism ... |
issubmgm 18625 | Expand definition of a sub... |
issubmgm2 18626 | Submagmas are subsets that... |
rabsubmgmd 18627 | Deduction for proving that... |
submgmss 18628 | Submagmas are subsets of t... |
submgmid 18629 | Every magma is trivially a... |
submgmcl 18630 | Submagmas are closed under... |
submgmmgm 18631 | Submagmas are themselves m... |
submgmbas 18632 | The base set of a submagma... |
subsubmgm 18633 | A submagma of a submagma i... |
resmgmhm 18634 | Restriction of a magma hom... |
resmgmhm2 18635 | One direction of ~ resmgmh... |
resmgmhm2b 18636 | Restriction of the codomai... |
mgmhmco 18637 | The composition of magma h... |
mgmhmima 18638 | The homomorphic image of a... |
mgmhmeql 18639 | The equalizer of two magma... |
submgmacs 18640 | Submagmas are an algebraic... |
issgrp 18643 | The predicate "is a semigr... |
issgrpv 18644 | The predicate "is a semigr... |
issgrpn0 18645 | The predicate "is a semigr... |
isnsgrp 18646 | A condition for a structur... |
sgrpmgm 18647 | A semigroup is a magma. (... |
sgrpass 18648 | A semigroup operation is a... |
sgrpcl 18649 | Closure of the operation o... |
sgrp0 18650 | Any set with an empty base... |
sgrp0b 18651 | The structure with an empt... |
sgrp1 18652 | The structure with one ele... |
issgrpd 18653 | Deduce a semigroup from it... |
sgrppropd 18654 | If two structures are sets... |
prdsplusgsgrpcl 18655 | Structure product pointwis... |
prdssgrpd 18656 | The product of a family of... |
ismnddef 18659 | The predicate "is a monoid... |
ismnd 18660 | The predicate "is a monoid... |
isnmnd 18661 | A condition for a structur... |
sgrpidmnd 18662 | A semigroup with an identi... |
mndsgrp 18663 | A monoid is a semigroup. ... |
mndmgm 18664 | A monoid is a magma. (Con... |
mndcl 18665 | Closure of the operation o... |
mndass 18666 | A monoid operation is asso... |
mndid 18667 | A monoid has a two-sided i... |
mndideu 18668 | The two-sided identity ele... |
mnd32g 18669 | Commutative/associative la... |
mnd12g 18670 | Commutative/associative la... |
mnd4g 18671 | Commutative/associative la... |
mndidcl 18672 | The identity element of a ... |
mndbn0 18673 | The base set of a monoid i... |
hashfinmndnn 18674 | A finite monoid has positi... |
mndplusf 18675 | The group addition operati... |
mndlrid 18676 | A monoid's identity elemen... |
mndlid 18677 | The identity element of a ... |
mndrid 18678 | The identity element of a ... |
ismndd 18679 | Deduce a monoid from its p... |
mndpfo 18680 | The addition operation of ... |
mndfo 18681 | The addition operation of ... |
mndpropd 18682 | If two structures have the... |
mndprop 18683 | If two structures have the... |
issubmnd 18684 | Characterize a submonoid b... |
ress0g 18685 | ` 0g ` is unaffected by re... |
submnd0 18686 | The zero of a submonoid is... |
mndinvmod 18687 | Uniqueness of an inverse e... |
prdsplusgcl 18688 | Structure product pointwis... |
prdsidlem 18689 | Characterization of identi... |
prdsmndd 18690 | The product of a family of... |
prds0g 18691 | Zero in a product of monoi... |
pwsmnd 18692 | The structure power of a m... |
pws0g 18693 | Zero in a structure power ... |
imasmnd2 18694 | The image structure of a m... |
imasmnd 18695 | The image structure of a m... |
imasmndf1 18696 | The image of a monoid unde... |
xpsmnd 18697 | The binary product of mono... |
xpsmnd0 18698 | The identity element of a ... |
mnd1 18699 | The (smallest) structure r... |
mnd1id 18700 | The singleton element of a... |
ismhm 18705 | Property of a monoid homom... |
ismhmd 18706 | Deduction version of ~ ism... |
mhmrcl1 18707 | Reverse closure of a monoi... |
mhmrcl2 18708 | Reverse closure of a monoi... |
mhmf 18709 | A monoid homomorphism is a... |
ismhm0 18710 | Property of a monoid homom... |
mhmismgmhm 18711 | Each monoid homomorphism i... |
mhmpropd 18712 | Monoid homomorphism depend... |
mhmlin 18713 | A monoid homomorphism comm... |
mhm0 18714 | A monoid homomorphism pres... |
idmhm 18715 | The identity homomorphism ... |
mhmf1o 18716 | A monoid homomorphism is b... |
submrcl 18717 | Reverse closure for submon... |
issubm 18718 | Expand definition of a sub... |
issubm2 18719 | Submonoids are subsets tha... |
issubmndb 18720 | The submonoid predicate. ... |
issubmd 18721 | Deduction for proving a su... |
mndissubm 18722 | If the base set of a monoi... |
resmndismnd 18723 | If the base set of a monoi... |
submss 18724 | Submonoids are subsets of ... |
submid 18725 | Every monoid is trivially ... |
subm0cl 18726 | Submonoids contain zero. ... |
submcl 18727 | Submonoids are closed unde... |
submmnd 18728 | Submonoids are themselves ... |
submbas 18729 | The base set of a submonoi... |
subm0 18730 | Submonoids have the same i... |
subsubm 18731 | A submonoid of a submonoid... |
0subm 18732 | The zero submonoid of an a... |
insubm 18733 | The intersection of two su... |
0mhm 18734 | The constant zero linear f... |
resmhm 18735 | Restriction of a monoid ho... |
resmhm2 18736 | One direction of ~ resmhm2... |
resmhm2b 18737 | Restriction of the codomai... |
mhmco 18738 | The composition of monoid ... |
mhmimalem 18739 | Lemma for ~ mhmima and sim... |
mhmima 18740 | The homomorphic image of a... |
mhmeql 18741 | The equalizer of two monoi... |
submacs 18742 | Submonoids are an algebrai... |
mndind 18743 | Induction in a monoid. In... |
prdspjmhm 18744 | A projection from a produc... |
pwspjmhm 18745 | A projection from a struct... |
pwsdiagmhm 18746 | Diagonal monoid homomorphi... |
pwsco1mhm 18747 | Right composition with a f... |
pwsco2mhm 18748 | Left composition with a mo... |
gsumvallem2 18749 | Lemma for properties of th... |
gsumsubm 18750 | Evaluate a group sum in a ... |
gsumz 18751 | Value of a group sum over ... |
gsumwsubmcl 18752 | Closure of the composite i... |
gsumws1 18753 | A singleton composite reco... |
gsumwcl 18754 | Closure of the composite o... |
gsumsgrpccat 18755 | Homomorphic property of no... |
gsumccat 18756 | Homomorphic property of co... |
gsumws2 18757 | Valuation of a pair in a m... |
gsumccatsn 18758 | Homomorphic property of co... |
gsumspl 18759 | The primary purpose of the... |
gsumwmhm 18760 | Behavior of homomorphisms ... |
gsumwspan 18761 | The submonoid generated by... |
frmdval 18766 | Value of the free monoid c... |
frmdbas 18767 | The base set of a free mon... |
frmdelbas 18768 | An element of the base set... |
frmdplusg 18769 | The monoid operation of a ... |
frmdadd 18770 | Value of the monoid operat... |
vrmdfval 18771 | The canonical injection fr... |
vrmdval 18772 | The value of the generatin... |
vrmdf 18773 | The mapping from the index... |
frmdmnd 18774 | A free monoid is a monoid.... |
frmd0 18775 | The identity of the free m... |
frmdsssubm 18776 | The set of words taking va... |
frmdgsum 18777 | Any word in a free monoid ... |
frmdss2 18778 | A subset of generators is ... |
frmdup1 18779 | Any assignment of the gene... |
frmdup2 18780 | The evaluation map has the... |
frmdup3lem 18781 | Lemma for ~ frmdup3 . (Co... |
frmdup3 18782 | Universal property of the ... |
efmnd 18785 | The monoid of endofunction... |
efmndbas 18786 | The base set of the monoid... |
efmndbasabf 18787 | The base set of the monoid... |
elefmndbas 18788 | Two ways of saying a funct... |
elefmndbas2 18789 | Two ways of saying a funct... |
efmndbasf 18790 | Elements in the monoid of ... |
efmndhash 18791 | The monoid of endofunction... |
efmndbasfi 18792 | The monoid of endofunction... |
efmndfv 18793 | The function value of an e... |
efmndtset 18794 | The topology of the monoid... |
efmndplusg 18795 | The group operation of a m... |
efmndov 18796 | The value of the group ope... |
efmndcl 18797 | The group operation of the... |
efmndtopn 18798 | The topology of the monoid... |
symggrplem 18799 | Lemma for ~ symggrp and ~ ... |
efmndmgm 18800 | The monoid of endofunction... |
efmndsgrp 18801 | The monoid of endofunction... |
ielefmnd 18802 | The identity function rest... |
efmndid 18803 | The identity function rest... |
efmndmnd 18804 | The monoid of endofunction... |
efmnd0nmnd 18805 | Even the monoid of endofun... |
efmndbas0 18806 | The base set of the monoid... |
efmnd1hash 18807 | The monoid of endofunction... |
efmnd1bas 18808 | The monoid of endofunction... |
efmnd2hash 18809 | The monoid of endofunction... |
submefmnd 18810 | If the base set of a monoi... |
sursubmefmnd 18811 | The set of surjective endo... |
injsubmefmnd 18812 | The set of injective endof... |
idressubmefmnd 18813 | The singleton containing o... |
idresefmnd 18814 | The structure with the sin... |
smndex1ibas 18815 | The modulo function ` I ` ... |
smndex1iidm 18816 | The modulo function ` I ` ... |
smndex1gbas 18817 | The constant functions ` (... |
smndex1gid 18818 | The composition of a const... |
smndex1igid 18819 | The composition of the mod... |
smndex1basss 18820 | The modulo function ` I ` ... |
smndex1bas 18821 | The base set of the monoid... |
smndex1mgm 18822 | The monoid of endofunction... |
smndex1sgrp 18823 | The monoid of endofunction... |
smndex1mndlem 18824 | Lemma for ~ smndex1mnd and... |
smndex1mnd 18825 | The monoid of endofunction... |
smndex1id 18826 | The modulo function ` I ` ... |
smndex1n0mnd 18827 | The identity of the monoid... |
nsmndex1 18828 | The base set ` B ` of the ... |
smndex2dbas 18829 | The doubling function ` D ... |
smndex2dnrinv 18830 | The doubling function ` D ... |
smndex2hbas 18831 | The halving functions ` H ... |
smndex2dlinvh 18832 | The halving functions ` H ... |
mgm2nsgrplem1 18833 | Lemma 1 for ~ mgm2nsgrp : ... |
mgm2nsgrplem2 18834 | Lemma 2 for ~ mgm2nsgrp . ... |
mgm2nsgrplem3 18835 | Lemma 3 for ~ mgm2nsgrp . ... |
mgm2nsgrplem4 18836 | Lemma 4 for ~ mgm2nsgrp : ... |
mgm2nsgrp 18837 | A small magma (with two el... |
sgrp2nmndlem1 18838 | Lemma 1 for ~ sgrp2nmnd : ... |
sgrp2nmndlem2 18839 | Lemma 2 for ~ sgrp2nmnd . ... |
sgrp2nmndlem3 18840 | Lemma 3 for ~ sgrp2nmnd . ... |
sgrp2rid2 18841 | A small semigroup (with tw... |
sgrp2rid2ex 18842 | A small semigroup (with tw... |
sgrp2nmndlem4 18843 | Lemma 4 for ~ sgrp2nmnd : ... |
sgrp2nmndlem5 18844 | Lemma 5 for ~ sgrp2nmnd : ... |
sgrp2nmnd 18845 | A small semigroup (with tw... |
mgmnsgrpex 18846 | There is a magma which is ... |
sgrpnmndex 18847 | There is a semigroup which... |
sgrpssmgm 18848 | The class of all semigroup... |
mndsssgrp 18849 | The class of all monoids i... |
pwmndgplus 18850 | The operation of the monoi... |
pwmndid 18851 | The identity of the monoid... |
pwmnd 18852 | The power set of a class `... |
isgrp 18859 | The predicate "is a group"... |
grpmnd 18860 | A group is a monoid. (Con... |
grpcl 18861 | Closure of the operation o... |
grpass 18862 | A group operation is assoc... |
grpinvex 18863 | Every member of a group ha... |
grpideu 18864 | The two-sided identity ele... |
grpassd 18865 | A group operation is assoc... |
grpmndd 18866 | A group is a monoid. (Con... |
grpcld 18867 | Closure of the operation o... |
grpplusf 18868 | The group addition operati... |
grpplusfo 18869 | The group addition operati... |
resgrpplusfrn 18870 | The underlying set of a gr... |
grppropd 18871 | If two structures have the... |
grpprop 18872 | If two structures have the... |
grppropstr 18873 | Generalize a specific 2-el... |
grpss 18874 | Show that a structure exte... |
isgrpd2e 18875 | Deduce a group from its pr... |
isgrpd2 18876 | Deduce a group from its pr... |
isgrpde 18877 | Deduce a group from its pr... |
isgrpd 18878 | Deduce a group from its pr... |
isgrpi 18879 | Properties that determine ... |
grpsgrp 18880 | A group is a semigroup. (... |
grpmgmd 18881 | A group is a magma, deduct... |
dfgrp2 18882 | Alternate definition of a ... |
dfgrp2e 18883 | Alternate definition of a ... |
isgrpix 18884 | Properties that determine ... |
grpidcl 18885 | The identity element of a ... |
grpbn0 18886 | The base set of a group is... |
grplid 18887 | The identity element of a ... |
grprid 18888 | The identity element of a ... |
grplidd 18889 | The identity element of a ... |
grpridd 18890 | The identity element of a ... |
grpn0 18891 | A group is not empty. (Co... |
hashfingrpnn 18892 | A finite group has positiv... |
grprcan 18893 | Right cancellation law for... |
grpinveu 18894 | The left inverse element o... |
grpid 18895 | Two ways of saying that an... |
isgrpid2 18896 | Properties showing that an... |
grpidd2 18897 | Deduce the identity elemen... |
grpinvfval 18898 | The inverse function of a ... |
grpinvfvalALT 18899 | Shorter proof of ~ grpinvf... |
grpinvval 18900 | The inverse of a group ele... |
grpinvfn 18901 | Functionality of the group... |
grpinvfvi 18902 | The group inverse function... |
grpsubfval 18903 | Group subtraction (divisio... |
grpsubfvalALT 18904 | Shorter proof of ~ grpsubf... |
grpsubval 18905 | Group subtraction (divisio... |
grpinvf 18906 | The group inversion operat... |
grpinvcl 18907 | A group element's inverse ... |
grpinvcld 18908 | A group element's inverse ... |
grplinv 18909 | The left inverse of a grou... |
grprinv 18910 | The right inverse of a gro... |
grpinvid1 18911 | The inverse of a group ele... |
grpinvid2 18912 | The inverse of a group ele... |
isgrpinv 18913 | Properties showing that a ... |
grplinvd 18914 | The left inverse of a grou... |
grprinvd 18915 | The right inverse of a gro... |
grplrinv 18916 | In a group, every member h... |
grpidinv2 18917 | A group's properties using... |
grpidinv 18918 | A group has a left and rig... |
grpinvid 18919 | The inverse of the identit... |
grplcan 18920 | Left cancellation law for ... |
grpasscan1 18921 | An associative cancellatio... |
grpasscan2 18922 | An associative cancellatio... |
grpidrcan 18923 | If right adding an element... |
grpidlcan 18924 | If left adding an element ... |
grpinvinv 18925 | Double inverse law for gro... |
grpinvcnv 18926 | The group inverse is its o... |
grpinv11 18927 | The group inverse is one-t... |
grpinvf1o 18928 | The group inverse is a one... |
grpinvnz 18929 | The inverse of a nonzero g... |
grpinvnzcl 18930 | The inverse of a nonzero g... |
grpsubinv 18931 | Subtraction of an inverse.... |
grplmulf1o 18932 | Left multiplication by a g... |
grpinvpropd 18933 | If two structures have the... |
grpidssd 18934 | If the base set of a group... |
grpinvssd 18935 | If the base set of a group... |
grpinvadd 18936 | The inverse of the group o... |
grpsubf 18937 | Functionality of group sub... |
grpsubcl 18938 | Closure of group subtracti... |
grpsubrcan 18939 | Right cancellation law for... |
grpinvsub 18940 | Inverse of a group subtrac... |
grpinvval2 18941 | A ~ df-neg -like equation ... |
grpsubid 18942 | Subtraction of a group ele... |
grpsubid1 18943 | Subtraction of the identit... |
grpsubeq0 18944 | If the difference between ... |
grpsubadd0sub 18945 | Subtraction expressed as a... |
grpsubadd 18946 | Relationship between group... |
grpsubsub 18947 | Double group subtraction. ... |
grpaddsubass 18948 | Associative-type law for g... |
grppncan 18949 | Cancellation law for subtr... |
grpnpcan 18950 | Cancellation law for subtr... |
grpsubsub4 18951 | Double group subtraction (... |
grppnpcan2 18952 | Cancellation law for mixed... |
grpnpncan 18953 | Cancellation law for group... |
grpnpncan0 18954 | Cancellation law for group... |
grpnnncan2 18955 | Cancellation law for group... |
dfgrp3lem 18956 | Lemma for ~ dfgrp3 . (Con... |
dfgrp3 18957 | Alternate definition of a ... |
dfgrp3e 18958 | Alternate definition of a ... |
grplactfval 18959 | The left group action of e... |
grplactval 18960 | The value of the left grou... |
grplactcnv 18961 | The left group action of e... |
grplactf1o 18962 | The left group action of e... |
grpsubpropd 18963 | Weak property deduction fo... |
grpsubpropd2 18964 | Strong property deduction ... |
grp1 18965 | The (smallest) structure r... |
grp1inv 18966 | The inverse function of th... |
prdsinvlem 18967 | Characterization of invers... |
prdsgrpd 18968 | The product of a family of... |
prdsinvgd 18969 | Negation in a product of g... |
pwsgrp 18970 | A structure power of a gro... |
pwsinvg 18971 | Negation in a group power.... |
pwssub 18972 | Subtraction in a group pow... |
imasgrp2 18973 | The image structure of a g... |
imasgrp 18974 | The image structure of a g... |
imasgrpf1 18975 | The image of a group under... |
qusgrp2 18976 | Prove that a quotient stru... |
xpsgrp 18977 | The binary product of grou... |
xpsinv 18978 | Value of the negation oper... |
xpsgrpsub 18979 | Value of the subtraction o... |
mhmlem 18980 | Lemma for ~ mhmmnd and ~ g... |
mhmid 18981 | A surjective monoid morphi... |
mhmmnd 18982 | The image of a monoid ` G ... |
mhmfmhm 18983 | The function fulfilling th... |
ghmgrp 18984 | The image of a group ` G `... |
mulgfval 18987 | Group multiple (exponentia... |
mulgfvalALT 18988 | Shorter proof of ~ mulgfva... |
mulgval 18989 | Value of the group multipl... |
mulgfn 18990 | Functionality of the group... |
mulgfvi 18991 | The group multiple operati... |
mulg0 18992 | Group multiple (exponentia... |
mulgnn 18993 | Group multiple (exponentia... |
ressmulgnn 18994 | Values for the group multi... |
ressmulgnn0 18995 | Values for the group multi... |
mulgnngsum 18996 | Group multiple (exponentia... |
mulgnn0gsum 18997 | Group multiple (exponentia... |
mulg1 18998 | Group multiple (exponentia... |
mulgnnp1 18999 | Group multiple (exponentia... |
mulg2 19000 | Group multiple (exponentia... |
mulgnegnn 19001 | Group multiple (exponentia... |
mulgnn0p1 19002 | Group multiple (exponentia... |
mulgnnsubcl 19003 | Closure of the group multi... |
mulgnn0subcl 19004 | Closure of the group multi... |
mulgsubcl 19005 | Closure of the group multi... |
mulgnncl 19006 | Closure of the group multi... |
mulgnn0cl 19007 | Closure of the group multi... |
mulgcl 19008 | Closure of the group multi... |
mulgneg 19009 | Group multiple (exponentia... |
mulgnegneg 19010 | The inverse of a negative ... |
mulgm1 19011 | Group multiple (exponentia... |
mulgnn0cld 19012 | Closure of the group multi... |
mulgcld 19013 | Deduction associated with ... |
mulgaddcomlem 19014 | Lemma for ~ mulgaddcom . ... |
mulgaddcom 19015 | The group multiple operato... |
mulginvcom 19016 | The group multiple operato... |
mulginvinv 19017 | The group multiple operato... |
mulgnn0z 19018 | A group multiple of the id... |
mulgz 19019 | A group multiple of the id... |
mulgnndir 19020 | Sum of group multiples, fo... |
mulgnn0dir 19021 | Sum of group multiples, ge... |
mulgdirlem 19022 | Lemma for ~ mulgdir . (Co... |
mulgdir 19023 | Sum of group multiples, ge... |
mulgp1 19024 | Group multiple (exponentia... |
mulgneg2 19025 | Group multiple (exponentia... |
mulgnnass 19026 | Product of group multiples... |
mulgnn0ass 19027 | Product of group multiples... |
mulgass 19028 | Product of group multiples... |
mulgassr 19029 | Reversed product of group ... |
mulgmodid 19030 | Casting out multiples of t... |
mulgsubdir 19031 | Distribution of group mult... |
mhmmulg 19032 | A homomorphism of monoids ... |
mulgpropd 19033 | Two structures with the sa... |
submmulgcl 19034 | Closure of the group multi... |
submmulg 19035 | A group multiple is the sa... |
pwsmulg 19036 | Value of a group multiple ... |
issubg 19043 | The subgroup predicate. (... |
subgss 19044 | A subgroup is a subset. (... |
subgid 19045 | A group is a subgroup of i... |
subggrp 19046 | A subgroup is a group. (C... |
subgbas 19047 | The base of the restricted... |
subgrcl 19048 | Reverse closure for the su... |
subg0 19049 | A subgroup of a group must... |
subginv 19050 | The inverse of an element ... |
subg0cl 19051 | The group identity is an e... |
subginvcl 19052 | The inverse of an element ... |
subgcl 19053 | A subgroup is closed under... |
subgsubcl 19054 | A subgroup is closed under... |
subgsub 19055 | The subtraction of element... |
subgmulgcl 19056 | Closure of the group multi... |
subgmulg 19057 | A group multiple is the sa... |
issubg2 19058 | Characterize the subgroups... |
issubgrpd2 19059 | Prove a subgroup by closur... |
issubgrpd 19060 | Prove a subgroup by closur... |
issubg3 19061 | A subgroup is a symmetric ... |
issubg4 19062 | A subgroup is a nonempty s... |
grpissubg 19063 | If the base set of a group... |
resgrpisgrp 19064 | If the base set of a group... |
subgsubm 19065 | A subgroup is a submonoid.... |
subsubg 19066 | A subgroup of a subgroup i... |
subgint 19067 | The intersection of a none... |
0subg 19068 | The zero subgroup of an ar... |
0subgOLD 19069 | Obsolete version of ~ 0sub... |
trivsubgd 19070 | The only subgroup of a tri... |
trivsubgsnd 19071 | The only subgroup of a tri... |
isnsg 19072 | Property of being a normal... |
isnsg2 19073 | Weaken the condition of ~ ... |
nsgbi 19074 | Defining property of a nor... |
nsgsubg 19075 | A normal subgroup is a sub... |
nsgconj 19076 | The conjugation of an elem... |
isnsg3 19077 | A subgroup is normal iff t... |
subgacs 19078 | Subgroups are an algebraic... |
nsgacs 19079 | Normal subgroups form an a... |
elnmz 19080 | Elementhood in the normali... |
nmzbi 19081 | Defining property of the n... |
nmzsubg 19082 | The normalizer N_G(S) of a... |
ssnmz 19083 | A subgroup is a subset of ... |
isnsg4 19084 | A subgroup is normal iff i... |
nmznsg 19085 | Any subgroup is a normal s... |
0nsg 19086 | The zero subgroup is norma... |
nsgid 19087 | The whole group is a norma... |
0idnsgd 19088 | The whole group and the ze... |
trivnsgd 19089 | The only normal subgroup o... |
triv1nsgd 19090 | A trivial group has exactl... |
1nsgtrivd 19091 | A group with exactly one n... |
releqg 19092 | The left coset equivalence... |
eqgfval 19093 | Value of the subgroup left... |
eqgval 19094 | Value of the subgroup left... |
eqger 19095 | The subgroup coset equival... |
eqglact 19096 | A left coset can be expres... |
eqgid 19097 | The left coset containing ... |
eqgen 19098 | Each coset is equipotent t... |
eqgcpbl 19099 | The subgroup coset equival... |
quselbas 19100 | Membership in the base set... |
quseccl0 19101 | Closure of the quotient ma... |
qusgrp 19102 | If ` Y ` is a normal subgr... |
quseccl 19103 | Closure of the quotient ma... |
qusadd 19104 | Value of the group operati... |
qus0 19105 | Value of the group identit... |
qusinv 19106 | Value of the group inverse... |
qussub 19107 | Value of the group subtrac... |
ecqusaddd 19108 | Addition of equivalence cl... |
ecqusaddcl 19109 | Closure of the addition in... |
lagsubg2 19110 | Lagrange's theorem for fin... |
lagsubg 19111 | Lagrange's theorem for Gro... |
eqg0subg 19112 | The coset equivalence rela... |
eqg0subgecsn 19113 | The equivalence classes mo... |
qus0subgbas 19114 | The base set of a quotient... |
qus0subgadd 19115 | The addition in a quotient... |
cycsubmel 19116 | Characterization of an ele... |
cycsubmcl 19117 | The set of nonnegative int... |
cycsubm 19118 | The set of nonnegative int... |
cyccom 19119 | Condition for an operation... |
cycsubmcom 19120 | The operation of a monoid ... |
cycsubggend 19121 | The cyclic subgroup genera... |
cycsubgcl 19122 | The set of integer powers ... |
cycsubgss 19123 | The cyclic subgroup genera... |
cycsubg 19124 | The cyclic group generated... |
cycsubgcld 19125 | The cyclic subgroup genera... |
cycsubg2 19126 | The subgroup generated by ... |
cycsubg2cl 19127 | Any multiple of an element... |
reldmghm 19130 | Lemma for group homomorphi... |
isghm 19131 | Property of being a homomo... |
isghm3 19132 | Property of a group homomo... |
ghmgrp1 19133 | A group homomorphism is on... |
ghmgrp2 19134 | A group homomorphism is on... |
ghmf 19135 | A group homomorphism is a ... |
ghmlin 19136 | A homomorphism of groups i... |
ghmid 19137 | A homomorphism of groups p... |
ghminv 19138 | A homomorphism of groups p... |
ghmsub 19139 | Linearity of subtraction t... |
isghmd 19140 | Deduction for a group homo... |
ghmmhm 19141 | A group homomorphism is a ... |
ghmmhmb 19142 | Group homomorphisms and mo... |
ghmmulg 19143 | A homomorphism of monoids ... |
ghmrn 19144 | The range of a homomorphis... |
0ghm 19145 | The constant zero linear f... |
idghm 19146 | The identity homomorphism ... |
resghm 19147 | Restriction of a homomorph... |
resghm2 19148 | One direction of ~ resghm2... |
resghm2b 19149 | Restriction of the codomai... |
ghmghmrn 19150 | A group homomorphism from ... |
ghmco 19151 | The composition of group h... |
ghmima 19152 | The image of a subgroup un... |
ghmpreima 19153 | The inverse image of a sub... |
ghmeql 19154 | The equalizer of two group... |
ghmnsgima 19155 | The image of a normal subg... |
ghmnsgpreima 19156 | The inverse image of a nor... |
ghmker 19157 | The kernel of a homomorphi... |
ghmeqker 19158 | Two source points map to t... |
pwsdiagghm 19159 | Diagonal homomorphism into... |
f1ghm0to0 19160 | If a group homomorphism ` ... |
ghmf1 19161 | Two ways of saying a group... |
kerf1ghm 19162 | A group homomorphism ` F `... |
ghmf1o 19163 | A bijective group homomorp... |
conjghm 19164 | Conjugation is an automorp... |
conjsubg 19165 | A conjugated subgroup is a... |
conjsubgen 19166 | A conjugated subgroup is e... |
conjnmz 19167 | A subgroup is unchanged un... |
conjnmzb 19168 | Alternative condition for ... |
conjnsg 19169 | A normal subgroup is uncha... |
qusghm 19170 | If ` Y ` is a normal subgr... |
ghmpropd 19171 | Group homomorphism depends... |
gimfn 19176 | The group isomorphism func... |
isgim 19177 | An isomorphism of groups i... |
gimf1o 19178 | An isomorphism of groups i... |
gimghm 19179 | An isomorphism of groups i... |
isgim2 19180 | A group isomorphism is a h... |
subggim 19181 | Behavior of subgroups unde... |
gimcnv 19182 | The converse of a bijectiv... |
gimco 19183 | The composition of group i... |
gim0to0 19184 | A group isomorphism maps t... |
brgic 19185 | The relation "is isomorphi... |
brgici 19186 | Prove isomorphic by an exp... |
gicref 19187 | Isomorphism is reflexive. ... |
giclcl 19188 | Isomorphism implies the le... |
gicrcl 19189 | Isomorphism implies the ri... |
gicsym 19190 | Isomorphism is symmetric. ... |
gictr 19191 | Isomorphism is transitive.... |
gicer 19192 | Isomorphism is an equivale... |
gicen 19193 | Isomorphic groups have equ... |
gicsubgen 19194 | A less trivial example of ... |
isga 19197 | The predicate "is a (left)... |
gagrp 19198 | The left argument of a gro... |
gaset 19199 | The right argument of a gr... |
gagrpid 19200 | The identity of the group ... |
gaf 19201 | The mapping of the group a... |
gafo 19202 | A group action is onto its... |
gaass 19203 | An "associative" property ... |
ga0 19204 | The action of a group on t... |
gaid 19205 | The trivial action of a gr... |
subgga 19206 | A subgroup acts on its par... |
gass 19207 | A subset of a group action... |
gasubg 19208 | The restriction of a group... |
gaid2 19209 | A group operation is a lef... |
galcan 19210 | The action of a particular... |
gacan 19211 | Group inverses cancel in a... |
gapm 19212 | The action of a particular... |
gaorb 19213 | The orbit equivalence rela... |
gaorber 19214 | The orbit equivalence rela... |
gastacl 19215 | The stabilizer subgroup in... |
gastacos 19216 | Write the coset relation f... |
orbstafun 19217 | Existence and uniqueness f... |
orbstaval 19218 | Value of the function at a... |
orbsta 19219 | The Orbit-Stabilizer theor... |
orbsta2 19220 | Relation between the size ... |
cntrval 19225 | Substitute definition of t... |
cntzfval 19226 | First level substitution f... |
cntzval 19227 | Definition substitution fo... |
elcntz 19228 | Elementhood in the central... |
cntzel 19229 | Membership in a centralize... |
cntzsnval 19230 | Special substitution for t... |
elcntzsn 19231 | Value of the centralizer o... |
sscntz 19232 | A centralizer expression f... |
cntzrcl 19233 | Reverse closure for elemen... |
cntzssv 19234 | The centralizer is uncondi... |
cntzi 19235 | Membership in a centralize... |
elcntr 19236 | Elementhood in the center ... |
cntrss 19237 | The center is a subset of ... |
cntri 19238 | Defining property of the c... |
resscntz 19239 | Centralizer in a substruct... |
cntzsgrpcl 19240 | Centralizers are closed un... |
cntz2ss 19241 | Centralizers reverse the s... |
cntzrec 19242 | Reciprocity relationship f... |
cntziinsn 19243 | Express any centralizer as... |
cntzsubm 19244 | Centralizers in a monoid a... |
cntzsubg 19245 | Centralizers in a group ar... |
cntzidss 19246 | If the elements of ` S ` c... |
cntzmhm 19247 | Centralizers in a monoid a... |
cntzmhm2 19248 | Centralizers in a monoid a... |
cntrsubgnsg 19249 | A central subgroup is norm... |
cntrnsg 19250 | The center of a group is a... |
oppgval 19253 | Value of the opposite grou... |
oppgplusfval 19254 | Value of the addition oper... |
oppgplus 19255 | Value of the addition oper... |
setsplusg 19256 | The other components of an... |
oppglemOLD 19257 | Obsolete version of ~ sets... |
oppgbas 19258 | Base set of an opposite gr... |
oppgbasOLD 19259 | Obsolete version of ~ oppg... |
oppgtset 19260 | Topology of an opposite gr... |
oppgtsetOLD 19261 | Obsolete version of ~ oppg... |
oppgtopn 19262 | Topology of an opposite gr... |
oppgmnd 19263 | The opposite of a monoid i... |
oppgmndb 19264 | Bidirectional form of ~ op... |
oppgid 19265 | Zero in a monoid is a symm... |
oppggrp 19266 | The opposite of a group is... |
oppggrpb 19267 | Bidirectional form of ~ op... |
oppginv 19268 | Inverses in a group are a ... |
invoppggim 19269 | The inverse is an antiauto... |
oppggic 19270 | Every group is (naturally)... |
oppgsubm 19271 | Being a submonoid is a sym... |
oppgsubg 19272 | Being a subgroup is a symm... |
oppgcntz 19273 | A centralizer in a group i... |
oppgcntr 19274 | The center of a group is t... |
gsumwrev 19275 | A sum in an opposite monoi... |
symgval 19278 | The value of the symmetric... |
permsetexOLD 19279 | Obsolete version of ~ f1os... |
symgbas 19280 | The base set of the symmet... |
symgbasexOLD 19281 | Obsolete as of 8-Aug-2024.... |
elsymgbas2 19282 | Two ways of saying a funct... |
elsymgbas 19283 | Two ways of saying a funct... |
symgbasf1o 19284 | Elements in the symmetric ... |
symgbasf 19285 | A permutation (element of ... |
symgbasmap 19286 | A permutation (element of ... |
symghash 19287 | The symmetric group on ` n... |
symgbasfi 19288 | The symmetric group on a f... |
symgfv 19289 | The function value of a pe... |
symgfvne 19290 | The function values of a p... |
symgressbas 19291 | The symmetric group on ` A... |
symgplusg 19292 | The group operation of a s... |
symgov 19293 | The value of the group ope... |
symgcl 19294 | The group operation of the... |
idresperm 19295 | The identity function rest... |
symgmov1 19296 | For a permutation of a set... |
symgmov2 19297 | For a permutation of a set... |
symgbas0 19298 | The base set of the symmet... |
symg1hash 19299 | The symmetric group on a s... |
symg1bas 19300 | The symmetric group on a s... |
symg2hash 19301 | The symmetric group on a (... |
symg2bas 19302 | The symmetric group on a p... |
0symgefmndeq 19303 | The symmetric group on the... |
snsymgefmndeq 19304 | The symmetric group on a s... |
symgpssefmnd 19305 | For a set ` A ` with more ... |
symgvalstruct 19306 | The value of the symmetric... |
symgvalstructOLD 19307 | Obsolete proof of ~ symgva... |
symgsubmefmnd 19308 | The symmetric group on a s... |
symgtset 19309 | The topology of the symmet... |
symggrp 19310 | The symmetric group on a s... |
symgid 19311 | The group identity element... |
symginv 19312 | The group inverse in the s... |
symgsubmefmndALT 19313 | The symmetric group on a s... |
galactghm 19314 | The currying of a group ac... |
lactghmga 19315 | The converse of ~ galactgh... |
symgtopn 19316 | The topology of the symmet... |
symgga 19317 | The symmetric group induce... |
pgrpsubgsymgbi 19318 | Every permutation group is... |
pgrpsubgsymg 19319 | Every permutation group is... |
idressubgsymg 19320 | The singleton containing o... |
idrespermg 19321 | The structure with the sin... |
cayleylem1 19322 | Lemma for ~ cayley . (Con... |
cayleylem2 19323 | Lemma for ~ cayley . (Con... |
cayley 19324 | Cayley's Theorem (construc... |
cayleyth 19325 | Cayley's Theorem (existenc... |
symgfix2 19326 | If a permutation does not ... |
symgextf 19327 | The extension of a permuta... |
symgextfv 19328 | The function value of the ... |
symgextfve 19329 | The function value of the ... |
symgextf1lem 19330 | Lemma for ~ symgextf1 . (... |
symgextf1 19331 | The extension of a permuta... |
symgextfo 19332 | The extension of a permuta... |
symgextf1o 19333 | The extension of a permuta... |
symgextsymg 19334 | The extension of a permuta... |
symgextres 19335 | The restriction of the ext... |
gsumccatsymgsn 19336 | Homomorphic property of co... |
gsmsymgrfixlem1 19337 | Lemma 1 for ~ gsmsymgrfix ... |
gsmsymgrfix 19338 | The composition of permuta... |
fvcosymgeq 19339 | The values of two composit... |
gsmsymgreqlem1 19340 | Lemma 1 for ~ gsmsymgreq .... |
gsmsymgreqlem2 19341 | Lemma 2 for ~ gsmsymgreq .... |
gsmsymgreq 19342 | Two combination of permuta... |
symgfixelq 19343 | A permutation of a set fix... |
symgfixels 19344 | The restriction of a permu... |
symgfixelsi 19345 | The restriction of a permu... |
symgfixf 19346 | The mapping of a permutati... |
symgfixf1 19347 | The mapping of a permutati... |
symgfixfolem1 19348 | Lemma 1 for ~ symgfixfo . ... |
symgfixfo 19349 | The mapping of a permutati... |
symgfixf1o 19350 | The mapping of a permutati... |
f1omvdmvd 19353 | A permutation of any class... |
f1omvdcnv 19354 | A permutation and its inve... |
mvdco 19355 | Composing two permutations... |
f1omvdconj 19356 | Conjugation of a permutati... |
f1otrspeq 19357 | A transposition is charact... |
f1omvdco2 19358 | If exactly one of two perm... |
f1omvdco3 19359 | If a point is moved by exa... |
pmtrfval 19360 | The function generating tr... |
pmtrval 19361 | A generated transposition,... |
pmtrfv 19362 | General value of mapping a... |
pmtrprfv 19363 | In a transposition of two ... |
pmtrprfv3 19364 | In a transposition of two ... |
pmtrf 19365 | Functionality of a transpo... |
pmtrmvd 19366 | A transposition moves prec... |
pmtrrn 19367 | Transposing two points giv... |
pmtrfrn 19368 | A transposition (as a kind... |
pmtrffv 19369 | Mapping of a point under a... |
pmtrrn2 19370 | For any transposition ther... |
pmtrfinv 19371 | A transposition function i... |
pmtrfmvdn0 19372 | A transposition moves at l... |
pmtrff1o 19373 | A transposition function i... |
pmtrfcnv 19374 | A transposition function i... |
pmtrfb 19375 | An intrinsic characterizat... |
pmtrfconj 19376 | Any conjugate of a transpo... |
symgsssg 19377 | The symmetric group has su... |
symgfisg 19378 | The symmetric group has a ... |
symgtrf 19379 | Transpositions are element... |
symggen 19380 | The span of the transposit... |
symggen2 19381 | A finite permutation group... |
symgtrinv 19382 | To invert a permutation re... |
pmtr3ncomlem1 19383 | Lemma 1 for ~ pmtr3ncom . ... |
pmtr3ncomlem2 19384 | Lemma 2 for ~ pmtr3ncom . ... |
pmtr3ncom 19385 | Transpositions over sets w... |
pmtrdifellem1 19386 | Lemma 1 for ~ pmtrdifel . ... |
pmtrdifellem2 19387 | Lemma 2 for ~ pmtrdifel . ... |
pmtrdifellem3 19388 | Lemma 3 for ~ pmtrdifel . ... |
pmtrdifellem4 19389 | Lemma 4 for ~ pmtrdifel . ... |
pmtrdifel 19390 | A transposition of element... |
pmtrdifwrdellem1 19391 | Lemma 1 for ~ pmtrdifwrdel... |
pmtrdifwrdellem2 19392 | Lemma 2 for ~ pmtrdifwrdel... |
pmtrdifwrdellem3 19393 | Lemma 3 for ~ pmtrdifwrdel... |
pmtrdifwrdel2lem1 19394 | Lemma 1 for ~ pmtrdifwrdel... |
pmtrdifwrdel 19395 | A sequence of transpositio... |
pmtrdifwrdel2 19396 | A sequence of transpositio... |
pmtrprfval 19397 | The transpositions on a pa... |
pmtrprfvalrn 19398 | The range of the transposi... |
psgnunilem1 19403 | Lemma for ~ psgnuni . Giv... |
psgnunilem5 19404 | Lemma for ~ psgnuni . It ... |
psgnunilem2 19405 | Lemma for ~ psgnuni . Ind... |
psgnunilem3 19406 | Lemma for ~ psgnuni . Any... |
psgnunilem4 19407 | Lemma for ~ psgnuni . An ... |
m1expaddsub 19408 | Addition and subtraction o... |
psgnuni 19409 | If the same permutation ca... |
psgnfval 19410 | Function definition of the... |
psgnfn 19411 | Functionality and domain o... |
psgndmsubg 19412 | The finitary permutations ... |
psgneldm 19413 | Property of being a finita... |
psgneldm2 19414 | The finitary permutations ... |
psgneldm2i 19415 | A sequence of transpositio... |
psgneu 19416 | A finitary permutation has... |
psgnval 19417 | Value of the permutation s... |
psgnvali 19418 | A finitary permutation has... |
psgnvalii 19419 | Any representation of a pe... |
psgnpmtr 19420 | All transpositions are odd... |
psgn0fv0 19421 | The permutation sign funct... |
sygbasnfpfi 19422 | The class of non-fixed poi... |
psgnfvalfi 19423 | Function definition of the... |
psgnvalfi 19424 | Value of the permutation s... |
psgnran 19425 | The range of the permutati... |
gsmtrcl 19426 | The group sum of transposi... |
psgnfitr 19427 | A permutation of a finite ... |
psgnfieu 19428 | A permutation of a finite ... |
pmtrsn 19429 | The value of the transposi... |
psgnsn 19430 | The permutation sign funct... |
psgnprfval 19431 | The permutation sign funct... |
psgnprfval1 19432 | The permutation sign of th... |
psgnprfval2 19433 | The permutation sign of th... |
odfval 19442 | Value of the order functio... |
odfvalALT 19443 | Shorter proof of ~ odfval ... |
odval 19444 | Second substitution for th... |
odlem1 19445 | The group element order is... |
odcl 19446 | The order of a group eleme... |
odf 19447 | Functionality of the group... |
odid 19448 | Any element to the power o... |
odlem2 19449 | Any positive annihilator o... |
odmodnn0 19450 | Reduce the argument of a g... |
mndodconglem 19451 | Lemma for ~ mndodcong . (... |
mndodcong 19452 | If two multipliers are con... |
mndodcongi 19453 | If two multipliers are con... |
oddvdsnn0 19454 | The only multiples of ` A ... |
odnncl 19455 | If a nonzero multiple of a... |
odmod 19456 | Reduce the argument of a g... |
oddvds 19457 | The only multiples of ` A ... |
oddvdsi 19458 | Any group element is annih... |
odcong 19459 | If two multipliers are con... |
odeq 19460 | The ~ oddvds property uniq... |
odval2 19461 | A non-conditional definiti... |
odcld 19462 | The order of a group eleme... |
odm1inv 19463 | The (order-1)th multiple o... |
odmulgid 19464 | A relationship between the... |
odmulg2 19465 | The order of a multiple di... |
odmulg 19466 | Relationship between the o... |
odmulgeq 19467 | A multiple of a point of f... |
odbezout 19468 | If ` N ` is coprime to the... |
od1 19469 | The order of the group ide... |
odeq1 19470 | The group identity is the ... |
odinv 19471 | The order of the inverse o... |
odf1 19472 | The multiples of an elemen... |
odinf 19473 | The multiples of an elemen... |
dfod2 19474 | An alternative definition ... |
odcl2 19475 | The order of an element of... |
oddvds2 19476 | The order of an element of... |
finodsubmsubg 19477 | A submonoid whose elements... |
0subgALT 19478 | A shorter proof of ~ 0subg... |
submod 19479 | The order of an element is... |
subgod 19480 | The order of an element is... |
odsubdvds 19481 | The order of an element of... |
odf1o1 19482 | An element with zero order... |
odf1o2 19483 | An element with nonzero or... |
odhash 19484 | An element of zero order g... |
odhash2 19485 | If an element has nonzero ... |
odhash3 19486 | An element which generates... |
odngen 19487 | A cyclic subgroup of size ... |
gexval 19488 | Value of the exponent of a... |
gexlem1 19489 | The group element order is... |
gexcl 19490 | The exponent of a group is... |
gexid 19491 | Any element to the power o... |
gexlem2 19492 | Any positive annihilator o... |
gexdvdsi 19493 | Any group element is annih... |
gexdvds 19494 | The only ` N ` that annihi... |
gexdvds2 19495 | An integer divides the gro... |
gexod 19496 | Any group element is annih... |
gexcl3 19497 | If the order of every grou... |
gexnnod 19498 | Every group element has fi... |
gexcl2 19499 | The exponent of a finite g... |
gexdvds3 19500 | The exponent of a finite g... |
gex1 19501 | A group or monoid has expo... |
ispgp 19502 | A group is a ` P ` -group ... |
pgpprm 19503 | Reverse closure for the fi... |
pgpgrp 19504 | Reverse closure for the se... |
pgpfi1 19505 | A finite group with order ... |
pgp0 19506 | The identity subgroup is a... |
subgpgp 19507 | A subgroup of a p-group is... |
sylow1lem1 19508 | Lemma for ~ sylow1 . The ... |
sylow1lem2 19509 | Lemma for ~ sylow1 . The ... |
sylow1lem3 19510 | Lemma for ~ sylow1 . One ... |
sylow1lem4 19511 | Lemma for ~ sylow1 . The ... |
sylow1lem5 19512 | Lemma for ~ sylow1 . Usin... |
sylow1 19513 | Sylow's first theorem. If... |
odcau 19514 | Cauchy's theorem for the o... |
pgpfi 19515 | The converse to ~ pgpfi1 .... |
pgpfi2 19516 | Alternate version of ~ pgp... |
pgphash 19517 | The order of a p-group. (... |
isslw 19518 | The property of being a Sy... |
slwprm 19519 | Reverse closure for the fi... |
slwsubg 19520 | A Sylow ` P ` -subgroup is... |
slwispgp 19521 | Defining property of a Syl... |
slwpss 19522 | A proper superset of a Syl... |
slwpgp 19523 | A Sylow ` P ` -subgroup is... |
pgpssslw 19524 | Every ` P ` -subgroup is c... |
slwn0 19525 | Every finite group contain... |
subgslw 19526 | A Sylow subgroup that is c... |
sylow2alem1 19527 | Lemma for ~ sylow2a . An ... |
sylow2alem2 19528 | Lemma for ~ sylow2a . All... |
sylow2a 19529 | A named lemma of Sylow's s... |
sylow2blem1 19530 | Lemma for ~ sylow2b . Eva... |
sylow2blem2 19531 | Lemma for ~ sylow2b . Lef... |
sylow2blem3 19532 | Sylow's second theorem. P... |
sylow2b 19533 | Sylow's second theorem. A... |
slwhash 19534 | A sylow subgroup has cardi... |
fislw 19535 | The sylow subgroups of a f... |
sylow2 19536 | Sylow's second theorem. S... |
sylow3lem1 19537 | Lemma for ~ sylow3 , first... |
sylow3lem2 19538 | Lemma for ~ sylow3 , first... |
sylow3lem3 19539 | Lemma for ~ sylow3 , first... |
sylow3lem4 19540 | Lemma for ~ sylow3 , first... |
sylow3lem5 19541 | Lemma for ~ sylow3 , secon... |
sylow3lem6 19542 | Lemma for ~ sylow3 , secon... |
sylow3 19543 | Sylow's third theorem. Th... |
lsmfval 19548 | The subgroup sum function ... |
lsmvalx 19549 | Subspace sum value (for a ... |
lsmelvalx 19550 | Subspace sum membership (f... |
lsmelvalix 19551 | Subspace sum membership (f... |
oppglsm 19552 | The subspace sum operation... |
lsmssv 19553 | Subgroup sum is a subset o... |
lsmless1x 19554 | Subset implies subgroup su... |
lsmless2x 19555 | Subset implies subgroup su... |
lsmub1x 19556 | Subgroup sum is an upper b... |
lsmub2x 19557 | Subgroup sum is an upper b... |
lsmval 19558 | Subgroup sum value (for a ... |
lsmelval 19559 | Subgroup sum membership (f... |
lsmelvali 19560 | Subgroup sum membership (f... |
lsmelvalm 19561 | Subgroup sum membership an... |
lsmelvalmi 19562 | Membership of vector subtr... |
lsmsubm 19563 | The sum of two commuting s... |
lsmsubg 19564 | The sum of two commuting s... |
lsmcom2 19565 | Subgroup sum commutes. (C... |
smndlsmidm 19566 | The direct product is idem... |
lsmub1 19567 | Subgroup sum is an upper b... |
lsmub2 19568 | Subgroup sum is an upper b... |
lsmunss 19569 | Union of subgroups is a su... |
lsmless1 19570 | Subset implies subgroup su... |
lsmless2 19571 | Subset implies subgroup su... |
lsmless12 19572 | Subset implies subgroup su... |
lsmidm 19573 | Subgroup sum is idempotent... |
lsmlub 19574 | The least upper bound prop... |
lsmss1 19575 | Subgroup sum with a subset... |
lsmss1b 19576 | Subgroup sum with a subset... |
lsmss2 19577 | Subgroup sum with a subset... |
lsmss2b 19578 | Subgroup sum with a subset... |
lsmass 19579 | Subgroup sum is associativ... |
mndlsmidm 19580 | Subgroup sum is idempotent... |
lsm01 19581 | Subgroup sum with the zero... |
lsm02 19582 | Subgroup sum with the zero... |
subglsm 19583 | The subgroup sum evaluated... |
lssnle 19584 | Equivalent expressions for... |
lsmmod 19585 | The modular law holds for ... |
lsmmod2 19586 | Modular law dual for subgr... |
lsmpropd 19587 | If two structures have the... |
cntzrecd 19588 | Commute the "subgroups com... |
lsmcntz 19589 | The "subgroups commute" pr... |
lsmcntzr 19590 | The "subgroups commute" pr... |
lsmdisj 19591 | Disjointness from a subgro... |
lsmdisj2 19592 | Association of the disjoin... |
lsmdisj3 19593 | Association of the disjoin... |
lsmdisjr 19594 | Disjointness from a subgro... |
lsmdisj2r 19595 | Association of the disjoin... |
lsmdisj3r 19596 | Association of the disjoin... |
lsmdisj2a 19597 | Association of the disjoin... |
lsmdisj2b 19598 | Association of the disjoin... |
lsmdisj3a 19599 | Association of the disjoin... |
lsmdisj3b 19600 | Association of the disjoin... |
subgdisj1 19601 | Vectors belonging to disjo... |
subgdisj2 19602 | Vectors belonging to disjo... |
subgdisjb 19603 | Vectors belonging to disjo... |
pj1fval 19604 | The left projection functi... |
pj1val 19605 | The left projection functi... |
pj1eu 19606 | Uniqueness of a left proje... |
pj1f 19607 | The left projection functi... |
pj2f 19608 | The right projection funct... |
pj1id 19609 | Any element of a direct su... |
pj1eq 19610 | Any element of a direct su... |
pj1lid 19611 | The left projection functi... |
pj1rid 19612 | The left projection functi... |
pj1ghm 19613 | The left projection functi... |
pj1ghm2 19614 | The left projection functi... |
lsmhash 19615 | The order of the direct pr... |
efgmval 19622 | Value of the formal invers... |
efgmf 19623 | The formal inverse operati... |
efgmnvl 19624 | The inversion function on ... |
efgrcl 19625 | Lemma for ~ efgval . (Con... |
efglem 19626 | Lemma for ~ efgval . (Con... |
efgval 19627 | Value of the free group co... |
efger 19628 | Value of the free group co... |
efgi 19629 | Value of the free group co... |
efgi0 19630 | Value of the free group co... |
efgi1 19631 | Value of the free group co... |
efgtf 19632 | Value of the free group co... |
efgtval 19633 | Value of the extension fun... |
efgval2 19634 | Value of the free group co... |
efgi2 19635 | Value of the free group co... |
efgtlen 19636 | Value of the free group co... |
efginvrel2 19637 | The inverse of the reverse... |
efginvrel1 19638 | The inverse of the reverse... |
efgsf 19639 | Value of the auxiliary fun... |
efgsdm 19640 | Elementhood in the domain ... |
efgsval 19641 | Value of the auxiliary fun... |
efgsdmi 19642 | Property of the last link ... |
efgsval2 19643 | Value of the auxiliary fun... |
efgsrel 19644 | The start and end of any e... |
efgs1 19645 | A singleton of an irreduci... |
efgs1b 19646 | Every extension sequence e... |
efgsp1 19647 | If ` F ` is an extension s... |
efgsres 19648 | An initial segment of an e... |
efgsfo 19649 | For any word, there is a s... |
efgredlema 19650 | The reduced word that form... |
efgredlemf 19651 | Lemma for ~ efgredleme . ... |
efgredlemg 19652 | Lemma for ~ efgred . (Con... |
efgredleme 19653 | Lemma for ~ efgred . (Con... |
efgredlemd 19654 | The reduced word that form... |
efgredlemc 19655 | The reduced word that form... |
efgredlemb 19656 | The reduced word that form... |
efgredlem 19657 | The reduced word that form... |
efgred 19658 | The reduced word that form... |
efgrelexlema 19659 | If two words ` A , B ` are... |
efgrelexlemb 19660 | If two words ` A , B ` are... |
efgrelex 19661 | If two words ` A , B ` are... |
efgredeu 19662 | There is a unique reduced ... |
efgred2 19663 | Two extension sequences ha... |
efgcpbllema 19664 | Lemma for ~ efgrelex . De... |
efgcpbllemb 19665 | Lemma for ~ efgrelex . Sh... |
efgcpbl 19666 | Two extension sequences ha... |
efgcpbl2 19667 | Two extension sequences ha... |
frgpval 19668 | Value of the free group co... |
frgpcpbl 19669 | Compatibility of the group... |
frgp0 19670 | The free group is a group.... |
frgpeccl 19671 | Closure of the quotient ma... |
frgpgrp 19672 | The free group is a group.... |
frgpadd 19673 | Addition in the free group... |
frgpinv 19674 | The inverse of an element ... |
frgpmhm 19675 | The "natural map" from wor... |
vrgpfval 19676 | The canonical injection fr... |
vrgpval 19677 | The value of the generatin... |
vrgpf 19678 | The mapping from the index... |
vrgpinv 19679 | The inverse of a generatin... |
frgpuptf 19680 | Any assignment of the gene... |
frgpuptinv 19681 | Any assignment of the gene... |
frgpuplem 19682 | Any assignment of the gene... |
frgpupf 19683 | Any assignment of the gene... |
frgpupval 19684 | Any assignment of the gene... |
frgpup1 19685 | Any assignment of the gene... |
frgpup2 19686 | The evaluation map has the... |
frgpup3lem 19687 | The evaluation map has the... |
frgpup3 19688 | Universal property of the ... |
0frgp 19689 | The free group on zero gen... |
isabl 19694 | The predicate "is an Abeli... |
ablgrp 19695 | An Abelian group is a grou... |
ablgrpd 19696 | An Abelian group is a grou... |
ablcmn 19697 | An Abelian group is a comm... |
ablcmnd 19698 | An Abelian group is a comm... |
iscmn 19699 | The predicate "is a commut... |
isabl2 19700 | The predicate "is an Abeli... |
cmnpropd 19701 | If two structures have the... |
ablpropd 19702 | If two structures have the... |
ablprop 19703 | If two structures have the... |
iscmnd 19704 | Properties that determine ... |
isabld 19705 | Properties that determine ... |
isabli 19706 | Properties that determine ... |
cmnmnd 19707 | A commutative monoid is a ... |
cmncom 19708 | A commutative monoid is co... |
ablcom 19709 | An Abelian group operation... |
cmn32 19710 | Commutative/associative la... |
cmn4 19711 | Commutative/associative la... |
cmn12 19712 | Commutative/associative la... |
abl32 19713 | Commutative/associative la... |
cmnmndd 19714 | A commutative monoid is a ... |
cmnbascntr 19715 | The base set of a commutat... |
rinvmod 19716 | Uniqueness of a right inve... |
ablinvadd 19717 | The inverse of an Abelian ... |
ablsub2inv 19718 | Abelian group subtraction ... |
ablsubadd 19719 | Relationship between Abeli... |
ablsub4 19720 | Commutative/associative su... |
abladdsub4 19721 | Abelian group addition/sub... |
abladdsub 19722 | Associative-type law for g... |
ablsubadd23 19723 | Commutative/associative la... |
ablsubaddsub 19724 | Double subtraction and add... |
ablpncan2 19725 | Cancellation law for subtr... |
ablpncan3 19726 | A cancellation law for Abe... |
ablsubsub 19727 | Law for double subtraction... |
ablsubsub4 19728 | Law for double subtraction... |
ablpnpcan 19729 | Cancellation law for mixed... |
ablnncan 19730 | Cancellation law for group... |
ablsub32 19731 | Swap the second and third ... |
ablnnncan 19732 | Cancellation law for group... |
ablnnncan1 19733 | Cancellation law for group... |
ablsubsub23 19734 | Swap subtrahend and result... |
mulgnn0di 19735 | Group multiple of a sum, f... |
mulgdi 19736 | Group multiple of a sum. ... |
mulgmhm 19737 | The map from ` x ` to ` n ... |
mulgghm 19738 | The map from ` x ` to ` n ... |
mulgsubdi 19739 | Group multiple of a differ... |
ghmfghm 19740 | The function fulfilling th... |
ghmcmn 19741 | The image of a commutative... |
ghmabl 19742 | The image of an abelian gr... |
invghm 19743 | The inversion map is a gro... |
eqgabl 19744 | Value of the subgroup cose... |
qusecsub 19745 | Two subgroup cosets are eq... |
subgabl 19746 | A subgroup of an abelian g... |
subcmn 19747 | A submonoid of a commutati... |
submcmn 19748 | A submonoid of a commutati... |
submcmn2 19749 | A submonoid is commutative... |
cntzcmn 19750 | The centralizer of any sub... |
cntzcmnss 19751 | Any subset in a commutativ... |
cntrcmnd 19752 | The center of a monoid is ... |
cntrabl 19753 | The center of a group is a... |
cntzspan 19754 | If the generators commute,... |
cntzcmnf 19755 | Discharge the centralizer ... |
ghmplusg 19756 | The pointwise sum of two l... |
ablnsg 19757 | Every subgroup of an abeli... |
odadd1 19758 | The order of a product in ... |
odadd2 19759 | The order of a product in ... |
odadd 19760 | The order of a product is ... |
gex2abl 19761 | A group with exponent 2 (o... |
gexexlem 19762 | Lemma for ~ gexex . (Cont... |
gexex 19763 | In an abelian group with f... |
torsubg 19764 | The set of all elements of... |
oddvdssubg 19765 | The set of all elements wh... |
lsmcomx 19766 | Subgroup sum commutes (ext... |
ablcntzd 19767 | All subgroups in an abelia... |
lsmcom 19768 | Subgroup sum commutes. (C... |
lsmsubg2 19769 | The sum of two subgroups i... |
lsm4 19770 | Commutative/associative la... |
prdscmnd 19771 | The product of a family of... |
prdsabld 19772 | The product of a family of... |
pwscmn 19773 | The structure power on a c... |
pwsabl 19774 | The structure power on an ... |
qusabl 19775 | If ` Y ` is a subgroup of ... |
abl1 19776 | The (smallest) structure r... |
abln0 19777 | Abelian groups (and theref... |
cnaddablx 19778 | The complex numbers are an... |
cnaddabl 19779 | The complex numbers are an... |
cnaddid 19780 | The group identity element... |
cnaddinv 19781 | Value of the group inverse... |
zaddablx 19782 | The integers are an Abelia... |
frgpnabllem1 19783 | Lemma for ~ frgpnabl . (C... |
frgpnabllem2 19784 | Lemma for ~ frgpnabl . (C... |
frgpnabl 19785 | The free group on two or m... |
imasabl 19786 | The image structure of an ... |
iscyg 19789 | Definition of a cyclic gro... |
iscyggen 19790 | The property of being a cy... |
iscyggen2 19791 | The property of being a cy... |
iscyg2 19792 | A cyclic group is a group ... |
cyggeninv 19793 | The inverse of a cyclic ge... |
cyggenod 19794 | An element is the generato... |
cyggenod2 19795 | In an infinite cyclic grou... |
iscyg3 19796 | Definition of a cyclic gro... |
iscygd 19797 | Definition of a cyclic gro... |
iscygodd 19798 | Show that a group with an ... |
cycsubmcmn 19799 | The set of nonnegative int... |
cyggrp 19800 | A cyclic group is a group.... |
cygabl 19801 | A cyclic group is abelian.... |
cygctb 19802 | A cyclic group is countabl... |
0cyg 19803 | The trivial group is cycli... |
prmcyg 19804 | A group with prime order i... |
lt6abl 19805 | A group with fewer than ` ... |
ghmcyg 19806 | The image of a cyclic grou... |
cyggex2 19807 | The exponent of a cyclic g... |
cyggex 19808 | The exponent of a finite c... |
cyggexb 19809 | A finite abelian group is ... |
giccyg 19810 | Cyclicity is a group prope... |
cycsubgcyg 19811 | The cyclic subgroup genera... |
cycsubgcyg2 19812 | The cyclic subgroup genera... |
gsumval3a 19813 | Value of the group sum ope... |
gsumval3eu 19814 | The group sum as defined i... |
gsumval3lem1 19815 | Lemma 1 for ~ gsumval3 . ... |
gsumval3lem2 19816 | Lemma 2 for ~ gsumval3 . ... |
gsumval3 19817 | Value of the group sum ope... |
gsumcllem 19818 | Lemma for ~ gsumcl and rel... |
gsumzres 19819 | Extend a finite group sum ... |
gsumzcl2 19820 | Closure of a finite group ... |
gsumzcl 19821 | Closure of a finite group ... |
gsumzf1o 19822 | Re-index a finite group su... |
gsumres 19823 | Extend a finite group sum ... |
gsumcl2 19824 | Closure of a finite group ... |
gsumcl 19825 | Closure of a finite group ... |
gsumf1o 19826 | Re-index a finite group su... |
gsumreidx 19827 | Re-index a finite group su... |
gsumzsubmcl 19828 | Closure of a group sum in ... |
gsumsubmcl 19829 | Closure of a group sum in ... |
gsumsubgcl 19830 | Closure of a group sum in ... |
gsumzaddlem 19831 | The sum of two group sums.... |
gsumzadd 19832 | The sum of two group sums.... |
gsumadd 19833 | The sum of two group sums.... |
gsummptfsadd 19834 | The sum of two group sums ... |
gsummptfidmadd 19835 | The sum of two group sums ... |
gsummptfidmadd2 19836 | The sum of two group sums ... |
gsumzsplit 19837 | Split a group sum into two... |
gsumsplit 19838 | Split a group sum into two... |
gsumsplit2 19839 | Split a group sum into two... |
gsummptfidmsplit 19840 | Split a group sum expresse... |
gsummptfidmsplitres 19841 | Split a group sum expresse... |
gsummptfzsplit 19842 | Split a group sum expresse... |
gsummptfzsplitl 19843 | Split a group sum expresse... |
gsumconst 19844 | Sum of a constant series. ... |
gsumconstf 19845 | Sum of a constant series. ... |
gsummptshft 19846 | Index shift of a finite gr... |
gsumzmhm 19847 | Apply a group homomorphism... |
gsummhm 19848 | Apply a group homomorphism... |
gsummhm2 19849 | Apply a group homomorphism... |
gsummptmhm 19850 | Apply a group homomorphism... |
gsummulglem 19851 | Lemma for ~ gsummulg and ~... |
gsummulg 19852 | Nonnegative multiple of a ... |
gsummulgz 19853 | Integer multiple of a grou... |
gsumzoppg 19854 | The opposite of a group su... |
gsumzinv 19855 | Inverse of a group sum. (... |
gsuminv 19856 | Inverse of a group sum. (... |
gsummptfidminv 19857 | Inverse of a group sum exp... |
gsumsub 19858 | The difference of two grou... |
gsummptfssub 19859 | The difference of two grou... |
gsummptfidmsub 19860 | The difference of two grou... |
gsumsnfd 19861 | Group sum of a singleton, ... |
gsumsnd 19862 | Group sum of a singleton, ... |
gsumsnf 19863 | Group sum of a singleton, ... |
gsumsn 19864 | Group sum of a singleton. ... |
gsumpr 19865 | Group sum of a pair. (Con... |
gsumzunsnd 19866 | Append an element to a fin... |
gsumunsnfd 19867 | Append an element to a fin... |
gsumunsnd 19868 | Append an element to a fin... |
gsumunsnf 19869 | Append an element to a fin... |
gsumunsn 19870 | Append an element to a fin... |
gsumdifsnd 19871 | Extract a summand from a f... |
gsumpt 19872 | Sum of a family that is no... |
gsummptf1o 19873 | Re-index a finite group su... |
gsummptun 19874 | Group sum of a disjoint un... |
gsummpt1n0 19875 | If only one summand in a f... |
gsummptif1n0 19876 | If only one summand in a f... |
gsummptcl 19877 | Closure of a finite group ... |
gsummptfif1o 19878 | Re-index a finite group su... |
gsummptfzcl 19879 | Closure of a finite group ... |
gsum2dlem1 19880 | Lemma 1 for ~ gsum2d . (C... |
gsum2dlem2 19881 | Lemma for ~ gsum2d . (Con... |
gsum2d 19882 | Write a sum over a two-dim... |
gsum2d2lem 19883 | Lemma for ~ gsum2d2 : show... |
gsum2d2 19884 | Write a group sum over a t... |
gsumcom2 19885 | Two-dimensional commutatio... |
gsumxp 19886 | Write a group sum over a c... |
gsumcom 19887 | Commute the arguments of a... |
gsumcom3 19888 | A commutative law for fini... |
gsumcom3fi 19889 | A commutative law for fini... |
gsumxp2 19890 | Write a group sum over a c... |
prdsgsum 19891 | Finite commutative sums in... |
pwsgsum 19892 | Finite commutative sums in... |
fsfnn0gsumfsffz 19893 | Replacing a finitely suppo... |
nn0gsumfz 19894 | Replacing a finitely suppo... |
nn0gsumfz0 19895 | Replacing a finitely suppo... |
gsummptnn0fz 19896 | A final group sum over a f... |
gsummptnn0fzfv 19897 | A final group sum over a f... |
telgsumfzslem 19898 | Lemma for ~ telgsumfzs (in... |
telgsumfzs 19899 | Telescoping group sum rang... |
telgsumfz 19900 | Telescoping group sum rang... |
telgsumfz0s 19901 | Telescoping finite group s... |
telgsumfz0 19902 | Telescoping finite group s... |
telgsums 19903 | Telescoping finitely suppo... |
telgsum 19904 | Telescoping finitely suppo... |
reldmdprd 19909 | The domain of the internal... |
dmdprd 19910 | The domain of definition o... |
dmdprdd 19911 | Show that a given family i... |
dprddomprc 19912 | A family of subgroups inde... |
dprddomcld 19913 | If a family of subgroups i... |
dprdval0prc 19914 | The internal direct produc... |
dprdval 19915 | The value of the internal ... |
eldprd 19916 | A class ` A ` is an intern... |
dprdgrp 19917 | Reverse closure for the in... |
dprdf 19918 | The function ` S ` is a fa... |
dprdf2 19919 | The function ` S ` is a fa... |
dprdcntz 19920 | The function ` S ` is a fa... |
dprddisj 19921 | The function ` S ` is a fa... |
dprdw 19922 | The property of being a fi... |
dprdwd 19923 | A mapping being a finitely... |
dprdff 19924 | A finitely supported funct... |
dprdfcl 19925 | A finitely supported funct... |
dprdffsupp 19926 | A finitely supported funct... |
dprdfcntz 19927 | A function on the elements... |
dprdssv 19928 | The internal direct produc... |
dprdfid 19929 | A function mapping all but... |
eldprdi 19930 | The domain of definition o... |
dprdfinv 19931 | Take the inverse of a grou... |
dprdfadd 19932 | Take the sum of group sums... |
dprdfsub 19933 | Take the difference of gro... |
dprdfeq0 19934 | The zero function is the o... |
dprdf11 19935 | Two group sums over a dire... |
dprdsubg 19936 | The internal direct produc... |
dprdub 19937 | Each factor is a subset of... |
dprdlub 19938 | The direct product is smal... |
dprdspan 19939 | The direct product is the ... |
dprdres 19940 | Restriction of a direct pr... |
dprdss 19941 | Create a direct product by... |
dprdz 19942 | A family consisting entire... |
dprd0 19943 | The empty family is an int... |
dprdf1o 19944 | Rearrange the index set of... |
dprdf1 19945 | Rearrange the index set of... |
subgdmdprd 19946 | A direct product in a subg... |
subgdprd 19947 | A direct product in a subg... |
dprdsn 19948 | A singleton family is an i... |
dmdprdsplitlem 19949 | Lemma for ~ dmdprdsplit . ... |
dprdcntz2 19950 | The function ` S ` is a fa... |
dprddisj2 19951 | The function ` S ` is a fa... |
dprd2dlem2 19952 | The direct product of a co... |
dprd2dlem1 19953 | The direct product of a co... |
dprd2da 19954 | The direct product of a co... |
dprd2db 19955 | The direct product of a co... |
dprd2d2 19956 | The direct product of a co... |
dmdprdsplit2lem 19957 | Lemma for ~ dmdprdsplit . ... |
dmdprdsplit2 19958 | The direct product splits ... |
dmdprdsplit 19959 | The direct product splits ... |
dprdsplit 19960 | The direct product is the ... |
dmdprdpr 19961 | A singleton family is an i... |
dprdpr 19962 | A singleton family is an i... |
dpjlem 19963 | Lemma for theorems about d... |
dpjcntz 19964 | The two subgroups that app... |
dpjdisj 19965 | The two subgroups that app... |
dpjlsm 19966 | The two subgroups that app... |
dpjfval 19967 | Value of the direct produc... |
dpjval 19968 | Value of the direct produc... |
dpjf 19969 | The ` X ` -th index projec... |
dpjidcl 19970 | The key property of projec... |
dpjeq 19971 | Decompose a group sum into... |
dpjid 19972 | The key property of projec... |
dpjlid 19973 | The ` X ` -th index projec... |
dpjrid 19974 | The ` Y ` -th index projec... |
dpjghm 19975 | The direct product is the ... |
dpjghm2 19976 | The direct product is the ... |
ablfacrplem 19977 | Lemma for ~ ablfacrp2 . (... |
ablfacrp 19978 | A finite abelian group who... |
ablfacrp2 19979 | The factors ` K , L ` of ~... |
ablfac1lem 19980 | Lemma for ~ ablfac1b . Sa... |
ablfac1a 19981 | The factors of ~ ablfac1b ... |
ablfac1b 19982 | Any abelian group is the d... |
ablfac1c 19983 | The factors of ~ ablfac1b ... |
ablfac1eulem 19984 | Lemma for ~ ablfac1eu . (... |
ablfac1eu 19985 | The factorization of ~ abl... |
pgpfac1lem1 19986 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem2 19987 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem3a 19988 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem3 19989 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem4 19990 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem5 19991 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1 19992 | Factorization of a finite ... |
pgpfaclem1 19993 | Lemma for ~ pgpfac . (Con... |
pgpfaclem2 19994 | Lemma for ~ pgpfac . (Con... |
pgpfaclem3 19995 | Lemma for ~ pgpfac . (Con... |
pgpfac 19996 | Full factorization of a fi... |
ablfaclem1 19997 | Lemma for ~ ablfac . (Con... |
ablfaclem2 19998 | Lemma for ~ ablfac . (Con... |
ablfaclem3 19999 | Lemma for ~ ablfac . (Con... |
ablfac 20000 | The Fundamental Theorem of... |
ablfac2 20001 | Choose generators for each... |
issimpg 20004 | The predicate "is a simple... |
issimpgd 20005 | Deduce a simple group from... |
simpggrp 20006 | A simple group is a group.... |
simpggrpd 20007 | A simple group is a group.... |
simpg2nsg 20008 | A simple group has two nor... |
trivnsimpgd 20009 | Trivial groups are not sim... |
simpgntrivd 20010 | Simple groups are nontrivi... |
simpgnideld 20011 | A simple group contains a ... |
simpgnsgd 20012 | The only normal subgroups ... |
simpgnsgeqd 20013 | A normal subgroup of a sim... |
2nsgsimpgd 20014 | If any normal subgroup of ... |
simpgnsgbid 20015 | A nontrivial group is simp... |
ablsimpnosubgd 20016 | A subgroup of an abelian s... |
ablsimpg1gend 20017 | An abelian simple group is... |
ablsimpgcygd 20018 | An abelian simple group is... |
ablsimpgfindlem1 20019 | Lemma for ~ ablsimpgfind .... |
ablsimpgfindlem2 20020 | Lemma for ~ ablsimpgfind .... |
cycsubggenodd 20021 | Relationship between the o... |
ablsimpgfind 20022 | An abelian simple group is... |
fincygsubgd 20023 | The subgroup referenced in... |
fincygsubgodd 20024 | Calculate the order of a s... |
fincygsubgodexd 20025 | A finite cyclic group has ... |
prmgrpsimpgd 20026 | A group of prime order is ... |
ablsimpgprmd 20027 | An abelian simple group ha... |
ablsimpgd 20028 | An abelian group is simple... |
fnmgp 20031 | The multiplicative group o... |
mgpval 20032 | Value of the multiplicatio... |
mgpplusg 20033 | Value of the group operati... |
mgplemOLD 20034 | Obsolete version of ~ sets... |
mgpbas 20035 | Base set of the multiplica... |
mgpbasOLD 20036 | Obsolete version of ~ mgpb... |
mgpsca 20037 | The multiplication monoid ... |
mgpscaOLD 20038 | Obsolete version of ~ mgps... |
mgptset 20039 | Topology component of the ... |
mgptsetOLD 20040 | Obsolete version of ~ mgpt... |
mgptopn 20041 | Topology of the multiplica... |
mgpds 20042 | Distance function of the m... |
mgpdsOLD 20043 | Obsolete version of ~ mgpd... |
mgpress 20044 | Subgroup commutes with the... |
mgpressOLD 20045 | Obsolete version of ~ mgpr... |
prdsmgp 20046 | The multiplicative monoid ... |
isrng 20049 | The predicate "is a non-un... |
rngabl 20050 | A non-unital ring is an (a... |
rngmgp 20051 | A non-unital ring is a sem... |
rngmgpf 20052 | Restricted functionality o... |
rnggrp 20053 | A non-unital ring is a (ad... |
rngass 20054 | Associative law for the mu... |
rngdi 20055 | Distributive law for the m... |
rngdir 20056 | Distributive law for the m... |
rngacl 20057 | Closure of the addition op... |
rng0cl 20058 | The zero element of a non-... |
rngcl 20059 | Closure of the multiplicat... |
rnglz 20060 | The zero of a non-unital r... |
rngrz 20061 | The zero of a non-unital r... |
rngmneg1 20062 | Negation of a product in a... |
rngmneg2 20063 | Negation of a product in a... |
rngm2neg 20064 | Double negation of a produ... |
rngansg 20065 | Every additive subgroup of... |
rngsubdi 20066 | Ring multiplication distri... |
rngsubdir 20067 | Ring multiplication distri... |
isrngd 20068 | Properties that determine ... |
rngpropd 20069 | If two structures have the... |
prdsmulrngcl 20070 | Closure of the multiplicat... |
prdsrngd 20071 | A product of non-unital ri... |
imasrng 20072 | The image structure of a n... |
imasrngf1 20073 | The image of a non-unital ... |
xpsrngd 20074 | A product of two non-unita... |
qusrng 20075 | The quotient structure of ... |
ringidval 20078 | The value of the unity ele... |
dfur2 20079 | The multiplicative identit... |
ringurd 20080 | Deduce the unity element o... |
issrg 20083 | The predicate "is a semiri... |
srgcmn 20084 | A semiring is a commutativ... |
srgmnd 20085 | A semiring is a monoid. (... |
srgmgp 20086 | A semiring is a monoid und... |
srgdilem 20087 | Lemma for ~ srgdi and ~ sr... |
srgcl 20088 | Closure of the multiplicat... |
srgass 20089 | Associative law for the mu... |
srgideu 20090 | The unity element of a sem... |
srgfcl 20091 | Functionality of the multi... |
srgdi 20092 | Distributive law for the m... |
srgdir 20093 | Distributive law for the m... |
srgidcl 20094 | The unity element of a sem... |
srg0cl 20095 | The zero element of a semi... |
srgidmlem 20096 | Lemma for ~ srglidm and ~ ... |
srglidm 20097 | The unity element of a sem... |
srgridm 20098 | The unity element of a sem... |
issrgid 20099 | Properties showing that an... |
srgacl 20100 | Closure of the addition op... |
srgcom 20101 | Commutativity of the addit... |
srgrz 20102 | The zero of a semiring is ... |
srglz 20103 | The zero of a semiring is ... |
srgisid 20104 | In a semiring, the only le... |
o2timesd 20105 | An element of a ring-like ... |
rglcom4d 20106 | Restricted commutativity o... |
srgo2times 20107 | A semiring element plus it... |
srgcom4lem 20108 | Lemma for ~ srgcom4 . Thi... |
srgcom4 20109 | Restricted commutativity o... |
srg1zr 20110 | The only semiring with a b... |
srgen1zr 20111 | The only semiring with one... |
srgmulgass 20112 | An associative property be... |
srgpcomp 20113 | If two elements of a semir... |
srgpcompp 20114 | If two elements of a semir... |
srgpcomppsc 20115 | If two elements of a semir... |
srglmhm 20116 | Left-multiplication in a s... |
srgrmhm 20117 | Right-multiplication in a ... |
srgsummulcr 20118 | A finite semiring sum mult... |
sgsummulcl 20119 | A finite semiring sum mult... |
srg1expzeq1 20120 | The exponentiation (by a n... |
srgbinomlem1 20121 | Lemma 1 for ~ srgbinomlem ... |
srgbinomlem2 20122 | Lemma 2 for ~ srgbinomlem ... |
srgbinomlem3 20123 | Lemma 3 for ~ srgbinomlem ... |
srgbinomlem4 20124 | Lemma 4 for ~ srgbinomlem ... |
srgbinomlem 20125 | Lemma for ~ srgbinom . In... |
srgbinom 20126 | The binomial theorem for c... |
csrgbinom 20127 | The binomial theorem for c... |
isring 20132 | The predicate "is a (unita... |
ringgrp 20133 | A ring is a group. (Contr... |
ringmgp 20134 | A ring is a monoid under m... |
iscrng 20135 | A commutative ring is a ri... |
crngmgp 20136 | A commutative ring's multi... |
ringgrpd 20137 | A ring is a group. (Contr... |
ringmnd 20138 | A ring is a monoid under a... |
ringmgm 20139 | A ring is a magma. (Contr... |
crngring 20140 | A commutative ring is a ri... |
crngringd 20141 | A commutative ring is a ri... |
crnggrpd 20142 | A commutative ring is a gr... |
mgpf 20143 | Restricted functionality o... |
ringdilem 20144 | Properties of a unital rin... |
ringcl 20145 | Closure of the multiplicat... |
crngcom 20146 | A commutative ring's multi... |
iscrng2 20147 | A commutative ring is a ri... |
ringass 20148 | Associative law for multip... |
ringideu 20149 | The unity element of a rin... |
crngbascntr 20150 | The base set of a commutat... |
ringassd 20151 | Associative law for multip... |
ringcld 20152 | Closure of the multiplicat... |
ringdi 20153 | Distributive law for the m... |
ringdir 20154 | Distributive law for the m... |
ringidcl 20155 | The unity element of a rin... |
ring0cl 20156 | The zero element of a ring... |
ringidmlem 20157 | Lemma for ~ ringlidm and ~... |
ringlidm 20158 | The unity element of a rin... |
ringridm 20159 | The unity element of a rin... |
isringid 20160 | Properties showing that an... |
ringlidmd 20161 | The unity element of a rin... |
ringridmd 20162 | The unity element of a rin... |
ringid 20163 | The multiplication operati... |
ringo2times 20164 | A ring element plus itself... |
ringadd2 20165 | A ring element plus itself... |
ringidss 20166 | A subset of the multiplica... |
ringacl 20167 | Closure of the addition op... |
ringcomlem 20168 | Lemma for ~ ringcom . Thi... |
ringcom 20169 | Commutativity of the addit... |
ringabl 20170 | A ring is an Abelian group... |
ringcmn 20171 | A ring is a commutative mo... |
ringabld 20172 | A ring is an Abelian group... |
ringcmnd 20173 | A ring is a commutative mo... |
ringrng 20174 | A unital ring is a non-uni... |
ringssrng 20175 | The unital rings are non-u... |
isringrng 20176 | The predicate "is a unital... |
ringpropd 20177 | If two structures have the... |
crngpropd 20178 | If two structures have the... |
ringprop 20179 | If two structures have the... |
isringd 20180 | Properties that determine ... |
iscrngd 20181 | Properties that determine ... |
ringlz 20182 | The zero of a unital ring ... |
ringrz 20183 | The zero of a unital ring ... |
ringlzd 20184 | The zero of a unital ring ... |
ringrzd 20185 | The zero of a unital ring ... |
ringsrg 20186 | Any ring is also a semirin... |
ring1eq0 20187 | If one and zero are equal,... |
ring1ne0 20188 | If a ring has at least two... |
ringinvnz1ne0 20189 | In a unital ring, a left i... |
ringinvnzdiv 20190 | In a unital ring, a left i... |
ringnegl 20191 | Negation in a ring is the ... |
ringnegr 20192 | Negation in a ring is the ... |
ringmneg1 20193 | Negation of a product in a... |
ringmneg2 20194 | Negation of a product in a... |
ringm2neg 20195 | Double negation of a produ... |
ringsubdi 20196 | Ring multiplication distri... |
ringsubdir 20197 | Ring multiplication distri... |
mulgass2 20198 | An associative property be... |
ring1 20199 | The (smallest) structure r... |
ringn0 20200 | Rings exist. (Contributed... |
ringlghm 20201 | Left-multiplication in a r... |
ringrghm 20202 | Right-multiplication in a ... |
gsummulc1OLD 20203 | Obsolete version of ~ gsum... |
gsummulc2OLD 20204 | Obsolete version of ~ gsum... |
gsummulc1 20205 | A finite ring sum multipli... |
gsummulc2 20206 | A finite ring sum multipli... |
gsummgp0 20207 | If one factor in a finite ... |
gsumdixp 20208 | Distribute a binary produc... |
prdsmulrcl 20209 | A structure product of rin... |
prdsringd 20210 | A product of rings is a ri... |
prdscrngd 20211 | A product of commutative r... |
prds1 20212 | Value of the ring unity in... |
pwsring 20213 | A structure power of a rin... |
pws1 20214 | Value of the ring unity in... |
pwscrng 20215 | A structure power of a com... |
pwsmgp 20216 | The multiplicative group o... |
pwspjmhmmgpd 20217 | The projection given by ~ ... |
pwsexpg 20218 | Value of a group exponenti... |
imasring 20219 | The image structure of a r... |
imasringf1 20220 | The image of a ring under ... |
xpsringd 20221 | A product of two rings is ... |
xpsring1d 20222 | The multiplicative identit... |
qusring2 20223 | The quotient structure of ... |
crngbinom 20224 | The binomial theorem for c... |
opprval 20227 | Value of the opposite ring... |
opprmulfval 20228 | Value of the multiplicatio... |
opprmul 20229 | Value of the multiplicatio... |
crngoppr 20230 | In a commutative ring, the... |
opprlem 20231 | Lemma for ~ opprbas and ~ ... |
opprlemOLD 20232 | Obsolete version of ~ oppr... |
opprbas 20233 | Base set of an opposite ri... |
opprbasOLD 20234 | Obsolete proof of ~ opprba... |
oppradd 20235 | Addition operation of an o... |
oppraddOLD 20236 | Obsolete proof of ~ opprba... |
opprrng 20237 | An opposite non-unital rin... |
opprrngb 20238 | A class is a non-unital ri... |
opprring 20239 | An opposite ring is a ring... |
opprringb 20240 | Bidirectional form of ~ op... |
oppr0 20241 | Additive identity of an op... |
oppr1 20242 | Multiplicative identity of... |
opprneg 20243 | The negative function in a... |
opprsubg 20244 | Being a subgroup is a symm... |
mulgass3 20245 | An associative property be... |
reldvdsr 20252 | The divides relation is a ... |
dvdsrval 20253 | Value of the divides relat... |
dvdsr 20254 | Value of the divides relat... |
dvdsr2 20255 | Value of the divides relat... |
dvdsrmul 20256 | A left-multiple of ` X ` i... |
dvdsrcl 20257 | Closure of a dividing elem... |
dvdsrcl2 20258 | Closure of a dividing elem... |
dvdsrid 20259 | An element in a (unital) r... |
dvdsrtr 20260 | Divisibility is transitive... |
dvdsrmul1 20261 | The divisibility relation ... |
dvdsrneg 20262 | An element divides its neg... |
dvdsr01 20263 | In a ring, zero is divisib... |
dvdsr02 20264 | Only zero is divisible by ... |
isunit 20265 | Property of being a unit o... |
1unit 20266 | The multiplicative identit... |
unitcl 20267 | A unit is an element of th... |
unitss 20268 | The set of units is contai... |
opprunit 20269 | Being a unit is a symmetri... |
crngunit 20270 | Property of being a unit i... |
dvdsunit 20271 | A divisor of a unit is a u... |
unitmulcl 20272 | The product of units is a ... |
unitmulclb 20273 | Reversal of ~ unitmulcl in... |
unitgrpbas 20274 | The base set of the group ... |
unitgrp 20275 | The group of units is a gr... |
unitabl 20276 | The group of units of a co... |
unitgrpid 20277 | The identity of the group ... |
unitsubm 20278 | The group of units is a su... |
invrfval 20281 | Multiplicative inverse fun... |
unitinvcl 20282 | The inverse of a unit exis... |
unitinvinv 20283 | The inverse of the inverse... |
ringinvcl 20284 | The inverse of a unit is a... |
unitlinv 20285 | A unit times its inverse i... |
unitrinv 20286 | A unit times its inverse i... |
1rinv 20287 | The inverse of the ring un... |
0unit 20288 | The additive identity is a... |
unitnegcl 20289 | The negative of a unit is ... |
ringunitnzdiv 20290 | In a unitary ring, a unit ... |
ring1nzdiv 20291 | In a unitary ring, the rin... |
dvrfval 20294 | Division operation in a ri... |
dvrval 20295 | Division operation in a ri... |
dvrcl 20296 | Closure of division operat... |
unitdvcl 20297 | The units are closed under... |
dvrid 20298 | A ring element divided by ... |
dvr1 20299 | A ring element divided by ... |
dvrass 20300 | An associative law for div... |
dvrcan1 20301 | A cancellation law for div... |
dvrcan3 20302 | A cancellation law for div... |
dvreq1 20303 | Equality in terms of ratio... |
dvrdir 20304 | Distributive law for the d... |
rdivmuldivd 20305 | Multiplication of two rati... |
ringinvdv 20306 | Write the inverse function... |
rngidpropd 20307 | The ring unity depends onl... |
dvdsrpropd 20308 | The divisibility relation ... |
unitpropd 20309 | The set of units depends o... |
invrpropd 20310 | The ring inverse function ... |
isirred 20311 | An irreducible element of ... |
isnirred 20312 | The property of being a no... |
isirred2 20313 | Expand out the class diffe... |
opprirred 20314 | Irreducibility is symmetri... |
irredn0 20315 | The additive identity is n... |
irredcl 20316 | An irreducible element is ... |
irrednu 20317 | An irreducible element is ... |
irredn1 20318 | The multiplicative identit... |
irredrmul 20319 | The product of an irreduci... |
irredlmul 20320 | The product of a unit and ... |
irredmul 20321 | If product of two elements... |
irredneg 20322 | The negative of an irreduc... |
irrednegb 20323 | An element is irreducible ... |
rnghmrcl 20330 | Reverse closure of a non-u... |
rnghmfn 20331 | The mapping of two non-uni... |
rnghmval 20332 | The set of the non-unital ... |
isrnghm 20333 | A function is a non-unital... |
isrnghmmul 20334 | A function is a non-unital... |
rnghmmgmhm 20335 | A non-unital ring homomorp... |
rnghmval2 20336 | The non-unital ring homomo... |
isrngim 20337 | An isomorphism of non-unit... |
rngimrcl 20338 | Reverse closure for an iso... |
rnghmghm 20339 | A non-unital ring homomorp... |
rnghmf 20340 | A ring homomorphism is a f... |
rnghmmul 20341 | A homomorphism of non-unit... |
isrnghm2d 20342 | Demonstration of non-unita... |
isrnghmd 20343 | Demonstration of non-unita... |
rnghmf1o 20344 | A non-unital ring homomorp... |
isrngim2 20345 | An isomorphism of non-unit... |
rngimf1o 20346 | An isomorphism of non-unit... |
rngimrnghm 20347 | An isomorphism of non-unit... |
rngimcnv 20348 | The converse of an isomorp... |
rnghmco 20349 | The composition of non-uni... |
idrnghm 20350 | The identity homomorphism ... |
c0mgm 20351 | The constant mapping to ze... |
c0mhm 20352 | The constant mapping to ze... |
c0ghm 20353 | The constant mapping to ze... |
c0snmgmhm 20354 | The constant mapping to ze... |
c0snmhm 20355 | The constant mapping to ze... |
c0snghm 20356 | The constant mapping to ze... |
rngisomfv1 20357 | If there is a non-unital r... |
rngisom1 20358 | If there is a non-unital r... |
rngisomring 20359 | If there is a non-unital r... |
rngisomring1 20360 | If there is a non-unital r... |
dfrhm2 20366 | The property of a ring hom... |
rhmrcl1 20368 | Reverse closure of a ring ... |
rhmrcl2 20369 | Reverse closure of a ring ... |
isrhm 20370 | A function is a ring homom... |
rhmmhm 20371 | A ring homomorphism is a h... |
rhmisrnghm 20372 | Each unital ring homomorph... |
isrim0OLD 20373 | Obsolete version of ~ isri... |
rimrcl 20374 | Reverse closure for an iso... |
isrim0 20375 | A ring isomorphism is a ho... |
rhmghm 20376 | A ring homomorphism is an ... |
rhmf 20377 | A ring homomorphism is a f... |
rhmmul 20378 | A homomorphism of rings pr... |
isrhm2d 20379 | Demonstration of ring homo... |
isrhmd 20380 | Demonstration of ring homo... |
rhm1 20381 | Ring homomorphisms are req... |
idrhm 20382 | The identity homomorphism ... |
rhmf1o 20383 | A ring homomorphism is bij... |
isrim 20384 | An isomorphism of rings is... |
isrimOLD 20385 | Obsolete version of ~ isri... |
rimf1o 20386 | An isomorphism of rings is... |
rimrhmOLD 20387 | Obsolete version of ~ rimr... |
rimrhm 20388 | A ring isomorphism is a ho... |
rimgim 20389 | An isomorphism of rings is... |
rimisrngim 20390 | Each unital ring isomorphi... |
rhmfn 20391 | The mapping of two rings t... |
rhmval 20392 | The ring homomorphisms bet... |
rhmco 20393 | The composition of ring ho... |
pwsco1rhm 20394 | Right composition with a f... |
pwsco2rhm 20395 | Left composition with a ri... |
brric 20396 | The relation "is isomorphi... |
brrici 20397 | Prove isomorphic by an exp... |
brric2 20398 | The relation "is isomorphi... |
ricgic 20399 | If two rings are (ring) is... |
rhmdvdsr 20400 | A ring homomorphism preser... |
rhmopp 20401 | A ring homomorphism is als... |
elrhmunit 20402 | Ring homomorphisms preserv... |
rhmunitinv 20403 | Ring homomorphisms preserv... |
isnzr 20406 | Property of a nonzero ring... |
nzrnz 20407 | One and zero are different... |
nzrring 20408 | A nonzero ring is a ring. ... |
nzrringOLD 20409 | Obsolete version of ~ nzrr... |
isnzr2 20410 | Equivalent characterizatio... |
isnzr2hash 20411 | Equivalent characterizatio... |
opprnzr 20412 | The opposite of a nonzero ... |
ringelnzr 20413 | A ring is nonzero if it ha... |
nzrunit 20414 | A unit is nonzero in any n... |
0ringnnzr 20415 | A ring is a zero ring iff ... |
0ring 20416 | If a ring has only one ele... |
0ringdif 20417 | A zero ring is a ring whic... |
0ringbas 20418 | The base set of a zero rin... |
0ring01eq 20419 | In a ring with only one el... |
01eq0ring 20420 | If the zero and the identi... |
01eq0ringOLD 20421 | Obsolete version of ~ 01eq... |
0ring01eqbi 20422 | In a unital ring the zero ... |
0ring1eq0 20423 | In a zero ring, a ring whi... |
c0rhm 20424 | The constant mapping to ze... |
c0rnghm 20425 | The constant mapping to ze... |
zrrnghm 20426 | The constant mapping to ze... |
nrhmzr 20427 | There is no ring homomorph... |
islring 20430 | The predicate "is a local ... |
lringnzr 20431 | A local ring is a nonzero ... |
lringring 20432 | A local ring is a ring. (... |
lringnz 20433 | A local ring is a nonzero ... |
lringuplu 20434 | If the sum of two elements... |
issubrng 20437 | The subring of non-unital ... |
subrngss 20438 | A subring is a subset. (C... |
subrngid 20439 | Every non-unital ring is a... |
subrngrng 20440 | A subring is a non-unital ... |
subrngrcl 20441 | Reverse closure for a subr... |
subrngsubg 20442 | A subring is a subgroup. ... |
subrngringnsg 20443 | A subring is a normal subg... |
subrngbas 20444 | Base set of a subring stru... |
subrng0 20445 | A subring always has the s... |
subrngacl 20446 | A subring is closed under ... |
subrngmcl 20447 | A subgroup is closed under... |
issubrng2 20448 | Characterize the subrings ... |
opprsubrng 20449 | Being a subring is a symme... |
subrngint 20450 | The intersection of a none... |
subrngin 20451 | The intersection of two su... |
subrngmre 20452 | The subrings of a non-unit... |
subsubrng 20453 | A subring of a subring is ... |
subsubrng2 20454 | The set of subrings of a s... |
rhmimasubrnglem 20455 | Lemma for ~ rhmimasubrng :... |
rhmimasubrng 20456 | The homomorphic image of a... |
cntzsubrng 20457 | Centralizers in a non-unit... |
subrngpropd 20458 | If two structures have the... |
issubrg 20463 | The subring predicate. (C... |
subrgss 20464 | A subring is a subset. (C... |
subrgid 20465 | Every ring is a subring of... |
subrgring 20466 | A subring is a ring. (Con... |
subrgcrng 20467 | A subring of a commutative... |
subrgrcl 20468 | Reverse closure for a subr... |
subrgsubg 20469 | A subring is a subgroup. ... |
subrgsubrng 20470 | A subring of a unital ring... |
subrg0 20471 | A subring always has the s... |
subrg1cl 20472 | A subring contains the mul... |
subrgbas 20473 | Base set of a subring stru... |
subrg1 20474 | A subring always has the s... |
subrgacl 20475 | A subring is closed under ... |
subrgmcl 20476 | A subgroup is closed under... |
subrgsubm 20477 | A subring is a submonoid o... |
subrgdvds 20478 | If an element divides anot... |
subrguss 20479 | A unit of a subring is a u... |
subrginv 20480 | A subring always has the s... |
subrgdv 20481 | A subring always has the s... |
subrgunit 20482 | An element of a ring is a ... |
subrgugrp 20483 | The units of a subring for... |
issubrg2 20484 | Characterize the subrings ... |
opprsubrg 20485 | Being a subring is a symme... |
subrgnzr 20486 | A subring of a nonzero rin... |
subrgint 20487 | The intersection of a none... |
subrgin 20488 | The intersection of two su... |
subrgmre 20489 | The subrings of a ring are... |
subsubrg 20490 | A subring of a subring is ... |
subsubrg2 20491 | The set of subrings of a s... |
issubrg3 20492 | A subring is an additive s... |
resrhm 20493 | Restriction of a ring homo... |
resrhm2b 20494 | Restriction of the codomai... |
rhmeql 20495 | The equalizer of two ring ... |
rhmima 20496 | The homomorphic image of a... |
rnrhmsubrg 20497 | The range of a ring homomo... |
cntzsubr 20498 | Centralizers in a ring are... |
pwsdiagrhm 20499 | Diagonal homomorphism into... |
subrgpropd 20500 | If two structures have the... |
rhmpropd 20501 | Ring homomorphism depends ... |
rngcval 20504 | Value of the category of n... |
rnghmresfn 20505 | The class of non-unital ri... |
rnghmresel 20506 | An element of the non-unit... |
rngcbas 20507 | Set of objects of the cate... |
rngchomfval 20508 | Set of arrows of the categ... |
rngchom 20509 | Set of arrows of the categ... |
elrngchom 20510 | A morphism of non-unital r... |
rngchomfeqhom 20511 | The functionalized Hom-set... |
rngccofval 20512 | Composition in the categor... |
rngcco 20513 | Composition in the categor... |
dfrngc2 20514 | Alternate definition of th... |
rnghmsscmap2 20515 | The non-unital ring homomo... |
rnghmsscmap 20516 | The non-unital ring homomo... |
rnghmsubcsetclem1 20517 | Lemma 1 for ~ rnghmsubcset... |
rnghmsubcsetclem2 20518 | Lemma 2 for ~ rnghmsubcset... |
rnghmsubcsetc 20519 | The non-unital ring homomo... |
rngccat 20520 | The category of non-unital... |
rngcid 20521 | The identity arrow in the ... |
rngcsect 20522 | A section in the category ... |
rngcinv 20523 | An inverse in the category... |
rngciso 20524 | An isomorphism in the cate... |
rngcifuestrc 20525 | The "inclusion functor" fr... |
funcrngcsetc 20526 | The "natural forgetful fun... |
funcrngcsetcALT 20527 | Alternate proof of ~ funcr... |
zrinitorngc 20528 | The zero ring is an initia... |
zrtermorngc 20529 | The zero ring is a termina... |
zrzeroorngc 20530 | The zero ring is a zero ob... |
ringcval 20533 | Value of the category of u... |
rhmresfn 20534 | The class of unital ring h... |
rhmresel 20535 | An element of the unital r... |
ringcbas 20536 | Set of objects of the cate... |
ringchomfval 20537 | Set of arrows of the categ... |
ringchom 20538 | Set of arrows of the categ... |
elringchom 20539 | A morphism of unital rings... |
ringchomfeqhom 20540 | The functionalized Hom-set... |
ringccofval 20541 | Composition in the categor... |
ringcco 20542 | Composition in the categor... |
dfringc2 20543 | Alternate definition of th... |
rhmsscmap2 20544 | The unital ring homomorphi... |
rhmsscmap 20545 | The unital ring homomorphi... |
rhmsubcsetclem1 20546 | Lemma 1 for ~ rhmsubcsetc ... |
rhmsubcsetclem2 20547 | Lemma 2 for ~ rhmsubcsetc ... |
rhmsubcsetc 20548 | The unital ring homomorphi... |
ringccat 20549 | The category of unital rin... |
ringcid 20550 | The identity arrow in the ... |
rhmsscrnghm 20551 | The unital ring homomorphi... |
rhmsubcrngclem1 20552 | Lemma 1 for ~ rhmsubcrngc ... |
rhmsubcrngclem2 20553 | Lemma 2 for ~ rhmsubcrngc ... |
rhmsubcrngc 20554 | The unital ring homomorphi... |
rngcresringcat 20555 | The restriction of the cat... |
ringcsect 20556 | A section in the category ... |
ringcinv 20557 | An inverse in the category... |
ringciso 20558 | An isomorphism in the cate... |
ringcbasbas 20559 | An element of the base set... |
funcringcsetc 20560 | The "natural forgetful fun... |
zrtermoringc 20561 | The zero ring is a termina... |
zrninitoringc 20562 | The zero ring is not an in... |
srhmsubclem1 20563 | Lemma 1 for ~ srhmsubc . ... |
srhmsubclem2 20564 | Lemma 2 for ~ srhmsubc . ... |
srhmsubclem3 20565 | Lemma 3 for ~ srhmsubc . ... |
srhmsubc 20566 | According to ~ df-subc , t... |
sringcat 20567 | The restriction of the cat... |
crhmsubc 20568 | According to ~ df-subc , t... |
cringcat 20569 | The restriction of the cat... |
rngcrescrhm 20570 | The category of non-unital... |
rhmsubclem1 20571 | Lemma 1 for ~ rhmsubc . (... |
rhmsubclem2 20572 | Lemma 2 for ~ rhmsubc . (... |
rhmsubclem3 20573 | Lemma 3 for ~ rhmsubc . (... |
rhmsubclem4 20574 | Lemma 4 for ~ rhmsubc . (... |
rhmsubc 20575 | According to ~ df-subc , t... |
rhmsubccat 20576 | The restriction of the cat... |
isdrng 20581 | The predicate "is a divisi... |
drngunit 20582 | Elementhood in the set of ... |
drngui 20583 | The set of units of a divi... |
drngring 20584 | A division ring is a ring.... |
drngringd 20585 | A division ring is a ring.... |
drnggrpd 20586 | A division ring is a group... |
drnggrp 20587 | A division ring is a group... |
isfld 20588 | A field is a commutative d... |
flddrngd 20589 | A field is a division ring... |
fldcrngd 20590 | A field is a commutative r... |
isdrng2 20591 | A division ring can equiva... |
drngprop 20592 | If two structures have the... |
drngmgp 20593 | A division ring contains a... |
drngmcl 20594 | The product of two nonzero... |
drngid 20595 | A division ring's unity is... |
drngunz 20596 | A division ring's unity is... |
drngnzr 20597 | All division rings are non... |
drngid2 20598 | Properties showing that an... |
drnginvrcl 20599 | Closure of the multiplicat... |
drnginvrn0 20600 | The multiplicative inverse... |
drnginvrcld 20601 | Closure of the multiplicat... |
drnginvrl 20602 | Property of the multiplica... |
drnginvrr 20603 | Property of the multiplica... |
drnginvrld 20604 | Property of the multiplica... |
drnginvrrd 20605 | Property of the multiplica... |
drngmul0or 20606 | A product is zero iff one ... |
drngmulne0 20607 | A product is nonzero iff b... |
drngmuleq0 20608 | An element is zero iff its... |
opprdrng 20609 | The opposite of a division... |
isdrngd 20610 | Properties that characteri... |
isdrngrd 20611 | Properties that characteri... |
isdrngdOLD 20612 | Obsolete version of ~ isdr... |
isdrngrdOLD 20613 | Obsolete version of ~ isdr... |
drngpropd 20614 | If two structures have the... |
fldpropd 20615 | If two structures have the... |
rng1nnzr 20616 | The (smallest) structure r... |
ring1zr 20617 | The only (unital) ring wit... |
rngen1zr 20618 | The only (unital) ring wit... |
ringen1zr 20619 | The only unital ring with ... |
rng1nfld 20620 | The zero ring is not a fie... |
issubdrg 20621 | Characterize the subfields... |
drhmsubc 20622 | According to ~ df-subc , t... |
drngcat 20623 | The restriction of the cat... |
fldcat 20624 | The restriction of the cat... |
fldc 20625 | The restriction of the cat... |
fldhmsubc 20626 | According to ~ df-subc , t... |
issdrg 20629 | Property of a division sub... |
sdrgrcl 20630 | Reverse closure for a sub-... |
sdrgdrng 20631 | A sub-division-ring is a d... |
sdrgsubrg 20632 | A sub-division-ring is a s... |
sdrgid 20633 | Every division ring is a d... |
sdrgss 20634 | A division subring is a su... |
sdrgbas 20635 | Base set of a sub-division... |
issdrg2 20636 | Property of a division sub... |
sdrgunit 20637 | A unit of a sub-division-r... |
imadrhmcl 20638 | The image of a (nontrivial... |
fldsdrgfld 20639 | A sub-division-ring of a f... |
acsfn1p 20640 | Construction of a closure ... |
subrgacs 20641 | Closure property of subrin... |
sdrgacs 20642 | Closure property of divisi... |
cntzsdrg 20643 | Centralizers in division r... |
subdrgint 20644 | The intersection of a none... |
sdrgint 20645 | The intersection of a none... |
primefld 20646 | The smallest sub division ... |
primefld0cl 20647 | The prime field contains t... |
primefld1cl 20648 | The prime field contains t... |
abvfval 20651 | Value of the set of absolu... |
isabv 20652 | Elementhood in the set of ... |
isabvd 20653 | Properties that determine ... |
abvrcl 20654 | Reverse closure for the ab... |
abvfge0 20655 | An absolute value is a fun... |
abvf 20656 | An absolute value is a fun... |
abvcl 20657 | An absolute value is a fun... |
abvge0 20658 | The absolute value of a nu... |
abveq0 20659 | The value of an absolute v... |
abvne0 20660 | The absolute value of a no... |
abvgt0 20661 | The absolute value of a no... |
abvmul 20662 | An absolute value distribu... |
abvtri 20663 | An absolute value satisfie... |
abv0 20664 | The absolute value of zero... |
abv1z 20665 | The absolute value of one ... |
abv1 20666 | The absolute value of one ... |
abvneg 20667 | The absolute value of a ne... |
abvsubtri 20668 | An absolute value satisfie... |
abvrec 20669 | The absolute value distrib... |
abvdiv 20670 | The absolute value distrib... |
abvdom 20671 | Any ring with an absolute ... |
abvres 20672 | The restriction of an abso... |
abvtrivd 20673 | The trivial absolute value... |
abvtriv 20674 | The trivial absolute value... |
abvpropd 20675 | If two structures have the... |
staffval 20680 | The functionalization of t... |
stafval 20681 | The functionalization of t... |
staffn 20682 | The functionalization is e... |
issrng 20683 | The predicate "is a star r... |
srngrhm 20684 | The involution function in... |
srngring 20685 | A star ring is a ring. (C... |
srngcnv 20686 | The involution function in... |
srngf1o 20687 | The involution function in... |
srngcl 20688 | The involution function in... |
srngnvl 20689 | The involution function in... |
srngadd 20690 | The involution function in... |
srngmul 20691 | The involution function in... |
srng1 20692 | The conjugate of the ring ... |
srng0 20693 | The conjugate of the ring ... |
issrngd 20694 | Properties that determine ... |
idsrngd 20695 | A commutative ring is a st... |
islmod 20700 | The predicate "is a left m... |
lmodlema 20701 | Lemma for properties of a ... |
islmodd 20702 | Properties that determine ... |
lmodgrp 20703 | A left module is a group. ... |
lmodring 20704 | The scalar component of a ... |
lmodfgrp 20705 | The scalar component of a ... |
lmodgrpd 20706 | A left module is a group. ... |
lmodbn0 20707 | The base set of a left mod... |
lmodacl 20708 | Closure of ring addition f... |
lmodmcl 20709 | Closure of ring multiplica... |
lmodsn0 20710 | The set of scalars in a le... |
lmodvacl 20711 | Closure of vector addition... |
lmodass 20712 | Left module vector sum is ... |
lmodlcan 20713 | Left cancellation law for ... |
lmodvscl 20714 | Closure of scalar product ... |
lmodvscld 20715 | Closure of scalar product ... |
scaffval 20716 | The scalar multiplication ... |
scafval 20717 | The scalar multiplication ... |
scafeq 20718 | If the scalar multiplicati... |
scaffn 20719 | The scalar multiplication ... |
lmodscaf 20720 | The scalar multiplication ... |
lmodvsdi 20721 | Distributive law for scala... |
lmodvsdir 20722 | Distributive law for scala... |
lmodvsass 20723 | Associative law for scalar... |
lmod0cl 20724 | The ring zero in a left mo... |
lmod1cl 20725 | The ring unity in a left m... |
lmodvs1 20726 | Scalar product with the ri... |
lmod0vcl 20727 | The zero vector is a vecto... |
lmod0vlid 20728 | Left identity law for the ... |
lmod0vrid 20729 | Right identity law for the... |
lmod0vid 20730 | Identity equivalent to the... |
lmod0vs 20731 | Zero times a vector is the... |
lmodvs0 20732 | Anything times the zero ve... |
lmodvsmmulgdi 20733 | Distributive law for a gro... |
lmodfopnelem1 20734 | Lemma 1 for ~ lmodfopne . ... |
lmodfopnelem2 20735 | Lemma 2 for ~ lmodfopne . ... |
lmodfopne 20736 | The (functionalized) opera... |
lcomf 20737 | A linear-combination sum i... |
lcomfsupp 20738 | A linear-combination sum i... |
lmodvnegcl 20739 | Closure of vector negative... |
lmodvnegid 20740 | Addition of a vector with ... |
lmodvneg1 20741 | Minus 1 times a vector is ... |
lmodvsneg 20742 | Multiplication of a vector... |
lmodvsubcl 20743 | Closure of vector subtract... |
lmodcom 20744 | Left module vector sum is ... |
lmodabl 20745 | A left module is an abelia... |
lmodcmn 20746 | A left module is a commuta... |
lmodnegadd 20747 | Distribute negation throug... |
lmod4 20748 | Commutative/associative la... |
lmodvsubadd 20749 | Relationship between vecto... |
lmodvaddsub4 20750 | Vector addition/subtractio... |
lmodvpncan 20751 | Addition/subtraction cance... |
lmodvnpcan 20752 | Cancellation law for vecto... |
lmodvsubval2 20753 | Value of vector subtractio... |
lmodsubvs 20754 | Subtraction of a scalar pr... |
lmodsubdi 20755 | Scalar multiplication dist... |
lmodsubdir 20756 | Scalar multiplication dist... |
lmodsubeq0 20757 | If the difference between ... |
lmodsubid 20758 | Subtraction of a vector fr... |
lmodvsghm 20759 | Scalar multiplication of t... |
lmodprop2d 20760 | If two structures have the... |
lmodpropd 20761 | If two structures have the... |
gsumvsmul 20762 | Pull a scalar multiplicati... |
mptscmfsupp0 20763 | A mapping to a scalar prod... |
mptscmfsuppd 20764 | A function mapping to a sc... |
rmodislmodlem 20765 | Lemma for ~ rmodislmod . ... |
rmodislmod 20766 | The right module ` R ` ind... |
rmodislmodOLD 20767 | Obsolete version of ~ rmod... |
lssset 20770 | The set of all (not necess... |
islss 20771 | The predicate "is a subspa... |
islssd 20772 | Properties that determine ... |
lssss 20773 | A subspace is a set of vec... |
lssel 20774 | A subspace member is a vec... |
lss1 20775 | The set of vectors in a le... |
lssuni 20776 | The union of all subspaces... |
lssn0 20777 | A subspace is not empty. ... |
00lss 20778 | The empty structure has no... |
lsscl 20779 | Closure property of a subs... |
lssvacl 20780 | Closure of vector addition... |
lssvsubcl 20781 | Closure of vector subtract... |
lssvancl1 20782 | Non-closure: if one vector... |
lssvancl2 20783 | Non-closure: if one vector... |
lss0cl 20784 | The zero vector belongs to... |
lsssn0 20785 | The singleton of the zero ... |
lss0ss 20786 | The zero subspace is inclu... |
lssle0 20787 | No subspace is smaller tha... |
lssne0 20788 | A nonzero subspace has a n... |
lssvneln0 20789 | A vector ` X ` which doesn... |
lssneln0 20790 | A vector ` X ` which doesn... |
lssssr 20791 | Conclude subspace ordering... |
lssvscl 20792 | Closure of scalar product ... |
lssvnegcl 20793 | Closure of negative vector... |
lsssubg 20794 | All subspaces are subgroup... |
lsssssubg 20795 | All subspaces are subgroup... |
islss3 20796 | A linear subspace of a mod... |
lsslmod 20797 | A submodule is a module. ... |
lsslss 20798 | The subspaces of a subspac... |
islss4 20799 | A linear subspace is a sub... |
lss1d 20800 | One-dimensional subspace (... |
lssintcl 20801 | The intersection of a none... |
lssincl 20802 | The intersection of two su... |
lssmre 20803 | The subspaces of a module ... |
lssacs 20804 | Submodules are an algebrai... |
prdsvscacl 20805 | Pointwise scalar multiplic... |
prdslmodd 20806 | The product of a family of... |
pwslmod 20807 | A structure power of a lef... |
lspfval 20810 | The span function for a le... |
lspf 20811 | The span function on a lef... |
lspval 20812 | The span of a set of vecto... |
lspcl 20813 | The span of a set of vecto... |
lspsncl 20814 | The span of a singleton is... |
lspprcl 20815 | The span of a pair is a su... |
lsptpcl 20816 | The span of an unordered t... |
lspsnsubg 20817 | The span of a singleton is... |
00lsp 20818 | ~ fvco4i lemma for linear ... |
lspid 20819 | The span of a subspace is ... |
lspssv 20820 | A span is a set of vectors... |
lspss 20821 | Span preserves subset orde... |
lspssid 20822 | A set of vectors is a subs... |
lspidm 20823 | The span of a set of vecto... |
lspun 20824 | The span of union is the s... |
lspssp 20825 | If a set of vectors is a s... |
mrclsp 20826 | Moore closure generalizes ... |
lspsnss 20827 | The span of the singleton ... |
lspsnel3 20828 | A member of the span of th... |
lspprss 20829 | The span of a pair of vect... |
lspsnid 20830 | A vector belongs to the sp... |
lspsnel6 20831 | Relationship between a vec... |
lspsnel5 20832 | Relationship between a vec... |
lspsnel5a 20833 | Relationship between a vec... |
lspprid1 20834 | A member of a pair of vect... |
lspprid2 20835 | A member of a pair of vect... |
lspprvacl 20836 | The sum of two vectors bel... |
lssats2 20837 | A way to express atomistic... |
lspsneli 20838 | A scalar product with a ve... |
lspsn 20839 | Span of the singleton of a... |
lspsnel 20840 | Member of span of the sing... |
lspsnvsi 20841 | Span of a scalar product o... |
lspsnss2 20842 | Comparable spans of single... |
lspsnneg 20843 | Negation does not change t... |
lspsnsub 20844 | Swapping subtraction order... |
lspsn0 20845 | Span of the singleton of t... |
lsp0 20846 | Span of the empty set. (C... |
lspuni0 20847 | Union of the span of the e... |
lspun0 20848 | The span of a union with t... |
lspsneq0 20849 | Span of the singleton is t... |
lspsneq0b 20850 | Equal singleton spans impl... |
lmodindp1 20851 | Two independent (non-colin... |
lsslsp 20852 | Spans in submodules corres... |
lsslspOLD 20853 | Obsolete version of ~ lssl... |
lss0v 20854 | The zero vector in a submo... |
lsspropd 20855 | If two structures have the... |
lsppropd 20856 | If two structures have the... |
reldmlmhm 20863 | Lemma for module homomorph... |
lmimfn 20864 | Lemma for module isomorphi... |
islmhm 20865 | Property of being a homomo... |
islmhm3 20866 | Property of a module homom... |
lmhmlem 20867 | Non-quantified consequence... |
lmhmsca 20868 | A homomorphism of left mod... |
lmghm 20869 | A homomorphism of left mod... |
lmhmlmod2 20870 | A homomorphism of left mod... |
lmhmlmod1 20871 | A homomorphism of left mod... |
lmhmf 20872 | A homomorphism of left mod... |
lmhmlin 20873 | A homomorphism of left mod... |
lmodvsinv 20874 | Multiplication of a vector... |
lmodvsinv2 20875 | Multiplying a negated vect... |
islmhm2 20876 | A one-equation proof of li... |
islmhmd 20877 | Deduction for a module hom... |
0lmhm 20878 | The constant zero linear f... |
idlmhm 20879 | The identity function on a... |
invlmhm 20880 | The negative function on a... |
lmhmco 20881 | The composition of two mod... |
lmhmplusg 20882 | The pointwise sum of two l... |
lmhmvsca 20883 | The pointwise scalar produ... |
lmhmf1o 20884 | A bijective module homomor... |
lmhmima 20885 | The image of a subspace un... |
lmhmpreima 20886 | The inverse image of a sub... |
lmhmlsp 20887 | Homomorphisms preserve spa... |
lmhmrnlss 20888 | The range of a homomorphis... |
lmhmkerlss 20889 | The kernel of a homomorphi... |
reslmhm 20890 | Restriction of a homomorph... |
reslmhm2 20891 | Expansion of the codomain ... |
reslmhm2b 20892 | Expansion of the codomain ... |
lmhmeql 20893 | The equalizer of two modul... |
lspextmo 20894 | A linear function is compl... |
pwsdiaglmhm 20895 | Diagonal homomorphism into... |
pwssplit0 20896 | Splitting for structure po... |
pwssplit1 20897 | Splitting for structure po... |
pwssplit2 20898 | Splitting for structure po... |
pwssplit3 20899 | Splitting for structure po... |
islmim 20900 | An isomorphism of left mod... |
lmimf1o 20901 | An isomorphism of left mod... |
lmimlmhm 20902 | An isomorphism of modules ... |
lmimgim 20903 | An isomorphism of modules ... |
islmim2 20904 | An isomorphism of left mod... |
lmimcnv 20905 | The converse of a bijectiv... |
brlmic 20906 | The relation "is isomorphi... |
brlmici 20907 | Prove isomorphic by an exp... |
lmiclcl 20908 | Isomorphism implies the le... |
lmicrcl 20909 | Isomorphism implies the ri... |
lmicsym 20910 | Module isomorphism is symm... |
lmhmpropd 20911 | Module homomorphism depend... |
islbs 20914 | The predicate " ` B ` is a... |
lbsss 20915 | A basis is a set of vector... |
lbsel 20916 | An element of a basis is a... |
lbssp 20917 | The span of a basis is the... |
lbsind 20918 | A basis is linearly indepe... |
lbsind2 20919 | A basis is linearly indepe... |
lbspss 20920 | No proper subset of a basi... |
lsmcl 20921 | The sum of two subspaces i... |
lsmspsn 20922 | Member of subspace sum of ... |
lsmelval2 20923 | Subspace sum membership in... |
lsmsp 20924 | Subspace sum in terms of s... |
lsmsp2 20925 | Subspace sum of spans of s... |
lsmssspx 20926 | Subspace sum (in its exten... |
lsmpr 20927 | The span of a pair of vect... |
lsppreli 20928 | A vector expressed as a su... |
lsmelpr 20929 | Two ways to say that a vec... |
lsppr0 20930 | The span of a vector paire... |
lsppr 20931 | Span of a pair of vectors.... |
lspprel 20932 | Member of the span of a pa... |
lspprabs 20933 | Absorption of vector sum i... |
lspvadd 20934 | The span of a vector sum i... |
lspsntri 20935 | Triangle-type inequality f... |
lspsntrim 20936 | Triangle-type inequality f... |
lbspropd 20937 | If two structures have the... |
pj1lmhm 20938 | The left projection functi... |
pj1lmhm2 20939 | The left projection functi... |
islvec 20942 | The predicate "is a left v... |
lvecdrng 20943 | The set of scalars of a le... |
lveclmod 20944 | A left vector space is a l... |
lveclmodd 20945 | A vector space is a left m... |
lvecgrpd 20946 | A vector space is a group.... |
lsslvec 20947 | A vector subspace is a vec... |
lmhmlvec 20948 | The property for modules t... |
lvecvs0or 20949 | If a scalar product is zer... |
lvecvsn0 20950 | A scalar product is nonzer... |
lssvs0or 20951 | If a scalar product belong... |
lvecvscan 20952 | Cancellation law for scala... |
lvecvscan2 20953 | Cancellation law for scala... |
lvecinv 20954 | Invert coefficient of scal... |
lspsnvs 20955 | A nonzero scalar product d... |
lspsneleq 20956 | Membership relation that i... |
lspsncmp 20957 | Comparable spans of nonzer... |
lspsnne1 20958 | Two ways to express that v... |
lspsnne2 20959 | Two ways to express that v... |
lspsnnecom 20960 | Swap two vectors with diff... |
lspabs2 20961 | Absorption law for span of... |
lspabs3 20962 | Absorption law for span of... |
lspsneq 20963 | Equal spans of singletons ... |
lspsneu 20964 | Nonzero vectors with equal... |
lspsnel4 20965 | A member of the span of th... |
lspdisj 20966 | The span of a vector not i... |
lspdisjb 20967 | A nonzero vector is not in... |
lspdisj2 20968 | Unequal spans are disjoint... |
lspfixed 20969 | Show membership in the spa... |
lspexch 20970 | Exchange property for span... |
lspexchn1 20971 | Exchange property for span... |
lspexchn2 20972 | Exchange property for span... |
lspindpi 20973 | Partial independence prope... |
lspindp1 20974 | Alternate way to say 3 vec... |
lspindp2l 20975 | Alternate way to say 3 vec... |
lspindp2 20976 | Alternate way to say 3 vec... |
lspindp3 20977 | Independence of 2 vectors ... |
lspindp4 20978 | (Partial) independence of ... |
lvecindp 20979 | Compute the ` X ` coeffici... |
lvecindp2 20980 | Sums of independent vector... |
lspsnsubn0 20981 | Unequal singleton spans im... |
lsmcv 20982 | Subspace sum has the cover... |
lspsolvlem 20983 | Lemma for ~ lspsolv . (Co... |
lspsolv 20984 | If ` X ` is in the span of... |
lssacsex 20985 | In a vector space, subspac... |
lspsnat 20986 | There is no subspace stric... |
lspsncv0 20987 | The span of a singleton co... |
lsppratlem1 20988 | Lemma for ~ lspprat . Let... |
lsppratlem2 20989 | Lemma for ~ lspprat . Sho... |
lsppratlem3 20990 | Lemma for ~ lspprat . In ... |
lsppratlem4 20991 | Lemma for ~ lspprat . In ... |
lsppratlem5 20992 | Lemma for ~ lspprat . Com... |
lsppratlem6 20993 | Lemma for ~ lspprat . Neg... |
lspprat 20994 | A proper subspace of the s... |
islbs2 20995 | An equivalent formulation ... |
islbs3 20996 | An equivalent formulation ... |
lbsacsbs 20997 | Being a basis in a vector ... |
lvecdim 20998 | The dimension theorem for ... |
lbsextlem1 20999 | Lemma for ~ lbsext . The ... |
lbsextlem2 21000 | Lemma for ~ lbsext . Sinc... |
lbsextlem3 21001 | Lemma for ~ lbsext . A ch... |
lbsextlem4 21002 | Lemma for ~ lbsext . ~ lbs... |
lbsextg 21003 | For any linearly independe... |
lbsext 21004 | For any linearly independe... |
lbsexg 21005 | Every vector space has a b... |
lbsex 21006 | Every vector space has a b... |
lvecprop2d 21007 | If two structures have the... |
lvecpropd 21008 | If two structures have the... |
sraval 21013 | Lemma for ~ srabase throug... |
sralem 21014 | Lemma for ~ srabase and si... |
sralemOLD 21015 | Obsolete version of ~ sral... |
srabase 21016 | Base set of a subring alge... |
srabaseOLD 21017 | Obsolete proof of ~ srabas... |
sraaddg 21018 | Additive operation of a su... |
sraaddgOLD 21019 | Obsolete proof of ~ sraadd... |
sramulr 21020 | Multiplicative operation o... |
sramulrOLD 21021 | Obsolete proof of ~ sramul... |
srasca 21022 | The set of scalars of a su... |
srascaOLD 21023 | Obsolete proof of ~ srasca... |
sravsca 21024 | The scalar product operati... |
sravscaOLD 21025 | Obsolete proof of ~ sravsc... |
sraip 21026 | The inner product operatio... |
sratset 21027 | Topology component of a su... |
sratsetOLD 21028 | Obsolete proof of ~ sratse... |
sratopn 21029 | Topology component of a su... |
srads 21030 | Distance function of a sub... |
sradsOLD 21031 | Obsolete proof of ~ srads ... |
sraring 21032 | Condition for a subring al... |
sralmod 21033 | The subring algebra is a l... |
sralmod0 21034 | The subring module inherit... |
issubrgd 21035 | Prove a subring by closure... |
rlmfn 21036 | ` ringLMod ` is a function... |
rlmval 21037 | Value of the ring module. ... |
rlmval2 21038 | Value of the ring module e... |
rlmbas 21039 | Base set of the ring modul... |
rlmplusg 21040 | Vector addition in the rin... |
rlm0 21041 | Zero vector in the ring mo... |
rlmsub 21042 | Subtraction in the ring mo... |
rlmmulr 21043 | Ring multiplication in the... |
rlmsca 21044 | Scalars in the ring module... |
rlmsca2 21045 | Scalars in the ring module... |
rlmvsca 21046 | Scalar multiplication in t... |
rlmtopn 21047 | Topology component of the ... |
rlmds 21048 | Metric component of the ri... |
rlmlmod 21049 | The ring module is a modul... |
rlmlvec 21050 | The ring module over a div... |
rlmlsm 21051 | Subgroup sum of the ring m... |
rlmvneg 21052 | Vector negation in the rin... |
rlmscaf 21053 | Functionalized scalar mult... |
ixpsnbasval 21054 | The value of an infinite C... |
lidlval 21059 | Value of the set of ring i... |
rspval 21060 | Value of the ring span fun... |
lidlss 21061 | An ideal is a subset of th... |
lidlssbas 21062 | The base set of the restri... |
lidlbas 21063 | A (left) ideal of a ring i... |
islidl 21064 | Predicate of being a (left... |
rnglidlmcl 21065 | A (left) ideal containing ... |
rngridlmcl 21066 | A right ideal (which is a ... |
dflidl2rng 21067 | Alternate (the usual textb... |
isridlrng 21068 | A right ideal is a left id... |
lidl0cl 21069 | An ideal contains 0. (Con... |
lidlacl 21070 | An ideal is closed under a... |
lidlnegcl 21071 | An ideal contains negative... |
lidlsubg 21072 | An ideal is a subgroup of ... |
lidlsubcl 21073 | An ideal is closed under s... |
lidlmcl 21074 | An ideal is closed under l... |
lidl1el 21075 | An ideal contains 1 iff it... |
dflidl2 21076 | Alternate (the usual textb... |
lidl0ALT 21077 | Alternate proof for ~ lidl... |
rnglidl0 21078 | Every non-unital ring cont... |
lidl0 21079 | Every ring contains a zero... |
lidl1ALT 21080 | Alternate proof for ~ lidl... |
rnglidl1 21081 | The base set of every non-... |
lidl1 21082 | Every ring contains a unit... |
lidlacs 21083 | The ideal system is an alg... |
rspcl 21084 | The span of a set of ring ... |
rspssid 21085 | The span of a set of ring ... |
rsp1 21086 | The span of the identity e... |
rsp0 21087 | The span of the zero eleme... |
rspssp 21088 | The ideal span of a set of... |
mrcrsp 21089 | Moore closure generalizes ... |
lidlnz 21090 | A nonzero ideal contains a... |
drngnidl 21091 | A division ring has only t... |
lidlrsppropd 21092 | The left ideals and ring s... |
rnglidlmmgm 21093 | The multiplicative group o... |
rnglidlmsgrp 21094 | The multiplicative group o... |
rnglidlrng 21095 | A (left) ideal of a non-un... |
2idlval 21098 | Definition of a two-sided ... |
isridl 21099 | A right ideal is a left id... |
2idlelb 21100 | Membership in a two-sided ... |
2idllidld 21101 | A two-sided ideal is a lef... |
2idlridld 21102 | A two-sided ideal is a rig... |
df2idl2rng 21103 | Alternate (the usual textb... |
df2idl2 21104 | Alternate (the usual textb... |
ridl0 21105 | Every ring contains a zero... |
ridl1 21106 | Every ring contains a unit... |
2idl0 21107 | Every ring contains a zero... |
2idl1 21108 | Every ring contains a unit... |
2idlss 21109 | A two-sided ideal is a sub... |
2idlbas 21110 | The base set of a two-side... |
2idlelbas 21111 | The base set of a two-side... |
rng2idlsubrng 21112 | A two-sided ideal of a non... |
rng2idlnsg 21113 | A two-sided ideal of a non... |
rng2idl0 21114 | The zero (additive identit... |
rng2idlsubgsubrng 21115 | A two-sided ideal of a non... |
rng2idlsubgnsg 21116 | A two-sided ideal of a non... |
rng2idlsubg0 21117 | The zero (additive identit... |
2idlcpblrng 21118 | The coset equivalence rela... |
2idlcpbl 21119 | The coset equivalence rela... |
qus2idrng 21120 | The quotient of a non-unit... |
qus1 21121 | The multiplicative identit... |
qusring 21122 | If ` S ` is a two-sided id... |
qusrhm 21123 | If ` S ` is a two-sided id... |
qusmul2 21124 | Value of the ring operatio... |
crngridl 21125 | In a commutative ring, the... |
crng2idl 21126 | In a commutative ring, a t... |
qusmulrng 21127 | Value of the multiplicatio... |
quscrng 21128 | The quotient of a commutat... |
rngqiprng1elbas 21129 | The ring unity of a two-si... |
rngqiprngghmlem1 21130 | Lemma 1 for ~ rngqiprngghm... |
rngqiprngghmlem2 21131 | Lemma 2 for ~ rngqiprngghm... |
rngqiprngghmlem3 21132 | Lemma 3 for ~ rngqiprngghm... |
rngqiprngimfolem 21133 | Lemma for ~ rngqiprngimfo ... |
rngqiprnglinlem1 21134 | Lemma 1 for ~ rngqiprnglin... |
rngqiprnglinlem2 21135 | Lemma 2 for ~ rngqiprnglin... |
rngqiprnglinlem3 21136 | Lemma 3 for ~ rngqiprnglin... |
rngqiprngimf1lem 21137 | Lemma for ~ rngqiprngimf1 ... |
rngqipbas 21138 | The base set of the produc... |
rngqiprng 21139 | The product of the quotien... |
rngqiprngimf 21140 | ` F ` is a function from (... |
rngqiprngimfv 21141 | The value of the function ... |
rngqiprngghm 21142 | ` F ` is a homomorphism of... |
rngqiprngimf1 21143 | ` F ` is a one-to-one func... |
rngqiprngimfo 21144 | ` F ` is a function from (... |
rngqiprnglin 21145 | ` F ` is linear with respe... |
rngqiprngho 21146 | ` F ` is a homomorphism of... |
rngqiprngim 21147 | ` F ` is an isomorphism of... |
rng2idl1cntr 21148 | The unity of a two-sided i... |
rngringbdlem1 21149 | In a unital ring, the quot... |
rngringbdlem2 21150 | A non-unital ring is unita... |
rngringbd 21151 | A non-unital ring is unita... |
ring2idlqus 21152 | For every unital ring ther... |
ring2idlqusb 21153 | A non-unital ring is unita... |
rngqiprngfulem1 21154 | Lemma 1 for ~ rngqiprngfu ... |
rngqiprngfulem2 21155 | Lemma 2 for ~ rngqiprngfu ... |
rngqiprngfulem3 21156 | Lemma 3 for ~ rngqiprngfu ... |
rngqiprngfulem4 21157 | Lemma 4 for ~ rngqiprngfu ... |
rngqiprngfulem5 21158 | Lemma 5 for ~ rngqiprngfu ... |
rngqipring1 21159 | The ring unity of the prod... |
rngqiprngfu 21160 | The function value of ` F ... |
rngqiprngu 21161 | If a non-unital ring has a... |
ring2idlqus1 21162 | If a non-unital ring has a... |
lpival 21167 | Value of the set of princi... |
islpidl 21168 | Property of being a princi... |
lpi0 21169 | The zero ideal is always p... |
lpi1 21170 | The unit ideal is always p... |
islpir 21171 | Principal ideal rings are ... |
lpiss 21172 | Principal ideals are a sub... |
islpir2 21173 | Principal ideal rings are ... |
lpirring 21174 | Principal ideal rings are ... |
drnglpir 21175 | Division rings are princip... |
rspsn 21176 | Membership in principal id... |
lidldvgen 21177 | An element generates an id... |
lpigen 21178 | An ideal is principal iff ... |
rrgval 21187 | Value of the set or left-r... |
isrrg 21188 | Membership in the set of l... |
rrgeq0i 21189 | Property of a left-regular... |
rrgeq0 21190 | Left-multiplication by a l... |
rrgsupp 21191 | Left multiplication by a l... |
rrgss 21192 | Left-regular elements are ... |
unitrrg 21193 | Units are regular elements... |
isdomn 21194 | Expand definition of a dom... |
domnnzr 21195 | A domain is a nonzero ring... |
domnring 21196 | A domain is a ring. (Cont... |
domneq0 21197 | In a domain, a product is ... |
domnmuln0 21198 | In a domain, a product of ... |
isdomn2 21199 | A ring is a domain iff all... |
domnrrg 21200 | In a domain, any nonzero e... |
isdomn5 21201 | The right conjunct in the ... |
isdomn4 21202 | A ring is a domain iff it ... |
opprdomn 21203 | The opposite of a domain i... |
abvn0b 21204 | Another characterization o... |
drngdomn 21205 | A division ring is a domai... |
isidom 21206 | An integral domain is a co... |
fldidom 21207 | A field is an integral dom... |
fldidomOLD 21208 | Obsolete version of ~ fldi... |
fidomndrnglem 21209 | Lemma for ~ fidomndrng . ... |
fidomndrng 21210 | A finite domain is a divis... |
fiidomfld 21211 | A finite integral domain i... |
cnfldstr 21230 | The field of complex numbe... |
cnfldex 21231 | The field of complex numbe... |
cnfldbas 21232 | The base set of the field ... |
cnfldadd 21233 | The addition operation of ... |
cnfldmul 21234 | The multiplication operati... |
cnfldcj 21235 | The conjugation operation ... |
cnfldtset 21236 | The topology component of ... |
cnfldle 21237 | The ordering of the field ... |
cnfldds 21238 | The metric of the field of... |
cnfldunif 21239 | The uniform structure comp... |
cnfldfun 21240 | The field of complex numbe... |
cnfldfunALT 21241 | The field of complex numbe... |
cnfldfunALTOLD 21242 | Obsolete proof of ~ cnfldf... |
xrsstr 21243 | The extended real structur... |
xrsex 21244 | The extended real structur... |
xrsbas 21245 | The base set of the extend... |
xrsadd 21246 | The addition operation of ... |
xrsmul 21247 | The multiplication operati... |
xrstset 21248 | The topology component of ... |
xrsle 21249 | The ordering of the extend... |
cncrng 21250 | The complex numbers form a... |
cnring 21251 | The complex numbers form a... |
xrsmcmn 21252 | The "multiplicative group"... |
cnfld0 21253 | Zero is the zero element o... |
cnfld1 21254 | One is the unity element o... |
cnfldneg 21255 | The additive inverse in th... |
cnfldplusf 21256 | The functionalized additio... |
cnfldsub 21257 | The subtraction operator i... |
cndrng 21258 | The complex numbers form a... |
cnflddiv 21259 | The division operation in ... |
cnfldinv 21260 | The multiplicative inverse... |
cnfldmulg 21261 | The group multiple functio... |
cnfldexp 21262 | The exponentiation operato... |
cnsrng 21263 | The complex numbers form a... |
xrsmgm 21264 | The "additive group" of th... |
xrsnsgrp 21265 | The "additive group" of th... |
xrsmgmdifsgrp 21266 | The "additive group" of th... |
xrs1mnd 21267 | The extended real numbers,... |
xrs10 21268 | The zero of the extended r... |
xrs1cmn 21269 | The extended real numbers ... |
xrge0subm 21270 | The nonnegative extended r... |
xrge0cmn 21271 | The nonnegative extended r... |
xrsds 21272 | The metric of the extended... |
xrsdsval 21273 | The metric of the extended... |
xrsdsreval 21274 | The metric of the extended... |
xrsdsreclblem 21275 | Lemma for ~ xrsdsreclb . ... |
xrsdsreclb 21276 | The metric of the extended... |
cnsubmlem 21277 | Lemma for ~ nn0subm and fr... |
cnsubglem 21278 | Lemma for ~ resubdrg and f... |
cnsubrglem 21279 | Lemma for ~ resubdrg and f... |
cnsubdrglem 21280 | Lemma for ~ resubdrg and f... |
qsubdrg 21281 | The rational numbers form ... |
zsubrg 21282 | The integers form a subrin... |
gzsubrg 21283 | The gaussian integers form... |
nn0subm 21284 | The nonnegative integers f... |
rege0subm 21285 | The nonnegative reals form... |
absabv 21286 | The regular absolute value... |
zsssubrg 21287 | The integers are a subset ... |
qsssubdrg 21288 | The rational numbers are a... |
cnsubrg 21289 | There are no subrings of t... |
cnmgpabl 21290 | The unit group of the comp... |
cnmgpid 21291 | The group identity element... |
cnmsubglem 21292 | Lemma for ~ rpmsubg and fr... |
rpmsubg 21293 | The positive reals form a ... |
gzrngunitlem 21294 | Lemma for ~ gzrngunit . (... |
gzrngunit 21295 | The units on ` ZZ [ _i ] `... |
gsumfsum 21296 | Relate a group sum on ` CC... |
regsumfsum 21297 | Relate a group sum on ` ( ... |
expmhm 21298 | Exponentiation is a monoid... |
nn0srg 21299 | The nonnegative integers f... |
rge0srg 21300 | The nonnegative real numbe... |
zringcrng 21303 | The ring of integers is a ... |
zringring 21304 | The ring of integers is a ... |
zringrng 21305 | The ring of integers is a ... |
zringabl 21306 | The ring of integers is an... |
zringgrp 21307 | The ring of integers is an... |
zringbas 21308 | The integers are the base ... |
zringplusg 21309 | The addition operation of ... |
zringsub 21310 | The subtraction of element... |
zringmulg 21311 | The multiplication (group ... |
zringmulr 21312 | The multiplication operati... |
zring0 21313 | The zero element of the ri... |
zring1 21314 | The unity element of the r... |
zringnzr 21315 | The ring of integers is a ... |
dvdsrzring 21316 | Ring divisibility in the r... |
zringlpirlem1 21317 | Lemma for ~ zringlpir . A... |
zringlpirlem2 21318 | Lemma for ~ zringlpir . A... |
zringlpirlem3 21319 | Lemma for ~ zringlpir . A... |
zringinvg 21320 | The additive inverse of an... |
zringunit 21321 | The units of ` ZZ ` are th... |
zringlpir 21322 | The integers are a princip... |
zringndrg 21323 | The integers are not a div... |
zringcyg 21324 | The integers are a cyclic ... |
zringsubgval 21325 | Subtraction in the ring of... |
zringmpg 21326 | The multiplicative group o... |
prmirredlem 21327 | A positive integer is irre... |
dfprm2 21328 | The positive irreducible e... |
prmirred 21329 | The irreducible elements o... |
expghm 21330 | Exponentiation is a group ... |
mulgghm2 21331 | The powers of a group elem... |
mulgrhm 21332 | The powers of the element ... |
mulgrhm2 21333 | The powers of the element ... |
irinitoringc 21334 | The ring of integers is an... |
nzerooringczr 21335 | There is no zero object in... |
pzriprnglem1 21336 | Lemma 1 for ~ pzriprng : `... |
pzriprnglem2 21337 | Lemma 2 for ~ pzriprng : ... |
pzriprnglem3 21338 | Lemma 3 for ~ pzriprng : ... |
pzriprnglem4 21339 | Lemma 4 for ~ pzriprng : `... |
pzriprnglem5 21340 | Lemma 5 for ~ pzriprng : `... |
pzriprnglem6 21341 | Lemma 6 for ~ pzriprng : `... |
pzriprnglem7 21342 | Lemma 7 for ~ pzriprng : `... |
pzriprnglem8 21343 | Lemma 8 for ~ pzriprng : `... |
pzriprnglem9 21344 | Lemma 9 for ~ pzriprng : ... |
pzriprnglem10 21345 | Lemma 10 for ~ pzriprng : ... |
pzriprnglem11 21346 | Lemma 11 for ~ pzriprng : ... |
pzriprnglem12 21347 | Lemma 12 for ~ pzriprng : ... |
pzriprnglem13 21348 | Lemma 13 for ~ pzriprng : ... |
pzriprnglem14 21349 | Lemma 14 for ~ pzriprng : ... |
pzriprngALT 21350 | The non-unital ring ` ( ZZ... |
pzriprng1ALT 21351 | The ring unity of the ring... |
pzriprng 21352 | The non-unital ring ` ( ZZ... |
pzriprng1 21353 | The ring unity of the ring... |
zrhval 21362 | Define the unique homomorp... |
zrhval2 21363 | Alternate value of the ` Z... |
zrhmulg 21364 | Value of the ` ZRHom ` hom... |
zrhrhmb 21365 | The ` ZRHom ` homomorphism... |
zrhrhm 21366 | The ` ZRHom ` homomorphism... |
zrh1 21367 | Interpretation of 1 in a r... |
zrh0 21368 | Interpretation of 0 in a r... |
zrhpropd 21369 | The ` ZZ ` ring homomorphi... |
zlmval 21370 | Augment an abelian group w... |
zlmlem 21371 | Lemma for ~ zlmbas and ~ z... |
zlmlemOLD 21372 | Obsolete version of ~ zlml... |
zlmbas 21373 | Base set of a ` ZZ ` -modu... |
zlmbasOLD 21374 | Obsolete version of ~ zlmb... |
zlmplusg 21375 | Group operation of a ` ZZ ... |
zlmplusgOLD 21376 | Obsolete version of ~ zlmb... |
zlmmulr 21377 | Ring operation of a ` ZZ `... |
zlmmulrOLD 21378 | Obsolete version of ~ zlmb... |
zlmsca 21379 | Scalar ring of a ` ZZ ` -m... |
zlmvsca 21380 | Scalar multiplication oper... |
zlmlmod 21381 | The ` ZZ ` -module operati... |
chrval 21382 | Definition substitution of... |
chrcl 21383 | Closure of the characteris... |
chrid 21384 | The canonical ` ZZ ` ring ... |
chrdvds 21385 | The ` ZZ ` ring homomorphi... |
chrcong 21386 | If two integers are congru... |
dvdschrmulg 21387 | In a ring, any multiple of... |
fermltlchr 21388 | A generalization of Fermat... |
chrnzr 21389 | Nonzero rings are precisel... |
chrrhm 21390 | The characteristic restric... |
domnchr 21391 | The characteristic of a do... |
znlidl 21392 | The set ` n ZZ ` is an ide... |
zncrng2 21393 | The value of the ` Z/nZ ` ... |
znval 21394 | The value of the ` Z/nZ ` ... |
znle 21395 | The value of the ` Z/nZ ` ... |
znval2 21396 | Self-referential expressio... |
znbaslem 21397 | Lemma for ~ znbas . (Cont... |
znbaslemOLD 21398 | Obsolete version of ~ znba... |
znbas2 21399 | The base set of ` Z/nZ ` i... |
znbas2OLD 21400 | Obsolete version of ~ znba... |
znadd 21401 | The additive structure of ... |
znaddOLD 21402 | Obsolete version of ~ znad... |
znmul 21403 | The multiplicative structu... |
znmulOLD 21404 | Obsolete version of ~ znad... |
znzrh 21405 | The ` ZZ ` ring homomorphi... |
znbas 21406 | The base set of ` Z/nZ ` s... |
zncrng 21407 | ` Z/nZ ` is a commutative ... |
znzrh2 21408 | The ` ZZ ` ring homomorphi... |
znzrhval 21409 | The ` ZZ ` ring homomorphi... |
znzrhfo 21410 | The ` ZZ ` ring homomorphi... |
zncyg 21411 | The group ` ZZ / n ZZ ` is... |
zndvds 21412 | Express equality of equiva... |
zndvds0 21413 | Special case of ~ zndvds w... |
znf1o 21414 | The function ` F ` enumera... |
zzngim 21415 | The ` ZZ ` ring homomorphi... |
znle2 21416 | The ordering of the ` Z/nZ... |
znleval 21417 | The ordering of the ` Z/nZ... |
znleval2 21418 | The ordering of the ` Z/nZ... |
zntoslem 21419 | Lemma for ~ zntos . (Cont... |
zntos 21420 | The ` Z/nZ ` structure is ... |
znhash 21421 | The ` Z/nZ ` structure has... |
znfi 21422 | The ` Z/nZ ` structure is ... |
znfld 21423 | The ` Z/nZ ` structure is ... |
znidomb 21424 | The ` Z/nZ ` structure is ... |
znchr 21425 | Cyclic rings are defined b... |
znunit 21426 | The units of ` Z/nZ ` are ... |
znunithash 21427 | The size of the unit group... |
znrrg 21428 | The regular elements of ` ... |
cygznlem1 21429 | Lemma for ~ cygzn . (Cont... |
cygznlem2a 21430 | Lemma for ~ cygzn . (Cont... |
cygznlem2 21431 | Lemma for ~ cygzn . (Cont... |
cygznlem3 21432 | A cyclic group with ` n ` ... |
cygzn 21433 | A cyclic group with ` n ` ... |
cygth 21434 | The "fundamental theorem o... |
cyggic 21435 | Cyclic groups are isomorph... |
frgpcyg 21436 | A free group is cyclic iff... |
freshmansdream 21437 | For a prime number ` P ` ,... |
cnmsgnsubg 21438 | The signs form a multiplic... |
cnmsgnbas 21439 | The base set of the sign s... |
cnmsgngrp 21440 | The group of signs under m... |
psgnghm 21441 | The sign is a homomorphism... |
psgnghm2 21442 | The sign is a homomorphism... |
psgninv 21443 | The sign of a permutation ... |
psgnco 21444 | Multiplicativity of the pe... |
zrhpsgnmhm 21445 | Embedding of permutation s... |
zrhpsgninv 21446 | The embedded sign of a per... |
evpmss 21447 | Even permutations are perm... |
psgnevpmb 21448 | A class is an even permuta... |
psgnodpm 21449 | A permutation which is odd... |
psgnevpm 21450 | A permutation which is eve... |
psgnodpmr 21451 | If a permutation has sign ... |
zrhpsgnevpm 21452 | The sign of an even permut... |
zrhpsgnodpm 21453 | The sign of an odd permuta... |
cofipsgn 21454 | Composition of any class `... |
zrhpsgnelbas 21455 | Embedding of permutation s... |
zrhcopsgnelbas 21456 | Embedding of permutation s... |
evpmodpmf1o 21457 | The function for performin... |
pmtrodpm 21458 | A transposition is an odd ... |
psgnfix1 21459 | A permutation of a finite ... |
psgnfix2 21460 | A permutation of a finite ... |
psgndiflemB 21461 | Lemma 1 for ~ psgndif . (... |
psgndiflemA 21462 | Lemma 2 for ~ psgndif . (... |
psgndif 21463 | Embedding of permutation s... |
copsgndif 21464 | Embedding of permutation s... |
rebase 21467 | The base of the field of r... |
remulg 21468 | The multiplication (group ... |
resubdrg 21469 | The real numbers form a di... |
resubgval 21470 | Subtraction in the field o... |
replusg 21471 | The addition operation of ... |
remulr 21472 | The multiplication operati... |
re0g 21473 | The zero element of the fi... |
re1r 21474 | The unity element of the f... |
rele2 21475 | The ordering relation of t... |
relt 21476 | The ordering relation of t... |
reds 21477 | The distance of the field ... |
redvr 21478 | The division operation of ... |
retos 21479 | The real numbers are a tot... |
refld 21480 | The real numbers form a fi... |
refldcj 21481 | The conjugation operation ... |
resrng 21482 | The real numbers form a st... |
regsumsupp 21483 | The group sum over the rea... |
rzgrp 21484 | The quotient group ` RR / ... |
isphl 21489 | The predicate "is a genera... |
phllvec 21490 | A pre-Hilbert space is a l... |
phllmod 21491 | A pre-Hilbert space is a l... |
phlsrng 21492 | The scalar ring of a pre-H... |
phllmhm 21493 | The inner product of a pre... |
ipcl 21494 | Closure of the inner produ... |
ipcj 21495 | Conjugate of an inner prod... |
iporthcom 21496 | Orthogonality (meaning inn... |
ip0l 21497 | Inner product with a zero ... |
ip0r 21498 | Inner product with a zero ... |
ipeq0 21499 | The inner product of a vec... |
ipdir 21500 | Distributive law for inner... |
ipdi 21501 | Distributive law for inner... |
ip2di 21502 | Distributive law for inner... |
ipsubdir 21503 | Distributive law for inner... |
ipsubdi 21504 | Distributive law for inner... |
ip2subdi 21505 | Distributive law for inner... |
ipass 21506 | Associative law for inner ... |
ipassr 21507 | "Associative" law for seco... |
ipassr2 21508 | "Associative" law for inne... |
ipffval 21509 | The inner product operatio... |
ipfval 21510 | The inner product operatio... |
ipfeq 21511 | If the inner product opera... |
ipffn 21512 | The inner product operatio... |
phlipf 21513 | The inner product operatio... |
ip2eq 21514 | Two vectors are equal iff ... |
isphld 21515 | Properties that determine ... |
phlpropd 21516 | If two structures have the... |
ssipeq 21517 | The inner product on a sub... |
phssipval 21518 | The inner product on a sub... |
phssip 21519 | The inner product (as a fu... |
phlssphl 21520 | A subspace of an inner pro... |
ocvfval 21527 | The orthocomplement operat... |
ocvval 21528 | Value of the orthocompleme... |
elocv 21529 | Elementhood in the orthoco... |
ocvi 21530 | Property of a member of th... |
ocvss 21531 | The orthocomplement of a s... |
ocvocv 21532 | A set is contained in its ... |
ocvlss 21533 | The orthocomplement of a s... |
ocv2ss 21534 | Orthocomplements reverse s... |
ocvin 21535 | An orthocomplement has tri... |
ocvsscon 21536 | Two ways to say that ` S `... |
ocvlsp 21537 | The orthocomplement of a l... |
ocv0 21538 | The orthocomplement of the... |
ocvz 21539 | The orthocomplement of the... |
ocv1 21540 | The orthocomplement of the... |
unocv 21541 | The orthocomplement of a u... |
iunocv 21542 | The orthocomplement of an ... |
cssval 21543 | The set of closed subspace... |
iscss 21544 | The predicate "is a closed... |
cssi 21545 | Property of a closed subsp... |
cssss 21546 | A closed subspace is a sub... |
iscss2 21547 | It is sufficient to prove ... |
ocvcss 21548 | The orthocomplement of any... |
cssincl 21549 | The zero subspace is a clo... |
css0 21550 | The zero subspace is a clo... |
css1 21551 | The whole space is a close... |
csslss 21552 | A closed subspace of a pre... |
lsmcss 21553 | A subset of a pre-Hilbert ... |
cssmre 21554 | The closed subspaces of a ... |
mrccss 21555 | The Moore closure correspo... |
thlval 21556 | Value of the Hilbert latti... |
thlbas 21557 | Base set of the Hilbert la... |
thlbasOLD 21558 | Obsolete proof of ~ thlbas... |
thlle 21559 | Ordering on the Hilbert la... |
thlleOLD 21560 | Obsolete proof of ~ thlle ... |
thlleval 21561 | Ordering on the Hilbert la... |
thloc 21562 | Orthocomplement on the Hil... |
pjfval 21569 | The value of the projectio... |
pjdm 21570 | A subspace is in the domai... |
pjpm 21571 | The projection map is a pa... |
pjfval2 21572 | Value of the projection ma... |
pjval 21573 | Value of the projection ma... |
pjdm2 21574 | A subspace is in the domai... |
pjff 21575 | A projection is a linear o... |
pjf 21576 | A projection is a function... |
pjf2 21577 | A projection is a function... |
pjfo 21578 | A projection is a surjecti... |
pjcss 21579 | A projection subspace is a... |
ocvpj 21580 | The orthocomplement of a p... |
ishil 21581 | The predicate "is a Hilber... |
ishil2 21582 | The predicate "is a Hilber... |
isobs 21583 | The predicate "is an ortho... |
obsip 21584 | The inner product of two e... |
obsipid 21585 | A basis element has length... |
obsrcl 21586 | Reverse closure for an ort... |
obsss 21587 | An orthonormal basis is a ... |
obsne0 21588 | A basis element is nonzero... |
obsocv 21589 | An orthonormal basis has t... |
obs2ocv 21590 | The double orthocomplement... |
obselocv 21591 | A basis element is in the ... |
obs2ss 21592 | A basis has no proper subs... |
obslbs 21593 | An orthogonal basis is a l... |
reldmdsmm 21596 | The direct sum is a well-b... |
dsmmval 21597 | Value of the module direct... |
dsmmbase 21598 | Base set of the module dir... |
dsmmval2 21599 | Self-referential definitio... |
dsmmbas2 21600 | Base set of the direct sum... |
dsmmfi 21601 | For finite products, the d... |
dsmmelbas 21602 | Membership in the finitely... |
dsmm0cl 21603 | The all-zero vector is con... |
dsmmacl 21604 | The finite hull is closed ... |
prdsinvgd2 21605 | Negation of a single coord... |
dsmmsubg 21606 | The finite hull of a produ... |
dsmmlss 21607 | The finite hull of a produ... |
dsmmlmod 21608 | The direct sum of a family... |
frlmval 21611 | Value of the "free module"... |
frlmlmod 21612 | The free module is a modul... |
frlmpws 21613 | The free module as a restr... |
frlmlss 21614 | The base set of the free m... |
frlmpwsfi 21615 | The finite free module is ... |
frlmsca 21616 | The ring of scalars of a f... |
frlm0 21617 | Zero in a free module (rin... |
frlmbas 21618 | Base set of the free modul... |
frlmelbas 21619 | Membership in the base set... |
frlmrcl 21620 | If a free module is inhabi... |
frlmbasfsupp 21621 | Elements of the free modul... |
frlmbasmap 21622 | Elements of the free modul... |
frlmbasf 21623 | Elements of the free modul... |
frlmlvec 21624 | The free module over a div... |
frlmfibas 21625 | The base set of the finite... |
elfrlmbasn0 21626 | If the dimension of a free... |
frlmplusgval 21627 | Addition in a free module.... |
frlmsubgval 21628 | Subtraction in a free modu... |
frlmvscafval 21629 | Scalar multiplication in a... |
frlmvplusgvalc 21630 | Coordinates of a sum with ... |
frlmvscaval 21631 | Coordinates of a scalar mu... |
frlmplusgvalb 21632 | Addition in a free module ... |
frlmvscavalb 21633 | Scalar multiplication in a... |
frlmvplusgscavalb 21634 | Addition combined with sca... |
frlmgsum 21635 | Finite commutative sums in... |
frlmsplit2 21636 | Restriction is homomorphic... |
frlmsslss 21637 | A subset of a free module ... |
frlmsslss2 21638 | A subset of a free module ... |
frlmbas3 21639 | An element of the base set... |
mpofrlmd 21640 | Elements of the free modul... |
frlmip 21641 | The inner product of a fre... |
frlmipval 21642 | The inner product of a fre... |
frlmphllem 21643 | Lemma for ~ frlmphl . (Co... |
frlmphl 21644 | Conditions for a free modu... |
uvcfval 21647 | Value of the unit-vector g... |
uvcval 21648 | Value of a single unit vec... |
uvcvval 21649 | Value of a unit vector coo... |
uvcvvcl 21650 | A coordinate of a unit vec... |
uvcvvcl2 21651 | A unit vector coordinate i... |
uvcvv1 21652 | The unit vector is one at ... |
uvcvv0 21653 | The unit vector is zero at... |
uvcff 21654 | Domain and codomain of the... |
uvcf1 21655 | In a nonzero ring, each un... |
uvcresum 21656 | Any element of a free modu... |
frlmssuvc1 21657 | A scalar multiple of a uni... |
frlmssuvc2 21658 | A nonzero scalar multiple ... |
frlmsslsp 21659 | A subset of a free module ... |
frlmlbs 21660 | The unit vectors comprise ... |
frlmup1 21661 | Any assignment of unit vec... |
frlmup2 21662 | The evaluation map has the... |
frlmup3 21663 | The range of such an evalu... |
frlmup4 21664 | Universal property of the ... |
ellspd 21665 | The elements of the span o... |
elfilspd 21666 | Simplified version of ~ el... |
rellindf 21671 | The independent-family pre... |
islinds 21672 | Property of an independent... |
linds1 21673 | An independent set of vect... |
linds2 21674 | An independent set of vect... |
islindf 21675 | Property of an independent... |
islinds2 21676 | Expanded property of an in... |
islindf2 21677 | Property of an independent... |
lindff 21678 | Functional property of a l... |
lindfind 21679 | A linearly independent fam... |
lindsind 21680 | A linearly independent set... |
lindfind2 21681 | In a linearly independent ... |
lindsind2 21682 | In a linearly independent ... |
lindff1 21683 | A linearly independent fam... |
lindfrn 21684 | The range of an independen... |
f1lindf 21685 | Rearranging and deleting e... |
lindfres 21686 | Any restriction of an inde... |
lindsss 21687 | Any subset of an independe... |
f1linds 21688 | A family constructed from ... |
islindf3 21689 | In a nonzero ring, indepen... |
lindfmm 21690 | Linear independence of a f... |
lindsmm 21691 | Linear independence of a s... |
lindsmm2 21692 | The monomorphic image of a... |
lsslindf 21693 | Linear independence is unc... |
lsslinds 21694 | Linear independence is unc... |
islbs4 21695 | A basis is an independent ... |
lbslinds 21696 | A basis is independent. (... |
islinds3 21697 | A subset is linearly indep... |
islinds4 21698 | A set is independent in a ... |
lmimlbs 21699 | The isomorphic image of a ... |
lmiclbs 21700 | Having a basis is an isomo... |
islindf4 21701 | A family is independent if... |
islindf5 21702 | A family is independent if... |
indlcim 21703 | An independent, spanning f... |
lbslcic 21704 | A module with a basis is i... |
lmisfree 21705 | A module has a basis iff i... |
lvecisfrlm 21706 | Every vector space is isom... |
lmimco 21707 | The composition of two iso... |
lmictra 21708 | Module isomorphism is tran... |
uvcf1o 21709 | In a nonzero ring, the map... |
uvcendim 21710 | In a nonzero ring, the num... |
frlmisfrlm 21711 | A free module is isomorphi... |
frlmiscvec 21712 | Every free module is isomo... |
isassa 21719 | The properties of an assoc... |
assalem 21720 | The properties of an assoc... |
assaass 21721 | Left-associative property ... |
assaassr 21722 | Right-associative property... |
assalmod 21723 | An associative algebra is ... |
assaring 21724 | An associative algebra is ... |
assasca 21725 | The scalars of an associat... |
assa2ass 21726 | Left- and right-associativ... |
isassad 21727 | Sufficient condition for b... |
issubassa3 21728 | A subring that is also a s... |
issubassa 21729 | The subalgebras of an asso... |
sraassab 21730 | A subring algebra is an as... |
sraassa 21731 | The subring algebra over a... |
sraassaOLD 21732 | Obsolete version of ~ sraa... |
rlmassa 21733 | The ring module over a com... |
assapropd 21734 | If two structures have the... |
aspval 21735 | Value of the algebraic clo... |
asplss 21736 | The algebraic span of a se... |
aspid 21737 | The algebraic span of a su... |
aspsubrg 21738 | The algebraic span of a se... |
aspss 21739 | Span preserves subset orde... |
aspssid 21740 | A set of vectors is a subs... |
asclfval 21741 | Function value of the alge... |
asclval 21742 | Value of a mapped algebra ... |
asclfn 21743 | Unconditional functionalit... |
asclf 21744 | The algebra scalars functi... |
asclghm 21745 | The algebra scalars functi... |
ascl0 21746 | The scalar 0 embedded into... |
ascl1 21747 | The scalar 1 embedded into... |
asclmul1 21748 | Left multiplication by a l... |
asclmul2 21749 | Right multiplication by a ... |
ascldimul 21750 | The algebra scalars functi... |
asclinvg 21751 | The group inverse (negatio... |
asclrhm 21752 | The scalar injection is a ... |
rnascl 21753 | The set of injected scalar... |
issubassa2 21754 | A subring of a unital alge... |
rnasclsubrg 21755 | The scalar multiples of th... |
rnasclmulcl 21756 | (Vector) multiplication is... |
rnasclassa 21757 | The scalar multiples of th... |
ressascl 21758 | The injection of scalars i... |
asclpropd 21759 | If two structures have the... |
aspval2 21760 | The algebraic closure is t... |
assamulgscmlem1 21761 | Lemma 1 for ~ assamulgscm ... |
assamulgscmlem2 21762 | Lemma for ~ assamulgscm (i... |
assamulgscm 21763 | Exponentiation of a scalar... |
asclmulg 21764 | Apply group multiplication... |
zlmassa 21765 | The ` ZZ ` -module operati... |
reldmpsr 21776 | The multivariate power ser... |
psrval 21777 | Value of the multivariate ... |
psrvalstr 21778 | The multivariate power ser... |
psrbag 21779 | Elementhood in the set of ... |
psrbagf 21780 | A finite bag is a function... |
psrbagfOLD 21781 | Obsolete version of ~ psrb... |
psrbagfsupp 21782 | Finite bags have finite su... |
psrbagfsuppOLD 21783 | Obsolete version of ~ psrb... |
snifpsrbag 21784 | A bag containing one eleme... |
fczpsrbag 21785 | The constant function equa... |
psrbaglesupp 21786 | The support of a dominated... |
psrbaglesuppOLD 21787 | Obsolete version of ~ psrb... |
psrbaglecl 21788 | The set of finite bags is ... |
psrbagleclOLD 21789 | Obsolete version of ~ psrb... |
psrbagaddcl 21790 | The sum of two finite bags... |
psrbagaddclOLD 21791 | Obsolete version of ~ psrb... |
psrbagcon 21792 | The analogue of the statem... |
psrbagconOLD 21793 | Obsolete version of ~ psrb... |
psrbaglefi 21794 | There are finitely many ba... |
psrbaglefiOLD 21795 | Obsolete version of ~ psrb... |
psrbagconcl 21796 | The complement of a bag is... |
psrbagconclOLD 21797 | Obsolete version of ~ psrb... |
psrbagconf1o 21798 | Bag complementation is a b... |
psrbagconf1oOLD 21799 | Obsolete version of ~ psrb... |
gsumbagdiaglemOLD 21800 | Obsolete version of ~ gsum... |
gsumbagdiagOLD 21801 | Obsolete version of ~ gsum... |
psrass1lemOLD 21802 | Obsolete version of ~ psra... |
gsumbagdiaglem 21803 | Lemma for ~ gsumbagdiag . ... |
gsumbagdiag 21804 | Two-dimensional commutatio... |
psrass1lem 21805 | A group sum commutation us... |
psrbas 21806 | The base set of the multiv... |
psrelbas 21807 | An element of the set of p... |
psrelbasfun 21808 | An element of the set of p... |
psrplusg 21809 | The addition operation of ... |
psradd 21810 | The addition operation of ... |
psraddcl 21811 | Closure of the power serie... |
psraddclOLD 21812 | Obsolete version of ~ psra... |
psrmulr 21813 | The multiplication operati... |
psrmulfval 21814 | The multiplication operati... |
psrmulval 21815 | The multiplication operati... |
psrmulcllem 21816 | Closure of the power serie... |
psrmulcl 21817 | Closure of the power serie... |
psrsca 21818 | The scalar field of the mu... |
psrvscafval 21819 | The scalar multiplication ... |
psrvsca 21820 | The scalar multiplication ... |
psrvscaval 21821 | The scalar multiplication ... |
psrvscacl 21822 | Closure of the power serie... |
psr0cl 21823 | The zero element of the ri... |
psr0lid 21824 | The zero element of the ri... |
psrnegcl 21825 | The negative function in t... |
psrlinv 21826 | The negative function in t... |
psrgrp 21827 | The ring of power series i... |
psrgrpOLD 21828 | Obsolete proof of ~ psrgrp... |
psr0 21829 | The zero element of the ri... |
psrneg 21830 | The negative function of t... |
psrlmod 21831 | The ring of power series i... |
psr1cl 21832 | The identity element of th... |
psrlidm 21833 | The identity element of th... |
psrridm 21834 | The identity element of th... |
psrass1 21835 | Associative identity for t... |
psrdi 21836 | Distributive law for the r... |
psrdir 21837 | Distributive law for the r... |
psrass23l 21838 | Associative identity for t... |
psrcom 21839 | Commutative law for the ri... |
psrass23 21840 | Associative identities for... |
psrring 21841 | The ring of power series i... |
psr1 21842 | The identity element of th... |
psrcrng 21843 | The ring of power series i... |
psrassa 21844 | The ring of power series i... |
resspsrbas 21845 | A restricted power series ... |
resspsradd 21846 | A restricted power series ... |
resspsrmul 21847 | A restricted power series ... |
resspsrvsca 21848 | A restricted power series ... |
subrgpsr 21849 | A subring of the base ring... |
mvrfval 21850 | Value of the generating el... |
mvrval 21851 | Value of the generating el... |
mvrval2 21852 | Value of the generating el... |
mvrid 21853 | The ` X i ` -th coefficien... |
mvrf 21854 | The power series variable ... |
mvrf1 21855 | The power series variable ... |
mvrcl2 21856 | A power series variable is... |
reldmmpl 21857 | The multivariate polynomia... |
mplval 21858 | Value of the set of multiv... |
mplbas 21859 | Base set of the set of mul... |
mplelbas 21860 | Property of being a polyno... |
mvrcl 21861 | A power series variable is... |
mvrf2 21862 | The power series/polynomia... |
mplrcl 21863 | Reverse closure for the po... |
mplelsfi 21864 | A polynomial treated as a ... |
mplval2 21865 | Self-referential expressio... |
mplbasss 21866 | The set of polynomials is ... |
mplelf 21867 | A polynomial is defined as... |
mplsubglem 21868 | If ` A ` is an ideal of se... |
mpllsslem 21869 | If ` A ` is an ideal of su... |
mplsubglem2 21870 | Lemma for ~ mplsubg and ~ ... |
mplsubg 21871 | The set of polynomials is ... |
mpllss 21872 | The set of polynomials is ... |
mplsubrglem 21873 | Lemma for ~ mplsubrg . (C... |
mplsubrg 21874 | The set of polynomials is ... |
mpl0 21875 | The zero polynomial. (Con... |
mplplusg 21876 | Value of addition in a pol... |
mplmulr 21877 | Value of multiplication in... |
mpladd 21878 | The addition operation on ... |
mplneg 21879 | The negative function on m... |
mplmul 21880 | The multiplication operati... |
mpl1 21881 | The identity element of th... |
mplsca 21882 | The scalar field of a mult... |
mplvsca2 21883 | The scalar multiplication ... |
mplvsca 21884 | The scalar multiplication ... |
mplvscaval 21885 | The scalar multiplication ... |
mplgrp 21886 | The polynomial ring is a g... |
mpllmod 21887 | The polynomial ring is a l... |
mplring 21888 | The polynomial ring is a r... |
mpllvec 21889 | The polynomial ring is a v... |
mplcrng 21890 | The polynomial ring is a c... |
mplassa 21891 | The polynomial ring is an ... |
ressmplbas2 21892 | The base set of a restrict... |
ressmplbas 21893 | A restricted polynomial al... |
ressmpladd 21894 | A restricted polynomial al... |
ressmplmul 21895 | A restricted polynomial al... |
ressmplvsca 21896 | A restricted power series ... |
subrgmpl 21897 | A subring of the base ring... |
subrgmvr 21898 | The variables in a subring... |
subrgmvrf 21899 | The variables in a polynom... |
mplmon 21900 | A monomial is a polynomial... |
mplmonmul 21901 | The product of two monomia... |
mplcoe1 21902 | Decompose a polynomial int... |
mplcoe3 21903 | Decompose a monomial in on... |
mplcoe5lem 21904 | Lemma for ~ mplcoe4 . (Co... |
mplcoe5 21905 | Decompose a monomial into ... |
mplcoe2 21906 | Decompose a monomial into ... |
mplbas2 21907 | An alternative expression ... |
ltbval 21908 | Value of the well-order on... |
ltbwe 21909 | The finite bag order is a ... |
reldmopsr 21910 | Lemma for ordered power se... |
opsrval 21911 | The value of the "ordered ... |
opsrle 21912 | An alternative expression ... |
opsrval2 21913 | Self-referential expressio... |
opsrbaslem 21914 | Get a component of the ord... |
opsrbaslemOLD 21915 | Obsolete version of ~ opsr... |
opsrbas 21916 | The base set of the ordere... |
opsrbasOLD 21917 | Obsolete version of ~ opsr... |
opsrplusg 21918 | The addition operation of ... |
opsrplusgOLD 21919 | Obsolete version of ~ opsr... |
opsrmulr 21920 | The multiplication operati... |
opsrmulrOLD 21921 | Obsolete version of ~ opsr... |
opsrvsca 21922 | The scalar product operati... |
opsrvscaOLD 21923 | Obsolete version of ~ opsr... |
opsrsca 21924 | The scalar ring of the ord... |
opsrscaOLD 21925 | Obsolete version of ~ opsr... |
opsrtoslem1 21926 | Lemma for ~ opsrtos . (Co... |
opsrtoslem2 21927 | Lemma for ~ opsrtos . (Co... |
opsrtos 21928 | The ordered power series s... |
opsrso 21929 | The ordered power series s... |
opsrcrng 21930 | The ring of ordered power ... |
opsrassa 21931 | The ring of ordered power ... |
mplmon2 21932 | Express a scaled monomial.... |
psrbag0 21933 | The empty bag is a bag. (... |
psrbagsn 21934 | A singleton bag is a bag. ... |
mplascl 21935 | Value of the scalar inject... |
mplasclf 21936 | The scalar injection is a ... |
subrgascl 21937 | The scalar injection funct... |
subrgasclcl 21938 | The scalars in a polynomia... |
mplmon2cl 21939 | A scaled monomial is a pol... |
mplmon2mul 21940 | Product of scaled monomial... |
mplind 21941 | Prove a property of polyno... |
mplcoe4 21942 | Decompose a polynomial int... |
evlslem4 21947 | The support of a tensor pr... |
psrbagev1 21948 | A bag of multipliers provi... |
psrbagev1OLD 21949 | Obsolete version of ~ psrb... |
psrbagev2 21950 | Closure of a sum using a b... |
psrbagev2OLD 21951 | Obsolete version of ~ psrb... |
evlslem2 21952 | A linear function on the p... |
evlslem3 21953 | Lemma for ~ evlseu . Poly... |
evlslem6 21954 | Lemma for ~ evlseu . Fini... |
evlslem1 21955 | Lemma for ~ evlseu , give ... |
evlseu 21956 | For a given interpretation... |
reldmevls 21957 | Well-behaved binary operat... |
mpfrcl 21958 | Reverse closure for the se... |
evlsval 21959 | Value of the polynomial ev... |
evlsval2 21960 | Characterizing properties ... |
evlsrhm 21961 | Polynomial evaluation is a... |
evlssca 21962 | Polynomial evaluation maps... |
evlsvar 21963 | Polynomial evaluation maps... |
evlsgsumadd 21964 | Polynomial evaluation maps... |
evlsgsummul 21965 | Polynomial evaluation maps... |
evlspw 21966 | Polynomial evaluation for ... |
evlsvarpw 21967 | Polynomial evaluation for ... |
evlval 21968 | Value of the simple/same r... |
evlrhm 21969 | The simple evaluation map ... |
evlsscasrng 21970 | The evaluation of a scalar... |
evlsca 21971 | Simple polynomial evaluati... |
evlsvarsrng 21972 | The evaluation of the vari... |
evlvar 21973 | Simple polynomial evaluati... |
mpfconst 21974 | Constants are multivariate... |
mpfproj 21975 | Projections are multivaria... |
mpfsubrg 21976 | Polynomial functions are a... |
mpff 21977 | Polynomial functions are f... |
mpfaddcl 21978 | The sum of multivariate po... |
mpfmulcl 21979 | The product of multivariat... |
mpfind 21980 | Prove a property of polyno... |
selvffval 21986 | Value of the "variable sel... |
selvfval 21987 | Value of the "variable sel... |
selvval 21988 | Value of the "variable sel... |
mhpfval 21990 | Value of the "homogeneous ... |
mhpval 21991 | Value of the "homogeneous ... |
ismhp 21992 | Property of being a homoge... |
ismhp2 21993 | Deduce a homogeneous polyn... |
ismhp3 21994 | A polynomial is homogeneou... |
mhpmpl 21995 | A homogeneous polynomial i... |
mhpdeg 21996 | All nonzero terms of a hom... |
mhp0cl 21997 | The zero polynomial is hom... |
mhpsclcl 21998 | A scalar (or constant) pol... |
mhpvarcl 21999 | A power series variable is... |
mhpmulcl 22000 | A product of homogeneous p... |
mhppwdeg 22001 | Degree of a homogeneous po... |
mhpaddcl 22002 | Homogeneous polynomials ar... |
mhpinvcl 22003 | Homogeneous polynomials ar... |
mhpsubg 22004 | Homogeneous polynomials fo... |
mhpvscacl 22005 | Homogeneous polynomials ar... |
mhplss 22006 | Homogeneous polynomials fo... |
psdffval 22008 | Value of the power series ... |
psdfval 22009 | Give a map between power s... |
psdval 22010 | Evaluate the partial deriv... |
psdcoef 22011 | Coefficient of a term of t... |
psdcl 22012 | The derivative of a power ... |
psdmplcl 22013 | The derivative of a polyno... |
psdadd 22014 | The derivative of a sum is... |
psdvsca 22015 | The derivative of a scaled... |
psr1baslem 22027 | The set of finite bags on ... |
psr1val 22028 | Value of the ring of univa... |
psr1crng 22029 | The ring of univariate pow... |
psr1assa 22030 | The ring of univariate pow... |
psr1tos 22031 | The ordered power series s... |
psr1bas2 22032 | The base set of the ring o... |
psr1bas 22033 | The base set of the ring o... |
vr1val 22034 | The value of the generator... |
vr1cl2 22035 | The variable ` X ` is a me... |
ply1val 22036 | The value of the set of un... |
ply1bas 22037 | The value of the base set ... |
ply1lss 22038 | Univariate polynomials for... |
ply1subrg 22039 | Univariate polynomials for... |
ply1crng 22040 | The ring of univariate pol... |
ply1assa 22041 | The ring of univariate pol... |
psr1bascl 22042 | A univariate power series ... |
psr1basf 22043 | Univariate power series ba... |
ply1basf 22044 | Univariate polynomial base... |
ply1bascl 22045 | A univariate polynomial is... |
ply1bascl2 22046 | A univariate polynomial is... |
coe1fval 22047 | Value of the univariate po... |
coe1fv 22048 | Value of an evaluated coef... |
fvcoe1 22049 | Value of a multivariate co... |
coe1fval3 22050 | Univariate power series co... |
coe1f2 22051 | Functionality of univariat... |
coe1fval2 22052 | Univariate polynomial coef... |
coe1f 22053 | Functionality of univariat... |
coe1fvalcl 22054 | A coefficient of a univari... |
coe1sfi 22055 | Finite support of univaria... |
coe1fsupp 22056 | The coefficient vector of ... |
mptcoe1fsupp 22057 | A mapping involving coeffi... |
coe1ae0 22058 | The coefficient vector of ... |
vr1cl 22059 | The generator of a univari... |
opsr0 22060 | Zero in the ordered power ... |
opsr1 22061 | One in the ordered power s... |
psr1plusg 22062 | Value of addition in a uni... |
psr1vsca 22063 | Value of scalar multiplica... |
psr1mulr 22064 | Value of multiplication in... |
ply1plusg 22065 | Value of addition in a uni... |
ply1vsca 22066 | Value of scalar multiplica... |
ply1mulr 22067 | Value of multiplication in... |
ply1ass23l 22068 | Associative identity with ... |
ressply1bas2 22069 | The base set of a restrict... |
ressply1bas 22070 | A restricted polynomial al... |
ressply1add 22071 | A restricted polynomial al... |
ressply1mul 22072 | A restricted polynomial al... |
ressply1vsca 22073 | A restricted power series ... |
subrgply1 22074 | A subring of the base ring... |
gsumply1subr 22075 | Evaluate a group sum in a ... |
psrbaspropd 22076 | Property deduction for pow... |
psrplusgpropd 22077 | Property deduction for pow... |
mplbaspropd 22078 | Property deduction for pol... |
psropprmul 22079 | Reversing multiplication i... |
ply1opprmul 22080 | Reversing multiplication i... |
00ply1bas 22081 | Lemma for ~ ply1basfvi and... |
ply1basfvi 22082 | Protection compatibility o... |
ply1plusgfvi 22083 | Protection compatibility o... |
ply1baspropd 22084 | Property deduction for uni... |
ply1plusgpropd 22085 | Property deduction for uni... |
opsrring 22086 | Ordered power series form ... |
opsrlmod 22087 | Ordered power series form ... |
psr1ring 22088 | Univariate power series fo... |
ply1ring 22089 | Univariate polynomials for... |
psr1lmod 22090 | Univariate power series fo... |
psr1sca 22091 | Scalars of a univariate po... |
psr1sca2 22092 | Scalars of a univariate po... |
ply1lmod 22093 | Univariate polynomials for... |
ply1sca 22094 | Scalars of a univariate po... |
ply1sca2 22095 | Scalars of a univariate po... |
ply1mpl0 22096 | The univariate polynomial ... |
ply10s0 22097 | Zero times a univariate po... |
ply1mpl1 22098 | The univariate polynomial ... |
ply1ascl 22099 | The univariate polynomial ... |
subrg1ascl 22100 | The scalar injection funct... |
subrg1asclcl 22101 | The scalars in a polynomia... |
subrgvr1 22102 | The variables in a subring... |
subrgvr1cl 22103 | The variables in a polynom... |
coe1z 22104 | The coefficient vector of ... |
coe1add 22105 | The coefficient vector of ... |
coe1addfv 22106 | A particular coefficient o... |
coe1subfv 22107 | A particular coefficient o... |
coe1mul2lem1 22108 | An equivalence for ~ coe1m... |
coe1mul2lem2 22109 | An equivalence for ~ coe1m... |
coe1mul2 22110 | The coefficient vector of ... |
coe1mul 22111 | The coefficient vector of ... |
ply1moncl 22112 | Closure of the expression ... |
ply1tmcl 22113 | Closure of the expression ... |
coe1tm 22114 | Coefficient vector of a po... |
coe1tmfv1 22115 | Nonzero coefficient of a p... |
coe1tmfv2 22116 | Zero coefficient of a poly... |
coe1tmmul2 22117 | Coefficient vector of a po... |
coe1tmmul 22118 | Coefficient vector of a po... |
coe1tmmul2fv 22119 | Function value of a right-... |
coe1pwmul 22120 | Coefficient vector of a po... |
coe1pwmulfv 22121 | Function value of a right-... |
ply1scltm 22122 | A scalar is a term with ze... |
coe1sclmul 22123 | Coefficient vector of a po... |
coe1sclmulfv 22124 | A single coefficient of a ... |
coe1sclmul2 22125 | Coefficient vector of a po... |
ply1sclf 22126 | A scalar polynomial is a p... |
ply1sclcl 22127 | The value of the algebra s... |
coe1scl 22128 | Coefficient vector of a sc... |
ply1sclid 22129 | Recover the base scalar fr... |
ply1sclf1 22130 | The polynomial scalar func... |
ply1scl0 22131 | The zero scalar is zero. ... |
ply1scl0OLD 22132 | Obsolete version of ~ ply1... |
ply1scln0 22133 | Nonzero scalars create non... |
ply1scl1 22134 | The one scalar is the unit... |
ply1scl1OLD 22135 | Obsolete version of ~ ply1... |
ply1idvr1 22136 | The identity of a polynomi... |
cply1mul 22137 | The product of two constan... |
ply1coefsupp 22138 | The decomposition of a uni... |
ply1coe 22139 | Decompose a univariate pol... |
eqcoe1ply1eq 22140 | Two polynomials over the s... |
ply1coe1eq 22141 | Two polynomials over the s... |
cply1coe0 22142 | All but the first coeffici... |
cply1coe0bi 22143 | A polynomial is constant (... |
coe1fzgsumdlem 22144 | Lemma for ~ coe1fzgsumd (i... |
coe1fzgsumd 22145 | Value of an evaluated coef... |
ply1scleq 22146 | Equality of a constant pol... |
ply1chr 22147 | The characteristic of a po... |
gsumsmonply1 22148 | A finite group sum of scal... |
gsummoncoe1 22149 | A coefficient of the polyn... |
gsumply1eq 22150 | Two univariate polynomials... |
lply1binom 22151 | The binomial theorem for l... |
lply1binomsc 22152 | The binomial theorem for l... |
ply1fermltlchr 22153 | Fermat's little theorem fo... |
reldmevls1 22158 | Well-behaved binary operat... |
ply1frcl 22159 | Reverse closure for the se... |
evls1fval 22160 | Value of the univariate po... |
evls1val 22161 | Value of the univariate po... |
evls1rhmlem 22162 | Lemma for ~ evl1rhm and ~ ... |
evls1rhm 22163 | Polynomial evaluation is a... |
evls1sca 22164 | Univariate polynomial eval... |
evls1gsumadd 22165 | Univariate polynomial eval... |
evls1gsummul 22166 | Univariate polynomial eval... |
evls1pw 22167 | Univariate polynomial eval... |
evls1varpw 22168 | Univariate polynomial eval... |
evl1fval 22169 | Value of the simple/same r... |
evl1val 22170 | Value of the simple/same r... |
evl1fval1lem 22171 | Lemma for ~ evl1fval1 . (... |
evl1fval1 22172 | Value of the simple/same r... |
evl1rhm 22173 | Polynomial evaluation is a... |
fveval1fvcl 22174 | The function value of the ... |
evl1sca 22175 | Polynomial evaluation maps... |
evl1scad 22176 | Polynomial evaluation buil... |
evl1var 22177 | Polynomial evaluation maps... |
evl1vard 22178 | Polynomial evaluation buil... |
evls1var 22179 | Univariate polynomial eval... |
evls1scasrng 22180 | The evaluation of a scalar... |
evls1varsrng 22181 | The evaluation of the vari... |
evl1addd 22182 | Polynomial evaluation buil... |
evl1subd 22183 | Polynomial evaluation buil... |
evl1muld 22184 | Polynomial evaluation buil... |
evl1vsd 22185 | Polynomial evaluation buil... |
evl1expd 22186 | Polynomial evaluation buil... |
pf1const 22187 | Constants are polynomial f... |
pf1id 22188 | The identity is a polynomi... |
pf1subrg 22189 | Polynomial functions are a... |
pf1rcl 22190 | Reverse closure for the se... |
pf1f 22191 | Polynomial functions are f... |
mpfpf1 22192 | Convert a multivariate pol... |
pf1mpf 22193 | Convert a univariate polyn... |
pf1addcl 22194 | The sum of multivariate po... |
pf1mulcl 22195 | The product of multivariat... |
pf1ind 22196 | Prove a property of polyno... |
evl1gsumdlem 22197 | Lemma for ~ evl1gsumd (ind... |
evl1gsumd 22198 | Polynomial evaluation buil... |
evl1gsumadd 22199 | Univariate polynomial eval... |
evl1gsumaddval 22200 | Value of a univariate poly... |
evl1gsummul 22201 | Univariate polynomial eval... |
evl1varpw 22202 | Univariate polynomial eval... |
evl1varpwval 22203 | Value of a univariate poly... |
evl1scvarpw 22204 | Univariate polynomial eval... |
evl1scvarpwval 22205 | Value of a univariate poly... |
evl1gsummon 22206 | Value of a univariate poly... |
mamufval 22209 | Functional value of the ma... |
mamuval 22210 | Multiplication of two matr... |
mamufv 22211 | A cell in the multiplicati... |
mamudm 22212 | The domain of the matrix m... |
mamufacex 22213 | Every solution of the equa... |
mamures 22214 | Rows in a matrix product a... |
mndvcl 22215 | Tuple-wise additive closur... |
mndvass 22216 | Tuple-wise associativity i... |
mndvlid 22217 | Tuple-wise left identity i... |
mndvrid 22218 | Tuple-wise right identity ... |
grpvlinv 22219 | Tuple-wise left inverse in... |
grpvrinv 22220 | Tuple-wise right inverse i... |
mhmvlin 22221 | Tuple extension of monoid ... |
ringvcl 22222 | Tuple-wise multiplication ... |
mamucl 22223 | Operation closure of matri... |
mamuass 22224 | Matrix multiplication is a... |
mamudi 22225 | Matrix multiplication dist... |
mamudir 22226 | Matrix multiplication dist... |
mamuvs1 22227 | Matrix multiplication dist... |
mamuvs2 22228 | Matrix multiplication dist... |
matbas0pc 22231 | There is no matrix with a ... |
matbas0 22232 | There is no matrix for a n... |
matval 22233 | Value of the matrix algebr... |
matrcl 22234 | Reverse closure for the ma... |
matbas 22235 | The matrix ring has the sa... |
matplusg 22236 | The matrix ring has the sa... |
matsca 22237 | The matrix ring has the sa... |
matscaOLD 22238 | Obsolete proof of ~ matsca... |
matvsca 22239 | The matrix ring has the sa... |
matvscaOLD 22240 | Obsolete proof of ~ matvsc... |
mat0 22241 | The matrix ring has the sa... |
matinvg 22242 | The matrix ring has the sa... |
mat0op 22243 | Value of a zero matrix as ... |
matsca2 22244 | The scalars of the matrix ... |
matbas2 22245 | The base set of the matrix... |
matbas2i 22246 | A matrix is a function. (... |
matbas2d 22247 | The base set of the matrix... |
eqmat 22248 | Two square matrices of the... |
matecl 22249 | Each entry (according to W... |
matecld 22250 | Each entry (according to W... |
matplusg2 22251 | Addition in the matrix rin... |
matvsca2 22252 | Scalar multiplication in t... |
matlmod 22253 | The matrix ring is a linea... |
matgrp 22254 | The matrix ring is a group... |
matvscl 22255 | Closure of the scalar mult... |
matsubg 22256 | The matrix ring has the sa... |
matplusgcell 22257 | Addition in the matrix rin... |
matsubgcell 22258 | Subtraction in the matrix ... |
matinvgcell 22259 | Additive inversion in the ... |
matvscacell 22260 | Scalar multiplication in t... |
matgsum 22261 | Finite commutative sums in... |
matmulr 22262 | Multiplication in the matr... |
mamumat1cl 22263 | The identity matrix (as op... |
mat1comp 22264 | The components of the iden... |
mamulid 22265 | The identity matrix (as op... |
mamurid 22266 | The identity matrix (as op... |
matring 22267 | Existence of the matrix ri... |
matassa 22268 | Existence of the matrix al... |
matmulcell 22269 | Multiplication in the matr... |
mpomatmul 22270 | Multiplication of two N x ... |
mat1 22271 | Value of an identity matri... |
mat1ov 22272 | Entries of an identity mat... |
mat1bas 22273 | The identity matrix is a m... |
matsc 22274 | The identity matrix multip... |
ofco2 22275 | Distribution law for the f... |
oftpos 22276 | The transposition of the v... |
mattposcl 22277 | The transpose of a square ... |
mattpostpos 22278 | The transpose of the trans... |
mattposvs 22279 | The transposition of a mat... |
mattpos1 22280 | The transposition of the i... |
tposmap 22281 | The transposition of an I ... |
mamutpos 22282 | Behavior of transposes in ... |
mattposm 22283 | Multiplying two transposed... |
matgsumcl 22284 | Closure of a group sum ove... |
madetsumid 22285 | The identity summand in th... |
matepmcl 22286 | Each entry of a matrix wit... |
matepm2cl 22287 | Each entry of a matrix wit... |
madetsmelbas 22288 | A summand of the determina... |
madetsmelbas2 22289 | A summand of the determina... |
mat0dimbas0 22290 | The empty set is the one a... |
mat0dim0 22291 | The zero of the algebra of... |
mat0dimid 22292 | The identity of the algebr... |
mat0dimscm 22293 | The scalar multiplication ... |
mat0dimcrng 22294 | The algebra of matrices wi... |
mat1dimelbas 22295 | A matrix with dimension 1 ... |
mat1dimbas 22296 | A matrix with dimension 1 ... |
mat1dim0 22297 | The zero of the algebra of... |
mat1dimid 22298 | The identity of the algebr... |
mat1dimscm 22299 | The scalar multiplication ... |
mat1dimmul 22300 | The ring multiplication in... |
mat1dimcrng 22301 | The algebra of matrices wi... |
mat1f1o 22302 | There is a 1-1 function fr... |
mat1rhmval 22303 | The value of the ring homo... |
mat1rhmelval 22304 | The value of the ring homo... |
mat1rhmcl 22305 | The value of the ring homo... |
mat1f 22306 | There is a function from a... |
mat1ghm 22307 | There is a group homomorph... |
mat1mhm 22308 | There is a monoid homomorp... |
mat1rhm 22309 | There is a ring homomorphi... |
mat1rngiso 22310 | There is a ring isomorphis... |
mat1ric 22311 | A ring is isomorphic to th... |
dmatval 22316 | The set of ` N ` x ` N ` d... |
dmatel 22317 | A ` N ` x ` N ` diagonal m... |
dmatmat 22318 | An ` N ` x ` N ` diagonal ... |
dmatid 22319 | The identity matrix is a d... |
dmatelnd 22320 | An extradiagonal entry of ... |
dmatmul 22321 | The product of two diagona... |
dmatsubcl 22322 | The difference of two diag... |
dmatsgrp 22323 | The set of diagonal matric... |
dmatmulcl 22324 | The product of two diagona... |
dmatsrng 22325 | The set of diagonal matric... |
dmatcrng 22326 | The subring of diagonal ma... |
dmatscmcl 22327 | The multiplication of a di... |
scmatval 22328 | The set of ` N ` x ` N ` s... |
scmatel 22329 | An ` N ` x ` N ` scalar ma... |
scmatscmid 22330 | A scalar matrix can be exp... |
scmatscmide 22331 | An entry of a scalar matri... |
scmatscmiddistr 22332 | Distributive law for scala... |
scmatmat 22333 | An ` N ` x ` N ` scalar ma... |
scmate 22334 | An entry of an ` N ` x ` N... |
scmatmats 22335 | The set of an ` N ` x ` N ... |
scmateALT 22336 | Alternate proof of ~ scmat... |
scmatscm 22337 | The multiplication of a ma... |
scmatid 22338 | The identity matrix is a s... |
scmatdmat 22339 | A scalar matrix is a diago... |
scmataddcl 22340 | The sum of two scalar matr... |
scmatsubcl 22341 | The difference of two scal... |
scmatmulcl 22342 | The product of two scalar ... |
scmatsgrp 22343 | The set of scalar matrices... |
scmatsrng 22344 | The set of scalar matrices... |
scmatcrng 22345 | The subring of scalar matr... |
scmatsgrp1 22346 | The set of scalar matrices... |
scmatsrng1 22347 | The set of scalar matrices... |
smatvscl 22348 | Closure of the scalar mult... |
scmatlss 22349 | The set of scalar matrices... |
scmatstrbas 22350 | The set of scalar matrices... |
scmatrhmval 22351 | The value of the ring homo... |
scmatrhmcl 22352 | The value of the ring homo... |
scmatf 22353 | There is a function from a... |
scmatfo 22354 | There is a function from a... |
scmatf1 22355 | There is a 1-1 function fr... |
scmatf1o 22356 | There is a bijection betwe... |
scmatghm 22357 | There is a group homomorph... |
scmatmhm 22358 | There is a monoid homomorp... |
scmatrhm 22359 | There is a ring homomorphi... |
scmatrngiso 22360 | There is a ring isomorphis... |
scmatric 22361 | A ring is isomorphic to ev... |
mat0scmat 22362 | The empty matrix over a ri... |
mat1scmat 22363 | A 1-dimensional matrix ove... |
mvmulfval 22366 | Functional value of the ma... |
mvmulval 22367 | Multiplication of a vector... |
mvmulfv 22368 | A cell/element in the vect... |
mavmulval 22369 | Multiplication of a vector... |
mavmulfv 22370 | A cell/element in the vect... |
mavmulcl 22371 | Multiplication of an NxN m... |
1mavmul 22372 | Multiplication of the iden... |
mavmulass 22373 | Associativity of the multi... |
mavmuldm 22374 | The domain of the matrix v... |
mavmulsolcl 22375 | Every solution of the equa... |
mavmul0 22376 | Multiplication of a 0-dime... |
mavmul0g 22377 | The result of the 0-dimens... |
mvmumamul1 22378 | The multiplication of an M... |
mavmumamul1 22379 | The multiplication of an N... |
marrepfval 22384 | First substitution for the... |
marrepval0 22385 | Second substitution for th... |
marrepval 22386 | Third substitution for the... |
marrepeval 22387 | An entry of a matrix with ... |
marrepcl 22388 | Closure of the row replace... |
marepvfval 22389 | First substitution for the... |
marepvval0 22390 | Second substitution for th... |
marepvval 22391 | Third substitution for the... |
marepveval 22392 | An entry of a matrix with ... |
marepvcl 22393 | Closure of the column repl... |
ma1repvcl 22394 | Closure of the column repl... |
ma1repveval 22395 | An entry of an identity ma... |
mulmarep1el 22396 | Element by element multipl... |
mulmarep1gsum1 22397 | The sum of element by elem... |
mulmarep1gsum2 22398 | The sum of element by elem... |
1marepvmarrepid 22399 | Replacing the ith row by 0... |
submabas 22402 | Any subset of the index se... |
submafval 22403 | First substitution for a s... |
submaval0 22404 | Second substitution for a ... |
submaval 22405 | Third substitution for a s... |
submaeval 22406 | An entry of a submatrix of... |
1marepvsma1 22407 | The submatrix of the ident... |
mdetfval 22410 | First substitution for the... |
mdetleib 22411 | Full substitution of our d... |
mdetleib2 22412 | Leibniz' formula can also ... |
nfimdetndef 22413 | The determinant is not def... |
mdetfval1 22414 | First substitution of an a... |
mdetleib1 22415 | Full substitution of an al... |
mdet0pr 22416 | The determinant function f... |
mdet0f1o 22417 | The determinant function f... |
mdet0fv0 22418 | The determinant of the emp... |
mdetf 22419 | Functionality of the deter... |
mdetcl 22420 | The determinant evaluates ... |
m1detdiag 22421 | The determinant of a 1-dim... |
mdetdiaglem 22422 | Lemma for ~ mdetdiag . Pr... |
mdetdiag 22423 | The determinant of a diago... |
mdetdiagid 22424 | The determinant of a diago... |
mdet1 22425 | The determinant of the ide... |
mdetrlin 22426 | The determinant function i... |
mdetrsca 22427 | The determinant function i... |
mdetrsca2 22428 | The determinant function i... |
mdetr0 22429 | The determinant of a matri... |
mdet0 22430 | The determinant of the zer... |
mdetrlin2 22431 | The determinant function i... |
mdetralt 22432 | The determinant function i... |
mdetralt2 22433 | The determinant function i... |
mdetero 22434 | The determinant function i... |
mdettpos 22435 | Determinant is invariant u... |
mdetunilem1 22436 | Lemma for ~ mdetuni . (Co... |
mdetunilem2 22437 | Lemma for ~ mdetuni . (Co... |
mdetunilem3 22438 | Lemma for ~ mdetuni . (Co... |
mdetunilem4 22439 | Lemma for ~ mdetuni . (Co... |
mdetunilem5 22440 | Lemma for ~ mdetuni . (Co... |
mdetunilem6 22441 | Lemma for ~ mdetuni . (Co... |
mdetunilem7 22442 | Lemma for ~ mdetuni . (Co... |
mdetunilem8 22443 | Lemma for ~ mdetuni . (Co... |
mdetunilem9 22444 | Lemma for ~ mdetuni . (Co... |
mdetuni0 22445 | Lemma for ~ mdetuni . (Co... |
mdetuni 22446 | According to the definitio... |
mdetmul 22447 | Multiplicativity of the de... |
m2detleiblem1 22448 | Lemma 1 for ~ m2detleib . ... |
m2detleiblem5 22449 | Lemma 5 for ~ m2detleib . ... |
m2detleiblem6 22450 | Lemma 6 for ~ m2detleib . ... |
m2detleiblem7 22451 | Lemma 7 for ~ m2detleib . ... |
m2detleiblem2 22452 | Lemma 2 for ~ m2detleib . ... |
m2detleiblem3 22453 | Lemma 3 for ~ m2detleib . ... |
m2detleiblem4 22454 | Lemma 4 for ~ m2detleib . ... |
m2detleib 22455 | Leibniz' Formula for 2x2-m... |
mndifsplit 22460 | Lemma for ~ maducoeval2 . ... |
madufval 22461 | First substitution for the... |
maduval 22462 | Second substitution for th... |
maducoeval 22463 | An entry of the adjunct (c... |
maducoeval2 22464 | An entry of the adjunct (c... |
maduf 22465 | Creating the adjunct of ma... |
madutpos 22466 | The adjuct of a transposed... |
madugsum 22467 | The determinant of a matri... |
madurid 22468 | Multiplying a matrix with ... |
madulid 22469 | Multiplying the adjunct of... |
minmar1fval 22470 | First substitution for the... |
minmar1val0 22471 | Second substitution for th... |
minmar1val 22472 | Third substitution for the... |
minmar1eval 22473 | An entry of a matrix for a... |
minmar1marrep 22474 | The minor matrix is a spec... |
minmar1cl 22475 | Closure of the row replace... |
maducoevalmin1 22476 | The coefficients of an adj... |
symgmatr01lem 22477 | Lemma for ~ symgmatr01 . ... |
symgmatr01 22478 | Applying a permutation tha... |
gsummatr01lem1 22479 | Lemma A for ~ gsummatr01 .... |
gsummatr01lem2 22480 | Lemma B for ~ gsummatr01 .... |
gsummatr01lem3 22481 | Lemma 1 for ~ gsummatr01 .... |
gsummatr01lem4 22482 | Lemma 2 for ~ gsummatr01 .... |
gsummatr01 22483 | Lemma 1 for ~ smadiadetlem... |
marep01ma 22484 | Replacing a row of a squar... |
smadiadetlem0 22485 | Lemma 0 for ~ smadiadet : ... |
smadiadetlem1 22486 | Lemma 1 for ~ smadiadet : ... |
smadiadetlem1a 22487 | Lemma 1a for ~ smadiadet :... |
smadiadetlem2 22488 | Lemma 2 for ~ smadiadet : ... |
smadiadetlem3lem0 22489 | Lemma 0 for ~ smadiadetlem... |
smadiadetlem3lem1 22490 | Lemma 1 for ~ smadiadetlem... |
smadiadetlem3lem2 22491 | Lemma 2 for ~ smadiadetlem... |
smadiadetlem3 22492 | Lemma 3 for ~ smadiadet . ... |
smadiadetlem4 22493 | Lemma 4 for ~ smadiadet . ... |
smadiadet 22494 | The determinant of a subma... |
smadiadetglem1 22495 | Lemma 1 for ~ smadiadetg .... |
smadiadetglem2 22496 | Lemma 2 for ~ smadiadetg .... |
smadiadetg 22497 | The determinant of a squar... |
smadiadetg0 22498 | Lemma for ~ smadiadetr : v... |
smadiadetr 22499 | The determinant of a squar... |
invrvald 22500 | If a matrix multiplied wit... |
matinv 22501 | The inverse of a matrix is... |
matunit 22502 | A matrix is a unit in the ... |
slesolvec 22503 | Every solution of a system... |
slesolinv 22504 | The solution of a system o... |
slesolinvbi 22505 | The solution of a system o... |
slesolex 22506 | Every system of linear equ... |
cramerimplem1 22507 | Lemma 1 for ~ cramerimp : ... |
cramerimplem2 22508 | Lemma 2 for ~ cramerimp : ... |
cramerimplem3 22509 | Lemma 3 for ~ cramerimp : ... |
cramerimp 22510 | One direction of Cramer's ... |
cramerlem1 22511 | Lemma 1 for ~ cramer . (C... |
cramerlem2 22512 | Lemma 2 for ~ cramer . (C... |
cramerlem3 22513 | Lemma 3 for ~ cramer . (C... |
cramer0 22514 | Special case of Cramer's r... |
cramer 22515 | Cramer's rule. According ... |
pmatring 22516 | The set of polynomial matr... |
pmatlmod 22517 | The set of polynomial matr... |
pmatassa 22518 | The set of polynomial matr... |
pmat0op 22519 | The zero polynomial matrix... |
pmat1op 22520 | The identity polynomial ma... |
pmat1ovd 22521 | Entries of the identity po... |
pmat0opsc 22522 | The zero polynomial matrix... |
pmat1opsc 22523 | The identity polynomial ma... |
pmat1ovscd 22524 | Entries of the identity po... |
pmatcoe1fsupp 22525 | For a polynomial matrix th... |
1pmatscmul 22526 | The scalar product of the ... |
cpmat 22533 | Value of the constructor o... |
cpmatpmat 22534 | A constant polynomial matr... |
cpmatel 22535 | Property of a constant pol... |
cpmatelimp 22536 | Implication of a set being... |
cpmatel2 22537 | Another property of a cons... |
cpmatelimp2 22538 | Another implication of a s... |
1elcpmat 22539 | The identity of the ring o... |
cpmatacl 22540 | The set of all constant po... |
cpmatinvcl 22541 | The set of all constant po... |
cpmatmcllem 22542 | Lemma for ~ cpmatmcl . (C... |
cpmatmcl 22543 | The set of all constant po... |
cpmatsubgpmat 22544 | The set of all constant po... |
cpmatsrgpmat 22545 | The set of all constant po... |
0elcpmat 22546 | The zero of the ring of al... |
mat2pmatfval 22547 | Value of the matrix transf... |
mat2pmatval 22548 | The result of a matrix tra... |
mat2pmatvalel 22549 | A (matrix) element of the ... |
mat2pmatbas 22550 | The result of a matrix tra... |
mat2pmatbas0 22551 | The result of a matrix tra... |
mat2pmatf 22552 | The matrix transformation ... |
mat2pmatf1 22553 | The matrix transformation ... |
mat2pmatghm 22554 | The transformation of matr... |
mat2pmatmul 22555 | The transformation of matr... |
mat2pmat1 22556 | The transformation of the ... |
mat2pmatmhm 22557 | The transformation of matr... |
mat2pmatrhm 22558 | The transformation of matr... |
mat2pmatlin 22559 | The transformation of matr... |
0mat2pmat 22560 | The transformed zero matri... |
idmatidpmat 22561 | The transformed identity m... |
d0mat2pmat 22562 | The transformed empty set ... |
d1mat2pmat 22563 | The transformation of a ma... |
mat2pmatscmxcl 22564 | A transformed matrix multi... |
m2cpm 22565 | The result of a matrix tra... |
m2cpmf 22566 | The matrix transformation ... |
m2cpmf1 22567 | The matrix transformation ... |
m2cpmghm 22568 | The transformation of matr... |
m2cpmmhm 22569 | The transformation of matr... |
m2cpmrhm 22570 | The transformation of matr... |
m2pmfzmap 22571 | The transformed values of ... |
m2pmfzgsumcl 22572 | Closure of the sum of scal... |
cpm2mfval 22573 | Value of the inverse matri... |
cpm2mval 22574 | The result of an inverse m... |
cpm2mvalel 22575 | A (matrix) element of the ... |
cpm2mf 22576 | The inverse matrix transfo... |
m2cpminvid 22577 | The inverse transformation... |
m2cpminvid2lem 22578 | Lemma for ~ m2cpminvid2 . ... |
m2cpminvid2 22579 | The transformation applied... |
m2cpmfo 22580 | The matrix transformation ... |
m2cpmf1o 22581 | The matrix transformation ... |
m2cpmrngiso 22582 | The transformation of matr... |
matcpmric 22583 | The ring of matrices over ... |
m2cpminv 22584 | The inverse matrix transfo... |
m2cpminv0 22585 | The inverse matrix transfo... |
decpmatval0 22588 | The matrix consisting of t... |
decpmatval 22589 | The matrix consisting of t... |
decpmate 22590 | An entry of the matrix con... |
decpmatcl 22591 | Closure of the decompositi... |
decpmataa0 22592 | The matrix consisting of t... |
decpmatfsupp 22593 | The mapping to the matrice... |
decpmatid 22594 | The matrix consisting of t... |
decpmatmullem 22595 | Lemma for ~ decpmatmul . ... |
decpmatmul 22596 | The matrix consisting of t... |
decpmatmulsumfsupp 22597 | Lemma 0 for ~ pm2mpmhm . ... |
pmatcollpw1lem1 22598 | Lemma 1 for ~ pmatcollpw1 ... |
pmatcollpw1lem2 22599 | Lemma 2 for ~ pmatcollpw1 ... |
pmatcollpw1 22600 | Write a polynomial matrix ... |
pmatcollpw2lem 22601 | Lemma for ~ pmatcollpw2 . ... |
pmatcollpw2 22602 | Write a polynomial matrix ... |
monmatcollpw 22603 | The matrix consisting of t... |
pmatcollpwlem 22604 | Lemma for ~ pmatcollpw . ... |
pmatcollpw 22605 | Write a polynomial matrix ... |
pmatcollpwfi 22606 | Write a polynomial matrix ... |
pmatcollpw3lem 22607 | Lemma for ~ pmatcollpw3 an... |
pmatcollpw3 22608 | Write a polynomial matrix ... |
pmatcollpw3fi 22609 | Write a polynomial matrix ... |
pmatcollpw3fi1lem1 22610 | Lemma 1 for ~ pmatcollpw3f... |
pmatcollpw3fi1lem2 22611 | Lemma 2 for ~ pmatcollpw3f... |
pmatcollpw3fi1 22612 | Write a polynomial matrix ... |
pmatcollpwscmatlem1 22613 | Lemma 1 for ~ pmatcollpwsc... |
pmatcollpwscmatlem2 22614 | Lemma 2 for ~ pmatcollpwsc... |
pmatcollpwscmat 22615 | Write a scalar matrix over... |
pm2mpf1lem 22618 | Lemma for ~ pm2mpf1 . (Co... |
pm2mpval 22619 | Value of the transformatio... |
pm2mpfval 22620 | A polynomial matrix transf... |
pm2mpcl 22621 | The transformation of poly... |
pm2mpf 22622 | The transformation of poly... |
pm2mpf1 22623 | The transformation of poly... |
pm2mpcoe1 22624 | A coefficient of the polyn... |
idpm2idmp 22625 | The transformation of the ... |
mptcoe1matfsupp 22626 | The mapping extracting the... |
mply1topmatcllem 22627 | Lemma for ~ mply1topmatcl ... |
mply1topmatval 22628 | A polynomial over matrices... |
mply1topmatcl 22629 | A polynomial over matrices... |
mp2pm2mplem1 22630 | Lemma 1 for ~ mp2pm2mp . ... |
mp2pm2mplem2 22631 | Lemma 2 for ~ mp2pm2mp . ... |
mp2pm2mplem3 22632 | Lemma 3 for ~ mp2pm2mp . ... |
mp2pm2mplem4 22633 | Lemma 4 for ~ mp2pm2mp . ... |
mp2pm2mplem5 22634 | Lemma 5 for ~ mp2pm2mp . ... |
mp2pm2mp 22635 | A polynomial over matrices... |
pm2mpghmlem2 22636 | Lemma 2 for ~ pm2mpghm . ... |
pm2mpghmlem1 22637 | Lemma 1 for pm2mpghm . (C... |
pm2mpfo 22638 | The transformation of poly... |
pm2mpf1o 22639 | The transformation of poly... |
pm2mpghm 22640 | The transformation of poly... |
pm2mpgrpiso 22641 | The transformation of poly... |
pm2mpmhmlem1 22642 | Lemma 1 for ~ pm2mpmhm . ... |
pm2mpmhmlem2 22643 | Lemma 2 for ~ pm2mpmhm . ... |
pm2mpmhm 22644 | The transformation of poly... |
pm2mprhm 22645 | The transformation of poly... |
pm2mprngiso 22646 | The transformation of poly... |
pmmpric 22647 | The ring of polynomial mat... |
monmat2matmon 22648 | The transformation of a po... |
pm2mp 22649 | The transformation of a su... |
chmatcl 22652 | Closure of the characteris... |
chmatval 22653 | The entries of the charact... |
chpmatfval 22654 | Value of the characteristi... |
chpmatval 22655 | The characteristic polynom... |
chpmatply1 22656 | The characteristic polynom... |
chpmatval2 22657 | The characteristic polynom... |
chpmat0d 22658 | The characteristic polynom... |
chpmat1dlem 22659 | Lemma for ~ chpmat1d . (C... |
chpmat1d 22660 | The characteristic polynom... |
chpdmatlem0 22661 | Lemma 0 for ~ chpdmat . (... |
chpdmatlem1 22662 | Lemma 1 for ~ chpdmat . (... |
chpdmatlem2 22663 | Lemma 2 for ~ chpdmat . (... |
chpdmatlem3 22664 | Lemma 3 for ~ chpdmat . (... |
chpdmat 22665 | The characteristic polynom... |
chpscmat 22666 | The characteristic polynom... |
chpscmat0 22667 | The characteristic polynom... |
chpscmatgsumbin 22668 | The characteristic polynom... |
chpscmatgsummon 22669 | The characteristic polynom... |
chp0mat 22670 | The characteristic polynom... |
chpidmat 22671 | The characteristic polynom... |
chmaidscmat 22672 | The characteristic polynom... |
fvmptnn04if 22673 | The function values of a m... |
fvmptnn04ifa 22674 | The function value of a ma... |
fvmptnn04ifb 22675 | The function value of a ma... |
fvmptnn04ifc 22676 | The function value of a ma... |
fvmptnn04ifd 22677 | The function value of a ma... |
chfacfisf 22678 | The "characteristic factor... |
chfacfisfcpmat 22679 | The "characteristic factor... |
chfacffsupp 22680 | The "characteristic factor... |
chfacfscmulcl 22681 | Closure of a scaled value ... |
chfacfscmul0 22682 | A scaled value of the "cha... |
chfacfscmulfsupp 22683 | A mapping of scaled values... |
chfacfscmulgsum 22684 | Breaking up a sum of value... |
chfacfpmmulcl 22685 | Closure of the value of th... |
chfacfpmmul0 22686 | The value of the "characte... |
chfacfpmmulfsupp 22687 | A mapping of values of the... |
chfacfpmmulgsum 22688 | Breaking up a sum of value... |
chfacfpmmulgsum2 22689 | Breaking up a sum of value... |
cayhamlem1 22690 | Lemma 1 for ~ cayleyhamilt... |
cpmadurid 22691 | The right-hand fundamental... |
cpmidgsum 22692 | Representation of the iden... |
cpmidgsumm2pm 22693 | Representation of the iden... |
cpmidpmatlem1 22694 | Lemma 1 for ~ cpmidpmat . ... |
cpmidpmatlem2 22695 | Lemma 2 for ~ cpmidpmat . ... |
cpmidpmatlem3 22696 | Lemma 3 for ~ cpmidpmat . ... |
cpmidpmat 22697 | Representation of the iden... |
cpmadugsumlemB 22698 | Lemma B for ~ cpmadugsum .... |
cpmadugsumlemC 22699 | Lemma C for ~ cpmadugsum .... |
cpmadugsumlemF 22700 | Lemma F for ~ cpmadugsum .... |
cpmadugsumfi 22701 | The product of the charact... |
cpmadugsum 22702 | The product of the charact... |
cpmidgsum2 22703 | Representation of the iden... |
cpmidg2sum 22704 | Equality of two sums repre... |
cpmadumatpolylem1 22705 | Lemma 1 for ~ cpmadumatpol... |
cpmadumatpolylem2 22706 | Lemma 2 for ~ cpmadumatpol... |
cpmadumatpoly 22707 | The product of the charact... |
cayhamlem2 22708 | Lemma for ~ cayhamlem3 . ... |
chcoeffeqlem 22709 | Lemma for ~ chcoeffeq . (... |
chcoeffeq 22710 | The coefficients of the ch... |
cayhamlem3 22711 | Lemma for ~ cayhamlem4 . ... |
cayhamlem4 22712 | Lemma for ~ cayleyhamilton... |
cayleyhamilton0 22713 | The Cayley-Hamilton theore... |
cayleyhamilton 22714 | The Cayley-Hamilton theore... |
cayleyhamiltonALT 22715 | Alternate proof of ~ cayle... |
cayleyhamilton1 22716 | The Cayley-Hamilton theore... |
istopg 22719 | Express the predicate " ` ... |
istop2g 22720 | Express the predicate " ` ... |
uniopn 22721 | The union of a subset of a... |
iunopn 22722 | The indexed union of a sub... |
inopn 22723 | The intersection of two op... |
fitop 22724 | A topology is closed under... |
fiinopn 22725 | The intersection of a none... |
iinopn 22726 | The intersection of a none... |
unopn 22727 | The union of two open sets... |
0opn 22728 | The empty set is an open s... |
0ntop 22729 | The empty set is not a top... |
topopn 22730 | The underlying set of a to... |
eltopss 22731 | A member of a topology is ... |
riinopn 22732 | A finite indexed relative ... |
rintopn 22733 | A finite relative intersec... |
istopon 22736 | Property of being a topolo... |
topontop 22737 | A topology on a given base... |
toponuni 22738 | The base set of a topology... |
topontopi 22739 | A topology on a given base... |
toponunii 22740 | The base set of a topology... |
toptopon 22741 | Alternative definition of ... |
toptopon2 22742 | A topology is the same thi... |
topontopon 22743 | A topology on a set is a t... |
funtopon 22744 | The class ` TopOn ` is a f... |
toponrestid 22745 | Given a topology on a set,... |
toponsspwpw 22746 | The set of topologies on a... |
dmtopon 22747 | The domain of ` TopOn ` is... |
fntopon 22748 | The class ` TopOn ` is a f... |
toprntopon 22749 | A topology is the same thi... |
toponmax 22750 | The base set of a topology... |
toponss 22751 | A member of a topology is ... |
toponcom 22752 | If ` K ` is a topology on ... |
toponcomb 22753 | Biconditional form of ~ to... |
topgele 22754 | The topologies over the sa... |
topsn 22755 | The only topology on a sin... |
istps 22758 | Express the predicate "is ... |
istps2 22759 | Express the predicate "is ... |
tpsuni 22760 | The base set of a topologi... |
tpstop 22761 | The topology extractor on ... |
tpspropd 22762 | A topological space depend... |
tpsprop2d 22763 | A topological space depend... |
topontopn 22764 | Express the predicate "is ... |
tsettps 22765 | If the topology component ... |
istpsi 22766 | Properties that determine ... |
eltpsg 22767 | Properties that determine ... |
eltpsgOLD 22768 | Obsolete version of ~ eltp... |
eltpsi 22769 | Properties that determine ... |
isbasisg 22772 | Express the predicate "the... |
isbasis2g 22773 | Express the predicate "the... |
isbasis3g 22774 | Express the predicate "the... |
basis1 22775 | Property of a basis. (Con... |
basis2 22776 | Property of a basis. (Con... |
fiinbas 22777 | If a set is closed under f... |
basdif0 22778 | A basis is not affected by... |
baspartn 22779 | A disjoint system of sets ... |
tgval 22780 | The topology generated by ... |
tgval2 22781 | Definition of a topology g... |
eltg 22782 | Membership in a topology g... |
eltg2 22783 | Membership in a topology g... |
eltg2b 22784 | Membership in a topology g... |
eltg4i 22785 | An open set in a topology ... |
eltg3i 22786 | The union of a set of basi... |
eltg3 22787 | Membership in a topology g... |
tgval3 22788 | Alternate expression for t... |
tg1 22789 | Property of a member of a ... |
tg2 22790 | Property of a member of a ... |
bastg 22791 | A member of a basis is a s... |
unitg 22792 | The topology generated by ... |
tgss 22793 | Subset relation for genera... |
tgcl 22794 | Show that a basis generate... |
tgclb 22795 | The property ~ tgcl can be... |
tgtopon 22796 | A basis generates a topolo... |
topbas 22797 | A topology is its own basi... |
tgtop 22798 | A topology is its own basi... |
eltop 22799 | Membership in a topology, ... |
eltop2 22800 | Membership in a topology. ... |
eltop3 22801 | Membership in a topology. ... |
fibas 22802 | A collection of finite int... |
tgdom 22803 | A space has no more open s... |
tgiun 22804 | The indexed union of a set... |
tgidm 22805 | The topology generator fun... |
bastop 22806 | Two ways to express that a... |
tgtop11 22807 | The topology generation fu... |
0top 22808 | The singleton of the empty... |
en1top 22809 | ` { (/) } ` is the only to... |
en2top 22810 | If a topology has two elem... |
tgss3 22811 | A criterion for determinin... |
tgss2 22812 | A criterion for determinin... |
basgen 22813 | Given a topology ` J ` , s... |
basgen2 22814 | Given a topology ` J ` , s... |
2basgen 22815 | Conditions that determine ... |
tgfiss 22816 | If a subbase is included i... |
tgdif0 22817 | A generated topology is no... |
bastop1 22818 | A subset of a topology is ... |
bastop2 22819 | A version of ~ bastop1 tha... |
distop 22820 | The discrete topology on a... |
topnex 22821 | The class of all topologie... |
distopon 22822 | The discrete topology on a... |
sn0topon 22823 | The singleton of the empty... |
sn0top 22824 | The singleton of the empty... |
indislem 22825 | A lemma to eliminate some ... |
indistopon 22826 | The indiscrete topology on... |
indistop 22827 | The indiscrete topology on... |
indisuni 22828 | The base set of the indisc... |
fctop 22829 | The finite complement topo... |
fctop2 22830 | The finite complement topo... |
cctop 22831 | The countable complement t... |
ppttop 22832 | The particular point topol... |
pptbas 22833 | The particular point topol... |
epttop 22834 | The excluded point topolog... |
indistpsx 22835 | The indiscrete topology on... |
indistps 22836 | The indiscrete topology on... |
indistps2 22837 | The indiscrete topology on... |
indistpsALT 22838 | The indiscrete topology on... |
indistpsALTOLD 22839 | Obsolete version of ~ indi... |
indistps2ALT 22840 | The indiscrete topology on... |
distps 22841 | The discrete topology on a... |
fncld 22848 | The closed-set generator i... |
cldval 22849 | The set of closed sets of ... |
ntrfval 22850 | The interior function on t... |
clsfval 22851 | The closure function on th... |
cldrcl 22852 | Reverse closure of the clo... |
iscld 22853 | The predicate "the class `... |
iscld2 22854 | A subset of the underlying... |
cldss 22855 | A closed set is a subset o... |
cldss2 22856 | The set of closed sets is ... |
cldopn 22857 | The complement of a closed... |
isopn2 22858 | A subset of the underlying... |
opncld 22859 | The complement of an open ... |
difopn 22860 | The difference of a closed... |
topcld 22861 | The underlying set of a to... |
ntrval 22862 | The interior of a subset o... |
clsval 22863 | The closure of a subset of... |
0cld 22864 | The empty set is closed. ... |
iincld 22865 | The indexed intersection o... |
intcld 22866 | The intersection of a set ... |
uncld 22867 | The union of two closed se... |
cldcls 22868 | A closed subset equals its... |
incld 22869 | The intersection of two cl... |
riincld 22870 | An indexed relative inters... |
iuncld 22871 | A finite indexed union of ... |
unicld 22872 | A finite union of closed s... |
clscld 22873 | The closure of a subset of... |
clsf 22874 | The closure function is a ... |
ntropn 22875 | The interior of a subset o... |
clsval2 22876 | Express closure in terms o... |
ntrval2 22877 | Interior expressed in term... |
ntrdif 22878 | An interior of a complemen... |
clsdif 22879 | A closure of a complement ... |
clsss 22880 | Subset relationship for cl... |
ntrss 22881 | Subset relationship for in... |
sscls 22882 | A subset of a topology's u... |
ntrss2 22883 | A subset includes its inte... |
ssntr 22884 | An open subset of a set is... |
clsss3 22885 | The closure of a subset of... |
ntrss3 22886 | The interior of a subset o... |
ntrin 22887 | A pairwise intersection of... |
cmclsopn 22888 | The complement of a closur... |
cmntrcld 22889 | The complement of an inter... |
iscld3 22890 | A subset is closed iff it ... |
iscld4 22891 | A subset is closed iff it ... |
isopn3 22892 | A subset is open iff it eq... |
clsidm 22893 | The closure operation is i... |
ntridm 22894 | The interior operation is ... |
clstop 22895 | The closure of a topology'... |
ntrtop 22896 | The interior of a topology... |
0ntr 22897 | A subset with an empty int... |
clsss2 22898 | If a subset is included in... |
elcls 22899 | Membership in a closure. ... |
elcls2 22900 | Membership in a closure. ... |
clsndisj 22901 | Any open set containing a ... |
ntrcls0 22902 | A subset whose closure has... |
ntreq0 22903 | Two ways to say that a sub... |
cldmre 22904 | The closed sets of a topol... |
mrccls 22905 | Moore closure generalizes ... |
cls0 22906 | The closure of the empty s... |
ntr0 22907 | The interior of the empty ... |
isopn3i 22908 | An open subset equals its ... |
elcls3 22909 | Membership in a closure in... |
opncldf1 22910 | A bijection useful for con... |
opncldf2 22911 | The values of the open-clo... |
opncldf3 22912 | The values of the converse... |
isclo 22913 | A set ` A ` is clopen iff ... |
isclo2 22914 | A set ` A ` is clopen iff ... |
discld 22915 | The open sets of a discret... |
sn0cld 22916 | The closed sets of the top... |
indiscld 22917 | The closed sets of an indi... |
mretopd 22918 | A Moore collection which i... |
toponmre 22919 | The topologies over a give... |
cldmreon 22920 | The closed sets of a topol... |
iscldtop 22921 | A family is the closed set... |
mreclatdemoBAD 22922 | The closed subspaces of a ... |
neifval 22925 | Value of the neighborhood ... |
neif 22926 | The neighborhood function ... |
neiss2 22927 | A set with a neighborhood ... |
neival 22928 | Value of the set of neighb... |
isnei 22929 | The predicate "the class `... |
neiint 22930 | An intuitive definition of... |
isneip 22931 | The predicate "the class `... |
neii1 22932 | A neighborhood is included... |
neisspw 22933 | The neighborhoods of any s... |
neii2 22934 | Property of a neighborhood... |
neiss 22935 | Any neighborhood of a set ... |
ssnei 22936 | A set is included in any o... |
elnei 22937 | A point belongs to any of ... |
0nnei 22938 | The empty set is not a nei... |
neips 22939 | A neighborhood of a set is... |
opnneissb 22940 | An open set is a neighborh... |
opnssneib 22941 | Any superset of an open se... |
ssnei2 22942 | Any subset ` M ` of ` X ` ... |
neindisj 22943 | Any neighborhood of an ele... |
opnneiss 22944 | An open set is a neighborh... |
opnneip 22945 | An open set is a neighborh... |
opnnei 22946 | A set is open iff it is a ... |
tpnei 22947 | The underlying set of a to... |
neiuni 22948 | The union of the neighborh... |
neindisj2 22949 | A point ` P ` belongs to t... |
topssnei 22950 | A finer topology has more ... |
innei 22951 | The intersection of two ne... |
opnneiid 22952 | Only an open set is a neig... |
neissex 22953 | For any neighborhood ` N `... |
0nei 22954 | The empty set is a neighbo... |
neipeltop 22955 | Lemma for ~ neiptopreu . ... |
neiptopuni 22956 | Lemma for ~ neiptopreu . ... |
neiptoptop 22957 | Lemma for ~ neiptopreu . ... |
neiptopnei 22958 | Lemma for ~ neiptopreu . ... |
neiptopreu 22959 | If, to each element ` P ` ... |
lpfval 22964 | The limit point function o... |
lpval 22965 | The set of limit points of... |
islp 22966 | The predicate "the class `... |
lpsscls 22967 | The limit points of a subs... |
lpss 22968 | The limit points of a subs... |
lpdifsn 22969 | ` P ` is a limit point of ... |
lpss3 22970 | Subset relationship for li... |
islp2 22971 | The predicate " ` P ` is a... |
islp3 22972 | The predicate " ` P ` is a... |
maxlp 22973 | A point is a limit point o... |
clslp 22974 | The closure of a subset of... |
islpi 22975 | A point belonging to a set... |
cldlp 22976 | A subset of a topological ... |
isperf 22977 | Definition of a perfect sp... |
isperf2 22978 | Definition of a perfect sp... |
isperf3 22979 | A perfect space is a topol... |
perflp 22980 | The limit points of a perf... |
perfi 22981 | Property of a perfect spac... |
perftop 22982 | A perfect space is a topol... |
restrcl 22983 | Reverse closure for the su... |
restbas 22984 | A subspace topology basis ... |
tgrest 22985 | A subspace can be generate... |
resttop 22986 | A subspace topology is a t... |
resttopon 22987 | A subspace topology is a t... |
restuni 22988 | The underlying set of a su... |
stoig 22989 | The topological space buil... |
restco 22990 | Composition of subspaces. ... |
restabs 22991 | Equivalence of being a sub... |
restin 22992 | When the subspace region i... |
restuni2 22993 | The underlying set of a su... |
resttopon2 22994 | The underlying set of a su... |
rest0 22995 | The subspace topology indu... |
restsn 22996 | The only subspace topology... |
restsn2 22997 | The subspace topology indu... |
restcld 22998 | A closed set of a subspace... |
restcldi 22999 | A closed set is closed in ... |
restcldr 23000 | A set which is closed in t... |
restopnb 23001 | If ` B ` is an open subset... |
ssrest 23002 | If ` K ` is a finer topolo... |
restopn2 23003 | If ` A ` is open, then ` B... |
restdis 23004 | A subspace of a discrete t... |
restfpw 23005 | The restriction of the set... |
neitr 23006 | The neighborhood of a trac... |
restcls 23007 | A closure in a subspace to... |
restntr 23008 | An interior in a subspace ... |
restlp 23009 | The limit points of a subs... |
restperf 23010 | Perfection of a subspace. ... |
perfopn 23011 | An open subset of a perfec... |
resstopn 23012 | The topology of a restrict... |
resstps 23013 | A restricted topological s... |
ordtbaslem 23014 | Lemma for ~ ordtbas . In ... |
ordtval 23015 | Value of the order topolog... |
ordtuni 23016 | Value of the order topolog... |
ordtbas2 23017 | Lemma for ~ ordtbas . (Co... |
ordtbas 23018 | In a total order, the fini... |
ordttopon 23019 | Value of the order topolog... |
ordtopn1 23020 | An upward ray ` ( P , +oo ... |
ordtopn2 23021 | A downward ray ` ( -oo , P... |
ordtopn3 23022 | An open interval ` ( A , B... |
ordtcld1 23023 | A downward ray ` ( -oo , P... |
ordtcld2 23024 | An upward ray ` [ P , +oo ... |
ordtcld3 23025 | A closed interval ` [ A , ... |
ordttop 23026 | The order topology is a to... |
ordtcnv 23027 | The order dual generates t... |
ordtrest 23028 | The subspace topology of a... |
ordtrest2lem 23029 | Lemma for ~ ordtrest2 . (... |
ordtrest2 23030 | An interval-closed set ` A... |
letopon 23031 | The topology of the extend... |
letop 23032 | The topology of the extend... |
letopuni 23033 | The topology of the extend... |
xrstopn 23034 | The topology component of ... |
xrstps 23035 | The extended real number s... |
leordtvallem1 23036 | Lemma for ~ leordtval . (... |
leordtvallem2 23037 | Lemma for ~ leordtval . (... |
leordtval2 23038 | The topology of the extend... |
leordtval 23039 | The topology of the extend... |
iccordt 23040 | A closed interval is close... |
iocpnfordt 23041 | An unbounded above open in... |
icomnfordt 23042 | An unbounded above open in... |
iooordt 23043 | An open interval is open i... |
reordt 23044 | The real numbers are an op... |
lecldbas 23045 | The set of closed interval... |
pnfnei 23046 | A neighborhood of ` +oo ` ... |
mnfnei 23047 | A neighborhood of ` -oo ` ... |
ordtrestixx 23048 | The restriction of the les... |
ordtresticc 23049 | The restriction of the les... |
lmrel 23056 | The topological space conv... |
lmrcl 23057 | Reverse closure for the co... |
lmfval 23058 | The relation "sequence ` f... |
cnfval 23059 | The set of all continuous ... |
cnpfval 23060 | The function mapping the p... |
iscn 23061 | The predicate "the class `... |
cnpval 23062 | The set of all functions f... |
iscnp 23063 | The predicate "the class `... |
iscn2 23064 | The predicate "the class `... |
iscnp2 23065 | The predicate "the class `... |
cntop1 23066 | Reverse closure for a cont... |
cntop2 23067 | Reverse closure for a cont... |
cnptop1 23068 | Reverse closure for a func... |
cnptop2 23069 | Reverse closure for a func... |
iscnp3 23070 | The predicate "the class `... |
cnprcl 23071 | Reverse closure for a func... |
cnf 23072 | A continuous function is a... |
cnpf 23073 | A continuous function at p... |
cnpcl 23074 | The value of a continuous ... |
cnf2 23075 | A continuous function is a... |
cnpf2 23076 | A continuous function at p... |
cnprcl2 23077 | Reverse closure for a func... |
tgcn 23078 | The continuity predicate w... |
tgcnp 23079 | The "continuous at a point... |
subbascn 23080 | The continuity predicate w... |
ssidcn 23081 | The identity function is a... |
cnpimaex 23082 | Property of a function con... |
idcn 23083 | A restricted identity func... |
lmbr 23084 | Express the binary relatio... |
lmbr2 23085 | Express the binary relatio... |
lmbrf 23086 | Express the binary relatio... |
lmconst 23087 | A constant sequence conver... |
lmcvg 23088 | Convergence property of a ... |
iscnp4 23089 | The predicate "the class `... |
cnpnei 23090 | A condition for continuity... |
cnima 23091 | An open subset of the codo... |
cnco 23092 | The composition of two con... |
cnpco 23093 | The composition of a funct... |
cnclima 23094 | A closed subset of the cod... |
iscncl 23095 | A characterization of a co... |
cncls2i 23096 | Property of the preimage o... |
cnntri 23097 | Property of the preimage o... |
cnclsi 23098 | Property of the image of a... |
cncls2 23099 | Continuity in terms of clo... |
cncls 23100 | Continuity in terms of clo... |
cnntr 23101 | Continuity in terms of int... |
cnss1 23102 | If the topology ` K ` is f... |
cnss2 23103 | If the topology ` K ` is f... |
cncnpi 23104 | A continuous function is c... |
cnsscnp 23105 | The set of continuous func... |
cncnp 23106 | A continuous function is c... |
cncnp2 23107 | A continuous function is c... |
cnnei 23108 | Continuity in terms of nei... |
cnconst2 23109 | A constant function is con... |
cnconst 23110 | A constant function is con... |
cnrest 23111 | Continuity of a restrictio... |
cnrest2 23112 | Equivalence of continuity ... |
cnrest2r 23113 | Equivalence of continuity ... |
cnpresti 23114 | One direction of ~ cnprest... |
cnprest 23115 | Equivalence of continuity ... |
cnprest2 23116 | Equivalence of point-conti... |
cndis 23117 | Every function is continuo... |
cnindis 23118 | Every function is continuo... |
cnpdis 23119 | If ` A ` is an isolated po... |
paste 23120 | Pasting lemma. If ` A ` a... |
lmfpm 23121 | If ` F ` converges, then `... |
lmfss 23122 | Inclusion of a function ha... |
lmcl 23123 | Closure of a limit. (Cont... |
lmss 23124 | Limit on a subspace. (Con... |
sslm 23125 | A finer topology has fewer... |
lmres 23126 | A function converges iff i... |
lmff 23127 | If ` F ` converges, there ... |
lmcls 23128 | Any convergent sequence of... |
lmcld 23129 | Any convergent sequence of... |
lmcnp 23130 | The image of a convergent ... |
lmcn 23131 | The image of a convergent ... |
ist0 23146 | The predicate "is a T_0 sp... |
ist1 23147 | The predicate "is a T_1 sp... |
ishaus 23148 | The predicate "is a Hausdo... |
iscnrm 23149 | The property of being comp... |
t0sep 23150 | Any two topologically indi... |
t0dist 23151 | Any two distinct points in... |
t1sncld 23152 | In a T_1 space, singletons... |
t1ficld 23153 | In a T_1 space, finite set... |
hausnei 23154 | Neighborhood property of a... |
t0top 23155 | A T_0 space is a topologic... |
t1top 23156 | A T_1 space is a topologic... |
haustop 23157 | A Hausdorff space is a top... |
isreg 23158 | The predicate "is a regula... |
regtop 23159 | A regular space is a topol... |
regsep 23160 | In a regular space, every ... |
isnrm 23161 | The predicate "is a normal... |
nrmtop 23162 | A normal space is a topolo... |
cnrmtop 23163 | A completely normal space ... |
iscnrm2 23164 | The property of being comp... |
ispnrm 23165 | The property of being perf... |
pnrmnrm 23166 | A perfectly normal space i... |
pnrmtop 23167 | A perfectly normal space i... |
pnrmcld 23168 | A closed set in a perfectl... |
pnrmopn 23169 | An open set in a perfectly... |
ist0-2 23170 | The predicate "is a T_0 sp... |
ist0-3 23171 | The predicate "is a T_0 sp... |
cnt0 23172 | The preimage of a T_0 topo... |
ist1-2 23173 | An alternate characterizat... |
t1t0 23174 | A T_1 space is a T_0 space... |
ist1-3 23175 | A space is T_1 iff every p... |
cnt1 23176 | The preimage of a T_1 topo... |
ishaus2 23177 | Express the predicate " ` ... |
haust1 23178 | A Hausdorff space is a T_1... |
hausnei2 23179 | The Hausdorff condition st... |
cnhaus 23180 | The preimage of a Hausdorf... |
nrmsep3 23181 | In a normal space, given a... |
nrmsep2 23182 | In a normal space, any two... |
nrmsep 23183 | In a normal space, disjoin... |
isnrm2 23184 | An alternate characterizat... |
isnrm3 23185 | A topological space is nor... |
cnrmi 23186 | A subspace of a completely... |
cnrmnrm 23187 | A completely normal space ... |
restcnrm 23188 | A subspace of a completely... |
resthauslem 23189 | Lemma for ~ resthaus and s... |
lpcls 23190 | The limit points of the cl... |
perfcls 23191 | A subset of a perfect spac... |
restt0 23192 | A subspace of a T_0 topolo... |
restt1 23193 | A subspace of a T_1 topolo... |
resthaus 23194 | A subspace of a Hausdorff ... |
t1sep2 23195 | Any two points in a T_1 sp... |
t1sep 23196 | Any two distinct points in... |
sncld 23197 | A singleton is closed in a... |
sshauslem 23198 | Lemma for ~ sshaus and sim... |
sst0 23199 | A topology finer than a T_... |
sst1 23200 | A topology finer than a T_... |
sshaus 23201 | A topology finer than a Ha... |
regsep2 23202 | In a regular space, a clos... |
isreg2 23203 | A topological space is reg... |
dnsconst 23204 | If a continuous mapping to... |
ordtt1 23205 | The order topology is T_1 ... |
lmmo 23206 | A sequence in a Hausdorff ... |
lmfun 23207 | The convergence relation i... |
dishaus 23208 | A discrete topology is Hau... |
ordthauslem 23209 | Lemma for ~ ordthaus . (C... |
ordthaus 23210 | The order topology of a to... |
xrhaus 23211 | The topology of the extend... |
iscmp 23214 | The predicate "is a compac... |
cmpcov 23215 | An open cover of a compact... |
cmpcov2 23216 | Rewrite ~ cmpcov for the c... |
cmpcovf 23217 | Combine ~ cmpcov with ~ ac... |
cncmp 23218 | Compactness is respected b... |
fincmp 23219 | A finite topology is compa... |
0cmp 23220 | The singleton of the empty... |
cmptop 23221 | A compact topology is a to... |
rncmp 23222 | The image of a compact set... |
imacmp 23223 | The image of a compact set... |
discmp 23224 | A discrete topology is com... |
cmpsublem 23225 | Lemma for ~ cmpsub . (Con... |
cmpsub 23226 | Two equivalent ways of des... |
tgcmp 23227 | A topology generated by a ... |
cmpcld 23228 | A closed subset of a compa... |
uncmp 23229 | The union of two compact s... |
fiuncmp 23230 | A finite union of compact ... |
sscmp 23231 | A subset of a compact topo... |
hauscmplem 23232 | Lemma for ~ hauscmp . (Co... |
hauscmp 23233 | A compact subspace of a T2... |
cmpfi 23234 | If a topology is compact a... |
cmpfii 23235 | In a compact topology, a s... |
bwth 23236 | The glorious Bolzano-Weier... |
isconn 23239 | The predicate ` J ` is a c... |
isconn2 23240 | The predicate ` J ` is a c... |
connclo 23241 | The only nonempty clopen s... |
conndisj 23242 | If a topology is connected... |
conntop 23243 | A connected topology is a ... |
indisconn 23244 | The indiscrete topology (o... |
dfconn2 23245 | An alternate definition of... |
connsuba 23246 | Connectedness for a subspa... |
connsub 23247 | Two equivalent ways of say... |
cnconn 23248 | Connectedness is respected... |
nconnsubb 23249 | Disconnectedness for a sub... |
connsubclo 23250 | If a clopen set meets a co... |
connima 23251 | The image of a connected s... |
conncn 23252 | A continuous function from... |
iunconnlem 23253 | Lemma for ~ iunconn . (Co... |
iunconn 23254 | The indexed union of conne... |
unconn 23255 | The union of two connected... |
clsconn 23256 | The closure of a connected... |
conncompid 23257 | The connected component co... |
conncompconn 23258 | The connected component co... |
conncompss 23259 | The connected component co... |
conncompcld 23260 | The connected component co... |
conncompclo 23261 | The connected component co... |
t1connperf 23262 | A connected T_1 space is p... |
is1stc 23267 | The predicate "is a first-... |
is1stc2 23268 | An equivalent way of sayin... |
1stctop 23269 | A first-countable topology... |
1stcclb 23270 | A property of points in a ... |
1stcfb 23271 | For any point ` A ` in a f... |
is2ndc 23272 | The property of being seco... |
2ndctop 23273 | A second-countable topolog... |
2ndci 23274 | A countable basis generate... |
2ndcsb 23275 | Having a countable subbase... |
2ndcredom 23276 | A second-countable space h... |
2ndc1stc 23277 | A second-countable space i... |
1stcrestlem 23278 | Lemma for ~ 1stcrest . (C... |
1stcrest 23279 | A subspace of a first-coun... |
2ndcrest 23280 | A subspace of a second-cou... |
2ndcctbss 23281 | If a topology is second-co... |
2ndcdisj 23282 | Any disjoint family of ope... |
2ndcdisj2 23283 | Any disjoint collection of... |
2ndcomap 23284 | A surjective continuous op... |
2ndcsep 23285 | A second-countable topolog... |
dis2ndc 23286 | A discrete space is second... |
1stcelcls 23287 | A point belongs to the clo... |
1stccnp 23288 | A mapping is continuous at... |
1stccn 23289 | A mapping ` X --> Y ` , wh... |
islly 23294 | The property of being a lo... |
isnlly 23295 | The property of being an n... |
llyeq 23296 | Equality theorem for the `... |
nllyeq 23297 | Equality theorem for the `... |
llytop 23298 | A locally ` A ` space is a... |
nllytop 23299 | A locally ` A ` space is a... |
llyi 23300 | The property of a locally ... |
nllyi 23301 | The property of an n-local... |
nlly2i 23302 | Eliminate the neighborhood... |
llynlly 23303 | A locally ` A ` space is n... |
llyssnlly 23304 | A locally ` A ` space is n... |
llyss 23305 | The "locally" predicate re... |
nllyss 23306 | The "n-locally" predicate ... |
subislly 23307 | The property of a subspace... |
restnlly 23308 | If the property ` A ` pass... |
restlly 23309 | If the property ` A ` pass... |
islly2 23310 | An alternative expression ... |
llyrest 23311 | An open subspace of a loca... |
nllyrest 23312 | An open subspace of an n-l... |
loclly 23313 | If ` A ` is a local proper... |
llyidm 23314 | Idempotence of the "locall... |
nllyidm 23315 | Idempotence of the "n-loca... |
toplly 23316 | A topology is locally a to... |
topnlly 23317 | A topology is n-locally a ... |
hauslly 23318 | A Hausdorff space is local... |
hausnlly 23319 | A Hausdorff space is n-loc... |
hausllycmp 23320 | A compact Hausdorff space ... |
cldllycmp 23321 | A closed subspace of a loc... |
lly1stc 23322 | First-countability is a lo... |
dislly 23323 | The discrete space ` ~P X ... |
disllycmp 23324 | A discrete space is locall... |
dis1stc 23325 | A discrete space is first-... |
hausmapdom 23326 | If ` X ` is a first-counta... |
hauspwdom 23327 | Simplify the cardinal ` A ... |
refrel 23334 | Refinement is a relation. ... |
isref 23335 | The property of being a re... |
refbas 23336 | A refinement covers the sa... |
refssex 23337 | Every set in a refinement ... |
ssref 23338 | A subcover is a refinement... |
refref 23339 | Reflexivity of refinement.... |
reftr 23340 | Refinement is transitive. ... |
refun0 23341 | Adding the empty set prese... |
isptfin 23342 | The statement "is a point-... |
islocfin 23343 | The statement "is a locall... |
finptfin 23344 | A finite cover is a point-... |
ptfinfin 23345 | A point covered by a point... |
finlocfin 23346 | A finite cover of a topolo... |
locfintop 23347 | A locally finite cover cov... |
locfinbas 23348 | A locally finite cover mus... |
locfinnei 23349 | A point covered by a local... |
lfinpfin 23350 | A locally finite cover is ... |
lfinun 23351 | Adding a finite set preser... |
locfincmp 23352 | For a compact space, the l... |
unisngl 23353 | Taking the union of the se... |
dissnref 23354 | The set of singletons is a... |
dissnlocfin 23355 | The set of singletons is l... |
locfindis 23356 | The locally finite covers ... |
locfincf 23357 | A locally finite cover in ... |
comppfsc 23358 | A space where every open c... |
kgenval 23361 | Value of the compact gener... |
elkgen 23362 | Value of the compact gener... |
kgeni 23363 | Property of the open sets ... |
kgentopon 23364 | The compact generator gene... |
kgenuni 23365 | The base set of the compac... |
kgenftop 23366 | The compact generator gene... |
kgenf 23367 | The compact generator is a... |
kgentop 23368 | A compactly generated spac... |
kgenss 23369 | The compact generator gene... |
kgenhaus 23370 | The compact generator gene... |
kgencmp 23371 | The compact generator topo... |
kgencmp2 23372 | The compact generator topo... |
kgenidm 23373 | The compact generator is i... |
iskgen2 23374 | A space is compactly gener... |
iskgen3 23375 | Derive the usual definitio... |
llycmpkgen2 23376 | A locally compact space is... |
cmpkgen 23377 | A compact space is compact... |
llycmpkgen 23378 | A locally compact space is... |
1stckgenlem 23379 | The one-point compactifica... |
1stckgen 23380 | A first-countable space is... |
kgen2ss 23381 | The compact generator pres... |
kgencn 23382 | A function from a compactl... |
kgencn2 23383 | A function ` F : J --> K `... |
kgencn3 23384 | The set of continuous func... |
kgen2cn 23385 | A continuous function is a... |
txval 23390 | Value of the binary topolo... |
txuni2 23391 | The underlying set of the ... |
txbasex 23392 | The basis for the product ... |
txbas 23393 | The set of Cartesian produ... |
eltx 23394 | A set in a product is open... |
txtop 23395 | The product of two topolog... |
ptval 23396 | The value of the product t... |
ptpjpre1 23397 | The preimage of a projecti... |
elpt 23398 | Elementhood in the bases o... |
elptr 23399 | A basic open set in the pr... |
elptr2 23400 | A basic open set in the pr... |
ptbasid 23401 | The base set of the produc... |
ptuni2 23402 | The base set for the produ... |
ptbasin 23403 | The basis for a product to... |
ptbasin2 23404 | The basis for a product to... |
ptbas 23405 | The basis for a product to... |
ptpjpre2 23406 | The basis for a product to... |
ptbasfi 23407 | The basis for the product ... |
pttop 23408 | The product topology is a ... |
ptopn 23409 | A basic open set in the pr... |
ptopn2 23410 | A sub-basic open set in th... |
xkotf 23411 | Functionality of function ... |
xkobval 23412 | Alternative expression for... |
xkoval 23413 | Value of the compact-open ... |
xkotop 23414 | The compact-open topology ... |
xkoopn 23415 | A basic open set of the co... |
txtopi 23416 | The product of two topolog... |
txtopon 23417 | The underlying set of the ... |
txuni 23418 | The underlying set of the ... |
txunii 23419 | The underlying set of the ... |
ptuni 23420 | The base set for the produ... |
ptunimpt 23421 | Base set of a product topo... |
pttopon 23422 | The base set for the produ... |
pttoponconst 23423 | The base set for a product... |
ptuniconst 23424 | The base set for a product... |
xkouni 23425 | The base set of the compac... |
xkotopon 23426 | The base set of the compac... |
ptval2 23427 | The value of the product t... |
txopn 23428 | The product of two open se... |
txcld 23429 | The product of two closed ... |
txcls 23430 | Closure of a rectangle in ... |
txss12 23431 | Subset property of the top... |
txbasval 23432 | It is sufficient to consid... |
neitx 23433 | The Cartesian product of t... |
txcnpi 23434 | Continuity of a two-argume... |
tx1cn 23435 | Continuity of the first pr... |
tx2cn 23436 | Continuity of the second p... |
ptpjcn 23437 | Continuity of a projection... |
ptpjopn 23438 | The projection map is an o... |
ptcld 23439 | A closed box in the produc... |
ptcldmpt 23440 | A closed box in the produc... |
ptclsg 23441 | The closure of a box in th... |
ptcls 23442 | The closure of a box in th... |
dfac14lem 23443 | Lemma for ~ dfac14 . By e... |
dfac14 23444 | Theorem ~ ptcls is an equi... |
xkoccn 23445 | The "constant function" fu... |
txcnp 23446 | If two functions are conti... |
ptcnplem 23447 | Lemma for ~ ptcnp . (Cont... |
ptcnp 23448 | If every projection of a f... |
upxp 23449 | Universal property of the ... |
txcnmpt 23450 | A map into the product of ... |
uptx 23451 | Universal property of the ... |
txcn 23452 | A map into the product of ... |
ptcn 23453 | If every projection of a f... |
prdstopn 23454 | Topology of a structure pr... |
prdstps 23455 | A structure product of top... |
pwstps 23456 | A structure power of a top... |
txrest 23457 | The subspace of a topologi... |
txdis 23458 | The topological product of... |
txindislem 23459 | Lemma for ~ txindis . (Co... |
txindis 23460 | The topological product of... |
txdis1cn 23461 | A function is jointly cont... |
txlly 23462 | If the property ` A ` is p... |
txnlly 23463 | If the property ` A ` is p... |
pthaus 23464 | The product of a collectio... |
ptrescn 23465 | Restriction is a continuou... |
txtube 23466 | The "tube lemma". If ` X ... |
txcmplem1 23467 | Lemma for ~ txcmp . (Cont... |
txcmplem2 23468 | Lemma for ~ txcmp . (Cont... |
txcmp 23469 | The topological product of... |
txcmpb 23470 | The topological product of... |
hausdiag 23471 | A topology is Hausdorff if... |
hauseqlcld 23472 | In a Hausdorff topology, t... |
txhaus 23473 | The topological product of... |
txlm 23474 | Two sequences converge iff... |
lmcn2 23475 | The image of a convergent ... |
tx1stc 23476 | The topological product of... |
tx2ndc 23477 | The topological product of... |
txkgen 23478 | The topological product of... |
xkohaus 23479 | If the codomain space is H... |
xkoptsub 23480 | The compact-open topology ... |
xkopt 23481 | The compact-open topology ... |
xkopjcn 23482 | Continuity of a projection... |
xkoco1cn 23483 | If ` F ` is a continuous f... |
xkoco2cn 23484 | If ` F ` is a continuous f... |
xkococnlem 23485 | Continuity of the composit... |
xkococn 23486 | Continuity of the composit... |
cnmptid 23487 | The identity function is c... |
cnmptc 23488 | A constant function is con... |
cnmpt11 23489 | The composition of continu... |
cnmpt11f 23490 | The composition of continu... |
cnmpt1t 23491 | The composition of continu... |
cnmpt12f 23492 | The composition of continu... |
cnmpt12 23493 | The composition of continu... |
cnmpt1st 23494 | The projection onto the fi... |
cnmpt2nd 23495 | The projection onto the se... |
cnmpt2c 23496 | A constant function is con... |
cnmpt21 23497 | The composition of continu... |
cnmpt21f 23498 | The composition of continu... |
cnmpt2t 23499 | The composition of continu... |
cnmpt22 23500 | The composition of continu... |
cnmpt22f 23501 | The composition of continu... |
cnmpt1res 23502 | The restriction of a conti... |
cnmpt2res 23503 | The restriction of a conti... |
cnmptcom 23504 | The argument converse of a... |
cnmptkc 23505 | The curried first projecti... |
cnmptkp 23506 | The evaluation of the inne... |
cnmptk1 23507 | The composition of a curri... |
cnmpt1k 23508 | The composition of a one-a... |
cnmptkk 23509 | The composition of two cur... |
xkofvcn 23510 | Joint continuity of the fu... |
cnmptk1p 23511 | The evaluation of a currie... |
cnmptk2 23512 | The uncurrying of a currie... |
xkoinjcn 23513 | Continuity of "injection",... |
cnmpt2k 23514 | The currying of a two-argu... |
txconn 23515 | The topological product of... |
imasnopn 23516 | If a relation graph is ope... |
imasncld 23517 | If a relation graph is clo... |
imasncls 23518 | If a relation graph is clo... |
qtopval 23521 | Value of the quotient topo... |
qtopval2 23522 | Value of the quotient topo... |
elqtop 23523 | Value of the quotient topo... |
qtopres 23524 | The quotient topology is u... |
qtoptop2 23525 | The quotient topology is a... |
qtoptop 23526 | The quotient topology is a... |
elqtop2 23527 | Value of the quotient topo... |
qtopuni 23528 | The base set of the quotie... |
elqtop3 23529 | Value of the quotient topo... |
qtoptopon 23530 | The base set of the quotie... |
qtopid 23531 | A quotient map is a contin... |
idqtop 23532 | The quotient topology indu... |
qtopcmplem 23533 | Lemma for ~ qtopcmp and ~ ... |
qtopcmp 23534 | A quotient of a compact sp... |
qtopconn 23535 | A quotient of a connected ... |
qtopkgen 23536 | A quotient of a compactly ... |
basqtop 23537 | An injection maps bases to... |
tgqtop 23538 | An injection maps generate... |
qtopcld 23539 | The property of being a cl... |
qtopcn 23540 | Universal property of a qu... |
qtopss 23541 | A surjective continuous fu... |
qtopeu 23542 | Universal property of the ... |
qtoprest 23543 | If ` A ` is a saturated op... |
qtopomap 23544 | If ` F ` is a surjective c... |
qtopcmap 23545 | If ` F ` is a surjective c... |
imastopn 23546 | The topology of an image s... |
imastps 23547 | The image of a topological... |
qustps 23548 | A quotient structure is a ... |
kqfval 23549 | Value of the function appe... |
kqfeq 23550 | Two points in the Kolmogor... |
kqffn 23551 | The topological indistingu... |
kqval 23552 | Value of the quotient topo... |
kqtopon 23553 | The Kolmogorov quotient is... |
kqid 23554 | The topological indistingu... |
ist0-4 23555 | The topological indistingu... |
kqfvima 23556 | When the image set is open... |
kqsat 23557 | Any open set is saturated ... |
kqdisj 23558 | A version of ~ imain for t... |
kqcldsat 23559 | Any closed set is saturate... |
kqopn 23560 | The topological indistingu... |
kqcld 23561 | The topological indistingu... |
kqt0lem 23562 | Lemma for ~ kqt0 . (Contr... |
isr0 23563 | The property " ` J ` is an... |
r0cld 23564 | The analogue of the T_1 ax... |
regr1lem 23565 | Lemma for ~ regr1 . (Cont... |
regr1lem2 23566 | A Kolmogorov quotient of a... |
kqreglem1 23567 | A Kolmogorov quotient of a... |
kqreglem2 23568 | If the Kolmogorov quotient... |
kqnrmlem1 23569 | A Kolmogorov quotient of a... |
kqnrmlem2 23570 | If the Kolmogorov quotient... |
kqtop 23571 | The Kolmogorov quotient is... |
kqt0 23572 | The Kolmogorov quotient is... |
kqf 23573 | The Kolmogorov quotient is... |
r0sep 23574 | The separation property of... |
nrmr0reg 23575 | A normal R_0 space is also... |
regr1 23576 | A regular space is R_1, wh... |
kqreg 23577 | The Kolmogorov quotient of... |
kqnrm 23578 | The Kolmogorov quotient of... |
hmeofn 23583 | The set of homeomorphisms ... |
hmeofval 23584 | The set of all the homeomo... |
ishmeo 23585 | The predicate F is a homeo... |
hmeocn 23586 | A homeomorphism is continu... |
hmeocnvcn 23587 | The converse of a homeomor... |
hmeocnv 23588 | The converse of a homeomor... |
hmeof1o2 23589 | A homeomorphism is a 1-1-o... |
hmeof1o 23590 | A homeomorphism is a 1-1-o... |
hmeoima 23591 | The image of an open set b... |
hmeoopn 23592 | Homeomorphisms preserve op... |
hmeocld 23593 | Homeomorphisms preserve cl... |
hmeocls 23594 | Homeomorphisms preserve cl... |
hmeontr 23595 | Homeomorphisms preserve in... |
hmeoimaf1o 23596 | The function mapping open ... |
hmeores 23597 | The restriction of a homeo... |
hmeoco 23598 | The composite of two homeo... |
idhmeo 23599 | The identity function is a... |
hmeocnvb 23600 | The converse of a homeomor... |
hmeoqtop 23601 | A homeomorphism is a quoti... |
hmph 23602 | Express the predicate ` J ... |
hmphi 23603 | If there is a homeomorphis... |
hmphtop 23604 | Reverse closure for the ho... |
hmphtop1 23605 | The relation "being homeom... |
hmphtop2 23606 | The relation "being homeom... |
hmphref 23607 | "Is homeomorphic to" is re... |
hmphsym 23608 | "Is homeomorphic to" is sy... |
hmphtr 23609 | "Is homeomorphic to" is tr... |
hmpher 23610 | "Is homeomorphic to" is an... |
hmphen 23611 | Homeomorphisms preserve th... |
hmphsymb 23612 | "Is homeomorphic to" is sy... |
haushmphlem 23613 | Lemma for ~ haushmph and s... |
cmphmph 23614 | Compactness is a topologic... |
connhmph 23615 | Connectedness is a topolog... |
t0hmph 23616 | T_0 is a topological prope... |
t1hmph 23617 | T_1 is a topological prope... |
haushmph 23618 | Hausdorff-ness is a topolo... |
reghmph 23619 | Regularity is a topologica... |
nrmhmph 23620 | Normality is a topological... |
hmph0 23621 | A topology homeomorphic to... |
hmphdis 23622 | Homeomorphisms preserve to... |
hmphindis 23623 | Homeomorphisms preserve to... |
indishmph 23624 | Equinumerous sets equipped... |
hmphen2 23625 | Homeomorphisms preserve th... |
cmphaushmeo 23626 | A continuous bijection fro... |
ordthmeolem 23627 | Lemma for ~ ordthmeo . (C... |
ordthmeo 23628 | An order isomorphism is a ... |
txhmeo 23629 | Lift a pair of homeomorphi... |
txswaphmeolem 23630 | Show inverse for the "swap... |
txswaphmeo 23631 | There is a homeomorphism f... |
pt1hmeo 23632 | The canonical homeomorphis... |
ptuncnv 23633 | Exhibit the converse funct... |
ptunhmeo 23634 | Define a homeomorphism fro... |
xpstopnlem1 23635 | The function ` F ` used in... |
xpstps 23636 | A binary product of topolo... |
xpstopnlem2 23637 | Lemma for ~ xpstopn . (Co... |
xpstopn 23638 | The topology on a binary p... |
ptcmpfi 23639 | A topological product of f... |
xkocnv 23640 | The inverse of the "curryi... |
xkohmeo 23641 | The Exponential Law for to... |
qtopf1 23642 | If a quotient map is injec... |
qtophmeo 23643 | If two functions on a base... |
t0kq 23644 | A topological space is T_0... |
kqhmph 23645 | A topological space is T_0... |
ist1-5lem 23646 | Lemma for ~ ist1-5 and sim... |
t1r0 23647 | A T_1 space is R_0. That ... |
ist1-5 23648 | A topological space is T_1... |
ishaus3 23649 | A topological space is Hau... |
nrmreg 23650 | A normal T_1 space is regu... |
reghaus 23651 | A regular T_0 space is Hau... |
nrmhaus 23652 | A T_1 normal space is Haus... |
elmptrab 23653 | Membership in a one-parame... |
elmptrab2 23654 | Membership in a one-parame... |
isfbas 23655 | The predicate " ` F ` is a... |
fbasne0 23656 | There are no empty filter ... |
0nelfb 23657 | No filter base contains th... |
fbsspw 23658 | A filter base on a set is ... |
fbelss 23659 | An element of the filter b... |
fbdmn0 23660 | The domain of a filter bas... |
isfbas2 23661 | The predicate " ` F ` is a... |
fbasssin 23662 | A filter base contains sub... |
fbssfi 23663 | A filter base contains sub... |
fbssint 23664 | A filter base contains sub... |
fbncp 23665 | A filter base does not con... |
fbun 23666 | A necessary and sufficient... |
fbfinnfr 23667 | No filter base containing ... |
opnfbas 23668 | The collection of open sup... |
trfbas2 23669 | Conditions for the trace o... |
trfbas 23670 | Conditions for the trace o... |
isfil 23673 | The predicate "is a filter... |
filfbas 23674 | A filter is a filter base.... |
0nelfil 23675 | The empty set doesn't belo... |
fileln0 23676 | An element of a filter is ... |
filsspw 23677 | A filter is a subset of th... |
filelss 23678 | An element of a filter is ... |
filss 23679 | A filter is closed under t... |
filin 23680 | A filter is closed under t... |
filtop 23681 | The underlying set belongs... |
isfil2 23682 | Derive the standard axioms... |
isfildlem 23683 | Lemma for ~ isfild . (Con... |
isfild 23684 | Sufficient condition for a... |
filfi 23685 | A filter is closed under t... |
filinn0 23686 | The intersection of two el... |
filintn0 23687 | A filter has the finite in... |
filn0 23688 | The empty set is not a fil... |
infil 23689 | The intersection of two fi... |
snfil 23690 | A singleton is a filter. ... |
fbasweak 23691 | A filter base on any set i... |
snfbas 23692 | Condition for a singleton ... |
fsubbas 23693 | A condition for a set to g... |
fbasfip 23694 | A filter base has the fini... |
fbunfip 23695 | A helpful lemma for showin... |
fgval 23696 | The filter generating clas... |
elfg 23697 | A condition for elements o... |
ssfg 23698 | A filter base is a subset ... |
fgss 23699 | A bigger base generates a ... |
fgss2 23700 | A condition for a filter t... |
fgfil 23701 | A filter generates itself.... |
elfilss 23702 | An element belongs to a fi... |
filfinnfr 23703 | No filter containing a fin... |
fgcl 23704 | A generated filter is a fi... |
fgabs 23705 | Absorption law for filter ... |
neifil 23706 | The neighborhoods of a non... |
filunibas 23707 | Recover the base set from ... |
filunirn 23708 | Two ways to express a filt... |
filconn 23709 | A filter gives rise to a c... |
fbasrn 23710 | Given a filter on a domain... |
filuni 23711 | The union of a nonempty se... |
trfil1 23712 | Conditions for the trace o... |
trfil2 23713 | Conditions for the trace o... |
trfil3 23714 | Conditions for the trace o... |
trfilss 23715 | If ` A ` is a member of th... |
fgtr 23716 | If ` A ` is a member of th... |
trfg 23717 | The trace operation and th... |
trnei 23718 | The trace, over a set ` A ... |
cfinfil 23719 | Relative complements of th... |
csdfil 23720 | The set of all elements wh... |
supfil 23721 | The supersets of a nonempt... |
zfbas 23722 | The set of upper sets of i... |
uzrest 23723 | The restriction of the set... |
uzfbas 23724 | The set of upper sets of i... |
isufil 23729 | The property of being an u... |
ufilfil 23730 | An ultrafilter is a filter... |
ufilss 23731 | For any subset of the base... |
ufilb 23732 | The complement is in an ul... |
ufilmax 23733 | Any filter finer than an u... |
isufil2 23734 | The maximal property of an... |
ufprim 23735 | An ultrafilter is a prime ... |
trufil 23736 | Conditions for the trace o... |
filssufilg 23737 | A filter is contained in s... |
filssufil 23738 | A filter is contained in s... |
isufl 23739 | Define the (strong) ultraf... |
ufli 23740 | Property of a set that sat... |
numufl 23741 | Consequence of ~ filssufil... |
fiufl 23742 | A finite set satisfies the... |
acufl 23743 | The axiom of choice implie... |
ssufl 23744 | If ` Y ` is a subset of ` ... |
ufileu 23745 | If the ultrafilter contain... |
filufint 23746 | A filter is equal to the i... |
uffix 23747 | Lemma for ~ fixufil and ~ ... |
fixufil 23748 | The condition describing a... |
uffixfr 23749 | An ultrafilter is either f... |
uffix2 23750 | A classification of fixed ... |
uffixsn 23751 | The singleton of the gener... |
ufildom1 23752 | An ultrafilter is generate... |
uffinfix 23753 | An ultrafilter containing ... |
cfinufil 23754 | An ultrafilter is free iff... |
ufinffr 23755 | An infinite subset is cont... |
ufilen 23756 | Any infinite set has an ul... |
ufildr 23757 | An ultrafilter gives rise ... |
fin1aufil 23758 | There are no definable fre... |
fmval 23769 | Introduce a function that ... |
fmfil 23770 | A mapping filter is a filt... |
fmf 23771 | Pushing-forward via a func... |
fmss 23772 | A finer filter produces a ... |
elfm 23773 | An element of a mapping fi... |
elfm2 23774 | An element of a mapping fi... |
fmfg 23775 | The image filter of a filt... |
elfm3 23776 | An alternate formulation o... |
imaelfm 23777 | An image of a filter eleme... |
rnelfmlem 23778 | Lemma for ~ rnelfm . (Con... |
rnelfm 23779 | A condition for a filter t... |
fmfnfmlem1 23780 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem2 23781 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem3 23782 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem4 23783 | Lemma for ~ fmfnfm . (Con... |
fmfnfm 23784 | A filter finer than an ima... |
fmufil 23785 | An image filter of an ultr... |
fmid 23786 | The filter map applied to ... |
fmco 23787 | Composition of image filte... |
ufldom 23788 | The ultrafilter lemma prop... |
flimval 23789 | The set of limit points of... |
elflim2 23790 | The predicate "is a limit ... |
flimtop 23791 | Reverse closure for the li... |
flimneiss 23792 | A filter contains the neig... |
flimnei 23793 | A filter contains all of t... |
flimelbas 23794 | A limit point of a filter ... |
flimfil 23795 | Reverse closure for the li... |
flimtopon 23796 | Reverse closure for the li... |
elflim 23797 | The predicate "is a limit ... |
flimss2 23798 | A limit point of a filter ... |
flimss1 23799 | A limit point of a filter ... |
neiflim 23800 | A point is a limit point o... |
flimopn 23801 | The condition for being a ... |
fbflim 23802 | A condition for a filter t... |
fbflim2 23803 | A condition for a filter b... |
flimclsi 23804 | The convergent points of a... |
hausflimlem 23805 | If ` A ` and ` B ` are bot... |
hausflimi 23806 | One direction of ~ hausfli... |
hausflim 23807 | A condition for a topology... |
flimcf 23808 | Fineness is properly chara... |
flimrest 23809 | The set of limit points in... |
flimclslem 23810 | Lemma for ~ flimcls . (Co... |
flimcls 23811 | Closure in terms of filter... |
flimsncls 23812 | If ` A ` is a limit point ... |
hauspwpwf1 23813 | Lemma for ~ hauspwpwdom . ... |
hauspwpwdom 23814 | If ` X ` is a Hausdorff sp... |
flffval 23815 | Given a topology and a fil... |
flfval 23816 | Given a function from a fi... |
flfnei 23817 | The property of being a li... |
flfneii 23818 | A neighborhood of a limit ... |
isflf 23819 | The property of being a li... |
flfelbas 23820 | A limit point of a functio... |
flffbas 23821 | Limit points of a function... |
flftg 23822 | Limit points of a function... |
hausflf 23823 | If a function has its valu... |
hausflf2 23824 | If a convergent function h... |
cnpflfi 23825 | Forward direction of ~ cnp... |
cnpflf2 23826 | ` F ` is continuous at poi... |
cnpflf 23827 | Continuity of a function a... |
cnflf 23828 | A function is continuous i... |
cnflf2 23829 | A function is continuous i... |
flfcnp 23830 | A continuous function pres... |
lmflf 23831 | The topological limit rela... |
txflf 23832 | Two sequences converge in ... |
flfcnp2 23833 | The image of a convergent ... |
fclsval 23834 | The set of all cluster poi... |
isfcls 23835 | A cluster point of a filte... |
fclsfil 23836 | Reverse closure for the cl... |
fclstop 23837 | Reverse closure for the cl... |
fclstopon 23838 | Reverse closure for the cl... |
isfcls2 23839 | A cluster point of a filte... |
fclsopn 23840 | Write the cluster point co... |
fclsopni 23841 | An open neighborhood of a ... |
fclselbas 23842 | A cluster point is in the ... |
fclsneii 23843 | A neighborhood of a cluste... |
fclssscls 23844 | The set of cluster points ... |
fclsnei 23845 | Cluster points in terms of... |
supnfcls 23846 | The filter of supersets of... |
fclsbas 23847 | Cluster points in terms of... |
fclsss1 23848 | A finer topology has fewer... |
fclsss2 23849 | A finer filter has fewer c... |
fclsrest 23850 | The set of cluster points ... |
fclscf 23851 | Characterization of finene... |
flimfcls 23852 | A limit point is a cluster... |
fclsfnflim 23853 | A filter clusters at a poi... |
flimfnfcls 23854 | A filter converges to a po... |
fclscmpi 23855 | Forward direction of ~ fcl... |
fclscmp 23856 | A space is compact iff eve... |
uffclsflim 23857 | The cluster points of an u... |
ufilcmp 23858 | A space is compact iff eve... |
fcfval 23859 | The set of cluster points ... |
isfcf 23860 | The property of being a cl... |
fcfnei 23861 | The property of being a cl... |
fcfelbas 23862 | A cluster point of a funct... |
fcfneii 23863 | A neighborhood of a cluste... |
flfssfcf 23864 | A limit point of a functio... |
uffcfflf 23865 | If the domain filter is an... |
cnpfcfi 23866 | Lemma for ~ cnpfcf . If a... |
cnpfcf 23867 | A function ` F ` is contin... |
cnfcf 23868 | Continuity of a function i... |
flfcntr 23869 | A continuous function's va... |
alexsublem 23870 | Lemma for ~ alexsub . (Co... |
alexsub 23871 | The Alexander Subbase Theo... |
alexsubb 23872 | Biconditional form of the ... |
alexsubALTlem1 23873 | Lemma for ~ alexsubALT . ... |
alexsubALTlem2 23874 | Lemma for ~ alexsubALT . ... |
alexsubALTlem3 23875 | Lemma for ~ alexsubALT . ... |
alexsubALTlem4 23876 | Lemma for ~ alexsubALT . ... |
alexsubALT 23877 | The Alexander Subbase Theo... |
ptcmplem1 23878 | Lemma for ~ ptcmp . (Cont... |
ptcmplem2 23879 | Lemma for ~ ptcmp . (Cont... |
ptcmplem3 23880 | Lemma for ~ ptcmp . (Cont... |
ptcmplem4 23881 | Lemma for ~ ptcmp . (Cont... |
ptcmplem5 23882 | Lemma for ~ ptcmp . (Cont... |
ptcmpg 23883 | Tychonoff's theorem: The ... |
ptcmp 23884 | Tychonoff's theorem: The ... |
cnextval 23887 | The function applying cont... |
cnextfval 23888 | The continuous extension o... |
cnextrel 23889 | In the general case, a con... |
cnextfun 23890 | If the target space is Hau... |
cnextfvval 23891 | The value of the continuou... |
cnextf 23892 | Extension by continuity. ... |
cnextcn 23893 | Extension by continuity. ... |
cnextfres1 23894 | ` F ` and its extension by... |
cnextfres 23895 | ` F ` and its extension by... |
istmd 23900 | The predicate "is a topolo... |
tmdmnd 23901 | A topological monoid is a ... |
tmdtps 23902 | A topological monoid is a ... |
istgp 23903 | The predicate "is a topolo... |
tgpgrp 23904 | A topological group is a g... |
tgptmd 23905 | A topological group is a t... |
tgptps 23906 | A topological group is a t... |
tmdtopon 23907 | The topology of a topologi... |
tgptopon 23908 | The topology of a topologi... |
tmdcn 23909 | In a topological monoid, t... |
tgpcn 23910 | In a topological group, th... |
tgpinv 23911 | In a topological group, th... |
grpinvhmeo 23912 | The inverse function in a ... |
cnmpt1plusg 23913 | Continuity of the group su... |
cnmpt2plusg 23914 | Continuity of the group su... |
tmdcn2 23915 | Write out the definition o... |
tgpsubcn 23916 | In a topological group, th... |
istgp2 23917 | A group with a topology is... |
tmdmulg 23918 | In a topological monoid, t... |
tgpmulg 23919 | In a topological group, th... |
tgpmulg2 23920 | In a topological monoid, t... |
tmdgsum 23921 | In a topological monoid, t... |
tmdgsum2 23922 | For any neighborhood ` U `... |
oppgtmd 23923 | The opposite of a topologi... |
oppgtgp 23924 | The opposite of a topologi... |
distgp 23925 | Any group equipped with th... |
indistgp 23926 | Any group equipped with th... |
efmndtmd 23927 | The monoid of endofunction... |
tmdlactcn 23928 | The left group action of e... |
tgplacthmeo 23929 | The left group action of e... |
submtmd 23930 | A submonoid of a topologic... |
subgtgp 23931 | A subgroup of a topologica... |
symgtgp 23932 | The symmetric group is a t... |
subgntr 23933 | A subgroup of a topologica... |
opnsubg 23934 | An open subgroup of a topo... |
clssubg 23935 | The closure of a subgroup ... |
clsnsg 23936 | The closure of a normal su... |
cldsubg 23937 | A subgroup of finite index... |
tgpconncompeqg 23938 | The connected component co... |
tgpconncomp 23939 | The identity component, th... |
tgpconncompss 23940 | The identity component is ... |
ghmcnp 23941 | A group homomorphism on to... |
snclseqg 23942 | The coset of the closure o... |
tgphaus 23943 | A topological group is Hau... |
tgpt1 23944 | Hausdorff and T1 are equiv... |
tgpt0 23945 | Hausdorff and T0 are equiv... |
qustgpopn 23946 | A quotient map in a topolo... |
qustgplem 23947 | Lemma for ~ qustgp . (Con... |
qustgp 23948 | The quotient of a topologi... |
qustgphaus 23949 | The quotient of a topologi... |
prdstmdd 23950 | The product of a family of... |
prdstgpd 23951 | The product of a family of... |
tsmsfbas 23954 | The collection of all sets... |
tsmslem1 23955 | The finite partial sums of... |
tsmsval2 23956 | Definition of the topologi... |
tsmsval 23957 | Definition of the topologi... |
tsmspropd 23958 | The group sum depends only... |
eltsms 23959 | The property of being a su... |
tsmsi 23960 | The property of being a su... |
tsmscl 23961 | A sum in a topological gro... |
haustsms 23962 | In a Hausdorff topological... |
haustsms2 23963 | In a Hausdorff topological... |
tsmscls 23964 | One half of ~ tgptsmscls ,... |
tsmsgsum 23965 | The convergent points of a... |
tsmsid 23966 | If a sum is finite, the us... |
haustsmsid 23967 | In a Hausdorff topological... |
tsms0 23968 | The sum of zero is zero. ... |
tsmssubm 23969 | Evaluate an infinite group... |
tsmsres 23970 | Extend an infinite group s... |
tsmsf1o 23971 | Re-index an infinite group... |
tsmsmhm 23972 | Apply a continuous group h... |
tsmsadd 23973 | The sum of two infinite gr... |
tsmsinv 23974 | Inverse of an infinite gro... |
tsmssub 23975 | The difference of two infi... |
tgptsmscls 23976 | A sum in a topological gro... |
tgptsmscld 23977 | The set of limit points to... |
tsmssplit 23978 | Split a topological group ... |
tsmsxplem1 23979 | Lemma for ~ tsmsxp . (Con... |
tsmsxplem2 23980 | Lemma for ~ tsmsxp . (Con... |
tsmsxp 23981 | Write a sum over a two-dim... |
istrg 23990 | Express the predicate " ` ... |
trgtmd 23991 | The multiplicative monoid ... |
istdrg 23992 | Express the predicate " ` ... |
tdrgunit 23993 | The unit group of a topolo... |
trgtgp 23994 | A topological ring is a to... |
trgtmd2 23995 | A topological ring is a to... |
trgtps 23996 | A topological ring is a to... |
trgring 23997 | A topological ring is a ri... |
trggrp 23998 | A topological ring is a gr... |
tdrgtrg 23999 | A topological division rin... |
tdrgdrng 24000 | A topological division rin... |
tdrgring 24001 | A topological division rin... |
tdrgtmd 24002 | A topological division rin... |
tdrgtps 24003 | A topological division rin... |
istdrg2 24004 | A topological-ring divisio... |
mulrcn 24005 | The functionalization of t... |
invrcn2 24006 | The multiplicative inverse... |
invrcn 24007 | The multiplicative inverse... |
cnmpt1mulr 24008 | Continuity of ring multipl... |
cnmpt2mulr 24009 | Continuity of ring multipl... |
dvrcn 24010 | The division function is c... |
istlm 24011 | The predicate " ` W ` is a... |
vscacn 24012 | The scalar multiplication ... |
tlmtmd 24013 | A topological module is a ... |
tlmtps 24014 | A topological module is a ... |
tlmlmod 24015 | A topological module is a ... |
tlmtrg 24016 | The scalar ring of a topol... |
tlmscatps 24017 | The scalar ring of a topol... |
istvc 24018 | A topological vector space... |
tvctdrg 24019 | The scalar field of a topo... |
cnmpt1vsca 24020 | Continuity of scalar multi... |
cnmpt2vsca 24021 | Continuity of scalar multi... |
tlmtgp 24022 | A topological vector space... |
tvctlm 24023 | A topological vector space... |
tvclmod 24024 | A topological vector space... |
tvclvec 24025 | A topological vector space... |
ustfn 24028 | The defined uniform struct... |
ustval 24029 | The class of all uniform s... |
isust 24030 | The predicate " ` U ` is a... |
ustssxp 24031 | Entourages are subsets of ... |
ustssel 24032 | A uniform structure is upw... |
ustbasel 24033 | The full set is always an ... |
ustincl 24034 | A uniform structure is clo... |
ustdiag 24035 | The diagonal set is includ... |
ustinvel 24036 | If ` V ` is an entourage, ... |
ustexhalf 24037 | For each entourage ` V ` t... |
ustrel 24038 | The elements of uniform st... |
ustfilxp 24039 | A uniform structure on a n... |
ustne0 24040 | A uniform structure cannot... |
ustssco 24041 | In an uniform structure, a... |
ustexsym 24042 | In an uniform structure, f... |
ustex2sym 24043 | In an uniform structure, f... |
ustex3sym 24044 | In an uniform structure, f... |
ustref 24045 | Any element of the base se... |
ust0 24046 | The unique uniform structu... |
ustn0 24047 | The empty set is not an un... |
ustund 24048 | If two intersecting sets `... |
ustelimasn 24049 | Any point ` A ` is near en... |
ustneism 24050 | For a point ` A ` in ` X `... |
elrnustOLD 24051 | Obsolete version of ~ elfv... |
ustbas2 24052 | Second direction for ~ ust... |
ustuni 24053 | The set union of a uniform... |
ustbas 24054 | Recover the base of an uni... |
ustimasn 24055 | Lemma for ~ ustuqtop . (C... |
trust 24056 | The trace of a uniform str... |
utopval 24059 | The topology induced by a ... |
elutop 24060 | Open sets in the topology ... |
utoptop 24061 | The topology induced by a ... |
utopbas 24062 | The base of the topology i... |
utoptopon 24063 | Topology induced by a unif... |
restutop 24064 | Restriction of a topology ... |
restutopopn 24065 | The restriction of the top... |
ustuqtoplem 24066 | Lemma for ~ ustuqtop . (C... |
ustuqtop0 24067 | Lemma for ~ ustuqtop . (C... |
ustuqtop1 24068 | Lemma for ~ ustuqtop , sim... |
ustuqtop2 24069 | Lemma for ~ ustuqtop . (C... |
ustuqtop3 24070 | Lemma for ~ ustuqtop , sim... |
ustuqtop4 24071 | Lemma for ~ ustuqtop . (C... |
ustuqtop5 24072 | Lemma for ~ ustuqtop . (C... |
ustuqtop 24073 | For a given uniform struct... |
utopsnneiplem 24074 | The neighborhoods of a poi... |
utopsnneip 24075 | The neighborhoods of a poi... |
utopsnnei 24076 | Images of singletons by en... |
utop2nei 24077 | For any symmetrical entour... |
utop3cls 24078 | Relation between a topolog... |
utopreg 24079 | All Hausdorff uniform spac... |
ussval 24086 | The uniform structure on u... |
ussid 24087 | In case the base of the ` ... |
isusp 24088 | The predicate ` W ` is a u... |
ressuss 24089 | Value of the uniform struc... |
ressust 24090 | The uniform structure of a... |
ressusp 24091 | The restriction of a unifo... |
tusval 24092 | The value of the uniform s... |
tuslem 24093 | Lemma for ~ tusbas , ~ tus... |
tuslemOLD 24094 | Obsolete proof of ~ tuslem... |
tusbas 24095 | The base set of a construc... |
tusunif 24096 | The uniform structure of a... |
tususs 24097 | The uniform structure of a... |
tustopn 24098 | The topology induced by a ... |
tususp 24099 | A constructed uniform spac... |
tustps 24100 | A constructed uniform spac... |
uspreg 24101 | If a uniform space is Haus... |
ucnval 24104 | The set of all uniformly c... |
isucn 24105 | The predicate " ` F ` is a... |
isucn2 24106 | The predicate " ` F ` is a... |
ucnimalem 24107 | Reformulate the ` G ` func... |
ucnima 24108 | An equivalent statement of... |
ucnprima 24109 | The preimage by a uniforml... |
iducn 24110 | The identity is uniformly ... |
cstucnd 24111 | A constant function is uni... |
ucncn 24112 | Uniform continuity implies... |
iscfilu 24115 | The predicate " ` F ` is a... |
cfilufbas 24116 | A Cauchy filter base is a ... |
cfiluexsm 24117 | For a Cauchy filter base a... |
fmucndlem 24118 | Lemma for ~ fmucnd . (Con... |
fmucnd 24119 | The image of a Cauchy filt... |
cfilufg 24120 | The filter generated by a ... |
trcfilu 24121 | Condition for the trace of... |
cfiluweak 24122 | A Cauchy filter base is al... |
neipcfilu 24123 | In an uniform space, a nei... |
iscusp 24126 | The predicate " ` W ` is a... |
cuspusp 24127 | A complete uniform space i... |
cuspcvg 24128 | In a complete uniform spac... |
iscusp2 24129 | The predicate " ` W ` is a... |
cnextucn 24130 | Extension by continuity. ... |
ucnextcn 24131 | Extension by continuity. ... |
ispsmet 24132 | Express the predicate " ` ... |
psmetdmdm 24133 | Recover the base set from ... |
psmetf 24134 | The distance function of a... |
psmetcl 24135 | Closure of the distance fu... |
psmet0 24136 | The distance function of a... |
psmettri2 24137 | Triangle inequality for th... |
psmetsym 24138 | The distance function of a... |
psmettri 24139 | Triangle inequality for th... |
psmetge0 24140 | The distance function of a... |
psmetxrge0 24141 | The distance function of a... |
psmetres2 24142 | Restriction of a pseudomet... |
psmetlecl 24143 | Real closure of an extende... |
distspace 24144 | A set ` X ` together with ... |
ismet 24151 | Express the predicate " ` ... |
isxmet 24152 | Express the predicate " ` ... |
ismeti 24153 | Properties that determine ... |
isxmetd 24154 | Properties that determine ... |
isxmet2d 24155 | It is safe to only require... |
metflem 24156 | Lemma for ~ metf and other... |
xmetf 24157 | Mapping of the distance fu... |
metf 24158 | Mapping of the distance fu... |
xmetcl 24159 | Closure of the distance fu... |
metcl 24160 | Closure of the distance fu... |
ismet2 24161 | An extended metric is a me... |
metxmet 24162 | A metric is an extended me... |
xmetdmdm 24163 | Recover the base set from ... |
metdmdm 24164 | Recover the base set from ... |
xmetunirn 24165 | Two ways to express an ext... |
xmeteq0 24166 | The value of an extended m... |
meteq0 24167 | The value of a metric is z... |
xmettri2 24168 | Triangle inequality for th... |
mettri2 24169 | Triangle inequality for th... |
xmet0 24170 | The distance function of a... |
met0 24171 | The distance function of a... |
xmetge0 24172 | The distance function of a... |
metge0 24173 | The distance function of a... |
xmetlecl 24174 | Real closure of an extende... |
xmetsym 24175 | The distance function of a... |
xmetpsmet 24176 | An extended metric is a ps... |
xmettpos 24177 | The distance function of a... |
metsym 24178 | The distance function of a... |
xmettri 24179 | Triangle inequality for th... |
mettri 24180 | Triangle inequality for th... |
xmettri3 24181 | Triangle inequality for th... |
mettri3 24182 | Triangle inequality for th... |
xmetrtri 24183 | One half of the reverse tr... |
xmetrtri2 24184 | The reverse triangle inequ... |
metrtri 24185 | Reverse triangle inequalit... |
xmetgt0 24186 | The distance function of a... |
metgt0 24187 | The distance function of a... |
metn0 24188 | A metric space is nonempty... |
xmetres2 24189 | Restriction of an extended... |
metreslem 24190 | Lemma for ~ metres . (Con... |
metres2 24191 | Lemma for ~ metres . (Con... |
xmetres 24192 | A restriction of an extend... |
metres 24193 | A restriction of a metric ... |
0met 24194 | The empty metric. (Contri... |
prdsdsf 24195 | The product metric is a fu... |
prdsxmetlem 24196 | The product metric is an e... |
prdsxmet 24197 | The product metric is an e... |
prdsmet 24198 | The product metric is a me... |
ressprdsds 24199 | Restriction of a product m... |
resspwsds 24200 | Restriction of a power met... |
imasdsf1olem 24201 | Lemma for ~ imasdsf1o . (... |
imasdsf1o 24202 | The distance function is t... |
imasf1oxmet 24203 | The image of an extended m... |
imasf1omet 24204 | The image of a metric is a... |
xpsdsfn 24205 | Closure of the metric in a... |
xpsdsfn2 24206 | Closure of the metric in a... |
xpsxmetlem 24207 | Lemma for ~ xpsxmet . (Co... |
xpsxmet 24208 | A product metric of extend... |
xpsdsval 24209 | Value of the metric in a b... |
xpsmet 24210 | The direct product of two ... |
blfvalps 24211 | The value of the ball func... |
blfval 24212 | The value of the ball func... |
blvalps 24213 | The ball around a point ` ... |
blval 24214 | The ball around a point ` ... |
elblps 24215 | Membership in a ball. (Co... |
elbl 24216 | Membership in a ball. (Co... |
elbl2ps 24217 | Membership in a ball. (Co... |
elbl2 24218 | Membership in a ball. (Co... |
elbl3ps 24219 | Membership in a ball, with... |
elbl3 24220 | Membership in a ball, with... |
blcomps 24221 | Commute the arguments to t... |
blcom 24222 | Commute the arguments to t... |
xblpnfps 24223 | The infinity ball in an ex... |
xblpnf 24224 | The infinity ball in an ex... |
blpnf 24225 | The infinity ball in a sta... |
bldisj 24226 | Two balls are disjoint if ... |
blgt0 24227 | A nonempty ball implies th... |
bl2in 24228 | Two balls are disjoint if ... |
xblss2ps 24229 | One ball is contained in a... |
xblss2 24230 | One ball is contained in a... |
blss2ps 24231 | One ball is contained in a... |
blss2 24232 | One ball is contained in a... |
blhalf 24233 | A ball of radius ` R / 2 `... |
blfps 24234 | Mapping of a ball. (Contr... |
blf 24235 | Mapping of a ball. (Contr... |
blrnps 24236 | Membership in the range of... |
blrn 24237 | Membership in the range of... |
xblcntrps 24238 | A ball contains its center... |
xblcntr 24239 | A ball contains its center... |
blcntrps 24240 | A ball contains its center... |
blcntr 24241 | A ball contains its center... |
xbln0 24242 | A ball is nonempty iff the... |
bln0 24243 | A ball is not empty. (Con... |
blelrnps 24244 | A ball belongs to the set ... |
blelrn 24245 | A ball belongs to the set ... |
blssm 24246 | A ball is a subset of the ... |
unirnblps 24247 | The union of the set of ba... |
unirnbl 24248 | The union of the set of ba... |
blin 24249 | The intersection of two ba... |
ssblps 24250 | The size of a ball increas... |
ssbl 24251 | The size of a ball increas... |
blssps 24252 | Any point ` P ` in a ball ... |
blss 24253 | Any point ` P ` in a ball ... |
blssexps 24254 | Two ways to express the ex... |
blssex 24255 | Two ways to express the ex... |
ssblex 24256 | A nested ball exists whose... |
blin2 24257 | Given any two balls and a ... |
blbas 24258 | The balls of a metric spac... |
blres 24259 | A ball in a restricted met... |
xmeterval 24260 | Value of the "finitely sep... |
xmeter 24261 | The "finitely separated" r... |
xmetec 24262 | The equivalence classes un... |
blssec 24263 | A ball centered at ` P ` i... |
blpnfctr 24264 | The infinity ball in an ex... |
xmetresbl 24265 | An extended metric restric... |
mopnval 24266 | An open set is a subset of... |
mopntopon 24267 | The set of open sets of a ... |
mopntop 24268 | The set of open sets of a ... |
mopnuni 24269 | The union of all open sets... |
elmopn 24270 | The defining property of a... |
mopnfss 24271 | The family of open sets of... |
mopnm 24272 | The base set of a metric s... |
elmopn2 24273 | A defining property of an ... |
mopnss 24274 | An open set of a metric sp... |
isxms 24275 | Express the predicate " ` ... |
isxms2 24276 | Express the predicate " ` ... |
isms 24277 | Express the predicate " ` ... |
isms2 24278 | Express the predicate " ` ... |
xmstopn 24279 | The topology component of ... |
mstopn 24280 | The topology component of ... |
xmstps 24281 | An extended metric space i... |
msxms 24282 | A metric space is an exten... |
mstps 24283 | A metric space is a topolo... |
xmsxmet 24284 | The distance function, sui... |
msmet 24285 | The distance function, sui... |
msf 24286 | The distance function of a... |
xmsxmet2 24287 | The distance function, sui... |
msmet2 24288 | The distance function, sui... |
mscl 24289 | Closure of the distance fu... |
xmscl 24290 | Closure of the distance fu... |
xmsge0 24291 | The distance function in a... |
xmseq0 24292 | The distance between two p... |
xmssym 24293 | The distance function in a... |
xmstri2 24294 | Triangle inequality for th... |
mstri2 24295 | Triangle inequality for th... |
xmstri 24296 | Triangle inequality for th... |
mstri 24297 | Triangle inequality for th... |
xmstri3 24298 | Triangle inequality for th... |
mstri3 24299 | Triangle inequality for th... |
msrtri 24300 | Reverse triangle inequalit... |
xmspropd 24301 | Property deduction for an ... |
mspropd 24302 | Property deduction for a m... |
setsmsbas 24303 | The base set of a construc... |
setsmsbasOLD 24304 | Obsolete proof of ~ setsms... |
setsmsds 24305 | The distance function of a... |
setsmsdsOLD 24306 | Obsolete proof of ~ setsms... |
setsmstset 24307 | The topology of a construc... |
setsmstopn 24308 | The topology of a construc... |
setsxms 24309 | The constructed metric spa... |
setsms 24310 | The constructed metric spa... |
tmsval 24311 | For any metric there is an... |
tmslem 24312 | Lemma for ~ tmsbas , ~ tms... |
tmslemOLD 24313 | Obsolete version of ~ tmsl... |
tmsbas 24314 | The base set of a construc... |
tmsds 24315 | The metric of a constructe... |
tmstopn 24316 | The topology of a construc... |
tmsxms 24317 | The constructed metric spa... |
tmsms 24318 | The constructed metric spa... |
imasf1obl 24319 | The image of a metric spac... |
imasf1oxms 24320 | The image of a metric spac... |
imasf1oms 24321 | The image of a metric spac... |
prdsbl 24322 | A ball in the product metr... |
mopni 24323 | An open set of a metric sp... |
mopni2 24324 | An open set of a metric sp... |
mopni3 24325 | An open set of a metric sp... |
blssopn 24326 | The balls of a metric spac... |
unimopn 24327 | The union of a collection ... |
mopnin 24328 | The intersection of two op... |
mopn0 24329 | The empty set is an open s... |
rnblopn 24330 | A ball of a metric space i... |
blopn 24331 | A ball of a metric space i... |
neibl 24332 | The neighborhoods around a... |
blnei 24333 | A ball around a point is a... |
lpbl 24334 | Every ball around a limit ... |
blsscls2 24335 | A smaller closed ball is c... |
blcld 24336 | A "closed ball" in a metri... |
blcls 24337 | The closure of an open bal... |
blsscls 24338 | If two concentric balls ha... |
metss 24339 | Two ways of saying that me... |
metequiv 24340 | Two ways of saying that tw... |
metequiv2 24341 | If there is a sequence of ... |
metss2lem 24342 | Lemma for ~ metss2 . (Con... |
metss2 24343 | If the metric ` D ` is "st... |
comet 24344 | The composition of an exte... |
stdbdmetval 24345 | Value of the standard boun... |
stdbdxmet 24346 | The standard bounded metri... |
stdbdmet 24347 | The standard bounded metri... |
stdbdbl 24348 | The standard bounded metri... |
stdbdmopn 24349 | The standard bounded metri... |
mopnex 24350 | The topology generated by ... |
methaus 24351 | The topology generated by ... |
met1stc 24352 | The topology generated by ... |
met2ndci 24353 | A separable metric space (... |
met2ndc 24354 | A metric space is second-c... |
metrest 24355 | Two alternate formulations... |
ressxms 24356 | The restriction of a metri... |
ressms 24357 | The restriction of a metri... |
prdsmslem1 24358 | Lemma for ~ prdsms . The ... |
prdsxmslem1 24359 | Lemma for ~ prdsms . The ... |
prdsxmslem2 24360 | Lemma for ~ prdsxms . The... |
prdsxms 24361 | The indexed product struct... |
prdsms 24362 | The indexed product struct... |
pwsxms 24363 | A power of an extended met... |
pwsms 24364 | A power of a metric space ... |
xpsxms 24365 | A binary product of metric... |
xpsms 24366 | A binary product of metric... |
tmsxps 24367 | Express the product of two... |
tmsxpsmopn 24368 | Express the product of two... |
tmsxpsval 24369 | Value of the product of tw... |
tmsxpsval2 24370 | Value of the product of tw... |
metcnp3 24371 | Two ways to express that `... |
metcnp 24372 | Two ways to say a mapping ... |
metcnp2 24373 | Two ways to say a mapping ... |
metcn 24374 | Two ways to say a mapping ... |
metcnpi 24375 | Epsilon-delta property of ... |
metcnpi2 24376 | Epsilon-delta property of ... |
metcnpi3 24377 | Epsilon-delta property of ... |
txmetcnp 24378 | Continuity of a binary ope... |
txmetcn 24379 | Continuity of a binary ope... |
metuval 24380 | Value of the uniform struc... |
metustel 24381 | Define a filter base ` F `... |
metustss 24382 | Range of the elements of t... |
metustrel 24383 | Elements of the filter bas... |
metustto 24384 | Any two elements of the fi... |
metustid 24385 | The identity diagonal is i... |
metustsym 24386 | Elements of the filter bas... |
metustexhalf 24387 | For any element ` A ` of t... |
metustfbas 24388 | The filter base generated ... |
metust 24389 | The uniform structure gene... |
cfilucfil 24390 | Given a metric ` D ` and a... |
metuust 24391 | The uniform structure gene... |
cfilucfil2 24392 | Given a metric ` D ` and a... |
blval2 24393 | The ball around a point ` ... |
elbl4 24394 | Membership in a ball, alte... |
metuel 24395 | Elementhood in the uniform... |
metuel2 24396 | Elementhood in the uniform... |
metustbl 24397 | The "section" image of an ... |
psmetutop 24398 | The topology induced by a ... |
xmetutop 24399 | The topology induced by a ... |
xmsusp 24400 | If the uniform set of a me... |
restmetu 24401 | The uniform structure gene... |
metucn 24402 | Uniform continuity in metr... |
dscmet 24403 | The discrete metric on any... |
dscopn 24404 | The discrete metric genera... |
nrmmetd 24405 | Show that a group norm gen... |
abvmet 24406 | An absolute value ` F ` ge... |
nmfval 24419 | The value of the norm func... |
nmval 24420 | The value of the norm as t... |
nmfval0 24421 | The value of the norm func... |
nmfval2 24422 | The value of the norm func... |
nmval2 24423 | The value of the norm on a... |
nmf2 24424 | The norm on a metric group... |
nmpropd 24425 | Weak property deduction fo... |
nmpropd2 24426 | Strong property deduction ... |
isngp 24427 | The property of being a no... |
isngp2 24428 | The property of being a no... |
isngp3 24429 | The property of being a no... |
ngpgrp 24430 | A normed group is a group.... |
ngpms 24431 | A normed group is a metric... |
ngpxms 24432 | A normed group is an exten... |
ngptps 24433 | A normed group is a topolo... |
ngpmet 24434 | The (induced) metric of a ... |
ngpds 24435 | Value of the distance func... |
ngpdsr 24436 | Value of the distance func... |
ngpds2 24437 | Write the distance between... |
ngpds2r 24438 | Write the distance between... |
ngpds3 24439 | Write the distance between... |
ngpds3r 24440 | Write the distance between... |
ngprcan 24441 | Cancel right addition insi... |
ngplcan 24442 | Cancel left addition insid... |
isngp4 24443 | Express the property of be... |
ngpinvds 24444 | Two elements are the same ... |
ngpsubcan 24445 | Cancel right subtraction i... |
nmf 24446 | The norm on a normed group... |
nmcl 24447 | The norm of a normed group... |
nmge0 24448 | The norm of a normed group... |
nmeq0 24449 | The identity is the only e... |
nmne0 24450 | The norm of a nonzero elem... |
nmrpcl 24451 | The norm of a nonzero elem... |
nminv 24452 | The norm of a negated elem... |
nmmtri 24453 | The triangle inequality fo... |
nmsub 24454 | The norm of the difference... |
nmrtri 24455 | Reverse triangle inequalit... |
nm2dif 24456 | Inequality for the differe... |
nmtri 24457 | The triangle inequality fo... |
nmtri2 24458 | Triangle inequality for th... |
ngpi 24459 | The properties of a normed... |
nm0 24460 | Norm of the identity eleme... |
nmgt0 24461 | The norm of a nonzero elem... |
sgrim 24462 | The induced metric on a su... |
sgrimval 24463 | The induced metric on a su... |
subgnm 24464 | The norm in a subgroup. (... |
subgnm2 24465 | A substructure assigns the... |
subgngp 24466 | A normed group restricted ... |
ngptgp 24467 | A normed abelian group is ... |
ngppropd 24468 | Property deduction for a n... |
reldmtng 24469 | The function ` toNrmGrp ` ... |
tngval 24470 | Value of the function whic... |
tnglem 24471 | Lemma for ~ tngbas and sim... |
tnglemOLD 24472 | Obsolete version of ~ tngl... |
tngbas 24473 | The base set of a structur... |
tngbasOLD 24474 | Obsolete proof of ~ tngbas... |
tngplusg 24475 | The group addition of a st... |
tngplusgOLD 24476 | Obsolete proof of ~ tngplu... |
tng0 24477 | The group identity of a st... |
tngmulr 24478 | The ring multiplication of... |
tngmulrOLD 24479 | Obsolete proof of ~ tngmul... |
tngsca 24480 | The scalar ring of a struc... |
tngscaOLD 24481 | Obsolete proof of ~ tngsca... |
tngvsca 24482 | The scalar multiplication ... |
tngvscaOLD 24483 | Obsolete proof of ~ tngvsc... |
tngip 24484 | The inner product operatio... |
tngipOLD 24485 | Obsolete proof of ~ tngip ... |
tngds 24486 | The metric function of a s... |
tngdsOLD 24487 | Obsolete proof of ~ tngds ... |
tngtset 24488 | The topology generated by ... |
tngtopn 24489 | The topology generated by ... |
tngnm 24490 | The topology generated by ... |
tngngp2 24491 | A norm turns a group into ... |
tngngpd 24492 | Derive the axioms for a no... |
tngngp 24493 | Derive the axioms for a no... |
tnggrpr 24494 | If a structure equipped wi... |
tngngp3 24495 | Alternate definition of a ... |
nrmtngdist 24496 | The augmentation of a norm... |
nrmtngnrm 24497 | The augmentation of a norm... |
tngngpim 24498 | The induced metric of a no... |
isnrg 24499 | A normed ring is a ring wi... |
nrgabv 24500 | The norm of a normed ring ... |
nrgngp 24501 | A normed ring is a normed ... |
nrgring 24502 | A normed ring is a ring. ... |
nmmul 24503 | The norm of a product in a... |
nrgdsdi 24504 | Distribute a distance calc... |
nrgdsdir 24505 | Distribute a distance calc... |
nm1 24506 | The norm of one in a nonze... |
unitnmn0 24507 | The norm of a unit is nonz... |
nminvr 24508 | The norm of an inverse in ... |
nmdvr 24509 | The norm of a division in ... |
nrgdomn 24510 | A nonzero normed ring is a... |
nrgtgp 24511 | A normed ring is a topolog... |
subrgnrg 24512 | A normed ring restricted t... |
tngnrg 24513 | Given any absolute value o... |
isnlm 24514 | A normed (left) module is ... |
nmvs 24515 | Defining property of a nor... |
nlmngp 24516 | A normed module is a norme... |
nlmlmod 24517 | A normed module is a left ... |
nlmnrg 24518 | The scalar component of a ... |
nlmngp2 24519 | The scalar component of a ... |
nlmdsdi 24520 | Distribute a distance calc... |
nlmdsdir 24521 | Distribute a distance calc... |
nlmmul0or 24522 | If a scalar product is zer... |
sranlm 24523 | The subring algebra over a... |
nlmvscnlem2 24524 | Lemma for ~ nlmvscn . Com... |
nlmvscnlem1 24525 | Lemma for ~ nlmvscn . (Co... |
nlmvscn 24526 | The scalar multiplication ... |
rlmnlm 24527 | The ring module over a nor... |
rlmnm 24528 | The norm function in the r... |
nrgtrg 24529 | A normed ring is a topolog... |
nrginvrcnlem 24530 | Lemma for ~ nrginvrcn . C... |
nrginvrcn 24531 | The ring inverse function ... |
nrgtdrg 24532 | A normed division ring is ... |
nlmtlm 24533 | A normed module is a topol... |
isnvc 24534 | A normed vector space is j... |
nvcnlm 24535 | A normed vector space is a... |
nvclvec 24536 | A normed vector space is a... |
nvclmod 24537 | A normed vector space is a... |
isnvc2 24538 | A normed vector space is j... |
nvctvc 24539 | A normed vector space is a... |
lssnlm 24540 | A subspace of a normed mod... |
lssnvc 24541 | A subspace of a normed vec... |
rlmnvc 24542 | The ring module over a nor... |
ngpocelbl 24543 | Membership of an off-cente... |
nmoffn 24550 | The function producing ope... |
reldmnghm 24551 | Lemma for normed group hom... |
reldmnmhm 24552 | Lemma for module homomorph... |
nmofval 24553 | Value of the operator norm... |
nmoval 24554 | Value of the operator norm... |
nmogelb 24555 | Property of the operator n... |
nmolb 24556 | Any upper bound on the val... |
nmolb2d 24557 | Any upper bound on the val... |
nmof 24558 | The operator norm is a fun... |
nmocl 24559 | The operator norm of an op... |
nmoge0 24560 | The operator norm of an op... |
nghmfval 24561 | A normed group homomorphis... |
isnghm 24562 | A normed group homomorphis... |
isnghm2 24563 | A normed group homomorphis... |
isnghm3 24564 | A normed group homomorphis... |
bddnghm 24565 | A bounded group homomorphi... |
nghmcl 24566 | A normed group homomorphis... |
nmoi 24567 | The operator norm achieves... |
nmoix 24568 | The operator norm is a bou... |
nmoi2 24569 | The operator norm is a bou... |
nmoleub 24570 | The operator norm, defined... |
nghmrcl1 24571 | Reverse closure for a norm... |
nghmrcl2 24572 | Reverse closure for a norm... |
nghmghm 24573 | A normed group homomorphis... |
nmo0 24574 | The operator norm of the z... |
nmoeq0 24575 | The operator norm is zero ... |
nmoco 24576 | An upper bound on the oper... |
nghmco 24577 | The composition of normed ... |
nmotri 24578 | Triangle inequality for th... |
nghmplusg 24579 | The sum of two bounded lin... |
0nghm 24580 | The zero operator is a nor... |
nmoid 24581 | The operator norm of the i... |
idnghm 24582 | The identity operator is a... |
nmods 24583 | Upper bound for the distan... |
nghmcn 24584 | A normed group homomorphis... |
isnmhm 24585 | A normed module homomorphi... |
nmhmrcl1 24586 | Reverse closure for a norm... |
nmhmrcl2 24587 | Reverse closure for a norm... |
nmhmlmhm 24588 | A normed module homomorphi... |
nmhmnghm 24589 | A normed module homomorphi... |
nmhmghm 24590 | A normed module homomorphi... |
isnmhm2 24591 | A normed module homomorphi... |
nmhmcl 24592 | A normed module homomorphi... |
idnmhm 24593 | The identity operator is a... |
0nmhm 24594 | The zero operator is a bou... |
nmhmco 24595 | The composition of bounded... |
nmhmplusg 24596 | The sum of two bounded lin... |
qtopbaslem 24597 | The set of open intervals ... |
qtopbas 24598 | The set of open intervals ... |
retopbas 24599 | A basis for the standard t... |
retop 24600 | The standard topology on t... |
uniretop 24601 | The underlying set of the ... |
retopon 24602 | The standard topology on t... |
retps 24603 | The standard topological s... |
iooretop 24604 | Open intervals are open se... |
icccld 24605 | Closed intervals are close... |
icopnfcld 24606 | Right-unbounded closed int... |
iocmnfcld 24607 | Left-unbounded closed inte... |
qdensere 24608 | ` QQ ` is dense in the sta... |
cnmetdval 24609 | Value of the distance func... |
cnmet 24610 | The absolute value metric ... |
cnxmet 24611 | The absolute value metric ... |
cnbl0 24612 | Two ways to write the open... |
cnblcld 24613 | Two ways to write the clos... |
cnfldms 24614 | The complex number field i... |
cnfldxms 24615 | The complex number field i... |
cnfldtps 24616 | The complex number field i... |
cnfldnm 24617 | The norm of the field of c... |
cnngp 24618 | The complex numbers form a... |
cnnrg 24619 | The complex numbers form a... |
cnfldtopn 24620 | The topology of the comple... |
cnfldtopon 24621 | The topology of the comple... |
cnfldtop 24622 | The topology of the comple... |
cnfldhaus 24623 | The topology of the comple... |
unicntop 24624 | The underlying set of the ... |
cnopn 24625 | The set of complex numbers... |
zringnrg 24626 | The ring of integers is a ... |
remetdval 24627 | Value of the distance func... |
remet 24628 | The absolute value metric ... |
rexmet 24629 | The absolute value metric ... |
bl2ioo 24630 | A ball in terms of an open... |
ioo2bl 24631 | An open interval of reals ... |
ioo2blex 24632 | An open interval of reals ... |
blssioo 24633 | The balls of the standard ... |
tgioo 24634 | The topology generated by ... |
qdensere2 24635 | ` QQ ` is dense in ` RR ` ... |
blcvx 24636 | An open ball in the comple... |
rehaus 24637 | The standard topology on t... |
tgqioo 24638 | The topology generated by ... |
re2ndc 24639 | The standard topology on t... |
resubmet 24640 | The subspace topology indu... |
tgioo2 24641 | The standard topology on t... |
rerest 24642 | The subspace topology indu... |
tgioo3 24643 | The standard topology on t... |
xrtgioo 24644 | The topology on the extend... |
xrrest 24645 | The subspace topology indu... |
xrrest2 24646 | The subspace topology indu... |
xrsxmet 24647 | The metric on the extended... |
xrsdsre 24648 | The metric on the extended... |
xrsblre 24649 | Any ball of the metric of ... |
xrsmopn 24650 | The metric on the extended... |
zcld 24651 | The integers are a closed ... |
recld2 24652 | The real numbers are a clo... |
zcld2 24653 | The integers are a closed ... |
zdis 24654 | The integers are a discret... |
sszcld 24655 | Every subset of the intege... |
reperflem 24656 | A subset of the real numbe... |
reperf 24657 | The real numbers are a per... |
cnperf 24658 | The complex numbers are a ... |
iccntr 24659 | The interior of a closed i... |
icccmplem1 24660 | Lemma for ~ icccmp . (Con... |
icccmplem2 24661 | Lemma for ~ icccmp . (Con... |
icccmplem3 24662 | Lemma for ~ icccmp . (Con... |
icccmp 24663 | A closed interval in ` RR ... |
reconnlem1 24664 | Lemma for ~ reconn . Conn... |
reconnlem2 24665 | Lemma for ~ reconn . (Con... |
reconn 24666 | A subset of the reals is c... |
retopconn 24667 | Corollary of ~ reconn . T... |
iccconn 24668 | A closed interval is conne... |
opnreen 24669 | Every nonempty open set is... |
rectbntr0 24670 | A countable subset of the ... |
xrge0gsumle 24671 | A finite sum in the nonneg... |
xrge0tsms 24672 | Any finite or infinite sum... |
xrge0tsms2 24673 | Any finite or infinite sum... |
metdcnlem 24674 | The metric function of a m... |
xmetdcn2 24675 | The metric function of an ... |
xmetdcn 24676 | The metric function of an ... |
metdcn2 24677 | The metric function of a m... |
metdcn 24678 | The metric function of a m... |
msdcn 24679 | The metric function of a m... |
cnmpt1ds 24680 | Continuity of the metric f... |
cnmpt2ds 24681 | Continuity of the metric f... |
nmcn 24682 | The norm of a normed group... |
ngnmcncn 24683 | The norm of a normed group... |
abscn 24684 | The absolute value functio... |
metdsval 24685 | Value of the "distance to ... |
metdsf 24686 | The distance from a point ... |
metdsge 24687 | The distance from the poin... |
metds0 24688 | If a point is in a set, it... |
metdstri 24689 | A generalization of the tr... |
metdsle 24690 | The distance from a point ... |
metdsre 24691 | The distance from a point ... |
metdseq0 24692 | The distance from a point ... |
metdscnlem 24693 | Lemma for ~ metdscn . (Co... |
metdscn 24694 | The function ` F ` which g... |
metdscn2 24695 | The function ` F ` which g... |
metnrmlem1a 24696 | Lemma for ~ metnrm . (Con... |
metnrmlem1 24697 | Lemma for ~ metnrm . (Con... |
metnrmlem2 24698 | Lemma for ~ metnrm . (Con... |
metnrmlem3 24699 | Lemma for ~ metnrm . (Con... |
metnrm 24700 | A metric space is normal. ... |
metreg 24701 | A metric space is regular.... |
addcnlem 24702 | Lemma for ~ addcn , ~ subc... |
addcn 24703 | Complex number addition is... |
subcn 24704 | Complex number subtraction... |
mulcn 24705 | Complex number multiplicat... |
divcnOLD 24706 | Obsolete version of ~ divc... |
mpomulcn 24707 | Complex number multiplicat... |
divcn 24708 | Complex number division is... |
cnfldtgp 24709 | The complex numbers form a... |
fsumcn 24710 | A finite sum of functions ... |
fsum2cn 24711 | Version of ~ fsumcn for tw... |
expcn 24712 | The power function on comp... |
divccn 24713 | Division by a nonzero cons... |
expcnOLD 24714 | Obsolete version of ~ expc... |
divccnOLD 24715 | Obsolete version of ~ divc... |
sqcn 24716 | The square function on com... |
iitopon 24721 | The unit interval is a top... |
iitop 24722 | The unit interval is a top... |
iiuni 24723 | The base set of the unit i... |
dfii2 24724 | Alternate definition of th... |
dfii3 24725 | Alternate definition of th... |
dfii4 24726 | Alternate definition of th... |
dfii5 24727 | The unit interval expresse... |
iicmp 24728 | The unit interval is compa... |
iiconn 24729 | The unit interval is conne... |
cncfval 24730 | The value of the continuou... |
elcncf 24731 | Membership in the set of c... |
elcncf2 24732 | Version of ~ elcncf with a... |
cncfrss 24733 | Reverse closure of the con... |
cncfrss2 24734 | Reverse closure of the con... |
cncff 24735 | A continuous complex funct... |
cncfi 24736 | Defining property of a con... |
elcncf1di 24737 | Membership in the set of c... |
elcncf1ii 24738 | Membership in the set of c... |
rescncf 24739 | A continuous complex funct... |
cncfcdm 24740 | Change the codomain of a c... |
cncfss 24741 | The set of continuous func... |
climcncf 24742 | Image of a limit under a c... |
abscncf 24743 | Absolute value is continuo... |
recncf 24744 | Real part is continuous. ... |
imcncf 24745 | Imaginary part is continuo... |
cjcncf 24746 | Complex conjugate is conti... |
mulc1cncf 24747 | Multiplication by a consta... |
divccncf 24748 | Division by a constant is ... |
cncfco 24749 | The composition of two con... |
cncfcompt2 24750 | Composition of continuous ... |
cncfmet 24751 | Relate complex function co... |
cncfcn 24752 | Relate complex function co... |
cncfcn1 24753 | Relate complex function co... |
cncfmptc 24754 | A constant function is a c... |
cncfmptid 24755 | The identity function is a... |
cncfmpt1f 24756 | Composition of continuous ... |
cncfmpt2f 24757 | Composition of continuous ... |
cncfmpt2ss 24758 | Composition of continuous ... |
addccncf 24759 | Adding a constant is a con... |
idcncf 24760 | The identity function is a... |
sub1cncf 24761 | Subtracting a constant is ... |
sub2cncf 24762 | Subtraction from a constan... |
cdivcncf 24763 | Division with a constant n... |
negcncf 24764 | The negative function is c... |
negcncfOLD 24765 | Obsolete version of ~ negc... |
negfcncf 24766 | The negative of a continuo... |
abscncfALT 24767 | Absolute value is continuo... |
cncfcnvcn 24768 | Rewrite ~ cmphaushmeo for ... |
expcncf 24769 | The power function on comp... |
cnmptre 24770 | Lemma for ~ iirevcn and re... |
cnmpopc 24771 | Piecewise definition of a ... |
iirev 24772 | Reverse the unit interval.... |
iirevcn 24773 | The reversion function is ... |
iihalf1 24774 | Map the first half of ` II... |
iihalf1cn 24775 | The first half function is... |
iihalf1cnOLD 24776 | Obsolete version of ~ iiha... |
iihalf2 24777 | Map the second half of ` I... |
iihalf2cn 24778 | The second half function i... |
iihalf2cnOLD 24779 | Obsolete version of ~ iiha... |
elii1 24780 | Divide the unit interval i... |
elii2 24781 | Divide the unit interval i... |
iimulcl 24782 | The unit interval is close... |
iimulcn 24783 | Multiplication is a contin... |
iimulcnOLD 24784 | Obsolete version of ~ iimu... |
icoopnst 24785 | A half-open interval start... |
iocopnst 24786 | A half-open interval endin... |
icchmeo 24787 | The natural bijection from... |
icchmeoOLD 24788 | Obsolete version of ~ icch... |
icopnfcnv 24789 | Define a bijection from ` ... |
icopnfhmeo 24790 | The defined bijection from... |
iccpnfcnv 24791 | Define a bijection from ` ... |
iccpnfhmeo 24792 | The defined bijection from... |
xrhmeo 24793 | The bijection from ` [ -u ... |
xrhmph 24794 | The extended reals are hom... |
xrcmp 24795 | The topology of the extend... |
xrconn 24796 | The topology of the extend... |
icccvx 24797 | A linear combination of tw... |
oprpiece1res1 24798 | Restriction to the first p... |
oprpiece1res2 24799 | Restriction to the second ... |
cnrehmeo 24800 | The canonical bijection fr... |
cnrehmeoOLD 24801 | Obsolete version of ~ cnre... |
cnheiborlem 24802 | Lemma for ~ cnheibor . (C... |
cnheibor 24803 | Heine-Borel theorem for co... |
cnllycmp 24804 | The topology on the comple... |
rellycmp 24805 | The topology on the reals ... |
bndth 24806 | The Boundedness Theorem. ... |
evth 24807 | The Extreme Value Theorem.... |
evth2 24808 | The Extreme Value Theorem,... |
lebnumlem1 24809 | Lemma for ~ lebnum . The ... |
lebnumlem2 24810 | Lemma for ~ lebnum . As a... |
lebnumlem3 24811 | Lemma for ~ lebnum . By t... |
lebnum 24812 | The Lebesgue number lemma,... |
xlebnum 24813 | Generalize ~ lebnum to ext... |
lebnumii 24814 | Specialize the Lebesgue nu... |
ishtpy 24820 | Membership in the class of... |
htpycn 24821 | A homotopy is a continuous... |
htpyi 24822 | A homotopy evaluated at it... |
ishtpyd 24823 | Deduction for membership i... |
htpycom 24824 | Given a homotopy from ` F ... |
htpyid 24825 | A homotopy from a function... |
htpyco1 24826 | Compose a homotopy with a ... |
htpyco2 24827 | Compose a homotopy with a ... |
htpycc 24828 | Concatenate two homotopies... |
isphtpy 24829 | Membership in the class of... |
phtpyhtpy 24830 | A path homotopy is a homot... |
phtpycn 24831 | A path homotopy is a conti... |
phtpyi 24832 | Membership in the class of... |
phtpy01 24833 | Two path-homotopic paths h... |
isphtpyd 24834 | Deduction for membership i... |
isphtpy2d 24835 | Deduction for membership i... |
phtpycom 24836 | Given a homotopy from ` F ... |
phtpyid 24837 | A homotopy from a path to ... |
phtpyco2 24838 | Compose a path homotopy wi... |
phtpycc 24839 | Concatenate two path homot... |
phtpcrel 24841 | The path homotopy relation... |
isphtpc 24842 | The relation "is path homo... |
phtpcer 24843 | Path homotopy is an equiva... |
phtpc01 24844 | Path homotopic paths have ... |
reparphti 24845 | Lemma for ~ reparpht . (C... |
reparphtiOLD 24846 | Obsolete version of ~ repa... |
reparpht 24847 | Reparametrization lemma. ... |
phtpcco2 24848 | Compose a path homotopy wi... |
pcofval 24859 | The value of the path conc... |
pcoval 24860 | The concatenation of two p... |
pcovalg 24861 | Evaluate the concatenation... |
pcoval1 24862 | Evaluate the concatenation... |
pco0 24863 | The starting point of a pa... |
pco1 24864 | The ending point of a path... |
pcoval2 24865 | Evaluate the concatenation... |
pcocn 24866 | The concatenation of two p... |
copco 24867 | The composition of a conca... |
pcohtpylem 24868 | Lemma for ~ pcohtpy . (Co... |
pcohtpy 24869 | Homotopy invariance of pat... |
pcoptcl 24870 | A constant function is a p... |
pcopt 24871 | Concatenation with a point... |
pcopt2 24872 | Concatenation with a point... |
pcoass 24873 | Order of concatenation doe... |
pcorevcl 24874 | Closure for a reversed pat... |
pcorevlem 24875 | Lemma for ~ pcorev . Prov... |
pcorev 24876 | Concatenation with the rev... |
pcorev2 24877 | Concatenation with the rev... |
pcophtb 24878 | The path homotopy equivale... |
om1val 24879 | The definition of the loop... |
om1bas 24880 | The base set of the loop s... |
om1elbas 24881 | Elementhood in the base se... |
om1addcl 24882 | Closure of the group opera... |
om1plusg 24883 | The group operation (which... |
om1tset 24884 | The topology of the loop s... |
om1opn 24885 | The topology of the loop s... |
pi1val 24886 | The definition of the fund... |
pi1bas 24887 | The base set of the fundam... |
pi1blem 24888 | Lemma for ~ pi1buni . (Co... |
pi1buni 24889 | Another way to write the l... |
pi1bas2 24890 | The base set of the fundam... |
pi1eluni 24891 | Elementhood in the base se... |
pi1bas3 24892 | The base set of the fundam... |
pi1cpbl 24893 | The group operation, loop ... |
elpi1 24894 | The elements of the fundam... |
elpi1i 24895 | The elements of the fundam... |
pi1addf 24896 | The group operation of ` p... |
pi1addval 24897 | The concatenation of two p... |
pi1grplem 24898 | Lemma for ~ pi1grp . (Con... |
pi1grp 24899 | The fundamental group is a... |
pi1id 24900 | The identity element of th... |
pi1inv 24901 | An inverse in the fundamen... |
pi1xfrf 24902 | Functionality of the loop ... |
pi1xfrval 24903 | The value of the loop tran... |
pi1xfr 24904 | Given a path ` F ` and its... |
pi1xfrcnvlem 24905 | Given a path ` F ` between... |
pi1xfrcnv 24906 | Given a path ` F ` between... |
pi1xfrgim 24907 | The mapping ` G ` between ... |
pi1cof 24908 | Functionality of the loop ... |
pi1coval 24909 | The value of the loop tran... |
pi1coghm 24910 | The mapping ` G ` between ... |
isclm 24913 | A subcomplex module is a l... |
clmsca 24914 | The ring of scalars ` F ` ... |
clmsubrg 24915 | The base set of the ring o... |
clmlmod 24916 | A subcomplex module is a l... |
clmgrp 24917 | A subcomplex module is an ... |
clmabl 24918 | A subcomplex module is an ... |
clmring 24919 | The scalar ring of a subco... |
clmfgrp 24920 | The scalar ring of a subco... |
clm0 24921 | The zero of the scalar rin... |
clm1 24922 | The identity of the scalar... |
clmadd 24923 | The addition of the scalar... |
clmmul 24924 | The multiplication of the ... |
clmcj 24925 | The conjugation of the sca... |
isclmi 24926 | Reverse direction of ~ isc... |
clmzss 24927 | The scalar ring of a subco... |
clmsscn 24928 | The scalar ring of a subco... |
clmsub 24929 | Subtraction in the scalar ... |
clmneg 24930 | Negation in the scalar rin... |
clmneg1 24931 | Minus one is in the scalar... |
clmabs 24932 | Norm in the scalar ring of... |
clmacl 24933 | Closure of ring addition f... |
clmmcl 24934 | Closure of ring multiplica... |
clmsubcl 24935 | Closure of ring subtractio... |
lmhmclm 24936 | The domain of a linear ope... |
clmvscl 24937 | Closure of scalar product ... |
clmvsass 24938 | Associative law for scalar... |
clmvscom 24939 | Commutative law for the sc... |
clmvsdir 24940 | Distributive law for scala... |
clmvsdi 24941 | Distributive law for scala... |
clmvs1 24942 | Scalar product with ring u... |
clmvs2 24943 | A vector plus itself is tw... |
clm0vs 24944 | Zero times a vector is the... |
clmopfne 24945 | The (functionalized) opera... |
isclmp 24946 | The predicate "is a subcom... |
isclmi0 24947 | Properties that determine ... |
clmvneg1 24948 | Minus 1 times a vector is ... |
clmvsneg 24949 | Multiplication of a vector... |
clmmulg 24950 | The group multiple functio... |
clmsubdir 24951 | Scalar multiplication dist... |
clmpm1dir 24952 | Subtractive distributive l... |
clmnegneg 24953 | Double negative of a vecto... |
clmnegsubdi2 24954 | Distribution of negative o... |
clmsub4 24955 | Rearrangement of 4 terms i... |
clmvsrinv 24956 | A vector minus itself. (C... |
clmvslinv 24957 | Minus a vector plus itself... |
clmvsubval 24958 | Value of vector subtractio... |
clmvsubval2 24959 | Value of vector subtractio... |
clmvz 24960 | Two ways to express the ne... |
zlmclm 24961 | The ` ZZ ` -module operati... |
clmzlmvsca 24962 | The scalar product of a su... |
nmoleub2lem 24963 | Lemma for ~ nmoleub2a and ... |
nmoleub2lem3 24964 | Lemma for ~ nmoleub2a and ... |
nmoleub2lem2 24965 | Lemma for ~ nmoleub2a and ... |
nmoleub2a 24966 | The operator norm is the s... |
nmoleub2b 24967 | The operator norm is the s... |
nmoleub3 24968 | The operator norm is the s... |
nmhmcn 24969 | A linear operator over a n... |
cmodscexp 24970 | The powers of ` _i ` belon... |
cmodscmulexp 24971 | The scalar product of a ve... |
cvslvec 24974 | A subcomplex vector space ... |
cvsclm 24975 | A subcomplex vector space ... |
iscvs 24976 | A subcomplex vector space ... |
iscvsp 24977 | The predicate "is a subcom... |
iscvsi 24978 | Properties that determine ... |
cvsi 24979 | The properties of a subcom... |
cvsunit 24980 | Unit group of the scalar r... |
cvsdiv 24981 | Division of the scalar rin... |
cvsdivcl 24982 | The scalar field of a subc... |
cvsmuleqdivd 24983 | An equality involving rati... |
cvsdiveqd 24984 | An equality involving rati... |
cnlmodlem1 24985 | Lemma 1 for ~ cnlmod . (C... |
cnlmodlem2 24986 | Lemma 2 for ~ cnlmod . (C... |
cnlmodlem3 24987 | Lemma 3 for ~ cnlmod . (C... |
cnlmod4 24988 | Lemma 4 for ~ cnlmod . (C... |
cnlmod 24989 | The set of complex numbers... |
cnstrcvs 24990 | The set of complex numbers... |
cnrbas 24991 | The set of complex numbers... |
cnrlmod 24992 | The complex left module of... |
cnrlvec 24993 | The complex left module of... |
cncvs 24994 | The complex left module of... |
recvs 24995 | The field of the real numb... |
recvsOLD 24996 | Obsolete version of ~ recv... |
qcvs 24997 | The field of rational numb... |
zclmncvs 24998 | The ring of integers as le... |
isncvsngp 24999 | A normed subcomplex vector... |
isncvsngpd 25000 | Properties that determine ... |
ncvsi 25001 | The properties of a normed... |
ncvsprp 25002 | Proportionality property o... |
ncvsge0 25003 | The norm of a scalar produ... |
ncvsm1 25004 | The norm of the opposite o... |
ncvsdif 25005 | The norm of the difference... |
ncvspi 25006 | The norm of a vector plus ... |
ncvs1 25007 | From any nonzero vector of... |
cnrnvc 25008 | The module of complex numb... |
cnncvs 25009 | The module of complex numb... |
cnnm 25010 | The norm of the normed sub... |
ncvspds 25011 | Value of the distance func... |
cnindmet 25012 | The metric induced on the ... |
cnncvsaddassdemo 25013 | Derive the associative law... |
cnncvsmulassdemo 25014 | Derive the associative law... |
cnncvsabsnegdemo 25015 | Derive the absolute value ... |
iscph 25020 | A subcomplex pre-Hilbert s... |
cphphl 25021 | A subcomplex pre-Hilbert s... |
cphnlm 25022 | A subcomplex pre-Hilbert s... |
cphngp 25023 | A subcomplex pre-Hilbert s... |
cphlmod 25024 | A subcomplex pre-Hilbert s... |
cphlvec 25025 | A subcomplex pre-Hilbert s... |
cphnvc 25026 | A subcomplex pre-Hilbert s... |
cphsubrglem 25027 | Lemma for ~ cphsubrg . (C... |
cphreccllem 25028 | Lemma for ~ cphreccl . (C... |
cphsca 25029 | A subcomplex pre-Hilbert s... |
cphsubrg 25030 | The scalar field of a subc... |
cphreccl 25031 | The scalar field of a subc... |
cphdivcl 25032 | The scalar field of a subc... |
cphcjcl 25033 | The scalar field of a subc... |
cphsqrtcl 25034 | The scalar field of a subc... |
cphabscl 25035 | The scalar field of a subc... |
cphsqrtcl2 25036 | The scalar field of a subc... |
cphsqrtcl3 25037 | If the scalar field of a s... |
cphqss 25038 | The scalar field of a subc... |
cphclm 25039 | A subcomplex pre-Hilbert s... |
cphnmvs 25040 | Norm of a scalar product. ... |
cphipcl 25041 | An inner product is a memb... |
cphnmfval 25042 | The value of the norm in a... |
cphnm 25043 | The square of the norm is ... |
nmsq 25044 | The square of the norm is ... |
cphnmf 25045 | The norm of a vector is a ... |
cphnmcl 25046 | The norm of a vector is a ... |
reipcl 25047 | An inner product of an ele... |
ipge0 25048 | The inner product in a sub... |
cphipcj 25049 | Conjugate of an inner prod... |
cphipipcj 25050 | An inner product times its... |
cphorthcom 25051 | Orthogonality (meaning inn... |
cphip0l 25052 | Inner product with a zero ... |
cphip0r 25053 | Inner product with a zero ... |
cphipeq0 25054 | The inner product of a vec... |
cphdir 25055 | Distributive law for inner... |
cphdi 25056 | Distributive law for inner... |
cph2di 25057 | Distributive law for inner... |
cphsubdir 25058 | Distributive law for inner... |
cphsubdi 25059 | Distributive law for inner... |
cph2subdi 25060 | Distributive law for inner... |
cphass 25061 | Associative law for inner ... |
cphassr 25062 | "Associative" law for seco... |
cph2ass 25063 | Move scalar multiplication... |
cphassi 25064 | Associative law for the fi... |
cphassir 25065 | "Associative" law for the ... |
cphpyth 25066 | The pythagorean theorem fo... |
tcphex 25067 | Lemma for ~ tcphbas and si... |
tcphval 25068 | Define a function to augme... |
tcphbas 25069 | The base set of a subcompl... |
tchplusg 25070 | The addition operation of ... |
tcphsub 25071 | The subtraction operation ... |
tcphmulr 25072 | The ring operation of a su... |
tcphsca 25073 | The scalar field of a subc... |
tcphvsca 25074 | The scalar multiplication ... |
tcphip 25075 | The inner product of a sub... |
tcphtopn 25076 | The topology of a subcompl... |
tcphphl 25077 | Augmentation of a subcompl... |
tchnmfval 25078 | The norm of a subcomplex p... |
tcphnmval 25079 | The norm of a subcomplex p... |
cphtcphnm 25080 | The norm of a norm-augment... |
tcphds 25081 | The distance of a pre-Hilb... |
phclm 25082 | A pre-Hilbert space whose ... |
tcphcphlem3 25083 | Lemma for ~ tcphcph : real... |
ipcau2 25084 | The Cauchy-Schwarz inequal... |
tcphcphlem1 25085 | Lemma for ~ tcphcph : the ... |
tcphcphlem2 25086 | Lemma for ~ tcphcph : homo... |
tcphcph 25087 | The standard definition of... |
ipcau 25088 | The Cauchy-Schwarz inequal... |
nmparlem 25089 | Lemma for ~ nmpar . (Cont... |
nmpar 25090 | A subcomplex pre-Hilbert s... |
cphipval2 25091 | Value of the inner product... |
4cphipval2 25092 | Four times the inner produ... |
cphipval 25093 | Value of the inner product... |
ipcnlem2 25094 | The inner product operatio... |
ipcnlem1 25095 | The inner product operatio... |
ipcn 25096 | The inner product operatio... |
cnmpt1ip 25097 | Continuity of inner produc... |
cnmpt2ip 25098 | Continuity of inner produc... |
csscld 25099 | A "closed subspace" in a s... |
clsocv 25100 | The orthogonal complement ... |
cphsscph 25101 | A subspace of a subcomplex... |
lmmbr 25108 | Express the binary relatio... |
lmmbr2 25109 | Express the binary relatio... |
lmmbr3 25110 | Express the binary relatio... |
lmmcvg 25111 | Convergence property of a ... |
lmmbrf 25112 | Express the binary relatio... |
lmnn 25113 | A condition that implies c... |
cfilfval 25114 | The set of Cauchy filters ... |
iscfil 25115 | The property of being a Ca... |
iscfil2 25116 | The property of being a Ca... |
cfilfil 25117 | A Cauchy filter is a filte... |
cfili 25118 | Property of a Cauchy filte... |
cfil3i 25119 | A Cauchy filter contains b... |
cfilss 25120 | A filter finer than a Cauc... |
fgcfil 25121 | The Cauchy filter conditio... |
fmcfil 25122 | The Cauchy filter conditio... |
iscfil3 25123 | A filter is Cauchy iff it ... |
cfilfcls 25124 | Similar to ultrafilters ( ... |
caufval 25125 | The set of Cauchy sequence... |
iscau 25126 | Express the property " ` F... |
iscau2 25127 | Express the property " ` F... |
iscau3 25128 | Express the Cauchy sequenc... |
iscau4 25129 | Express the property " ` F... |
iscauf 25130 | Express the property " ` F... |
caun0 25131 | A metric with a Cauchy seq... |
caufpm 25132 | Inclusion of a Cauchy sequ... |
caucfil 25133 | A Cauchy sequence predicat... |
iscmet 25134 | The property " ` D ` is a ... |
cmetcvg 25135 | The convergence of a Cauch... |
cmetmet 25136 | A complete metric space is... |
cmetmeti 25137 | A complete metric space is... |
cmetcaulem 25138 | Lemma for ~ cmetcau . (Co... |
cmetcau 25139 | The convergence of a Cauch... |
iscmet3lem3 25140 | Lemma for ~ iscmet3 . (Co... |
iscmet3lem1 25141 | Lemma for ~ iscmet3 . (Co... |
iscmet3lem2 25142 | Lemma for ~ iscmet3 . (Co... |
iscmet3 25143 | The property " ` D ` is a ... |
iscmet2 25144 | A metric ` D ` is complete... |
cfilresi 25145 | A Cauchy filter on a metri... |
cfilres 25146 | Cauchy filter on a metric ... |
caussi 25147 | Cauchy sequence on a metri... |
causs 25148 | Cauchy sequence on a metri... |
equivcfil 25149 | If the metric ` D ` is "st... |
equivcau 25150 | If the metric ` D ` is "st... |
lmle 25151 | If the distance from each ... |
nglmle 25152 | If the norm of each member... |
lmclim 25153 | Relate a limit on the metr... |
lmclimf 25154 | Relate a limit on the metr... |
metelcls 25155 | A point belongs to the clo... |
metcld 25156 | A subset of a metric space... |
metcld2 25157 | A subset of a metric space... |
caubl 25158 | Sufficient condition to en... |
caublcls 25159 | The convergent point of a ... |
metcnp4 25160 | Two ways to say a mapping ... |
metcn4 25161 | Two ways to say a mapping ... |
iscmet3i 25162 | Properties that determine ... |
lmcau 25163 | Every convergent sequence ... |
flimcfil 25164 | Every convergent filter in... |
metsscmetcld 25165 | A complete subspace of a m... |
cmetss 25166 | A subspace of a complete m... |
equivcmet 25167 | If two metrics are strongl... |
relcmpcmet 25168 | If ` D ` is a metric space... |
cmpcmet 25169 | A compact metric space is ... |
cfilucfil3 25170 | Given a metric ` D ` and a... |
cfilucfil4 25171 | Given a metric ` D ` and a... |
cncmet 25172 | The set of complex numbers... |
recmet 25173 | The real numbers are a com... |
bcthlem1 25174 | Lemma for ~ bcth . Substi... |
bcthlem2 25175 | Lemma for ~ bcth . The ba... |
bcthlem3 25176 | Lemma for ~ bcth . The li... |
bcthlem4 25177 | Lemma for ~ bcth . Given ... |
bcthlem5 25178 | Lemma for ~ bcth . The pr... |
bcth 25179 | Baire's Category Theorem. ... |
bcth2 25180 | Baire's Category Theorem, ... |
bcth3 25181 | Baire's Category Theorem, ... |
isbn 25188 | A Banach space is a normed... |
bnsca 25189 | The scalar field of a Bana... |
bnnvc 25190 | A Banach space is a normed... |
bnnlm 25191 | A Banach space is a normed... |
bnngp 25192 | A Banach space is a normed... |
bnlmod 25193 | A Banach space is a left m... |
bncms 25194 | A Banach space is a comple... |
iscms 25195 | A complete metric space is... |
cmscmet 25196 | The induced metric on a co... |
bncmet 25197 | The induced metric on Bana... |
cmsms 25198 | A complete metric space is... |
cmspropd 25199 | Property deduction for a c... |
cmssmscld 25200 | The restriction of a metri... |
cmsss 25201 | The restriction of a compl... |
lssbn 25202 | A subspace of a Banach spa... |
cmetcusp1 25203 | If the uniform set of a co... |
cmetcusp 25204 | The uniform space generate... |
cncms 25205 | The field of complex numbe... |
cnflduss 25206 | The uniform structure of t... |
cnfldcusp 25207 | The field of complex numbe... |
resscdrg 25208 | The real numbers are a sub... |
cncdrg 25209 | The only complete subfield... |
srabn 25210 | The subring algebra over a... |
rlmbn 25211 | The ring module over a com... |
ishl 25212 | The predicate "is a subcom... |
hlbn 25213 | Every subcomplex Hilbert s... |
hlcph 25214 | Every subcomplex Hilbert s... |
hlphl 25215 | Every subcomplex Hilbert s... |
hlcms 25216 | Every subcomplex Hilbert s... |
hlprlem 25217 | Lemma for ~ hlpr . (Contr... |
hlress 25218 | The scalar field of a subc... |
hlpr 25219 | The scalar field of a subc... |
ishl2 25220 | A Hilbert space is a compl... |
cphssphl 25221 | A Banach subspace of a sub... |
cmslssbn 25222 | A complete linear subspace... |
cmscsscms 25223 | A closed subspace of a com... |
bncssbn 25224 | A closed subspace of a Ban... |
cssbn 25225 | A complete subspace of a n... |
csschl 25226 | A complete subspace of a c... |
cmslsschl 25227 | A complete linear subspace... |
chlcsschl 25228 | A closed subspace of a sub... |
retopn 25229 | The topology of the real n... |
recms 25230 | The real numbers form a co... |
reust 25231 | The Uniform structure of t... |
recusp 25232 | The real numbers form a co... |
rrxval 25237 | Value of the generalized E... |
rrxbase 25238 | The base of the generalize... |
rrxprds 25239 | Expand the definition of t... |
rrxip 25240 | The inner product of the g... |
rrxnm 25241 | The norm of the generalize... |
rrxcph 25242 | Generalized Euclidean real... |
rrxds 25243 | The distance over generali... |
rrxvsca 25244 | The scalar product over ge... |
rrxplusgvscavalb 25245 | The result of the addition... |
rrxsca 25246 | The field of real numbers ... |
rrx0 25247 | The zero ("origin") in a g... |
rrx0el 25248 | The zero ("origin") in a g... |
csbren 25249 | Cauchy-Schwarz-Bunjakovsky... |
trirn 25250 | Triangle inequality in R^n... |
rrxf 25251 | Euclidean vectors as funct... |
rrxfsupp 25252 | Euclidean vectors are of f... |
rrxsuppss 25253 | Support of Euclidean vecto... |
rrxmvallem 25254 | Support of the function us... |
rrxmval 25255 | The value of the Euclidean... |
rrxmfval 25256 | The value of the Euclidean... |
rrxmetlem 25257 | Lemma for ~ rrxmet . (Con... |
rrxmet 25258 | Euclidean space is a metri... |
rrxdstprj1 25259 | The distance between two p... |
rrxbasefi 25260 | The base of the generalize... |
rrxdsfi 25261 | The distance over generali... |
rrxmetfi 25262 | Euclidean space is a metri... |
rrxdsfival 25263 | The value of the Euclidean... |
ehlval 25264 | Value of the Euclidean spa... |
ehlbase 25265 | The base of the Euclidean ... |
ehl0base 25266 | The base of the Euclidean ... |
ehl0 25267 | The Euclidean space of dim... |
ehleudis 25268 | The Euclidean distance fun... |
ehleudisval 25269 | The value of the Euclidean... |
ehl1eudis 25270 | The Euclidean distance fun... |
ehl1eudisval 25271 | The value of the Euclidean... |
ehl2eudis 25272 | The Euclidean distance fun... |
ehl2eudisval 25273 | The value of the Euclidean... |
minveclem1 25274 | Lemma for ~ minvec . The ... |
minveclem4c 25275 | Lemma for ~ minvec . The ... |
minveclem2 25276 | Lemma for ~ minvec . Any ... |
minveclem3a 25277 | Lemma for ~ minvec . ` D `... |
minveclem3b 25278 | Lemma for ~ minvec . The ... |
minveclem3 25279 | Lemma for ~ minvec . The ... |
minveclem4a 25280 | Lemma for ~ minvec . ` F `... |
minveclem4b 25281 | Lemma for ~ minvec . The ... |
minveclem4 25282 | Lemma for ~ minvec . The ... |
minveclem5 25283 | Lemma for ~ minvec . Disc... |
minveclem6 25284 | Lemma for ~ minvec . Any ... |
minveclem7 25285 | Lemma for ~ minvec . Sinc... |
minvec 25286 | Minimizing vector theorem,... |
pjthlem1 25287 | Lemma for ~ pjth . (Contr... |
pjthlem2 25288 | Lemma for ~ pjth . (Contr... |
pjth 25289 | Projection Theorem: Any H... |
pjth2 25290 | Projection Theorem with ab... |
cldcss 25291 | Corollary of the Projectio... |
cldcss2 25292 | Corollary of the Projectio... |
hlhil 25293 | Corollary of the Projectio... |
addcncf 25294 | The addition of two contin... |
subcncf 25295 | The addition of two contin... |
mulcncf 25296 | The multiplication of two ... |
mulcncfOLD 25297 | Obsolete version of ~ mulc... |
divcncf 25298 | The quotient of two contin... |
pmltpclem1 25299 | Lemma for ~ pmltpc . (Con... |
pmltpclem2 25300 | Lemma for ~ pmltpc . (Con... |
pmltpc 25301 | Any function on the reals ... |
ivthlem1 25302 | Lemma for ~ ivth . The se... |
ivthlem2 25303 | Lemma for ~ ivth . Show t... |
ivthlem3 25304 | Lemma for ~ ivth , the int... |
ivth 25305 | The intermediate value the... |
ivth2 25306 | The intermediate value the... |
ivthle 25307 | The intermediate value the... |
ivthle2 25308 | The intermediate value the... |
ivthicc 25309 | The interval between any t... |
evthicc 25310 | Specialization of the Extr... |
evthicc2 25311 | Combine ~ ivthicc with ~ e... |
cniccbdd 25312 | A continuous function on a... |
ovolfcl 25317 | Closure for the interval e... |
ovolfioo 25318 | Unpack the interval coveri... |
ovolficc 25319 | Unpack the interval coveri... |
ovolficcss 25320 | Any (closed) interval cove... |
ovolfsval 25321 | The value of the interval ... |
ovolfsf 25322 | Closure for the interval l... |
ovolsf 25323 | Closure for the partial su... |
ovolval 25324 | The value of the outer mea... |
elovolmlem 25325 | Lemma for ~ elovolm and re... |
elovolm 25326 | Elementhood in the set ` M... |
elovolmr 25327 | Sufficient condition for e... |
ovolmge0 25328 | The set ` M ` is composed ... |
ovolcl 25329 | The volume of a set is an ... |
ovollb 25330 | The outer volume is a lowe... |
ovolgelb 25331 | The outer volume is the gr... |
ovolge0 25332 | The volume of a set is alw... |
ovolf 25333 | The domain and codomain of... |
ovollecl 25334 | If an outer volume is boun... |
ovolsslem 25335 | Lemma for ~ ovolss . (Con... |
ovolss 25336 | The volume of a set is mon... |
ovolsscl 25337 | If a set is contained in a... |
ovolssnul 25338 | A subset of a nullset is n... |
ovollb2lem 25339 | Lemma for ~ ovollb2 . (Co... |
ovollb2 25340 | It is often more convenien... |
ovolctb 25341 | The volume of a denumerabl... |
ovolq 25342 | The rational numbers have ... |
ovolctb2 25343 | The volume of a countable ... |
ovol0 25344 | The empty set has 0 outer ... |
ovolfi 25345 | A finite set has 0 outer L... |
ovolsn 25346 | A singleton has 0 outer Le... |
ovolunlem1a 25347 | Lemma for ~ ovolun . (Con... |
ovolunlem1 25348 | Lemma for ~ ovolun . (Con... |
ovolunlem2 25349 | Lemma for ~ ovolun . (Con... |
ovolun 25350 | The Lebesgue outer measure... |
ovolunnul 25351 | Adding a nullset does not ... |
ovolfiniun 25352 | The Lebesgue outer measure... |
ovoliunlem1 25353 | Lemma for ~ ovoliun . (Co... |
ovoliunlem2 25354 | Lemma for ~ ovoliun . (Co... |
ovoliunlem3 25355 | Lemma for ~ ovoliun . (Co... |
ovoliun 25356 | The Lebesgue outer measure... |
ovoliun2 25357 | The Lebesgue outer measure... |
ovoliunnul 25358 | A countable union of nulls... |
shft2rab 25359 | If ` B ` is a shift of ` A... |
ovolshftlem1 25360 | Lemma for ~ ovolshft . (C... |
ovolshftlem2 25361 | Lemma for ~ ovolshft . (C... |
ovolshft 25362 | The Lebesgue outer measure... |
sca2rab 25363 | If ` B ` is a scale of ` A... |
ovolscalem1 25364 | Lemma for ~ ovolsca . (Co... |
ovolscalem2 25365 | Lemma for ~ ovolshft . (C... |
ovolsca 25366 | The Lebesgue outer measure... |
ovolicc1 25367 | The measure of a closed in... |
ovolicc2lem1 25368 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem2 25369 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem3 25370 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem4 25371 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem5 25372 | Lemma for ~ ovolicc2 . (C... |
ovolicc2 25373 | The measure of a closed in... |
ovolicc 25374 | The measure of a closed in... |
ovolicopnf 25375 | The measure of a right-unb... |
ovolre 25376 | The measure of the real nu... |
ismbl 25377 | The predicate " ` A ` is L... |
ismbl2 25378 | From ~ ovolun , it suffice... |
volres 25379 | A self-referencing abbrevi... |
volf 25380 | The domain and codomain of... |
mblvol 25381 | The volume of a measurable... |
mblss 25382 | A measurable set is a subs... |
mblsplit 25383 | The defining property of m... |
volss 25384 | The Lebesgue measure is mo... |
cmmbl 25385 | The complement of a measur... |
nulmbl 25386 | A nullset is measurable. ... |
nulmbl2 25387 | A set of outer measure zer... |
unmbl 25388 | A union of measurable sets... |
shftmbl 25389 | A shift of a measurable se... |
0mbl 25390 | The empty set is measurabl... |
rembl 25391 | The set of all real number... |
unidmvol 25392 | The union of the Lebesgue ... |
inmbl 25393 | An intersection of measura... |
difmbl 25394 | A difference of measurable... |
finiunmbl 25395 | A finite union of measurab... |
volun 25396 | The Lebesgue measure funct... |
volinun 25397 | Addition of non-disjoint s... |
volfiniun 25398 | The volume of a disjoint f... |
iundisj 25399 | Rewrite a countable union ... |
iundisj2 25400 | A disjoint union is disjoi... |
voliunlem1 25401 | Lemma for ~ voliun . (Con... |
voliunlem2 25402 | Lemma for ~ voliun . (Con... |
voliunlem3 25403 | Lemma for ~ voliun . (Con... |
iunmbl 25404 | The measurable sets are cl... |
voliun 25405 | The Lebesgue measure funct... |
volsuplem 25406 | Lemma for ~ volsup . (Con... |
volsup 25407 | The volume of the limit of... |
iunmbl2 25408 | The measurable sets are cl... |
ioombl1lem1 25409 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem2 25410 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem3 25411 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem4 25412 | Lemma for ~ ioombl1 . (Co... |
ioombl1 25413 | An open right-unbounded in... |
icombl1 25414 | A closed unbounded-above i... |
icombl 25415 | A closed-below, open-above... |
ioombl 25416 | An open real interval is m... |
iccmbl 25417 | A closed real interval is ... |
iccvolcl 25418 | A closed real interval has... |
ovolioo 25419 | The measure of an open int... |
volioo 25420 | The measure of an open int... |
ioovolcl 25421 | An open real interval has ... |
ovolfs2 25422 | Alternative expression for... |
ioorcl2 25423 | An open interval with fini... |
ioorf 25424 | Define a function from ope... |
ioorval 25425 | Define a function from ope... |
ioorinv2 25426 | The function ` F ` is an "... |
ioorinv 25427 | The function ` F ` is an "... |
ioorcl 25428 | The function ` F ` does no... |
uniiccdif 25429 | A union of closed interval... |
uniioovol 25430 | A disjoint union of open i... |
uniiccvol 25431 | An almost-disjoint union o... |
uniioombllem1 25432 | Lemma for ~ uniioombl . (... |
uniioombllem2a 25433 | Lemma for ~ uniioombl . (... |
uniioombllem2 25434 | Lemma for ~ uniioombl . (... |
uniioombllem3a 25435 | Lemma for ~ uniioombl . (... |
uniioombllem3 25436 | Lemma for ~ uniioombl . (... |
uniioombllem4 25437 | Lemma for ~ uniioombl . (... |
uniioombllem5 25438 | Lemma for ~ uniioombl . (... |
uniioombllem6 25439 | Lemma for ~ uniioombl . (... |
uniioombl 25440 | A disjoint union of open i... |
uniiccmbl 25441 | An almost-disjoint union o... |
dyadf 25442 | The function ` F ` returns... |
dyadval 25443 | Value of the dyadic ration... |
dyadovol 25444 | Volume of a dyadic rationa... |
dyadss 25445 | Two closed dyadic rational... |
dyaddisjlem 25446 | Lemma for ~ dyaddisj . (C... |
dyaddisj 25447 | Two closed dyadic rational... |
dyadmaxlem 25448 | Lemma for ~ dyadmax . (Co... |
dyadmax 25449 | Any nonempty set of dyadic... |
dyadmbllem 25450 | Lemma for ~ dyadmbl . (Co... |
dyadmbl 25451 | Any union of dyadic ration... |
opnmbllem 25452 | Lemma for ~ opnmbl . (Con... |
opnmbl 25453 | All open sets are measurab... |
opnmblALT 25454 | All open sets are measurab... |
subopnmbl 25455 | Sets which are open in a m... |
volsup2 25456 | The volume of ` A ` is the... |
volcn 25457 | The function formed by res... |
volivth 25458 | The Intermediate Value The... |
vitalilem1 25459 | Lemma for ~ vitali . (Con... |
vitalilem2 25460 | Lemma for ~ vitali . (Con... |
vitalilem3 25461 | Lemma for ~ vitali . (Con... |
vitalilem4 25462 | Lemma for ~ vitali . (Con... |
vitalilem5 25463 | Lemma for ~ vitali . (Con... |
vitali 25464 | If the reals can be well-o... |
ismbf1 25475 | The predicate " ` F ` is a... |
mbff 25476 | A measurable function is a... |
mbfdm 25477 | The domain of a measurable... |
mbfconstlem 25478 | Lemma for ~ mbfconst and r... |
ismbf 25479 | The predicate " ` F ` is a... |
ismbfcn 25480 | A complex function is meas... |
mbfima 25481 | Definitional property of a... |
mbfimaicc 25482 | The preimage of any closed... |
mbfimasn 25483 | The preimage of a point un... |
mbfconst 25484 | A constant function is mea... |
mbf0 25485 | The empty function is meas... |
mbfid 25486 | The identity function is m... |
mbfmptcl 25487 | Lemma for the ` MblFn ` pr... |
mbfdm2 25488 | The domain of a measurable... |
ismbfcn2 25489 | A complex function is meas... |
ismbfd 25490 | Deduction to prove measura... |
ismbf2d 25491 | Deduction to prove measura... |
mbfeqalem1 25492 | Lemma for ~ mbfeqalem2 . ... |
mbfeqalem2 25493 | Lemma for ~ mbfeqa . (Con... |
mbfeqa 25494 | If two functions are equal... |
mbfres 25495 | The restriction of a measu... |
mbfres2 25496 | Measurability of a piecewi... |
mbfss 25497 | Change the domain of a mea... |
mbfmulc2lem 25498 | Multiplication by a consta... |
mbfmulc2re 25499 | Multiplication by a consta... |
mbfmax 25500 | The maximum of two functio... |
mbfneg 25501 | The negative of a measurab... |
mbfpos 25502 | The positive part of a mea... |
mbfposr 25503 | Converse to ~ mbfpos . (C... |
mbfposb 25504 | A function is measurable i... |
ismbf3d 25505 | Simplified form of ~ ismbf... |
mbfimaopnlem 25506 | Lemma for ~ mbfimaopn . (... |
mbfimaopn 25507 | The preimage of any open s... |
mbfimaopn2 25508 | The preimage of any set op... |
cncombf 25509 | The composition of a conti... |
cnmbf 25510 | A continuous function is m... |
mbfaddlem 25511 | The sum of two measurable ... |
mbfadd 25512 | The sum of two measurable ... |
mbfsub 25513 | The difference of two meas... |
mbfmulc2 25514 | A complex constant times a... |
mbfsup 25515 | The supremum of a sequence... |
mbfinf 25516 | The infimum of a sequence ... |
mbflimsup 25517 | The limit supremum of a se... |
mbflimlem 25518 | The pointwise limit of a s... |
mbflim 25519 | The pointwise limit of a s... |
0pval 25522 | The zero function evaluate... |
0plef 25523 | Two ways to say that the f... |
0pledm 25524 | Adjust the domain of the l... |
isi1f 25525 | The predicate " ` F ` is a... |
i1fmbf 25526 | Simple functions are measu... |
i1ff 25527 | A simple function is a fun... |
i1frn 25528 | A simple function has fini... |
i1fima 25529 | Any preimage of a simple f... |
i1fima2 25530 | Any preimage of a simple f... |
i1fima2sn 25531 | Preimage of a singleton. ... |
i1fd 25532 | A simplified set of assump... |
i1f0rn 25533 | Any simple function takes ... |
itg1val 25534 | The value of the integral ... |
itg1val2 25535 | The value of the integral ... |
itg1cl 25536 | Closure of the integral on... |
itg1ge0 25537 | Closure of the integral on... |
i1f0 25538 | The zero function is simpl... |
itg10 25539 | The zero function has zero... |
i1f1lem 25540 | Lemma for ~ i1f1 and ~ itg... |
i1f1 25541 | Base case simple functions... |
itg11 25542 | The integral of an indicat... |
itg1addlem1 25543 | Decompose a preimage, whic... |
i1faddlem 25544 | Decompose the preimage of ... |
i1fmullem 25545 | Decompose the preimage of ... |
i1fadd 25546 | The sum of two simple func... |
i1fmul 25547 | The pointwise product of t... |
itg1addlem2 25548 | Lemma for ~ itg1add . The... |
itg1addlem3 25549 | Lemma for ~ itg1add . (Co... |
itg1addlem4 25550 | Lemma for ~ itg1add . (Co... |
itg1addlem4OLD 25551 | Obsolete version of ~ itg1... |
itg1addlem5 25552 | Lemma for ~ itg1add . (Co... |
itg1add 25553 | The integral of a sum of s... |
i1fmulclem 25554 | Decompose the preimage of ... |
i1fmulc 25555 | A nonnegative constant tim... |
itg1mulc 25556 | The integral of a constant... |
i1fres 25557 | The "restriction" of a sim... |
i1fpos 25558 | The positive part of a sim... |
i1fposd 25559 | Deduction form of ~ i1fpos... |
i1fsub 25560 | The difference of two simp... |
itg1sub 25561 | The integral of a differen... |
itg10a 25562 | The integral of a simple f... |
itg1ge0a 25563 | The integral of an almost ... |
itg1lea 25564 | Approximate version of ~ i... |
itg1le 25565 | If one simple function dom... |
itg1climres 25566 | Restricting the simple fun... |
mbfi1fseqlem1 25567 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem2 25568 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem3 25569 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem4 25570 | Lemma for ~ mbfi1fseq . T... |
mbfi1fseqlem5 25571 | Lemma for ~ mbfi1fseq . V... |
mbfi1fseqlem6 25572 | Lemma for ~ mbfi1fseq . V... |
mbfi1fseq 25573 | A characterization of meas... |
mbfi1flimlem 25574 | Lemma for ~ mbfi1flim . (... |
mbfi1flim 25575 | Any real measurable functi... |
mbfmullem2 25576 | Lemma for ~ mbfmul . (Con... |
mbfmullem 25577 | Lemma for ~ mbfmul . (Con... |
mbfmul 25578 | The product of two measura... |
itg2lcl 25579 | The set of lower sums is a... |
itg2val 25580 | Value of the integral on n... |
itg2l 25581 | Elementhood in the set ` L... |
itg2lr 25582 | Sufficient condition for e... |
xrge0f 25583 | A real function is a nonne... |
itg2cl 25584 | The integral of a nonnegat... |
itg2ub 25585 | The integral of a nonnegat... |
itg2leub 25586 | Any upper bound on the int... |
itg2ge0 25587 | The integral of a nonnegat... |
itg2itg1 25588 | The integral of a nonnegat... |
itg20 25589 | The integral of the zero f... |
itg2lecl 25590 | If an ` S.2 ` integral is ... |
itg2le 25591 | If one function dominates ... |
itg2const 25592 | Integral of a constant fun... |
itg2const2 25593 | When the base set of a con... |
itg2seq 25594 | Definitional property of t... |
itg2uba 25595 | Approximate version of ~ i... |
itg2lea 25596 | Approximate version of ~ i... |
itg2eqa 25597 | Approximate equality of in... |
itg2mulclem 25598 | Lemma for ~ itg2mulc . (C... |
itg2mulc 25599 | The integral of a nonnegat... |
itg2splitlem 25600 | Lemma for ~ itg2split . (... |
itg2split 25601 | The ` S.2 ` integral split... |
itg2monolem1 25602 | Lemma for ~ itg2mono . We... |
itg2monolem2 25603 | Lemma for ~ itg2mono . (C... |
itg2monolem3 25604 | Lemma for ~ itg2mono . (C... |
itg2mono 25605 | The Monotone Convergence T... |
itg2i1fseqle 25606 | Subject to the conditions ... |
itg2i1fseq 25607 | Subject to the conditions ... |
itg2i1fseq2 25608 | In an extension to the res... |
itg2i1fseq3 25609 | Special case of ~ itg2i1fs... |
itg2addlem 25610 | Lemma for ~ itg2add . (Co... |
itg2add 25611 | The ` S.2 ` integral is li... |
itg2gt0 25612 | If the function ` F ` is s... |
itg2cnlem1 25613 | Lemma for ~ itgcn . (Cont... |
itg2cnlem2 25614 | Lemma for ~ itgcn . (Cont... |
itg2cn 25615 | A sort of absolute continu... |
ibllem 25616 | Conditioned equality theor... |
isibl 25617 | The predicate " ` F ` is i... |
isibl2 25618 | The predicate " ` F ` is i... |
iblmbf 25619 | An integrable function is ... |
iblitg 25620 | If a function is integrabl... |
dfitg 25621 | Evaluate the class substit... |
itgex 25622 | An integral is a set. (Co... |
itgeq1f 25623 | Equality theorem for an in... |
itgeq1 25624 | Equality theorem for an in... |
nfitg1 25625 | Bound-variable hypothesis ... |
nfitg 25626 | Bound-variable hypothesis ... |
cbvitg 25627 | Change bound variable in a... |
cbvitgv 25628 | Change bound variable in a... |
itgeq2 25629 | Equality theorem for an in... |
itgresr 25630 | The domain of an integral ... |
itg0 25631 | The integral of anything o... |
itgz 25632 | The integral of zero on an... |
itgeq2dv 25633 | Equality theorem for an in... |
itgmpt 25634 | Change bound variable in a... |
itgcl 25635 | The integral of an integra... |
itgvallem 25636 | Substitution lemma. (Cont... |
itgvallem3 25637 | Lemma for ~ itgposval and ... |
ibl0 25638 | The zero function is integ... |
iblcnlem1 25639 | Lemma for ~ iblcnlem . (C... |
iblcnlem 25640 | Expand out the universal q... |
itgcnlem 25641 | Expand out the sum in ~ df... |
iblrelem 25642 | Integrability of a real fu... |
iblposlem 25643 | Lemma for ~ iblpos . (Con... |
iblpos 25644 | Integrability of a nonnega... |
iblre 25645 | Integrability of a real fu... |
itgrevallem1 25646 | Lemma for ~ itgposval and ... |
itgposval 25647 | The integral of a nonnegat... |
itgreval 25648 | Decompose the integral of ... |
itgrecl 25649 | Real closure of an integra... |
iblcn 25650 | Integrability of a complex... |
itgcnval 25651 | Decompose the integral of ... |
itgre 25652 | Real part of an integral. ... |
itgim 25653 | Imaginary part of an integ... |
iblneg 25654 | The negative of an integra... |
itgneg 25655 | Negation of an integral. ... |
iblss 25656 | A subset of an integrable ... |
iblss2 25657 | Change the domain of an in... |
itgitg2 25658 | Transfer an integral using... |
i1fibl 25659 | A simple function is integ... |
itgitg1 25660 | Transfer an integral using... |
itgle 25661 | Monotonicity of an integra... |
itgge0 25662 | The integral of a positive... |
itgss 25663 | Expand the set of an integ... |
itgss2 25664 | Expand the set of an integ... |
itgeqa 25665 | Approximate equality of in... |
itgss3 25666 | Expand the set of an integ... |
itgioo 25667 | Equality of integrals on o... |
itgless 25668 | Expand the integral of a n... |
iblconst 25669 | A constant function is int... |
itgconst 25670 | Integral of a constant fun... |
ibladdlem 25671 | Lemma for ~ ibladd . (Con... |
ibladd 25672 | Add two integrals over the... |
iblsub 25673 | Subtract two integrals ove... |
itgaddlem1 25674 | Lemma for ~ itgadd . (Con... |
itgaddlem2 25675 | Lemma for ~ itgadd . (Con... |
itgadd 25676 | Add two integrals over the... |
itgsub 25677 | Subtract two integrals ove... |
itgfsum 25678 | Take a finite sum of integ... |
iblabslem 25679 | Lemma for ~ iblabs . (Con... |
iblabs 25680 | The absolute value of an i... |
iblabsr 25681 | A measurable function is i... |
iblmulc2 25682 | Multiply an integral by a ... |
itgmulc2lem1 25683 | Lemma for ~ itgmulc2 : pos... |
itgmulc2lem2 25684 | Lemma for ~ itgmulc2 : rea... |
itgmulc2 25685 | Multiply an integral by a ... |
itgabs 25686 | The triangle inequality fo... |
itgsplit 25687 | The ` S. ` integral splits... |
itgspliticc 25688 | The ` S. ` integral splits... |
itgsplitioo 25689 | The ` S. ` integral splits... |
bddmulibl 25690 | A bounded function times a... |
bddibl 25691 | A bounded function is inte... |
cniccibl 25692 | A continuous function on a... |
bddiblnc 25693 | Choice-free proof of ~ bdd... |
cnicciblnc 25694 | Choice-free proof of ~ cni... |
itggt0 25695 | The integral of a strictly... |
itgcn 25696 | Transfer ~ itg2cn to the f... |
ditgeq1 25699 | Equality theorem for the d... |
ditgeq2 25700 | Equality theorem for the d... |
ditgeq3 25701 | Equality theorem for the d... |
ditgeq3dv 25702 | Equality theorem for the d... |
ditgex 25703 | A directed integral is a s... |
ditg0 25704 | Value of the directed inte... |
cbvditg 25705 | Change bound variable in a... |
cbvditgv 25706 | Change bound variable in a... |
ditgpos 25707 | Value of the directed inte... |
ditgneg 25708 | Value of the directed inte... |
ditgcl 25709 | Closure of a directed inte... |
ditgswap 25710 | Reverse a directed integra... |
ditgsplitlem 25711 | Lemma for ~ ditgsplit . (... |
ditgsplit 25712 | This theorem is the raison... |
reldv 25721 | The derivative function is... |
limcvallem 25722 | Lemma for ~ ellimc . (Con... |
limcfval 25723 | Value and set bounds on th... |
ellimc 25724 | Value of the limit predica... |
limcrcl 25725 | Reverse closure for the li... |
limccl 25726 | Closure of the limit opera... |
limcdif 25727 | It suffices to consider fu... |
ellimc2 25728 | Write the definition of a ... |
limcnlp 25729 | If ` B ` is not a limit po... |
ellimc3 25730 | Write the epsilon-delta de... |
limcflflem 25731 | Lemma for ~ limcflf . (Co... |
limcflf 25732 | The limit operator can be ... |
limcmo 25733 | If ` B ` is a limit point ... |
limcmpt 25734 | Express the limit operator... |
limcmpt2 25735 | Express the limit operator... |
limcresi 25736 | Any limit of ` F ` is also... |
limcres 25737 | If ` B ` is an interior po... |
cnplimc 25738 | A function is continuous a... |
cnlimc 25739 | ` F ` is a continuous func... |
cnlimci 25740 | If ` F ` is a continuous f... |
cnmptlimc 25741 | If ` F ` is a continuous f... |
limccnp 25742 | If the limit of ` F ` at `... |
limccnp2 25743 | The image of a convergent ... |
limcco 25744 | Composition of two limits.... |
limciun 25745 | A point is a limit of ` F ... |
limcun 25746 | A point is a limit of ` F ... |
dvlem 25747 | Closure for a difference q... |
dvfval 25748 | Value and set bounds on th... |
eldv 25749 | The differentiable predica... |
dvcl 25750 | The derivative function ta... |
dvbssntr 25751 | The set of differentiable ... |
dvbss 25752 | The set of differentiable ... |
dvbsss 25753 | The set of differentiable ... |
perfdvf 25754 | The derivative is a functi... |
recnprss 25755 | Both ` RR ` and ` CC ` are... |
recnperf 25756 | Both ` RR ` and ` CC ` are... |
dvfg 25757 | Explicitly write out the f... |
dvf 25758 | The derivative is a functi... |
dvfcn 25759 | The derivative is a functi... |
dvreslem 25760 | Lemma for ~ dvres . (Cont... |
dvres2lem 25761 | Lemma for ~ dvres2 . (Con... |
dvres 25762 | Restriction of a derivativ... |
dvres2 25763 | Restriction of the base se... |
dvres3 25764 | Restriction of a complex d... |
dvres3a 25765 | Restriction of a complex d... |
dvidlem 25766 | Lemma for ~ dvid and ~ dvc... |
dvmptresicc 25767 | Derivative of a function r... |
dvconst 25768 | Derivative of a constant f... |
dvid 25769 | Derivative of the identity... |
dvcnp 25770 | The difference quotient is... |
dvcnp2 25771 | A function is continuous a... |
dvcnp2OLD 25772 | Obsolete version of ~ dvcn... |
dvcn 25773 | A differentiable function ... |
dvnfval 25774 | Value of the iterated deri... |
dvnff 25775 | The iterated derivative is... |
dvn0 25776 | Zero times iterated deriva... |
dvnp1 25777 | Successor iterated derivat... |
dvn1 25778 | One times iterated derivat... |
dvnf 25779 | The N-times derivative is ... |
dvnbss 25780 | The set of N-times differe... |
dvnadd 25781 | The ` N ` -th derivative o... |
dvn2bss 25782 | An N-times differentiable ... |
dvnres 25783 | Multiple derivative versio... |
cpnfval 25784 | Condition for n-times cont... |
fncpn 25785 | The ` C^n ` object is a fu... |
elcpn 25786 | Condition for n-times cont... |
cpnord 25787 | ` C^n ` conditions are ord... |
cpncn 25788 | A ` C^n ` function is cont... |
cpnres 25789 | The restriction of a ` C^n... |
dvaddbr 25790 | The sum rule for derivativ... |
dvmulbr 25791 | The product rule for deriv... |
dvmulbrOLD 25792 | Obsolete version of ~ dvmu... |
dvadd 25793 | The sum rule for derivativ... |
dvmul 25794 | The product rule for deriv... |
dvaddf 25795 | The sum rule for everywher... |
dvmulf 25796 | The product rule for every... |
dvcmul 25797 | The product rule when one ... |
dvcmulf 25798 | The product rule when one ... |
dvcobr 25799 | The chain rule for derivat... |
dvcobrOLD 25800 | Obsolete version of ~ dvco... |
dvco 25801 | The chain rule for derivat... |
dvcof 25802 | The chain rule for everywh... |
dvcjbr 25803 | The derivative of the conj... |
dvcj 25804 | The derivative of the conj... |
dvfre 25805 | The derivative of a real f... |
dvnfre 25806 | The ` N ` -th derivative o... |
dvexp 25807 | Derivative of a power func... |
dvexp2 25808 | Derivative of an exponenti... |
dvrec 25809 | Derivative of the reciproc... |
dvmptres3 25810 | Function-builder for deriv... |
dvmptid 25811 | Function-builder for deriv... |
dvmptc 25812 | Function-builder for deriv... |
dvmptcl 25813 | Closure lemma for ~ dvmptc... |
dvmptadd 25814 | Function-builder for deriv... |
dvmptmul 25815 | Function-builder for deriv... |
dvmptres2 25816 | Function-builder for deriv... |
dvmptres 25817 | Function-builder for deriv... |
dvmptcmul 25818 | Function-builder for deriv... |
dvmptdivc 25819 | Function-builder for deriv... |
dvmptneg 25820 | Function-builder for deriv... |
dvmptsub 25821 | Function-builder for deriv... |
dvmptcj 25822 | Function-builder for deriv... |
dvmptre 25823 | Function-builder for deriv... |
dvmptim 25824 | Function-builder for deriv... |
dvmptntr 25825 | Function-builder for deriv... |
dvmptco 25826 | Function-builder for deriv... |
dvrecg 25827 | Derivative of the reciproc... |
dvmptdiv 25828 | Function-builder for deriv... |
dvmptfsum 25829 | Function-builder for deriv... |
dvcnvlem 25830 | Lemma for ~ dvcnvre . (Co... |
dvcnv 25831 | A weak version of ~ dvcnvr... |
dvexp3 25832 | Derivative of an exponenti... |
dveflem 25833 | Derivative of the exponent... |
dvef 25834 | Derivative of the exponent... |
dvsincos 25835 | Derivative of the sine and... |
dvsin 25836 | Derivative of the sine fun... |
dvcos 25837 | Derivative of the cosine f... |
dvferm1lem 25838 | Lemma for ~ dvferm . (Con... |
dvferm1 25839 | One-sided version of ~ dvf... |
dvferm2lem 25840 | Lemma for ~ dvferm . (Con... |
dvferm2 25841 | One-sided version of ~ dvf... |
dvferm 25842 | Fermat's theorem on statio... |
rollelem 25843 | Lemma for ~ rolle . (Cont... |
rolle 25844 | Rolle's theorem. If ` F `... |
cmvth 25845 | Cauchy's Mean Value Theore... |
cmvthOLD 25846 | Obsolete version of ~ cmvt... |
mvth 25847 | The Mean Value Theorem. I... |
dvlip 25848 | A function with derivative... |
dvlipcn 25849 | A complex function with de... |
dvlip2 25850 | Combine the results of ~ d... |
c1liplem1 25851 | Lemma for ~ c1lip1 . (Con... |
c1lip1 25852 | C^1 functions are Lipschit... |
c1lip2 25853 | C^1 functions are Lipschit... |
c1lip3 25854 | C^1 functions are Lipschit... |
dveq0 25855 | If a continuous function h... |
dv11cn 25856 | Two functions defined on a... |
dvgt0lem1 25857 | Lemma for ~ dvgt0 and ~ dv... |
dvgt0lem2 25858 | Lemma for ~ dvgt0 and ~ dv... |
dvgt0 25859 | A function on a closed int... |
dvlt0 25860 | A function on a closed int... |
dvge0 25861 | A function on a closed int... |
dvle 25862 | If ` A ( x ) , C ( x ) ` a... |
dvivthlem1 25863 | Lemma for ~ dvivth . (Con... |
dvivthlem2 25864 | Lemma for ~ dvivth . (Con... |
dvivth 25865 | Darboux' theorem, or the i... |
dvne0 25866 | A function on a closed int... |
dvne0f1 25867 | A function on a closed int... |
lhop1lem 25868 | Lemma for ~ lhop1 . (Cont... |
lhop1 25869 | L'Hôpital's Rule for... |
lhop2 25870 | L'Hôpital's Rule for... |
lhop 25871 | L'Hôpital's Rule. I... |
dvcnvrelem1 25872 | Lemma for ~ dvcnvre . (Co... |
dvcnvrelem2 25873 | Lemma for ~ dvcnvre . (Co... |
dvcnvre 25874 | The derivative rule for in... |
dvcvx 25875 | A real function with stric... |
dvfsumle 25876 | Compare a finite sum to an... |
dvfsumleOLD 25877 | Obsolete version of ~ dvfs... |
dvfsumge 25878 | Compare a finite sum to an... |
dvfsumabs 25879 | Compare a finite sum to an... |
dvmptrecl 25880 | Real closure of a derivati... |
dvfsumrlimf 25881 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem1 25882 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem2 25883 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem2OLD 25884 | Obsolete version of ~ dvfs... |
dvfsumlem3 25885 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem4 25886 | Lemma for ~ dvfsumrlim . ... |
dvfsumrlimge0 25887 | Lemma for ~ dvfsumrlim . ... |
dvfsumrlim 25888 | Compare a finite sum to an... |
dvfsumrlim2 25889 | Compare a finite sum to an... |
dvfsumrlim3 25890 | Conjoin the statements of ... |
dvfsum2 25891 | The reverse of ~ dvfsumrli... |
ftc1lem1 25892 | Lemma for ~ ftc1a and ~ ft... |
ftc1lem2 25893 | Lemma for ~ ftc1 . (Contr... |
ftc1a 25894 | The Fundamental Theorem of... |
ftc1lem3 25895 | Lemma for ~ ftc1 . (Contr... |
ftc1lem4 25896 | Lemma for ~ ftc1 . (Contr... |
ftc1lem5 25897 | Lemma for ~ ftc1 . (Contr... |
ftc1lem6 25898 | Lemma for ~ ftc1 . (Contr... |
ftc1 25899 | The Fundamental Theorem of... |
ftc1cn 25900 | Strengthen the assumptions... |
ftc2 25901 | The Fundamental Theorem of... |
ftc2ditglem 25902 | Lemma for ~ ftc2ditg . (C... |
ftc2ditg 25903 | Directed integral analogue... |
itgparts 25904 | Integration by parts. If ... |
itgsubstlem 25905 | Lemma for ~ itgsubst . (C... |
itgsubst 25906 | Integration by ` u ` -subs... |
itgpowd 25907 | The integral of a monomial... |
reldmmdeg 25912 | Multivariate degree is a b... |
tdeglem1 25913 | Functionality of the total... |
tdeglem1OLD 25914 | Obsolete version of ~ tdeg... |
tdeglem3 25915 | Additivity of the total de... |
tdeglem3OLD 25916 | Obsolete version of ~ tdeg... |
tdeglem4 25917 | There is only one multi-in... |
tdeglem4OLD 25918 | Obsolete version of ~ tdeg... |
tdeglem2 25919 | Simplification of total de... |
mdegfval 25920 | Value of the multivariate ... |
mdegval 25921 | Value of the multivariate ... |
mdegleb 25922 | Property of being of limit... |
mdeglt 25923 | If there is an upper limit... |
mdegldg 25924 | A nonzero polynomial has s... |
mdegxrcl 25925 | Closure of polynomial degr... |
mdegxrf 25926 | Functionality of polynomia... |
mdegcl 25927 | Sharp closure for multivar... |
mdeg0 25928 | Degree of the zero polynom... |
mdegnn0cl 25929 | Degree of a nonzero polyno... |
degltlem1 25930 | Theorem on arithmetic of e... |
degltp1le 25931 | Theorem on arithmetic of e... |
mdegaddle 25932 | The degree of a sum is at ... |
mdegvscale 25933 | The degree of a scalar mul... |
mdegvsca 25934 | The degree of a scalar mul... |
mdegle0 25935 | A polynomial has nonpositi... |
mdegmullem 25936 | Lemma for ~ mdegmulle2 . ... |
mdegmulle2 25937 | The multivariate degree of... |
deg1fval 25938 | Relate univariate polynomi... |
deg1xrf 25939 | Functionality of univariat... |
deg1xrcl 25940 | Closure of univariate poly... |
deg1cl 25941 | Sharp closure of univariat... |
mdegpropd 25942 | Property deduction for pol... |
deg1fvi 25943 | Univariate polynomial degr... |
deg1propd 25944 | Property deduction for pol... |
deg1z 25945 | Degree of the zero univari... |
deg1nn0cl 25946 | Degree of a nonzero univar... |
deg1n0ima 25947 | Degree image of a set of p... |
deg1nn0clb 25948 | A polynomial is nonzero if... |
deg1lt0 25949 | A polynomial is zero iff i... |
deg1ldg 25950 | A nonzero univariate polyn... |
deg1ldgn 25951 | An index at which a polyno... |
deg1ldgdomn 25952 | A nonzero univariate polyn... |
deg1leb 25953 | Property of being of limit... |
deg1val 25954 | Value of the univariate de... |
deg1lt 25955 | If the degree of a univari... |
deg1ge 25956 | Conversely, a nonzero coef... |
coe1mul3 25957 | The coefficient vector of ... |
coe1mul4 25958 | Value of the "leading" coe... |
deg1addle 25959 | The degree of a sum is at ... |
deg1addle2 25960 | If both factors have degre... |
deg1add 25961 | Exact degree of a sum of t... |
deg1vscale 25962 | The degree of a scalar tim... |
deg1vsca 25963 | The degree of a scalar tim... |
deg1invg 25964 | The degree of the negated ... |
deg1suble 25965 | The degree of a difference... |
deg1sub 25966 | Exact degree of a differen... |
deg1mulle2 25967 | Produce a bound on the pro... |
deg1sublt 25968 | Subtraction of two polynom... |
deg1le0 25969 | A polynomial has nonpositi... |
deg1sclle 25970 | A scalar polynomial has no... |
deg1scl 25971 | A nonzero scalar polynomia... |
deg1mul2 25972 | Degree of multiplication o... |
deg1mul3 25973 | Degree of multiplication o... |
deg1mul3le 25974 | Degree of multiplication o... |
deg1tmle 25975 | Limiting degree of a polyn... |
deg1tm 25976 | Exact degree of a polynomi... |
deg1pwle 25977 | Limiting degree of a varia... |
deg1pw 25978 | Exact degree of a variable... |
ply1nz 25979 | Univariate polynomials ove... |
ply1nzb 25980 | Univariate polynomials are... |
ply1domn 25981 | Corollary of ~ deg1mul2 : ... |
ply1idom 25982 | The ring of univariate pol... |
ply1divmo 25993 | Uniqueness of a quotient i... |
ply1divex 25994 | Lemma for ~ ply1divalg : e... |
ply1divalg 25995 | The division algorithm for... |
ply1divalg2 25996 | Reverse the order of multi... |
uc1pval 25997 | Value of the set of unitic... |
isuc1p 25998 | Being a unitic polynomial.... |
mon1pval 25999 | Value of the set of monic ... |
ismon1p 26000 | Being a monic polynomial. ... |
uc1pcl 26001 | Unitic polynomials are pol... |
mon1pcl 26002 | Monic polynomials are poly... |
uc1pn0 26003 | Unitic polynomials are not... |
mon1pn0 26004 | Monic polynomials are not ... |
uc1pdeg 26005 | Unitic polynomials have no... |
uc1pldg 26006 | Unitic polynomials have un... |
mon1pldg 26007 | Unitic polynomials have on... |
mon1puc1p 26008 | Monic polynomials are unit... |
uc1pmon1p 26009 | Make a unitic polynomial m... |
deg1submon1p 26010 | The difference of two moni... |
q1pval 26011 | Value of the univariate po... |
q1peqb 26012 | Characterizing property of... |
q1pcl 26013 | Closure of the quotient by... |
r1pval 26014 | Value of the polynomial re... |
r1pcl 26015 | Closure of remainder follo... |
r1pdeglt 26016 | The remainder has a degree... |
r1pid 26017 | Express the original polyn... |
dvdsq1p 26018 | Divisibility in a polynomi... |
dvdsr1p 26019 | Divisibility in a polynomi... |
ply1remlem 26020 | A term of the form ` x - N... |
ply1rem 26021 | The polynomial remainder t... |
facth1 26022 | The factor theorem and its... |
fta1glem1 26023 | Lemma for ~ fta1g . (Cont... |
fta1glem2 26024 | Lemma for ~ fta1g . (Cont... |
fta1g 26025 | The one-sided fundamental ... |
fta1blem 26026 | Lemma for ~ fta1b . (Cont... |
fta1b 26027 | The assumption that ` R ` ... |
drnguc1p 26028 | Over a division ring, all ... |
ig1peu 26029 | There is a unique monic po... |
ig1pval 26030 | Substitutions for the poly... |
ig1pval2 26031 | Generator of the zero idea... |
ig1pval3 26032 | Characterizing properties ... |
ig1pcl 26033 | The monic generator of an ... |
ig1pdvds 26034 | The monic generator of an ... |
ig1prsp 26035 | Any ideal of polynomials o... |
ply1lpir 26036 | The ring of polynomials ov... |
ply1pid 26037 | The polynomials over a fie... |
plyco0 26046 | Two ways to say that a fun... |
plyval 26047 | Value of the polynomial se... |
plybss 26048 | Reverse closure of the par... |
elply 26049 | Definition of a polynomial... |
elply2 26050 | The coefficient function c... |
plyun0 26051 | The set of polynomials is ... |
plyf 26052 | The polynomial is a functi... |
plyss 26053 | The polynomial set functio... |
plyssc 26054 | Every polynomial ring is c... |
elplyr 26055 | Sufficient condition for e... |
elplyd 26056 | Sufficient condition for e... |
ply1termlem 26057 | Lemma for ~ ply1term . (C... |
ply1term 26058 | A one-term polynomial. (C... |
plypow 26059 | A power is a polynomial. ... |
plyconst 26060 | A constant function is a p... |
ne0p 26061 | A test to show that a poly... |
ply0 26062 | The zero function is a pol... |
plyid 26063 | The identity function is a... |
plyeq0lem 26064 | Lemma for ~ plyeq0 . If `... |
plyeq0 26065 | If a polynomial is zero at... |
plypf1 26066 | Write the set of complex p... |
plyaddlem1 26067 | Derive the coefficient fun... |
plymullem1 26068 | Derive the coefficient fun... |
plyaddlem 26069 | Lemma for ~ plyadd . (Con... |
plymullem 26070 | Lemma for ~ plymul . (Con... |
plyadd 26071 | The sum of two polynomials... |
plymul 26072 | The product of two polynom... |
plysub 26073 | The difference of two poly... |
plyaddcl 26074 | The sum of two polynomials... |
plymulcl 26075 | The product of two polynom... |
plysubcl 26076 | The difference of two poly... |
coeval 26077 | Value of the coefficient f... |
coeeulem 26078 | Lemma for ~ coeeu . (Cont... |
coeeu 26079 | Uniqueness of the coeffici... |
coelem 26080 | Lemma for properties of th... |
coeeq 26081 | If ` A ` satisfies the pro... |
dgrval 26082 | Value of the degree functi... |
dgrlem 26083 | Lemma for ~ dgrcl and simi... |
coef 26084 | The domain and codomain of... |
coef2 26085 | The domain and codomain of... |
coef3 26086 | The domain and codomain of... |
dgrcl 26087 | The degree of any polynomi... |
dgrub 26088 | If the ` M ` -th coefficie... |
dgrub2 26089 | All the coefficients above... |
dgrlb 26090 | If all the coefficients ab... |
coeidlem 26091 | Lemma for ~ coeid . (Cont... |
coeid 26092 | Reconstruct a polynomial a... |
coeid2 26093 | Reconstruct a polynomial a... |
coeid3 26094 | Reconstruct a polynomial a... |
plyco 26095 | The composition of two pol... |
coeeq2 26096 | Compute the coefficient fu... |
dgrle 26097 | Given an explicit expressi... |
dgreq 26098 | If the highest term in a p... |
0dgr 26099 | A constant function has de... |
0dgrb 26100 | A function has degree zero... |
dgrnznn 26101 | A nonzero polynomial with ... |
coefv0 26102 | The result of evaluating a... |
coeaddlem 26103 | Lemma for ~ coeadd and ~ d... |
coemullem 26104 | Lemma for ~ coemul and ~ d... |
coeadd 26105 | The coefficient function o... |
coemul 26106 | A coefficient of a product... |
coe11 26107 | The coefficient function i... |
coemulhi 26108 | The leading coefficient of... |
coemulc 26109 | The coefficient function i... |
coe0 26110 | The coefficients of the ze... |
coesub 26111 | The coefficient function o... |
coe1termlem 26112 | The coefficient function o... |
coe1term 26113 | The coefficient function o... |
dgr1term 26114 | The degree of a monomial. ... |
plycn 26115 | A polynomial is a continuo... |
plycnOLD 26116 | Obsolete version of ~ plyc... |
dgr0 26117 | The degree of the zero pol... |
coeidp 26118 | The coefficients of the id... |
dgrid 26119 | The degree of the identity... |
dgreq0 26120 | The leading coefficient of... |
dgrlt 26121 | Two ways to say that the d... |
dgradd 26122 | The degree of a sum of pol... |
dgradd2 26123 | The degree of a sum of pol... |
dgrmul2 26124 | The degree of a product of... |
dgrmul 26125 | The degree of a product of... |
dgrmulc 26126 | Scalar multiplication by a... |
dgrsub 26127 | The degree of a difference... |
dgrcolem1 26128 | The degree of a compositio... |
dgrcolem2 26129 | Lemma for ~ dgrco . (Cont... |
dgrco 26130 | The degree of a compositio... |
plycjlem 26131 | Lemma for ~ plycj and ~ co... |
plycj 26132 | The double conjugation of ... |
coecj 26133 | Double conjugation of a po... |
plyrecj 26134 | A polynomial with real coe... |
plymul0or 26135 | Polynomial multiplication ... |
ofmulrt 26136 | The set of roots of a prod... |
plyreres 26137 | Real-coefficient polynomia... |
dvply1 26138 | Derivative of a polynomial... |
dvply2g 26139 | The derivative of a polyno... |
dvply2 26140 | The derivative of a polyno... |
dvnply2 26141 | Polynomials have polynomia... |
dvnply 26142 | Polynomials have polynomia... |
plycpn 26143 | Polynomials are smooth. (... |
quotval 26146 | Value of the quotient func... |
plydivlem1 26147 | Lemma for ~ plydivalg . (... |
plydivlem2 26148 | Lemma for ~ plydivalg . (... |
plydivlem3 26149 | Lemma for ~ plydivex . Ba... |
plydivlem4 26150 | Lemma for ~ plydivex . In... |
plydivex 26151 | Lemma for ~ plydivalg . (... |
plydiveu 26152 | Lemma for ~ plydivalg . (... |
plydivalg 26153 | The division algorithm on ... |
quotlem 26154 | Lemma for properties of th... |
quotcl 26155 | The quotient of two polyno... |
quotcl2 26156 | Closure of the quotient fu... |
quotdgr 26157 | Remainder property of the ... |
plyremlem 26158 | Closure of a linear factor... |
plyrem 26159 | The polynomial remainder t... |
facth 26160 | The factor theorem. If a ... |
fta1lem 26161 | Lemma for ~ fta1 . (Contr... |
fta1 26162 | The easy direction of the ... |
quotcan 26163 | Exact division with a mult... |
vieta1lem1 26164 | Lemma for ~ vieta1 . (Con... |
vieta1lem2 26165 | Lemma for ~ vieta1 : induc... |
vieta1 26166 | The first-order Vieta's fo... |
plyexmo 26167 | An infinite set of values ... |
elaa 26170 | Elementhood in the set of ... |
aacn 26171 | An algebraic number is a c... |
aasscn 26172 | The algebraic numbers are ... |
elqaalem1 26173 | Lemma for ~ elqaa . The f... |
elqaalem2 26174 | Lemma for ~ elqaa . (Cont... |
elqaalem3 26175 | Lemma for ~ elqaa . (Cont... |
elqaa 26176 | The set of numbers generat... |
qaa 26177 | Every rational number is a... |
qssaa 26178 | The rational numbers are c... |
iaa 26179 | The imaginary unit is alge... |
aareccl 26180 | The reciprocal of an algeb... |
aacjcl 26181 | The conjugate of an algebr... |
aannenlem1 26182 | Lemma for ~ aannen . (Con... |
aannenlem2 26183 | Lemma for ~ aannen . (Con... |
aannenlem3 26184 | The algebraic numbers are ... |
aannen 26185 | The algebraic numbers are ... |
aalioulem1 26186 | Lemma for ~ aaliou . An i... |
aalioulem2 26187 | Lemma for ~ aaliou . (Con... |
aalioulem3 26188 | Lemma for ~ aaliou . (Con... |
aalioulem4 26189 | Lemma for ~ aaliou . (Con... |
aalioulem5 26190 | Lemma for ~ aaliou . (Con... |
aalioulem6 26191 | Lemma for ~ aaliou . (Con... |
aaliou 26192 | Liouville's theorem on dio... |
geolim3 26193 | Geometric series convergen... |
aaliou2 26194 | Liouville's approximation ... |
aaliou2b 26195 | Liouville's approximation ... |
aaliou3lem1 26196 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem2 26197 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem3 26198 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem8 26199 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem4 26200 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem5 26201 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem6 26202 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem7 26203 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem9 26204 | Example of a "Liouville nu... |
aaliou3 26205 | Example of a "Liouville nu... |
taylfvallem1 26210 | Lemma for ~ taylfval . (C... |
taylfvallem 26211 | Lemma for ~ taylfval . (C... |
taylfval 26212 | Define the Taylor polynomi... |
eltayl 26213 | Value of the Taylor series... |
taylf 26214 | The Taylor series defines ... |
tayl0 26215 | The Taylor series is alway... |
taylplem1 26216 | Lemma for ~ taylpfval and ... |
taylplem2 26217 | Lemma for ~ taylpfval and ... |
taylpfval 26218 | Define the Taylor polynomi... |
taylpf 26219 | The Taylor polynomial is a... |
taylpval 26220 | Value of the Taylor polyno... |
taylply2 26221 | The Taylor polynomial is a... |
taylply 26222 | The Taylor polynomial is a... |
dvtaylp 26223 | The derivative of the Tayl... |
dvntaylp 26224 | The ` M ` -th derivative o... |
dvntaylp0 26225 | The first ` N ` derivative... |
taylthlem1 26226 | Lemma for ~ taylth . This... |
taylthlem2 26227 | Lemma for ~ taylth . (Con... |
taylth 26228 | Taylor's theorem. The Tay... |
ulmrel 26231 | The uniform limit relation... |
ulmscl 26232 | Closure of the base set in... |
ulmval 26233 | Express the predicate: Th... |
ulmcl 26234 | Closure of a uniform limit... |
ulmf 26235 | Closure of a uniform limit... |
ulmpm 26236 | Closure of a uniform limit... |
ulmf2 26237 | Closure of a uniform limit... |
ulm2 26238 | Simplify ~ ulmval when ` F... |
ulmi 26239 | The uniform limit property... |
ulmclm 26240 | A uniform limit of functio... |
ulmres 26241 | A sequence of functions co... |
ulmshftlem 26242 | Lemma for ~ ulmshft . (Co... |
ulmshft 26243 | A sequence of functions co... |
ulm0 26244 | Every function converges u... |
ulmuni 26245 | A sequence of functions un... |
ulmdm 26246 | Two ways to express that a... |
ulmcaulem 26247 | Lemma for ~ ulmcau and ~ u... |
ulmcau 26248 | A sequence of functions co... |
ulmcau2 26249 | A sequence of functions co... |
ulmss 26250 | A uniform limit of functio... |
ulmbdd 26251 | A uniform limit of bounded... |
ulmcn 26252 | A uniform limit of continu... |
ulmdvlem1 26253 | Lemma for ~ ulmdv . (Cont... |
ulmdvlem2 26254 | Lemma for ~ ulmdv . (Cont... |
ulmdvlem3 26255 | Lemma for ~ ulmdv . (Cont... |
ulmdv 26256 | If ` F ` is a sequence of ... |
mtest 26257 | The Weierstrass M-test. I... |
mtestbdd 26258 | Given the hypotheses of th... |
mbfulm 26259 | A uniform limit of measura... |
iblulm 26260 | A uniform limit of integra... |
itgulm 26261 | A uniform limit of integra... |
itgulm2 26262 | A uniform limit of integra... |
pserval 26263 | Value of the function ` G ... |
pserval2 26264 | Value of the function ` G ... |
psergf 26265 | The sequence of terms in t... |
radcnvlem1 26266 | Lemma for ~ radcnvlt1 , ~ ... |
radcnvlem2 26267 | Lemma for ~ radcnvlt1 , ~ ... |
radcnvlem3 26268 | Lemma for ~ radcnvlt1 , ~ ... |
radcnv0 26269 | Zero is always a convergen... |
radcnvcl 26270 | The radius of convergence ... |
radcnvlt1 26271 | If ` X ` is within the ope... |
radcnvlt2 26272 | If ` X ` is within the ope... |
radcnvle 26273 | If ` X ` is a convergent p... |
dvradcnv 26274 | The radius of convergence ... |
pserulm 26275 | If ` S ` is a region conta... |
psercn2 26276 | Since by ~ pserulm the ser... |
psercn2OLD 26277 | Obsolete version of ~ pser... |
psercnlem2 26278 | Lemma for ~ psercn . (Con... |
psercnlem1 26279 | Lemma for ~ psercn . (Con... |
psercn 26280 | An infinite series converg... |
pserdvlem1 26281 | Lemma for ~ pserdv . (Con... |
pserdvlem2 26282 | Lemma for ~ pserdv . (Con... |
pserdv 26283 | The derivative of a power ... |
pserdv2 26284 | The derivative of a power ... |
abelthlem1 26285 | Lemma for ~ abelth . (Con... |
abelthlem2 26286 | Lemma for ~ abelth . The ... |
abelthlem3 26287 | Lemma for ~ abelth . (Con... |
abelthlem4 26288 | Lemma for ~ abelth . (Con... |
abelthlem5 26289 | Lemma for ~ abelth . (Con... |
abelthlem6 26290 | Lemma for ~ abelth . (Con... |
abelthlem7a 26291 | Lemma for ~ abelth . (Con... |
abelthlem7 26292 | Lemma for ~ abelth . (Con... |
abelthlem8 26293 | Lemma for ~ abelth . (Con... |
abelthlem9 26294 | Lemma for ~ abelth . By a... |
abelth 26295 | Abel's theorem. If the po... |
abelth2 26296 | Abel's theorem, restricted... |
efcn 26297 | The exponential function i... |
sincn 26298 | Sine is continuous. (Cont... |
coscn 26299 | Cosine is continuous. (Co... |
reeff1olem 26300 | Lemma for ~ reeff1o . (Co... |
reeff1o 26301 | The real exponential funct... |
reefiso 26302 | The exponential function o... |
efcvx 26303 | The exponential function o... |
reefgim 26304 | The exponential function i... |
pilem1 26305 | Lemma for ~ pire , ~ pigt2... |
pilem2 26306 | Lemma for ~ pire , ~ pigt2... |
pilem3 26307 | Lemma for ~ pire , ~ pigt2... |
pigt2lt4 26308 | ` _pi ` is between 2 and 4... |
sinpi 26309 | The sine of ` _pi ` is 0. ... |
pire 26310 | ` _pi ` is a real number. ... |
picn 26311 | ` _pi ` is a complex numbe... |
pipos 26312 | ` _pi ` is positive. (Con... |
pirp 26313 | ` _pi ` is a positive real... |
negpicn 26314 | ` -u _pi ` is a real numbe... |
sinhalfpilem 26315 | Lemma for ~ sinhalfpi and ... |
halfpire 26316 | ` _pi / 2 ` is real. (Con... |
neghalfpire 26317 | ` -u _pi / 2 ` is real. (... |
neghalfpirx 26318 | ` -u _pi / 2 ` is an exten... |
pidiv2halves 26319 | Adding ` _pi / 2 ` to itse... |
sinhalfpi 26320 | The sine of ` _pi / 2 ` is... |
coshalfpi 26321 | The cosine of ` _pi / 2 ` ... |
cosneghalfpi 26322 | The cosine of ` -u _pi / 2... |
efhalfpi 26323 | The exponential of ` _i _p... |
cospi 26324 | The cosine of ` _pi ` is `... |
efipi 26325 | The exponential of ` _i x.... |
eulerid 26326 | Euler's identity. (Contri... |
sin2pi 26327 | The sine of ` 2 _pi ` is 0... |
cos2pi 26328 | The cosine of ` 2 _pi ` is... |
ef2pi 26329 | The exponential of ` 2 _pi... |
ef2kpi 26330 | If ` K ` is an integer, th... |
efper 26331 | The exponential function i... |
sinperlem 26332 | Lemma for ~ sinper and ~ c... |
sinper 26333 | The sine function is perio... |
cosper 26334 | The cosine function is per... |
sin2kpi 26335 | If ` K ` is an integer, th... |
cos2kpi 26336 | If ` K ` is an integer, th... |
sin2pim 26337 | Sine of a number subtracte... |
cos2pim 26338 | Cosine of a number subtrac... |
sinmpi 26339 | Sine of a number less ` _p... |
cosmpi 26340 | Cosine of a number less ` ... |
sinppi 26341 | Sine of a number plus ` _p... |
cosppi 26342 | Cosine of a number plus ` ... |
efimpi 26343 | The exponential function a... |
sinhalfpip 26344 | The sine of ` _pi / 2 ` pl... |
sinhalfpim 26345 | The sine of ` _pi / 2 ` mi... |
coshalfpip 26346 | The cosine of ` _pi / 2 ` ... |
coshalfpim 26347 | The cosine of ` _pi / 2 ` ... |
ptolemy 26348 | Ptolemy's Theorem. This t... |
sincosq1lem 26349 | Lemma for ~ sincosq1sgn . ... |
sincosq1sgn 26350 | The signs of the sine and ... |
sincosq2sgn 26351 | The signs of the sine and ... |
sincosq3sgn 26352 | The signs of the sine and ... |
sincosq4sgn 26353 | The signs of the sine and ... |
coseq00topi 26354 | Location of the zeroes of ... |
coseq0negpitopi 26355 | Location of the zeroes of ... |
tanrpcl 26356 | Positive real closure of t... |
tangtx 26357 | The tangent function is gr... |
tanabsge 26358 | The tangent function is gr... |
sinq12gt0 26359 | The sine of a number stric... |
sinq12ge0 26360 | The sine of a number betwe... |
sinq34lt0t 26361 | The sine of a number stric... |
cosq14gt0 26362 | The cosine of a number str... |
cosq14ge0 26363 | The cosine of a number bet... |
sincosq1eq 26364 | Complementarity of the sin... |
sincos4thpi 26365 | The sine and cosine of ` _... |
tan4thpi 26366 | The tangent of ` _pi / 4 `... |
sincos6thpi 26367 | The sine and cosine of ` _... |
sincos3rdpi 26368 | The sine and cosine of ` _... |
pigt3 26369 | ` _pi ` is greater than 3.... |
pige3 26370 | ` _pi ` is greater than or... |
pige3ALT 26371 | Alternate proof of ~ pige3... |
abssinper 26372 | The absolute value of sine... |
sinkpi 26373 | The sine of an integer mul... |
coskpi 26374 | The absolute value of the ... |
sineq0 26375 | A complex number whose sin... |
coseq1 26376 | A complex number whose cos... |
cos02pilt1 26377 | Cosine is less than one be... |
cosq34lt1 26378 | Cosine is less than one in... |
efeq1 26379 | A complex number whose exp... |
cosne0 26380 | The cosine function has no... |
cosordlem 26381 | Lemma for ~ cosord . (Con... |
cosord 26382 | Cosine is decreasing over ... |
cos0pilt1 26383 | Cosine is between minus on... |
cos11 26384 | Cosine is one-to-one over ... |
sinord 26385 | Sine is increasing over th... |
recosf1o 26386 | The cosine function is a b... |
resinf1o 26387 | The sine function is a bij... |
tanord1 26388 | The tangent function is st... |
tanord 26389 | The tangent function is st... |
tanregt0 26390 | The real part of the tange... |
negpitopissre 26391 | The interval ` ( -u _pi (,... |
efgh 26392 | The exponential function o... |
efif1olem1 26393 | Lemma for ~ efif1o . (Con... |
efif1olem2 26394 | Lemma for ~ efif1o . (Con... |
efif1olem3 26395 | Lemma for ~ efif1o . (Con... |
efif1olem4 26396 | The exponential function o... |
efif1o 26397 | The exponential function o... |
efifo 26398 | The exponential function o... |
eff1olem 26399 | The exponential function m... |
eff1o 26400 | The exponential function m... |
efabl 26401 | The image of a subgroup of... |
efsubm 26402 | The image of a subgroup of... |
circgrp 26403 | The circle group ` T ` is ... |
circsubm 26404 | The circle group ` T ` is ... |
logrn 26409 | The range of the natural l... |
ellogrn 26410 | Write out the property ` A... |
dflog2 26411 | The natural logarithm func... |
relogrn 26412 | The range of the natural l... |
logrncn 26413 | The range of the natural l... |
eff1o2 26414 | The exponential function r... |
logf1o 26415 | The natural logarithm func... |
dfrelog 26416 | The natural logarithm func... |
relogf1o 26417 | The natural logarithm func... |
logrncl 26418 | Closure of the natural log... |
logcl 26419 | Closure of the natural log... |
logimcl 26420 | Closure of the imaginary p... |
logcld 26421 | The logarithm of a nonzero... |
logimcld 26422 | The imaginary part of the ... |
logimclad 26423 | The imaginary part of the ... |
abslogimle 26424 | The imaginary part of the ... |
logrnaddcl 26425 | The range of the natural l... |
relogcl 26426 | Closure of the natural log... |
eflog 26427 | Relationship between the n... |
logeq0im1 26428 | If the logarithm of a numb... |
logccne0 26429 | The logarithm isn't 0 if i... |
logne0 26430 | Logarithm of a non-1 posit... |
reeflog 26431 | Relationship between the n... |
logef 26432 | Relationship between the n... |
relogef 26433 | Relationship between the n... |
logeftb 26434 | Relationship between the n... |
relogeftb 26435 | Relationship between the n... |
log1 26436 | The natural logarithm of `... |
loge 26437 | The natural logarithm of `... |
logneg 26438 | The natural logarithm of a... |
logm1 26439 | The natural logarithm of n... |
lognegb 26440 | If a number has imaginary ... |
relogoprlem 26441 | Lemma for ~ relogmul and ~... |
relogmul 26442 | The natural logarithm of t... |
relogdiv 26443 | The natural logarithm of t... |
explog 26444 | Exponentiation of a nonzer... |
reexplog 26445 | Exponentiation of a positi... |
relogexp 26446 | The natural logarithm of p... |
relog 26447 | Real part of a logarithm. ... |
relogiso 26448 | The natural logarithm func... |
reloggim 26449 | The natural logarithm is a... |
logltb 26450 | The natural logarithm func... |
logfac 26451 | The logarithm of a factori... |
eflogeq 26452 | Solve an equation involvin... |
logleb 26453 | Natural logarithm preserve... |
rplogcl 26454 | Closure of the logarithm f... |
logge0 26455 | The logarithm of a number ... |
logcj 26456 | The natural logarithm dist... |
efiarg 26457 | The exponential of the "ar... |
cosargd 26458 | The cosine of the argument... |
cosarg0d 26459 | The cosine of the argument... |
argregt0 26460 | Closure of the argument of... |
argrege0 26461 | Closure of the argument of... |
argimgt0 26462 | Closure of the argument of... |
argimlt0 26463 | Closure of the argument of... |
logimul 26464 | Multiplying a number by ` ... |
logneg2 26465 | The logarithm of the negat... |
logmul2 26466 | Generalization of ~ relogm... |
logdiv2 26467 | Generalization of ~ relogd... |
abslogle 26468 | Bound on the magnitude of ... |
tanarg 26469 | The basic relation between... |
logdivlti 26470 | The ` log x / x ` function... |
logdivlt 26471 | The ` log x / x ` function... |
logdivle 26472 | The ` log x / x ` function... |
relogcld 26473 | Closure of the natural log... |
reeflogd 26474 | Relationship between the n... |
relogmuld 26475 | The natural logarithm of t... |
relogdivd 26476 | The natural logarithm of t... |
logled 26477 | Natural logarithm preserve... |
relogefd 26478 | Relationship between the n... |
rplogcld 26479 | Closure of the logarithm f... |
logge0d 26480 | The logarithm of a number ... |
logge0b 26481 | The logarithm of a number ... |
loggt0b 26482 | The logarithm of a number ... |
logle1b 26483 | The logarithm of a number ... |
loglt1b 26484 | The logarithm of a number ... |
divlogrlim 26485 | The inverse logarithm func... |
logno1 26486 | The logarithm function is ... |
dvrelog 26487 | The derivative of the real... |
relogcn 26488 | The real logarithm functio... |
ellogdm 26489 | Elementhood in the "contin... |
logdmn0 26490 | A number in the continuous... |
logdmnrp 26491 | A number in the continuous... |
logdmss 26492 | The continuity domain of `... |
logcnlem2 26493 | Lemma for ~ logcn . (Cont... |
logcnlem3 26494 | Lemma for ~ logcn . (Cont... |
logcnlem4 26495 | Lemma for ~ logcn . (Cont... |
logcnlem5 26496 | Lemma for ~ logcn . (Cont... |
logcn 26497 | The logarithm function is ... |
dvloglem 26498 | Lemma for ~ dvlog . (Cont... |
logdmopn 26499 | The "continuous domain" of... |
logf1o2 26500 | The logarithm maps its con... |
dvlog 26501 | The derivative of the comp... |
dvlog2lem 26502 | Lemma for ~ dvlog2 . (Con... |
dvlog2 26503 | The derivative of the comp... |
advlog 26504 | The antiderivative of the ... |
advlogexp 26505 | The antiderivative of a po... |
efopnlem1 26506 | Lemma for ~ efopn . (Cont... |
efopnlem2 26507 | Lemma for ~ efopn . (Cont... |
efopn 26508 | The exponential map is an ... |
logtayllem 26509 | Lemma for ~ logtayl . (Co... |
logtayl 26510 | The Taylor series for ` -u... |
logtaylsum 26511 | The Taylor series for ` -u... |
logtayl2 26512 | Power series expression fo... |
logccv 26513 | The natural logarithm func... |
cxpval 26514 | Value of the complex power... |
cxpef 26515 | Value of the complex power... |
0cxp 26516 | Value of the complex power... |
cxpexpz 26517 | Relate the complex power f... |
cxpexp 26518 | Relate the complex power f... |
logcxp 26519 | Logarithm of a complex pow... |
cxp0 26520 | Value of the complex power... |
cxp1 26521 | Value of the complex power... |
1cxp 26522 | Value of the complex power... |
ecxp 26523 | Write the exponential func... |
cxpcl 26524 | Closure of the complex pow... |
recxpcl 26525 | Real closure of the comple... |
rpcxpcl 26526 | Positive real closure of t... |
cxpne0 26527 | Complex exponentiation is ... |
cxpeq0 26528 | Complex exponentiation is ... |
cxpadd 26529 | Sum of exponents law for c... |
cxpp1 26530 | Value of a nonzero complex... |
cxpneg 26531 | Value of a complex number ... |
cxpsub 26532 | Exponent subtraction law f... |
cxpge0 26533 | Nonnegative exponentiation... |
mulcxplem 26534 | Lemma for ~ mulcxp . (Con... |
mulcxp 26535 | Complex exponentiation of ... |
cxprec 26536 | Complex exponentiation of ... |
divcxp 26537 | Complex exponentiation of ... |
cxpmul 26538 | Product of exponents law f... |
cxpmul2 26539 | Product of exponents law f... |
cxproot 26540 | The complex power function... |
cxpmul2z 26541 | Generalize ~ cxpmul2 to ne... |
abscxp 26542 | Absolute value of a power,... |
abscxp2 26543 | Absolute value of a power,... |
cxplt 26544 | Ordering property for comp... |
cxple 26545 | Ordering property for comp... |
cxplea 26546 | Ordering property for comp... |
cxple2 26547 | Ordering property for comp... |
cxplt2 26548 | Ordering property for comp... |
cxple2a 26549 | Ordering property for comp... |
cxplt3 26550 | Ordering property for comp... |
cxple3 26551 | Ordering property for comp... |
cxpsqrtlem 26552 | Lemma for ~ cxpsqrt . (Co... |
cxpsqrt 26553 | The complex exponential fu... |
logsqrt 26554 | Logarithm of a square root... |
cxp0d 26555 | Value of the complex power... |
cxp1d 26556 | Value of the complex power... |
1cxpd 26557 | Value of the complex power... |
cxpcld 26558 | Closure of the complex pow... |
cxpmul2d 26559 | Product of exponents law f... |
0cxpd 26560 | Value of the complex power... |
cxpexpzd 26561 | Relate the complex power f... |
cxpefd 26562 | Value of the complex power... |
cxpne0d 26563 | Complex exponentiation is ... |
cxpp1d 26564 | Value of a nonzero complex... |
cxpnegd 26565 | Value of a complex number ... |
cxpmul2zd 26566 | Generalize ~ cxpmul2 to ne... |
cxpaddd 26567 | Sum of exponents law for c... |
cxpsubd 26568 | Exponent subtraction law f... |
cxpltd 26569 | Ordering property for comp... |
cxpled 26570 | Ordering property for comp... |
cxplead 26571 | Ordering property for comp... |
divcxpd 26572 | Complex exponentiation of ... |
recxpcld 26573 | Positive real closure of t... |
cxpge0d 26574 | Nonnegative exponentiation... |
cxple2ad 26575 | Ordering property for comp... |
cxplt2d 26576 | Ordering property for comp... |
cxple2d 26577 | Ordering property for comp... |
mulcxpd 26578 | Complex exponentiation of ... |
recxpf1lem 26579 | Complex exponentiation on ... |
cxpsqrtth 26580 | Square root theorem over t... |
2irrexpq 26581 | There exist irrational num... |
cxprecd 26582 | Complex exponentiation of ... |
rpcxpcld 26583 | Positive real closure of t... |
logcxpd 26584 | Logarithm of a complex pow... |
cxplt3d 26585 | Ordering property for comp... |
cxple3d 26586 | Ordering property for comp... |
cxpmuld 26587 | Product of exponents law f... |
cxpgt0d 26588 | A positive real raised to ... |
cxpcom 26589 | Commutative law for real e... |
dvcxp1 26590 | The derivative of a comple... |
dvcxp2 26591 | The derivative of a comple... |
dvsqrt 26592 | The derivative of the real... |
dvcncxp1 26593 | Derivative of complex powe... |
dvcnsqrt 26594 | Derivative of square root ... |
cxpcn 26595 | Domain of continuity of th... |
cxpcnOLD 26596 | Obsolete version of ~ cxpc... |
cxpcn2 26597 | Continuity of the complex ... |
cxpcn3lem 26598 | Lemma for ~ cxpcn3 . (Con... |
cxpcn3 26599 | Extend continuity of the c... |
resqrtcn 26600 | Continuity of the real squ... |
sqrtcn 26601 | Continuity of the square r... |
cxpaddlelem 26602 | Lemma for ~ cxpaddle . (C... |
cxpaddle 26603 | Ordering property for comp... |
abscxpbnd 26604 | Bound on the absolute valu... |
root1id 26605 | Property of an ` N ` -th r... |
root1eq1 26606 | The only powers of an ` N ... |
root1cj 26607 | Within the ` N ` -th roots... |
cxpeq 26608 | Solve an equation involvin... |
loglesqrt 26609 | An upper bound on the loga... |
logreclem 26610 | Symmetry of the natural lo... |
logrec 26611 | Logarithm of a reciprocal ... |
logbval 26614 | Define the value of the ` ... |
logbcl 26615 | General logarithm closure.... |
logbid1 26616 | General logarithm is 1 whe... |
logb1 26617 | The logarithm of ` 1 ` to ... |
elogb 26618 | The general logarithm of a... |
logbchbase 26619 | Change of base for logarit... |
relogbval 26620 | Value of the general logar... |
relogbcl 26621 | Closure of the general log... |
relogbzcl 26622 | Closure of the general log... |
relogbreexp 26623 | Power law for the general ... |
relogbzexp 26624 | Power law for the general ... |
relogbmul 26625 | The logarithm of the produ... |
relogbmulexp 26626 | The logarithm of the produ... |
relogbdiv 26627 | The logarithm of the quoti... |
relogbexp 26628 | Identity law for general l... |
nnlogbexp 26629 | Identity law for general l... |
logbrec 26630 | Logarithm of a reciprocal ... |
logbleb 26631 | The general logarithm func... |
logblt 26632 | The general logarithm func... |
relogbcxp 26633 | Identity law for the gener... |
cxplogb 26634 | Identity law for the gener... |
relogbcxpb 26635 | The logarithm is the inver... |
logbmpt 26636 | The general logarithm to a... |
logbf 26637 | The general logarithm to a... |
logbfval 26638 | The general logarithm of a... |
relogbf 26639 | The general logarithm to a... |
logblog 26640 | The general logarithm to t... |
logbgt0b 26641 | The logarithm of a positiv... |
logbgcd1irr 26642 | The logarithm of an intege... |
2logb9irr 26643 | Example for ~ logbgcd1irr ... |
logbprmirr 26644 | The logarithm of a prime t... |
2logb3irr 26645 | Example for ~ logbprmirr .... |
2logb9irrALT 26646 | Alternate proof of ~ 2logb... |
sqrt2cxp2logb9e3 26647 | The square root of two to ... |
2irrexpqALT 26648 | Alternate proof of ~ 2irre... |
angval 26649 | Define the angle function,... |
angcan 26650 | Cancel a constant multipli... |
angneg 26651 | Cancel a negative sign in ... |
angvald 26652 | The (signed) angle between... |
angcld 26653 | The (signed) angle between... |
angrteqvd 26654 | Two vectors are at a right... |
cosangneg2d 26655 | The cosine of the angle be... |
angrtmuld 26656 | Perpendicularity of two ve... |
ang180lem1 26657 | Lemma for ~ ang180 . Show... |
ang180lem2 26658 | Lemma for ~ ang180 . Show... |
ang180lem3 26659 | Lemma for ~ ang180 . Sinc... |
ang180lem4 26660 | Lemma for ~ ang180 . Redu... |
ang180lem5 26661 | Lemma for ~ ang180 : Redu... |
ang180 26662 | The sum of angles ` m A B ... |
lawcoslem1 26663 | Lemma for ~ lawcos . Here... |
lawcos 26664 | Law of cosines (also known... |
pythag 26665 | Pythagorean theorem. Give... |
isosctrlem1 26666 | Lemma for ~ isosctr . (Co... |
isosctrlem2 26667 | Lemma for ~ isosctr . Cor... |
isosctrlem3 26668 | Lemma for ~ isosctr . Cor... |
isosctr 26669 | Isosceles triangle theorem... |
ssscongptld 26670 | If two triangles have equa... |
affineequiv 26671 | Equivalence between two wa... |
affineequiv2 26672 | Equivalence between two wa... |
affineequiv3 26673 | Equivalence between two wa... |
affineequiv4 26674 | Equivalence between two wa... |
affineequivne 26675 | Equivalence between two wa... |
angpieqvdlem 26676 | Equivalence used in the pr... |
angpieqvdlem2 26677 | Equivalence used in ~ angp... |
angpined 26678 | If the angle at ABC is ` _... |
angpieqvd 26679 | The angle ABC is ` _pi ` i... |
chordthmlem 26680 | If ` M ` is the midpoint o... |
chordthmlem2 26681 | If M is the midpoint of AB... |
chordthmlem3 26682 | If M is the midpoint of AB... |
chordthmlem4 26683 | If P is on the segment AB ... |
chordthmlem5 26684 | If P is on the segment AB ... |
chordthm 26685 | The intersecting chords th... |
heron 26686 | Heron's formula gives the ... |
quad2 26687 | The quadratic equation, wi... |
quad 26688 | The quadratic equation. (... |
1cubrlem 26689 | The cube roots of unity. ... |
1cubr 26690 | The cube roots of unity. ... |
dcubic1lem 26691 | Lemma for ~ dcubic1 and ~ ... |
dcubic2 26692 | Reverse direction of ~ dcu... |
dcubic1 26693 | Forward direction of ~ dcu... |
dcubic 26694 | Solutions to the depressed... |
mcubic 26695 | Solutions to a monic cubic... |
cubic2 26696 | The solution to the genera... |
cubic 26697 | The cubic equation, which ... |
binom4 26698 | Work out a quartic binomia... |
dquartlem1 26699 | Lemma for ~ dquart . (Con... |
dquartlem2 26700 | Lemma for ~ dquart . (Con... |
dquart 26701 | Solve a depressed quartic ... |
quart1cl 26702 | Closure lemmas for ~ quart... |
quart1lem 26703 | Lemma for ~ quart1 . (Con... |
quart1 26704 | Depress a quartic equation... |
quartlem1 26705 | Lemma for ~ quart . (Cont... |
quartlem2 26706 | Closure lemmas for ~ quart... |
quartlem3 26707 | Closure lemmas for ~ quart... |
quartlem4 26708 | Closure lemmas for ~ quart... |
quart 26709 | The quartic equation, writ... |
asinlem 26716 | The argument to the logari... |
asinlem2 26717 | The argument to the logari... |
asinlem3a 26718 | Lemma for ~ asinlem3 . (C... |
asinlem3 26719 | The argument to the logari... |
asinf 26720 | Domain and codomain of the... |
asincl 26721 | Closure for the arcsin fun... |
acosf 26722 | Domain and codoamin of the... |
acoscl 26723 | Closure for the arccos fun... |
atandm 26724 | Since the property is a li... |
atandm2 26725 | This form of ~ atandm is a... |
atandm3 26726 | A compact form of ~ atandm... |
atandm4 26727 | A compact form of ~ atandm... |
atanf 26728 | Domain and codoamin of the... |
atancl 26729 | Closure for the arctan fun... |
asinval 26730 | Value of the arcsin functi... |
acosval 26731 | Value of the arccos functi... |
atanval 26732 | Value of the arctan functi... |
atanre 26733 | A real number is in the do... |
asinneg 26734 | The arcsine function is od... |
acosneg 26735 | The negative symmetry rela... |
efiasin 26736 | The exponential of the arc... |
sinasin 26737 | The arcsine function is an... |
cosacos 26738 | The arccosine function is ... |
asinsinlem 26739 | Lemma for ~ asinsin . (Co... |
asinsin 26740 | The arcsine function compo... |
acoscos 26741 | The arccosine function is ... |
asin1 26742 | The arcsine of ` 1 ` is ` ... |
acos1 26743 | The arccosine of ` 1 ` is ... |
reasinsin 26744 | The arcsine function compo... |
asinsinb 26745 | Relationship between sine ... |
acoscosb 26746 | Relationship between cosin... |
asinbnd 26747 | The arcsine function has r... |
acosbnd 26748 | The arccosine function has... |
asinrebnd 26749 | Bounds on the arcsine func... |
asinrecl 26750 | The arcsine function is re... |
acosrecl 26751 | The arccosine function is ... |
cosasin 26752 | The cosine of the arcsine ... |
sinacos 26753 | The sine of the arccosine ... |
atandmneg 26754 | The domain of the arctange... |
atanneg 26755 | The arctangent function is... |
atan0 26756 | The arctangent of zero is ... |
atandmcj 26757 | The arctangent function di... |
atancj 26758 | The arctangent function di... |
atanrecl 26759 | The arctangent function is... |
efiatan 26760 | Value of the exponential o... |
atanlogaddlem 26761 | Lemma for ~ atanlogadd . ... |
atanlogadd 26762 | The rule ` sqrt ( z w ) = ... |
atanlogsublem 26763 | Lemma for ~ atanlogsub . ... |
atanlogsub 26764 | A variation on ~ atanlogad... |
efiatan2 26765 | Value of the exponential o... |
2efiatan 26766 | Value of the exponential o... |
tanatan 26767 | The arctangent function is... |
atandmtan 26768 | The tangent function has r... |
cosatan 26769 | The cosine of an arctangen... |
cosatanne0 26770 | The arctangent function ha... |
atantan 26771 | The arctangent function is... |
atantanb 26772 | Relationship between tange... |
atanbndlem 26773 | Lemma for ~ atanbnd . (Co... |
atanbnd 26774 | The arctangent function is... |
atanord 26775 | The arctangent function is... |
atan1 26776 | The arctangent of ` 1 ` is... |
bndatandm 26777 | A point in the open unit d... |
atans 26778 | The "domain of continuity"... |
atans2 26779 | It suffices to show that `... |
atansopn 26780 | The domain of continuity o... |
atansssdm 26781 | The domain of continuity o... |
ressatans 26782 | The real number line is a ... |
dvatan 26783 | The derivative of the arct... |
atancn 26784 | The arctangent is a contin... |
atantayl 26785 | The Taylor series for ` ar... |
atantayl2 26786 | The Taylor series for ` ar... |
atantayl3 26787 | The Taylor series for ` ar... |
leibpilem1 26788 | Lemma for ~ leibpi . (Con... |
leibpilem2 26789 | The Leibniz formula for ` ... |
leibpi 26790 | The Leibniz formula for ` ... |
leibpisum 26791 | The Leibniz formula for ` ... |
log2cnv 26792 | Using the Taylor series fo... |
log2tlbnd 26793 | Bound the error term in th... |
log2ublem1 26794 | Lemma for ~ log2ub . The ... |
log2ublem2 26795 | Lemma for ~ log2ub . (Con... |
log2ublem3 26796 | Lemma for ~ log2ub . In d... |
log2ub 26797 | ` log 2 ` is less than ` 2... |
log2le1 26798 | ` log 2 ` is less than ` 1... |
birthdaylem1 26799 | Lemma for ~ birthday . (C... |
birthdaylem2 26800 | For general ` N ` and ` K ... |
birthdaylem3 26801 | For general ` N ` and ` K ... |
birthday 26802 | The Birthday Problem. The... |
dmarea 26805 | The domain of the area fun... |
areambl 26806 | The fibers of a measurable... |
areass 26807 | A measurable region is a s... |
dfarea 26808 | Rewrite ~ df-area self-ref... |
areaf 26809 | Area measurement is a func... |
areacl 26810 | The area of a measurable r... |
areage0 26811 | The area of a measurable r... |
areaval 26812 | The area of a measurable r... |
rlimcnp 26813 | Relate a limit of a real-v... |
rlimcnp2 26814 | Relate a limit of a real-v... |
rlimcnp3 26815 | Relate a limit of a real-v... |
xrlimcnp 26816 | Relate a limit of a real-v... |
efrlim 26817 | The limit of the sequence ... |
efrlimOLD 26818 | Obsolete version of ~ efrl... |
dfef2 26819 | The limit of the sequence ... |
cxplim 26820 | A power to a negative expo... |
sqrtlim 26821 | The inverse square root fu... |
rlimcxp 26822 | Any power to a positive ex... |
o1cxp 26823 | An eventually bounded func... |
cxp2limlem 26824 | A linear factor grows slow... |
cxp2lim 26825 | Any power grows slower tha... |
cxploglim 26826 | The logarithm grows slower... |
cxploglim2 26827 | Every power of the logarit... |
divsqrtsumlem 26828 | Lemma for ~ divsqrsum and ... |
divsqrsumf 26829 | The function ` F ` used in... |
divsqrsum 26830 | The sum ` sum_ n <_ x ( 1 ... |
divsqrtsum2 26831 | A bound on the distance of... |
divsqrtsumo1 26832 | The sum ` sum_ n <_ x ( 1 ... |
cvxcl 26833 | Closure of a 0-1 linear co... |
scvxcvx 26834 | A strictly convex function... |
jensenlem1 26835 | Lemma for ~ jensen . (Con... |
jensenlem2 26836 | Lemma for ~ jensen . (Con... |
jensen 26837 | Jensen's inequality, a fin... |
amgmlem 26838 | Lemma for ~ amgm . (Contr... |
amgm 26839 | Inequality of arithmetic a... |
logdifbnd 26842 | Bound on the difference of... |
logdiflbnd 26843 | Lower bound on the differe... |
emcllem1 26844 | Lemma for ~ emcl . The se... |
emcllem2 26845 | Lemma for ~ emcl . ` F ` i... |
emcllem3 26846 | Lemma for ~ emcl . The fu... |
emcllem4 26847 | Lemma for ~ emcl . The di... |
emcllem5 26848 | Lemma for ~ emcl . The pa... |
emcllem6 26849 | Lemma for ~ emcl . By the... |
emcllem7 26850 | Lemma for ~ emcl and ~ har... |
emcl 26851 | Closure and bounds for the... |
harmonicbnd 26852 | A bound on the harmonic se... |
harmonicbnd2 26853 | A bound on the harmonic se... |
emre 26854 | The Euler-Mascheroni const... |
emgt0 26855 | The Euler-Mascheroni const... |
harmonicbnd3 26856 | A bound on the harmonic se... |
harmoniclbnd 26857 | A bound on the harmonic se... |
harmonicubnd 26858 | A bound on the harmonic se... |
harmonicbnd4 26859 | The asymptotic behavior of... |
fsumharmonic 26860 | Bound a finite sum based o... |
zetacvg 26863 | The zeta series is converg... |
eldmgm 26870 | Elementhood in the set of ... |
dmgmaddn0 26871 | If ` A ` is not a nonposit... |
dmlogdmgm 26872 | If ` A ` is in the continu... |
rpdmgm 26873 | A positive real number is ... |
dmgmn0 26874 | If ` A ` is not a nonposit... |
dmgmaddnn0 26875 | If ` A ` is not a nonposit... |
dmgmdivn0 26876 | Lemma for ~ lgamf . (Cont... |
lgamgulmlem1 26877 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem2 26878 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem3 26879 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem4 26880 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem5 26881 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem6 26882 | The series ` G ` is unifor... |
lgamgulm 26883 | The series ` G ` is unifor... |
lgamgulm2 26884 | Rewrite the limit of the s... |
lgambdd 26885 | The log-Gamma function is ... |
lgamucov 26886 | The ` U ` regions used in ... |
lgamucov2 26887 | The ` U ` regions used in ... |
lgamcvglem 26888 | Lemma for ~ lgamf and ~ lg... |
lgamcl 26889 | The log-Gamma function is ... |
lgamf 26890 | The log-Gamma function is ... |
gamf 26891 | The Gamma function is a co... |
gamcl 26892 | The exponential of the log... |
eflgam 26893 | The exponential of the log... |
gamne0 26894 | The Gamma function is neve... |
igamval 26895 | Value of the inverse Gamma... |
igamz 26896 | Value of the inverse Gamma... |
igamgam 26897 | Value of the inverse Gamma... |
igamlgam 26898 | Value of the inverse Gamma... |
igamf 26899 | Closure of the inverse Gam... |
igamcl 26900 | Closure of the inverse Gam... |
gamigam 26901 | The Gamma function is the ... |
lgamcvg 26902 | The series ` G ` converges... |
lgamcvg2 26903 | The series ` G ` converges... |
gamcvg 26904 | The pointwise exponential ... |
lgamp1 26905 | The functional equation of... |
gamp1 26906 | The functional equation of... |
gamcvg2lem 26907 | Lemma for ~ gamcvg2 . (Co... |
gamcvg2 26908 | An infinite product expres... |
regamcl 26909 | The Gamma function is real... |
relgamcl 26910 | The log-Gamma function is ... |
rpgamcl 26911 | The log-Gamma function is ... |
lgam1 26912 | The log-Gamma function at ... |
gam1 26913 | The log-Gamma function at ... |
facgam 26914 | The Gamma function general... |
gamfac 26915 | The Gamma function general... |
wilthlem1 26916 | The only elements that are... |
wilthlem2 26917 | Lemma for ~ wilth : induct... |
wilthlem3 26918 | Lemma for ~ wilth . Here ... |
wilth 26919 | Wilson's theorem. A numbe... |
wilthimp 26920 | The forward implication of... |
ftalem1 26921 | Lemma for ~ fta : "growth... |
ftalem2 26922 | Lemma for ~ fta . There e... |
ftalem3 26923 | Lemma for ~ fta . There e... |
ftalem4 26924 | Lemma for ~ fta : Closure... |
ftalem5 26925 | Lemma for ~ fta : Main pr... |
ftalem6 26926 | Lemma for ~ fta : Dischar... |
ftalem7 26927 | Lemma for ~ fta . Shift t... |
fta 26928 | The Fundamental Theorem of... |
basellem1 26929 | Lemma for ~ basel . Closu... |
basellem2 26930 | Lemma for ~ basel . Show ... |
basellem3 26931 | Lemma for ~ basel . Using... |
basellem4 26932 | Lemma for ~ basel . By ~ ... |
basellem5 26933 | Lemma for ~ basel . Using... |
basellem6 26934 | Lemma for ~ basel . The f... |
basellem7 26935 | Lemma for ~ basel . The f... |
basellem8 26936 | Lemma for ~ basel . The f... |
basellem9 26937 | Lemma for ~ basel . Since... |
basel 26938 | The sum of the inverse squ... |
efnnfsumcl 26951 | Finite sum closure in the ... |
ppisval 26952 | The set of primes less tha... |
ppisval2 26953 | The set of primes less tha... |
ppifi 26954 | The set of primes less tha... |
prmdvdsfi 26955 | The set of prime divisors ... |
chtf 26956 | Domain and codoamin of the... |
chtcl 26957 | Real closure of the Chebys... |
chtval 26958 | Value of the Chebyshev fun... |
efchtcl 26959 | The Chebyshev function is ... |
chtge0 26960 | The Chebyshev function is ... |
vmaval 26961 | Value of the von Mangoldt ... |
isppw 26962 | Two ways to say that ` A `... |
isppw2 26963 | Two ways to say that ` A `... |
vmappw 26964 | Value of the von Mangoldt ... |
vmaprm 26965 | Value of the von Mangoldt ... |
vmacl 26966 | Closure for the von Mangol... |
vmaf 26967 | Functionality of the von M... |
efvmacl 26968 | The von Mangoldt is closed... |
vmage0 26969 | The von Mangoldt function ... |
chpval 26970 | Value of the second Chebys... |
chpf 26971 | Functionality of the secon... |
chpcl 26972 | Closure for the second Che... |
efchpcl 26973 | The second Chebyshev funct... |
chpge0 26974 | The second Chebyshev funct... |
ppival 26975 | Value of the prime-countin... |
ppival2 26976 | Value of the prime-countin... |
ppival2g 26977 | Value of the prime-countin... |
ppif 26978 | Domain and codomain of the... |
ppicl 26979 | Real closure of the prime-... |
muval 26980 | The value of the Möbi... |
muval1 26981 | The value of the Möbi... |
muval2 26982 | The value of the Möbi... |
isnsqf 26983 | Two ways to say that a num... |
issqf 26984 | Two ways to say that a num... |
sqfpc 26985 | The prime count of a squar... |
dvdssqf 26986 | A divisor of a squarefree ... |
sqf11 26987 | A squarefree number is com... |
muf 26988 | The Möbius function i... |
mucl 26989 | Closure of the Möbius... |
sgmval 26990 | The value of the divisor f... |
sgmval2 26991 | The value of the divisor f... |
0sgm 26992 | The value of the sum-of-di... |
sgmf 26993 | The divisor function is a ... |
sgmcl 26994 | Closure of the divisor fun... |
sgmnncl 26995 | Closure of the divisor fun... |
mule1 26996 | The Möbius function t... |
chtfl 26997 | The Chebyshev function doe... |
chpfl 26998 | The second Chebyshev funct... |
ppiprm 26999 | The prime-counting functio... |
ppinprm 27000 | The prime-counting functio... |
chtprm 27001 | The Chebyshev function at ... |
chtnprm 27002 | The Chebyshev function at ... |
chpp1 27003 | The second Chebyshev funct... |
chtwordi 27004 | The Chebyshev function is ... |
chpwordi 27005 | The second Chebyshev funct... |
chtdif 27006 | The difference of the Cheb... |
efchtdvds 27007 | The exponentiated Chebyshe... |
ppifl 27008 | The prime-counting functio... |
ppip1le 27009 | The prime-counting functio... |
ppiwordi 27010 | The prime-counting functio... |
ppidif 27011 | The difference of the prim... |
ppi1 27012 | The prime-counting functio... |
cht1 27013 | The Chebyshev function at ... |
vma1 27014 | The von Mangoldt function ... |
chp1 27015 | The second Chebyshev funct... |
ppi1i 27016 | Inference form of ~ ppiprm... |
ppi2i 27017 | Inference form of ~ ppinpr... |
ppi2 27018 | The prime-counting functio... |
ppi3 27019 | The prime-counting functio... |
cht2 27020 | The Chebyshev function at ... |
cht3 27021 | The Chebyshev function at ... |
ppinncl 27022 | Closure of the prime-count... |
chtrpcl 27023 | Closure of the Chebyshev f... |
ppieq0 27024 | The prime-counting functio... |
ppiltx 27025 | The prime-counting functio... |
prmorcht 27026 | Relate the primorial (prod... |
mumullem1 27027 | Lemma for ~ mumul . A mul... |
mumullem2 27028 | Lemma for ~ mumul . The p... |
mumul 27029 | The Möbius function i... |
sqff1o 27030 | There is a bijection from ... |
fsumdvdsdiaglem 27031 | A "diagonal commutation" o... |
fsumdvdsdiag 27032 | A "diagonal commutation" o... |
fsumdvdscom 27033 | A double commutation of di... |
dvdsppwf1o 27034 | A bijection from the divis... |
dvdsflf1o 27035 | A bijection from the numbe... |
dvdsflsumcom 27036 | A sum commutation from ` s... |
fsumfldivdiaglem 27037 | Lemma for ~ fsumfldivdiag ... |
fsumfldivdiag 27038 | The right-hand side of ~ d... |
musum 27039 | The sum of the Möbius... |
musumsum 27040 | Evaluate a collapsing sum ... |
muinv 27041 | The Möbius inversion ... |
mpodvdsmulf1o 27042 | If ` M ` and ` N ` are two... |
fsumdvdsmul 27043 | Product of two divisor sum... |
dvdsmulf1o 27044 | If ` M ` and ` N ` are two... |
fsumdvdsmulOLD 27045 | Obsolete version of ~ fsum... |
sgmppw 27046 | The value of the divisor f... |
0sgmppw 27047 | A prime power ` P ^ K ` ha... |
1sgmprm 27048 | The sum of divisors for a ... |
1sgm2ppw 27049 | The sum of the divisors of... |
sgmmul 27050 | The divisor function for f... |
ppiublem1 27051 | Lemma for ~ ppiub . (Cont... |
ppiublem2 27052 | A prime greater than ` 3 `... |
ppiub 27053 | An upper bound on the prim... |
vmalelog 27054 | The von Mangoldt function ... |
chtlepsi 27055 | The first Chebyshev functi... |
chprpcl 27056 | Closure of the second Cheb... |
chpeq0 27057 | The second Chebyshev funct... |
chteq0 27058 | The first Chebyshev functi... |
chtleppi 27059 | Upper bound on the ` theta... |
chtublem 27060 | Lemma for ~ chtub . (Cont... |
chtub 27061 | An upper bound on the Cheb... |
fsumvma 27062 | Rewrite a sum over the von... |
fsumvma2 27063 | Apply ~ fsumvma for the co... |
pclogsum 27064 | The logarithmic analogue o... |
vmasum 27065 | The sum of the von Mangold... |
logfac2 27066 | Another expression for the... |
chpval2 27067 | Express the second Chebysh... |
chpchtsum 27068 | The second Chebyshev funct... |
chpub 27069 | An upper bound on the seco... |
logfacubnd 27070 | A simple upper bound on th... |
logfaclbnd 27071 | A lower bound on the logar... |
logfacbnd3 27072 | Show the stronger statemen... |
logfacrlim 27073 | Combine the estimates ~ lo... |
logexprlim 27074 | The sum ` sum_ n <_ x , lo... |
logfacrlim2 27075 | Write out ~ logfacrlim as ... |
mersenne 27076 | A Mersenne prime is a prim... |
perfect1 27077 | Euclid's contribution to t... |
perfectlem1 27078 | Lemma for ~ perfect . (Co... |
perfectlem2 27079 | Lemma for ~ perfect . (Co... |
perfect 27080 | The Euclid-Euler theorem, ... |
dchrval 27083 | Value of the group of Diri... |
dchrbas 27084 | Base set of the group of D... |
dchrelbas 27085 | A Dirichlet character is a... |
dchrelbas2 27086 | A Dirichlet character is a... |
dchrelbas3 27087 | A Dirichlet character is a... |
dchrelbasd 27088 | A Dirichlet character is a... |
dchrrcl 27089 | Reverse closure for a Diri... |
dchrmhm 27090 | A Dirichlet character is a... |
dchrf 27091 | A Dirichlet character is a... |
dchrelbas4 27092 | A Dirichlet character is a... |
dchrzrh1 27093 | Value of a Dirichlet chara... |
dchrzrhcl 27094 | A Dirichlet character take... |
dchrzrhmul 27095 | A Dirichlet character is c... |
dchrplusg 27096 | Group operation on the gro... |
dchrmul 27097 | Group operation on the gro... |
dchrmulcl 27098 | Closure of the group opera... |
dchrn0 27099 | A Dirichlet character is n... |
dchr1cl 27100 | Closure of the principal D... |
dchrmullid 27101 | Left identity for the prin... |
dchrinvcl 27102 | Closure of the group inver... |
dchrabl 27103 | The set of Dirichlet chara... |
dchrfi 27104 | The group of Dirichlet cha... |
dchrghm 27105 | A Dirichlet character rest... |
dchr1 27106 | Value of the principal Dir... |
dchreq 27107 | A Dirichlet character is d... |
dchrresb 27108 | A Dirichlet character is d... |
dchrabs 27109 | A Dirichlet character take... |
dchrinv 27110 | The inverse of a Dirichlet... |
dchrabs2 27111 | A Dirichlet character take... |
dchr1re 27112 | The principal Dirichlet ch... |
dchrptlem1 27113 | Lemma for ~ dchrpt . (Con... |
dchrptlem2 27114 | Lemma for ~ dchrpt . (Con... |
dchrptlem3 27115 | Lemma for ~ dchrpt . (Con... |
dchrpt 27116 | For any element other than... |
dchrsum2 27117 | An orthogonality relation ... |
dchrsum 27118 | An orthogonality relation ... |
sumdchr2 27119 | Lemma for ~ sumdchr . (Co... |
dchrhash 27120 | There are exactly ` phi ( ... |
sumdchr 27121 | An orthogonality relation ... |
dchr2sum 27122 | An orthogonality relation ... |
sum2dchr 27123 | An orthogonality relation ... |
bcctr 27124 | Value of the central binom... |
pcbcctr 27125 | Prime count of a central b... |
bcmono 27126 | The binomial coefficient i... |
bcmax 27127 | The binomial coefficient t... |
bcp1ctr 27128 | Ratio of two central binom... |
bclbnd 27129 | A bound on the binomial co... |
efexple 27130 | Convert a bound on a power... |
bpos1lem 27131 | Lemma for ~ bpos1 . (Cont... |
bpos1 27132 | Bertrand's postulate, chec... |
bposlem1 27133 | An upper bound on the prim... |
bposlem2 27134 | There are no odd primes in... |
bposlem3 27135 | Lemma for ~ bpos . Since ... |
bposlem4 27136 | Lemma for ~ bpos . (Contr... |
bposlem5 27137 | Lemma for ~ bpos . Bound ... |
bposlem6 27138 | Lemma for ~ bpos . By usi... |
bposlem7 27139 | Lemma for ~ bpos . The fu... |
bposlem8 27140 | Lemma for ~ bpos . Evalua... |
bposlem9 27141 | Lemma for ~ bpos . Derive... |
bpos 27142 | Bertrand's postulate: ther... |
zabsle1 27145 | ` { -u 1 , 0 , 1 } ` is th... |
lgslem1 27146 | When ` a ` is coprime to t... |
lgslem2 27147 | The set ` Z ` of all integ... |
lgslem3 27148 | The set ` Z ` of all integ... |
lgslem4 27149 | Lemma for ~ lgsfcl2 . (Co... |
lgsval 27150 | Value of the Legendre symb... |
lgsfval 27151 | Value of the function ` F ... |
lgsfcl2 27152 | The function ` F ` is clos... |
lgscllem 27153 | The Legendre symbol is an ... |
lgsfcl 27154 | Closure of the function ` ... |
lgsfle1 27155 | The function ` F ` has mag... |
lgsval2lem 27156 | Lemma for ~ lgsval2 . (Co... |
lgsval4lem 27157 | Lemma for ~ lgsval4 . (Co... |
lgscl2 27158 | The Legendre symbol is an ... |
lgs0 27159 | The Legendre symbol when t... |
lgscl 27160 | The Legendre symbol is an ... |
lgsle1 27161 | The Legendre symbol has ab... |
lgsval2 27162 | The Legendre symbol at a p... |
lgs2 27163 | The Legendre symbol at ` 2... |
lgsval3 27164 | The Legendre symbol at an ... |
lgsvalmod 27165 | The Legendre symbol is equ... |
lgsval4 27166 | Restate ~ lgsval for nonze... |
lgsfcl3 27167 | Closure of the function ` ... |
lgsval4a 27168 | Same as ~ lgsval4 for posi... |
lgscl1 27169 | The value of the Legendre ... |
lgsneg 27170 | The Legendre symbol is eit... |
lgsneg1 27171 | The Legendre symbol for no... |
lgsmod 27172 | The Legendre (Jacobi) symb... |
lgsdilem 27173 | Lemma for ~ lgsdi and ~ lg... |
lgsdir2lem1 27174 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem2 27175 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem3 27176 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem4 27177 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem5 27178 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2 27179 | The Legendre symbol is com... |
lgsdirprm 27180 | The Legendre symbol is com... |
lgsdir 27181 | The Legendre symbol is com... |
lgsdilem2 27182 | Lemma for ~ lgsdi . (Cont... |
lgsdi 27183 | The Legendre symbol is com... |
lgsne0 27184 | The Legendre symbol is non... |
lgsabs1 27185 | The Legendre symbol is non... |
lgssq 27186 | The Legendre symbol at a s... |
lgssq2 27187 | The Legendre symbol at a s... |
lgsprme0 27188 | The Legendre symbol at any... |
1lgs 27189 | The Legendre symbol at ` 1... |
lgs1 27190 | The Legendre symbol at ` 1... |
lgsmodeq 27191 | The Legendre (Jacobi) symb... |
lgsmulsqcoprm 27192 | The Legendre (Jacobi) symb... |
lgsdirnn0 27193 | Variation on ~ lgsdir vali... |
lgsdinn0 27194 | Variation on ~ lgsdi valid... |
lgsqrlem1 27195 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem2 27196 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem3 27197 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem4 27198 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem5 27199 | Lemma for ~ lgsqr . (Cont... |
lgsqr 27200 | The Legendre symbol for od... |
lgsqrmod 27201 | If the Legendre symbol of ... |
lgsqrmodndvds 27202 | If the Legendre symbol of ... |
lgsdchrval 27203 | The Legendre symbol functi... |
lgsdchr 27204 | The Legendre symbol functi... |
gausslemma2dlem0a 27205 | Auxiliary lemma 1 for ~ ga... |
gausslemma2dlem0b 27206 | Auxiliary lemma 2 for ~ ga... |
gausslemma2dlem0c 27207 | Auxiliary lemma 3 for ~ ga... |
gausslemma2dlem0d 27208 | Auxiliary lemma 4 for ~ ga... |
gausslemma2dlem0e 27209 | Auxiliary lemma 5 for ~ ga... |
gausslemma2dlem0f 27210 | Auxiliary lemma 6 for ~ ga... |
gausslemma2dlem0g 27211 | Auxiliary lemma 7 for ~ ga... |
gausslemma2dlem0h 27212 | Auxiliary lemma 8 for ~ ga... |
gausslemma2dlem0i 27213 | Auxiliary lemma 9 for ~ ga... |
gausslemma2dlem1a 27214 | Lemma for ~ gausslemma2dle... |
gausslemma2dlem1 27215 | Lemma 1 for ~ gausslemma2d... |
gausslemma2dlem2 27216 | Lemma 2 for ~ gausslemma2d... |
gausslemma2dlem3 27217 | Lemma 3 for ~ gausslemma2d... |
gausslemma2dlem4 27218 | Lemma 4 for ~ gausslemma2d... |
gausslemma2dlem5a 27219 | Lemma for ~ gausslemma2dle... |
gausslemma2dlem5 27220 | Lemma 5 for ~ gausslemma2d... |
gausslemma2dlem6 27221 | Lemma 6 for ~ gausslemma2d... |
gausslemma2dlem7 27222 | Lemma 7 for ~ gausslemma2d... |
gausslemma2d 27223 | Gauss' Lemma (see also the... |
lgseisenlem1 27224 | Lemma for ~ lgseisen . If... |
lgseisenlem2 27225 | Lemma for ~ lgseisen . Th... |
lgseisenlem3 27226 | Lemma for ~ lgseisen . (C... |
lgseisenlem4 27227 | Lemma for ~ lgseisen . Th... |
lgseisen 27228 | Eisenstein's lemma, an exp... |
lgsquadlem1 27229 | Lemma for ~ lgsquad . Cou... |
lgsquadlem2 27230 | Lemma for ~ lgsquad . Cou... |
lgsquadlem3 27231 | Lemma for ~ lgsquad . (Co... |
lgsquad 27232 | The Law of Quadratic Recip... |
lgsquad2lem1 27233 | Lemma for ~ lgsquad2 . (C... |
lgsquad2lem2 27234 | Lemma for ~ lgsquad2 . (C... |
lgsquad2 27235 | Extend ~ lgsquad to coprim... |
lgsquad3 27236 | Extend ~ lgsquad2 to integ... |
m1lgs 27237 | The first supplement to th... |
2lgslem1a1 27238 | Lemma 1 for ~ 2lgslem1a . ... |
2lgslem1a2 27239 | Lemma 2 for ~ 2lgslem1a . ... |
2lgslem1a 27240 | Lemma 1 for ~ 2lgslem1 . ... |
2lgslem1b 27241 | Lemma 2 for ~ 2lgslem1 . ... |
2lgslem1c 27242 | Lemma 3 for ~ 2lgslem1 . ... |
2lgslem1 27243 | Lemma 1 for ~ 2lgs . (Con... |
2lgslem2 27244 | Lemma 2 for ~ 2lgs . (Con... |
2lgslem3a 27245 | Lemma for ~ 2lgslem3a1 . ... |
2lgslem3b 27246 | Lemma for ~ 2lgslem3b1 . ... |
2lgslem3c 27247 | Lemma for ~ 2lgslem3c1 . ... |
2lgslem3d 27248 | Lemma for ~ 2lgslem3d1 . ... |
2lgslem3a1 27249 | Lemma 1 for ~ 2lgslem3 . ... |
2lgslem3b1 27250 | Lemma 2 for ~ 2lgslem3 . ... |
2lgslem3c1 27251 | Lemma 3 for ~ 2lgslem3 . ... |
2lgslem3d1 27252 | Lemma 4 for ~ 2lgslem3 . ... |
2lgslem3 27253 | Lemma 3 for ~ 2lgs . (Con... |
2lgs2 27254 | The Legendre symbol for ` ... |
2lgslem4 27255 | Lemma 4 for ~ 2lgs : speci... |
2lgs 27256 | The second supplement to t... |
2lgsoddprmlem1 27257 | Lemma 1 for ~ 2lgsoddprm .... |
2lgsoddprmlem2 27258 | Lemma 2 for ~ 2lgsoddprm .... |
2lgsoddprmlem3a 27259 | Lemma 1 for ~ 2lgsoddprmle... |
2lgsoddprmlem3b 27260 | Lemma 2 for ~ 2lgsoddprmle... |
2lgsoddprmlem3c 27261 | Lemma 3 for ~ 2lgsoddprmle... |
2lgsoddprmlem3d 27262 | Lemma 4 for ~ 2lgsoddprmle... |
2lgsoddprmlem3 27263 | Lemma 3 for ~ 2lgsoddprm .... |
2lgsoddprmlem4 27264 | Lemma 4 for ~ 2lgsoddprm .... |
2lgsoddprm 27265 | The second supplement to t... |
2sqlem1 27266 | Lemma for ~ 2sq . (Contri... |
2sqlem2 27267 | Lemma for ~ 2sq . (Contri... |
mul2sq 27268 | Fibonacci's identity (actu... |
2sqlem3 27269 | Lemma for ~ 2sqlem5 . (Co... |
2sqlem4 27270 | Lemma for ~ 2sqlem5 . (Co... |
2sqlem5 27271 | Lemma for ~ 2sq . If a nu... |
2sqlem6 27272 | Lemma for ~ 2sq . If a nu... |
2sqlem7 27273 | Lemma for ~ 2sq . (Contri... |
2sqlem8a 27274 | Lemma for ~ 2sqlem8 . (Co... |
2sqlem8 27275 | Lemma for ~ 2sq . (Contri... |
2sqlem9 27276 | Lemma for ~ 2sq . (Contri... |
2sqlem10 27277 | Lemma for ~ 2sq . Every f... |
2sqlem11 27278 | Lemma for ~ 2sq . (Contri... |
2sq 27279 | All primes of the form ` 4... |
2sqblem 27280 | Lemma for ~ 2sqb . (Contr... |
2sqb 27281 | The converse to ~ 2sq . (... |
2sq2 27282 | ` 2 ` is the sum of square... |
2sqn0 27283 | If the sum of two squares ... |
2sqcoprm 27284 | If the sum of two squares ... |
2sqmod 27285 | Given two decompositions o... |
2sqmo 27286 | There exists at most one d... |
2sqnn0 27287 | All primes of the form ` 4... |
2sqnn 27288 | All primes of the form ` 4... |
addsq2reu 27289 | For each complex number ` ... |
addsqn2reu 27290 | For each complex number ` ... |
addsqrexnreu 27291 | For each complex number, t... |
addsqnreup 27292 | There is no unique decompo... |
addsq2nreurex 27293 | For each complex number ` ... |
addsqn2reurex2 27294 | For each complex number ` ... |
2sqreulem1 27295 | Lemma 1 for ~ 2sqreu . (C... |
2sqreultlem 27296 | Lemma for ~ 2sqreult . (C... |
2sqreultblem 27297 | Lemma for ~ 2sqreultb . (... |
2sqreunnlem1 27298 | Lemma 1 for ~ 2sqreunn . ... |
2sqreunnltlem 27299 | Lemma for ~ 2sqreunnlt . ... |
2sqreunnltblem 27300 | Lemma for ~ 2sqreunnltb . ... |
2sqreulem2 27301 | Lemma 2 for ~ 2sqreu etc. ... |
2sqreulem3 27302 | Lemma 3 for ~ 2sqreu etc. ... |
2sqreulem4 27303 | Lemma 4 for ~ 2sqreu et. ... |
2sqreunnlem2 27304 | Lemma 2 for ~ 2sqreunn . ... |
2sqreu 27305 | There exists a unique deco... |
2sqreunn 27306 | There exists a unique deco... |
2sqreult 27307 | There exists a unique deco... |
2sqreultb 27308 | There exists a unique deco... |
2sqreunnlt 27309 | There exists a unique deco... |
2sqreunnltb 27310 | There exists a unique deco... |
2sqreuop 27311 | There exists a unique deco... |
2sqreuopnn 27312 | There exists a unique deco... |
2sqreuoplt 27313 | There exists a unique deco... |
2sqreuopltb 27314 | There exists a unique deco... |
2sqreuopnnlt 27315 | There exists a unique deco... |
2sqreuopnnltb 27316 | There exists a unique deco... |
2sqreuopb 27317 | There exists a unique deco... |
chebbnd1lem1 27318 | Lemma for ~ chebbnd1 : sho... |
chebbnd1lem2 27319 | Lemma for ~ chebbnd1 : Sh... |
chebbnd1lem3 27320 | Lemma for ~ chebbnd1 : get... |
chebbnd1 27321 | The Chebyshev bound: The ... |
chtppilimlem1 27322 | Lemma for ~ chtppilim . (... |
chtppilimlem2 27323 | Lemma for ~ chtppilim . (... |
chtppilim 27324 | The ` theta ` function is ... |
chto1ub 27325 | The ` theta ` function is ... |
chebbnd2 27326 | The Chebyshev bound, part ... |
chto1lb 27327 | The ` theta ` function is ... |
chpchtlim 27328 | The ` psi ` and ` theta ` ... |
chpo1ub 27329 | The ` psi ` function is up... |
chpo1ubb 27330 | The ` psi ` function is up... |
vmadivsum 27331 | The sum of the von Mangold... |
vmadivsumb 27332 | Give a total bound on the ... |
rplogsumlem1 27333 | Lemma for ~ rplogsum . (C... |
rplogsumlem2 27334 | Lemma for ~ rplogsum . Eq... |
dchrisum0lem1a 27335 | Lemma for ~ dchrisum0lem1 ... |
rpvmasumlem 27336 | Lemma for ~ rpvmasum . Ca... |
dchrisumlema 27337 | Lemma for ~ dchrisum . Le... |
dchrisumlem1 27338 | Lemma for ~ dchrisum . Le... |
dchrisumlem2 27339 | Lemma for ~ dchrisum . Le... |
dchrisumlem3 27340 | Lemma for ~ dchrisum . Le... |
dchrisum 27341 | If ` n e. [ M , +oo ) |-> ... |
dchrmusumlema 27342 | Lemma for ~ dchrmusum and ... |
dchrmusum2 27343 | The sum of the Möbius... |
dchrvmasumlem1 27344 | An alternative expression ... |
dchrvmasum2lem 27345 | Give an expression for ` l... |
dchrvmasum2if 27346 | Combine the results of ~ d... |
dchrvmasumlem2 27347 | Lemma for ~ dchrvmasum . ... |
dchrvmasumlem3 27348 | Lemma for ~ dchrvmasum . ... |
dchrvmasumlema 27349 | Lemma for ~ dchrvmasum and... |
dchrvmasumiflem1 27350 | Lemma for ~ dchrvmasumif .... |
dchrvmasumiflem2 27351 | Lemma for ~ dchrvmasum . ... |
dchrvmasumif 27352 | An asymptotic approximatio... |
dchrvmaeq0 27353 | The set ` W ` is the colle... |
dchrisum0fval 27354 | Value of the function ` F ... |
dchrisum0fmul 27355 | The function ` F ` , the d... |
dchrisum0ff 27356 | The function ` F ` is a re... |
dchrisum0flblem1 27357 | Lemma for ~ dchrisum0flb .... |
dchrisum0flblem2 27358 | Lemma for ~ dchrisum0flb .... |
dchrisum0flb 27359 | The divisor sum of a real ... |
dchrisum0fno1 27360 | The sum ` sum_ k <_ x , F ... |
rpvmasum2 27361 | A partial result along the... |
dchrisum0re 27362 | Suppose ` X ` is a non-pri... |
dchrisum0lema 27363 | Lemma for ~ dchrisum0 . A... |
dchrisum0lem1b 27364 | Lemma for ~ dchrisum0lem1 ... |
dchrisum0lem1 27365 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem2a 27366 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem2 27367 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem3 27368 | Lemma for ~ dchrisum0 . (... |
dchrisum0 27369 | The sum ` sum_ n e. NN , X... |
dchrisumn0 27370 | The sum ` sum_ n e. NN , X... |
dchrmusumlem 27371 | The sum of the Möbius... |
dchrvmasumlem 27372 | The sum of the Möbius... |
dchrmusum 27373 | The sum of the Möbius... |
dchrvmasum 27374 | The sum of the von Mangold... |
rpvmasum 27375 | The sum of the von Mangold... |
rplogsum 27376 | The sum of ` log p / p ` o... |
dirith2 27377 | Dirichlet's theorem: there... |
dirith 27378 | Dirichlet's theorem: there... |
mudivsum 27379 | Asymptotic formula for ` s... |
mulogsumlem 27380 | Lemma for ~ mulogsum . (C... |
mulogsum 27381 | Asymptotic formula for ... |
logdivsum 27382 | Asymptotic analysis of ... |
mulog2sumlem1 27383 | Asymptotic formula for ... |
mulog2sumlem2 27384 | Lemma for ~ mulog2sum . (... |
mulog2sumlem3 27385 | Lemma for ~ mulog2sum . (... |
mulog2sum 27386 | Asymptotic formula for ... |
vmalogdivsum2 27387 | The sum ` sum_ n <_ x , La... |
vmalogdivsum 27388 | The sum ` sum_ n <_ x , La... |
2vmadivsumlem 27389 | Lemma for ~ 2vmadivsum . ... |
2vmadivsum 27390 | The sum ` sum_ m n <_ x , ... |
logsqvma 27391 | A formula for ` log ^ 2 ( ... |
logsqvma2 27392 | The Möbius inverse of... |
log2sumbnd 27393 | Bound on the difference be... |
selberglem1 27394 | Lemma for ~ selberg . Est... |
selberglem2 27395 | Lemma for ~ selberg . (Co... |
selberglem3 27396 | Lemma for ~ selberg . Est... |
selberg 27397 | Selberg's symmetry formula... |
selbergb 27398 | Convert eventual boundedne... |
selberg2lem 27399 | Lemma for ~ selberg2 . Eq... |
selberg2 27400 | Selberg's symmetry formula... |
selberg2b 27401 | Convert eventual boundedne... |
chpdifbndlem1 27402 | Lemma for ~ chpdifbnd . (... |
chpdifbndlem2 27403 | Lemma for ~ chpdifbnd . (... |
chpdifbnd 27404 | A bound on the difference ... |
logdivbnd 27405 | A bound on a sum of logs, ... |
selberg3lem1 27406 | Introduce a log weighting ... |
selberg3lem2 27407 | Lemma for ~ selberg3 . Eq... |
selberg3 27408 | Introduce a log weighting ... |
selberg4lem1 27409 | Lemma for ~ selberg4 . Eq... |
selberg4 27410 | The Selberg symmetry formu... |
pntrval 27411 | Define the residual of the... |
pntrf 27412 | Functionality of the resid... |
pntrmax 27413 | There is a bound on the re... |
pntrsumo1 27414 | A bound on a sum over ` R ... |
pntrsumbnd 27415 | A bound on a sum over ` R ... |
pntrsumbnd2 27416 | A bound on a sum over ` R ... |
selbergr 27417 | Selberg's symmetry formula... |
selberg3r 27418 | Selberg's symmetry formula... |
selberg4r 27419 | Selberg's symmetry formula... |
selberg34r 27420 | The sum of ~ selberg3r and... |
pntsval 27421 | Define the "Selberg functi... |
pntsf 27422 | Functionality of the Selbe... |
selbergs 27423 | Selberg's symmetry formula... |
selbergsb 27424 | Selberg's symmetry formula... |
pntsval2 27425 | The Selberg function can b... |
pntrlog2bndlem1 27426 | The sum of ~ selberg3r and... |
pntrlog2bndlem2 27427 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem3 27428 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem4 27429 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem5 27430 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem6a 27431 | Lemma for ~ pntrlog2bndlem... |
pntrlog2bndlem6 27432 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bnd 27433 | A bound on ` R ( x ) log ^... |
pntpbnd1a 27434 | Lemma for ~ pntpbnd . (Co... |
pntpbnd1 27435 | Lemma for ~ pntpbnd . (Co... |
pntpbnd2 27436 | Lemma for ~ pntpbnd . (Co... |
pntpbnd 27437 | Lemma for ~ pnt . Establi... |
pntibndlem1 27438 | Lemma for ~ pntibnd . (Co... |
pntibndlem2a 27439 | Lemma for ~ pntibndlem2 . ... |
pntibndlem2 27440 | Lemma for ~ pntibnd . The... |
pntibndlem3 27441 | Lemma for ~ pntibnd . Pac... |
pntibnd 27442 | Lemma for ~ pnt . Establi... |
pntlemd 27443 | Lemma for ~ pnt . Closure... |
pntlemc 27444 | Lemma for ~ pnt . Closure... |
pntlema 27445 | Lemma for ~ pnt . Closure... |
pntlemb 27446 | Lemma for ~ pnt . Unpack ... |
pntlemg 27447 | Lemma for ~ pnt . Closure... |
pntlemh 27448 | Lemma for ~ pnt . Bounds ... |
pntlemn 27449 | Lemma for ~ pnt . The "na... |
pntlemq 27450 | Lemma for ~ pntlemj . (Co... |
pntlemr 27451 | Lemma for ~ pntlemj . (Co... |
pntlemj 27452 | Lemma for ~ pnt . The ind... |
pntlemi 27453 | Lemma for ~ pnt . Elimina... |
pntlemf 27454 | Lemma for ~ pnt . Add up ... |
pntlemk 27455 | Lemma for ~ pnt . Evaluat... |
pntlemo 27456 | Lemma for ~ pnt . Combine... |
pntleme 27457 | Lemma for ~ pnt . Package... |
pntlem3 27458 | Lemma for ~ pnt . Equatio... |
pntlemp 27459 | Lemma for ~ pnt . Wrappin... |
pntleml 27460 | Lemma for ~ pnt . Equatio... |
pnt3 27461 | The Prime Number Theorem, ... |
pnt2 27462 | The Prime Number Theorem, ... |
pnt 27463 | The Prime Number Theorem: ... |
abvcxp 27464 | Raising an absolute value ... |
padicfval 27465 | Value of the p-adic absolu... |
padicval 27466 | Value of the p-adic absolu... |
ostth2lem1 27467 | Lemma for ~ ostth2 , altho... |
qrngbas 27468 | The base set of the field ... |
qdrng 27469 | The rationals form a divis... |
qrng0 27470 | The zero element of the fi... |
qrng1 27471 | The unity element of the f... |
qrngneg 27472 | The additive inverse in th... |
qrngdiv 27473 | The division operation in ... |
qabvle 27474 | By using induction on ` N ... |
qabvexp 27475 | Induct the product rule ~ ... |
ostthlem1 27476 | Lemma for ~ ostth . If tw... |
ostthlem2 27477 | Lemma for ~ ostth . Refin... |
qabsabv 27478 | The regular absolute value... |
padicabv 27479 | The p-adic absolute value ... |
padicabvf 27480 | The p-adic absolute value ... |
padicabvcxp 27481 | All positive powers of the... |
ostth1 27482 | - Lemma for ~ ostth : triv... |
ostth2lem2 27483 | Lemma for ~ ostth2 . (Con... |
ostth2lem3 27484 | Lemma for ~ ostth2 . (Con... |
ostth2lem4 27485 | Lemma for ~ ostth2 . (Con... |
ostth2 27486 | - Lemma for ~ ostth : regu... |
ostth3 27487 | - Lemma for ~ ostth : p-ad... |
ostth 27488 | Ostrowski's theorem, which... |
elno 27495 | Membership in the surreals... |
sltval 27496 | The value of the surreal l... |
bdayval 27497 | The value of the birthday ... |
nofun 27498 | A surreal is a function. ... |
nodmon 27499 | The domain of a surreal is... |
norn 27500 | The range of a surreal is ... |
nofnbday 27501 | A surreal is a function ov... |
nodmord 27502 | The domain of a surreal ha... |
elno2 27503 | An alternative condition f... |
elno3 27504 | Another condition for memb... |
sltval2 27505 | Alternate expression for s... |
nofv 27506 | The function value of a su... |
nosgnn0 27507 | ` (/) ` is not a surreal s... |
nosgnn0i 27508 | If ` X ` is a surreal sign... |
noreson 27509 | The restriction of a surre... |
sltintdifex 27510 |
If ` A |
sltres 27511 | If the restrictions of two... |
noxp1o 27512 | The Cartesian product of a... |
noseponlem 27513 | Lemma for ~ nosepon . Con... |
nosepon 27514 | Given two unequal surreals... |
noextend 27515 | Extending a surreal by one... |
noextendseq 27516 | Extend a surreal by a sequ... |
noextenddif 27517 | Calculate the place where ... |
noextendlt 27518 | Extending a surreal with a... |
noextendgt 27519 | Extending a surreal with a... |
nolesgn2o 27520 | Given ` A ` less-than or e... |
nolesgn2ores 27521 | Given ` A ` less-than or e... |
nogesgn1o 27522 | Given ` A ` greater than o... |
nogesgn1ores 27523 | Given ` A ` greater than o... |
sltsolem1 27524 | Lemma for ~ sltso . The "... |
sltso 27525 | Less-than totally orders t... |
bdayfo 27526 | The birthday function maps... |
fvnobday 27527 | The value of a surreal at ... |
nosepnelem 27528 | Lemma for ~ nosepne . (Co... |
nosepne 27529 | The value of two non-equal... |
nosep1o 27530 | If the value of a surreal ... |
nosep2o 27531 | If the value of a surreal ... |
nosepdmlem 27532 | Lemma for ~ nosepdm . (Co... |
nosepdm 27533 | The first place two surrea... |
nosepeq 27534 | The values of two surreals... |
nosepssdm 27535 | Given two non-equal surrea... |
nodenselem4 27536 | Lemma for ~ nodense . Sho... |
nodenselem5 27537 | Lemma for ~ nodense . If ... |
nodenselem6 27538 | The restriction of a surre... |
nodenselem7 27539 | Lemma for ~ nodense . ` A ... |
nodenselem8 27540 | Lemma for ~ nodense . Giv... |
nodense 27541 | Given two distinct surreal... |
bdayimaon 27542 | Lemma for full-eta propert... |
nolt02olem 27543 | Lemma for ~ nolt02o . If ... |
nolt02o 27544 | Given ` A ` less-than ` B ... |
nogt01o 27545 | Given ` A ` greater than `... |
noresle 27546 | Restriction law for surrea... |
nomaxmo 27547 | A class of surreals has at... |
nominmo 27548 | A class of surreals has at... |
nosupprefixmo 27549 | In any class of surreals, ... |
noinfprefixmo 27550 | In any class of surreals, ... |
nosupcbv 27551 | Lemma to change bound vari... |
nosupno 27552 | The next several theorems ... |
nosupdm 27553 | The domain of the surreal ... |
nosupbday 27554 | Birthday bounding law for ... |
nosupfv 27555 | The value of surreal supre... |
nosupres 27556 | A restriction law for surr... |
nosupbnd1lem1 27557 | Lemma for ~ nosupbnd1 . E... |
nosupbnd1lem2 27558 | Lemma for ~ nosupbnd1 . W... |
nosupbnd1lem3 27559 | Lemma for ~ nosupbnd1 . I... |
nosupbnd1lem4 27560 | Lemma for ~ nosupbnd1 . I... |
nosupbnd1lem5 27561 | Lemma for ~ nosupbnd1 . I... |
nosupbnd1lem6 27562 | Lemma for ~ nosupbnd1 . E... |
nosupbnd1 27563 | Bounding law from below fo... |
nosupbnd2lem1 27564 | Bounding law from above wh... |
nosupbnd2 27565 | Bounding law from above fo... |
noinfcbv 27566 | Change bound variables for... |
noinfno 27567 | The next several theorems ... |
noinfdm 27568 | Next, we calculate the dom... |
noinfbday 27569 | Birthday bounding law for ... |
noinffv 27570 | The value of surreal infim... |
noinfres 27571 | The restriction of surreal... |
noinfbnd1lem1 27572 | Lemma for ~ noinfbnd1 . E... |
noinfbnd1lem2 27573 | Lemma for ~ noinfbnd1 . W... |
noinfbnd1lem3 27574 | Lemma for ~ noinfbnd1 . I... |
noinfbnd1lem4 27575 | Lemma for ~ noinfbnd1 . I... |
noinfbnd1lem5 27576 | Lemma for ~ noinfbnd1 . I... |
noinfbnd1lem6 27577 | Lemma for ~ noinfbnd1 . E... |
noinfbnd1 27578 | Bounding law from above fo... |
noinfbnd2lem1 27579 | Bounding law from below wh... |
noinfbnd2 27580 | Bounding law from below fo... |
nosupinfsep 27581 | Given two sets of surreals... |
noetasuplem1 27582 | Lemma for ~ noeta . Estab... |
noetasuplem2 27583 | Lemma for ~ noeta . The r... |
noetasuplem3 27584 | Lemma for ~ noeta . ` Z ` ... |
noetasuplem4 27585 | Lemma for ~ noeta . When ... |
noetainflem1 27586 | Lemma for ~ noeta . Estab... |
noetainflem2 27587 | Lemma for ~ noeta . The r... |
noetainflem3 27588 | Lemma for ~ noeta . ` W ` ... |
noetainflem4 27589 | Lemma for ~ noeta . If ` ... |
noetalem1 27590 | Lemma for ~ noeta . Eithe... |
noetalem2 27591 | Lemma for ~ noeta . The f... |
noeta 27592 | The full-eta axiom for the... |
sltirr 27595 | Surreal less-than is irref... |
slttr 27596 | Surreal less-than is trans... |
sltasym 27597 | Surreal less-than is asymm... |
sltlin 27598 | Surreal less-than obeys tr... |
slttrieq2 27599 | Trichotomy law for surreal... |
slttrine 27600 | Trichotomy law for surreal... |
slenlt 27601 | Surreal less-than or equal... |
sltnle 27602 | Surreal less-than in terms... |
sleloe 27603 | Surreal less-than or equal... |
sletri3 27604 | Trichotomy law for surreal... |
sltletr 27605 | Surreal transitive law. (... |
slelttr 27606 | Surreal transitive law. (... |
sletr 27607 | Surreal transitive law. (... |
slttrd 27608 | Surreal less-than is trans... |
sltletrd 27609 | Surreal less-than is trans... |
slelttrd 27610 | Surreal less-than is trans... |
sletrd 27611 | Surreal less-than or equal... |
slerflex 27612 | Surreal less-than or equal... |
sletric 27613 | Surreal trichotomy law. (... |
maxs1 27614 | A surreal is less than or ... |
maxs2 27615 | A surreal is less than or ... |
mins1 27616 | The minimum of two surreal... |
mins2 27617 | The minimum of two surreal... |
sltled 27618 | Surreal less-than implies ... |
sltne 27619 | Surreal less-than implies ... |
sltlend 27620 | Surreal less-than in terms... |
bdayfun 27621 | The birthday function is a... |
bdayfn 27622 | The birthday function is a... |
bdaydm 27623 | The birthday function's do... |
bdayrn 27624 | The birthday function's ra... |
bdayelon 27625 | The value of the birthday ... |
nocvxminlem 27626 | Lemma for ~ nocvxmin . Gi... |
nocvxmin 27627 | Given a nonempty convex cl... |
noprc 27628 | The surreal numbers are a ... |
noeta2 27633 | A version of ~ noeta with ... |
brsslt 27634 | Binary relation form of th... |
ssltex1 27635 | The first argument of surr... |
ssltex2 27636 | The second argument of sur... |
ssltss1 27637 | The first argument of surr... |
ssltss2 27638 | The second argument of sur... |
ssltsep 27639 | The separation property of... |
ssltd 27640 | Deduce surreal set less-th... |
ssltsn 27641 | Surreal set less-than of t... |
ssltsepc 27642 | Two elements of separated ... |
ssltsepcd 27643 | Two elements of separated ... |
sssslt1 27644 | Relation between surreal s... |
sssslt2 27645 | Relation between surreal s... |
nulsslt 27646 | The empty set is less-than... |
nulssgt 27647 | The empty set is greater t... |
conway 27648 | Conway's Simplicity Theore... |
scutval 27649 | The value of the surreal c... |
scutcut 27650 | Cut properties of the surr... |
scutcl 27651 | Closure law for surreal cu... |
scutcld 27652 | Closure law for surreal cu... |
scutbday 27653 | The birthday of the surrea... |
eqscut 27654 | Condition for equality to ... |
eqscut2 27655 | Condition for equality to ... |
sslttr 27656 | Transitive law for surreal... |
ssltun1 27657 | Union law for surreal set ... |
ssltun2 27658 | Union law for surreal set ... |
scutun12 27659 | Union law for surreal cuts... |
dmscut 27660 | The domain of the surreal ... |
scutf 27661 | Functionality statement fo... |
etasslt 27662 | A restatement of ~ noeta u... |
etasslt2 27663 | A version of ~ etasslt wit... |
scutbdaybnd 27664 | An upper bound on the birt... |
scutbdaybnd2 27665 | An upper bound on the birt... |
scutbdaybnd2lim 27666 | An upper bound on the birt... |
scutbdaylt 27667 | If a surreal lies in a gap... |
slerec 27668 | A comparison law for surre... |
sltrec 27669 | A comparison law for surre... |
ssltdisj 27670 | If ` A ` preceeds ` B ` , ... |
0sno 27675 | Surreal zero is a surreal.... |
1sno 27676 | Surreal one is a surreal. ... |
bday0s 27677 | Calculate the birthday of ... |
0slt1s 27678 | Surreal zero is less than ... |
bday0b 27679 | The only surreal with birt... |
bday1s 27680 | The birthday of surreal on... |
cuteq0 27681 | Condition for a surreal cu... |
cuteq1 27682 | Condition for a surreal cu... |
sgt0ne0 27683 | A positive surreal is not ... |
sgt0ne0d 27684 | A positive surreal is not ... |
madeval 27695 | The value of the made by f... |
madeval2 27696 | Alternative characterizati... |
oldval 27697 | The value of the old optio... |
newval 27698 | The value of the new optio... |
madef 27699 | The made function is a fun... |
oldf 27700 | The older function is a fu... |
newf 27701 | The new function is a func... |
old0 27702 | No surreal is older than `... |
madessno 27703 | Made sets are surreals. (... |
oldssno 27704 | Old sets are surreals. (C... |
newssno 27705 | New sets are surreals. (C... |
leftval 27706 | The value of the left opti... |
rightval 27707 | The value of the right opt... |
leftf 27708 | The functionality of the l... |
rightf 27709 | The functionality of the r... |
elmade 27710 | Membership in the made fun... |
elmade2 27711 | Membership in the made fun... |
elold 27712 | Membership in an old set. ... |
ssltleft 27713 | A surreal is greater than ... |
ssltright 27714 | A surreal is less than its... |
lltropt 27715 | The left options of a surr... |
made0 27716 | The only surreal made on d... |
new0 27717 | The only surreal new on da... |
old1 27718 | The only surreal older tha... |
madess 27719 | If ` A ` is less than or e... |
oldssmade 27720 | The older-than set is a su... |
leftssold 27721 | The left options are a sub... |
rightssold 27722 | The right options are a su... |
leftssno 27723 | The left set of a surreal ... |
rightssno 27724 | The right set of a surreal... |
madecut 27725 | Given a section that is a ... |
madeun 27726 | The made set is the union ... |
madeoldsuc 27727 | The made set is the old se... |
oldsuc 27728 | The value of the old set a... |
oldlim 27729 | The value of the old set a... |
madebdayim 27730 | If a surreal is a member o... |
oldbdayim 27731 | If ` X ` is in the old set... |
oldirr 27732 | No surreal is a member of ... |
leftirr 27733 | No surreal is a member of ... |
rightirr 27734 | No surreal is a member of ... |
left0s 27735 | The left set of ` 0s ` is ... |
right0s 27736 | The right set of ` 0s ` is... |
left1s 27737 | The left set of ` 1s ` is ... |
right1s 27738 | The right set of ` 1s ` is... |
lrold 27739 | The union of the left and ... |
madebdaylemold 27740 | Lemma for ~ madebday . If... |
madebdaylemlrcut 27741 | Lemma for ~ madebday . If... |
madebday 27742 | A surreal is part of the s... |
oldbday 27743 | A surreal is part of the s... |
newbday 27744 | A surreal is an element of... |
lrcut 27745 | A surreal is equal to the ... |
scutfo 27746 | The surreal cut function i... |
sltn0 27747 | If ` X ` is less than ` Y ... |
lruneq 27748 | If two surreals share a bi... |
sltlpss 27749 | If two surreals share a bi... |
slelss 27750 | If two surreals ` A ` and ... |
0elold 27751 | Zero is in the old set of ... |
0elleft 27752 | Zero is in the left set of... |
0elright 27753 | Zero is in the right set o... |
cofsslt 27754 | If every element of ` A ` ... |
coinitsslt 27755 | If ` B ` is coinitial with... |
cofcut1 27756 | If ` C ` is cofinal with `... |
cofcut1d 27757 | If ` C ` is cofinal with `... |
cofcut2 27758 | If ` A ` and ` C ` are mut... |
cofcut2d 27759 | If ` A ` and ` C ` are mut... |
cofcutr 27760 | If ` X ` is the cut of ` A... |
cofcutr1d 27761 | If ` X ` is the cut of ` A... |
cofcutr2d 27762 | If ` X ` is the cut of ` A... |
cofcutrtime 27763 | If ` X ` is the cut of ` A... |
cofcutrtime1d 27764 | If ` X ` is a timely cut o... |
cofcutrtime2d 27765 | If ` X ` is a timely cut o... |
cofss 27766 | Cofinality for a subset. ... |
coiniss 27767 | Coinitiality for a subset.... |
cutlt 27768 | Eliminating all elements b... |
cutpos 27769 | Reduce the elements of a c... |
lrrecval 27772 | The next step in the devel... |
lrrecval2 27773 | Next, we establish an alte... |
lrrecpo 27774 | Now, we establish that ` R... |
lrrecse 27775 | Next, we show that ` R ` i... |
lrrecfr 27776 | Now we show that ` R ` is ... |
lrrecpred 27777 | Finally, we calculate the ... |
noinds 27778 | Induction principle for a ... |
norecfn 27779 | Surreal recursion over one... |
norecov 27780 | Calculate the value of the... |
noxpordpo 27783 | To get through most of the... |
noxpordfr 27784 | Next we establish the foun... |
noxpordse 27785 | Next we establish the set-... |
noxpordpred 27786 | Next we calculate the pred... |
no2indslem 27787 | Double induction on surrea... |
no2inds 27788 | Double induction on surrea... |
norec2fn 27789 | The double-recursion opera... |
norec2ov 27790 | The value of the double-re... |
no3inds 27791 | Triple induction over surr... |
addsfn 27794 | Surreal addition is a func... |
addsval 27795 | The value of surreal addit... |
addsval2 27796 | The value of surreal addit... |
addsrid 27797 | Surreal addition to zero i... |
addsridd 27798 | Surreal addition to zero i... |
addscom 27799 | Surreal addition commutes.... |
addscomd 27800 | Surreal addition commutes.... |
addslid 27801 | Surreal addition to zero i... |
addsproplem1 27802 | Lemma for surreal addition... |
addsproplem2 27803 | Lemma for surreal addition... |
addsproplem3 27804 | Lemma for surreal addition... |
addsproplem4 27805 | Lemma for surreal addition... |
addsproplem5 27806 | Lemma for surreal addition... |
addsproplem6 27807 | Lemma for surreal addition... |
addsproplem7 27808 | Lemma for surreal addition... |
addsprop 27809 | Inductively show that surr... |
addscutlem 27810 | Lemma for ~ addscut . Sho... |
addscut 27811 | Demonstrate the cut proper... |
addscut2 27812 | Show that the cut involved... |
addscld 27813 | Surreal numbers are closed... |
addscl 27814 | Surreal numbers are closed... |
addsf 27815 | Function statement for sur... |
addsfo 27816 | Surreal addition is onto. ... |
peano2no 27817 | A theorem for surreals tha... |
sltadd1im 27818 | Surreal less-than is prese... |
sltadd2im 27819 | Surreal less-than is prese... |
sleadd1im 27820 | Surreal less-than or equal... |
sleadd2im 27821 | Surreal less-than or equal... |
sleadd1 27822 | Addition to both sides of ... |
sleadd2 27823 | Addition to both sides of ... |
sltadd2 27824 | Addition to both sides of ... |
sltadd1 27825 | Addition to both sides of ... |
addscan2 27826 | Cancellation law for surre... |
addscan1 27827 | Cancellation law for surre... |
sleadd1d 27828 | Addition to both sides of ... |
sleadd2d 27829 | Addition to both sides of ... |
sltadd2d 27830 | Addition to both sides of ... |
sltadd1d 27831 | Addition to both sides of ... |
addscan2d 27832 | Cancellation law for surre... |
addscan1d 27833 | Cancellation law for surre... |
addsuniflem 27834 | Lemma for ~ addsunif . St... |
addsunif 27835 | Uniformity theorem for sur... |
addsasslem1 27836 | Lemma for addition associa... |
addsasslem2 27837 | Lemma for addition associa... |
addsass 27838 | Surreal addition is associ... |
addsassd 27839 | Surreal addition is associ... |
adds32d 27840 | Commutative/associative la... |
adds12d 27841 | Commutative/associative la... |
adds4d 27842 | Rearrangement of four term... |
adds42d 27843 | Rearrangement of four term... |
sltaddpos1d 27844 | Addition of a positive num... |
sltaddpos2d 27845 | Addition of a positive num... |
slt2addd 27846 | Adding both sides of two s... |
addsgt0d 27847 | The sum of two positive su... |
negsfn 27852 | Surreal negation is a func... |
subsfn 27853 | Surreal subtraction is a f... |
negsval 27854 | The value of the surreal n... |
negs0s 27855 | Negative surreal zero is s... |
negsproplem1 27856 | Lemma for surreal negation... |
negsproplem2 27857 | Lemma for surreal negation... |
negsproplem3 27858 | Lemma for surreal negation... |
negsproplem4 27859 | Lemma for surreal negation... |
negsproplem5 27860 | Lemma for surreal negation... |
negsproplem6 27861 | Lemma for surreal negation... |
negsproplem7 27862 | Lemma for surreal negation... |
negsprop 27863 | Show closure and ordering ... |
negscl 27864 | The surreals are closed un... |
negscld 27865 | The surreals are closed un... |
sltnegim 27866 | The forward direction of t... |
negscut 27867 | The cut properties of surr... |
negscut2 27868 | The cut that defines surre... |
negsid 27869 | Surreal addition of a numb... |
negsidd 27870 | Surreal addition of a numb... |
negsex 27871 | Every surreal has a negati... |
negnegs 27872 | A surreal is equal to the ... |
sltneg 27873 | Negative of both sides of ... |
sleneg 27874 | Negative of both sides of ... |
sltnegd 27875 | Negative of both sides of ... |
slenegd 27876 | Negative of both sides of ... |
negs11 27877 | Surreal negation is one-to... |
negsdi 27878 | Distribution of surreal ne... |
slt0neg2d 27879 | Comparison of a surreal an... |
negsf 27880 | Function statement for sur... |
negsfo 27881 | Function statement for sur... |
negsf1o 27882 | Surreal negation is a bije... |
negsunif 27883 | Uniformity property for su... |
negsbdaylem 27884 | Lemma for ~ negsbday . Bo... |
negsbday 27885 | Negation of a surreal numb... |
subsval 27886 | The value of surreal subtr... |
subsvald 27887 | The value of surreal subtr... |
subscl 27888 | Closure law for surreal su... |
subscld 27889 | Closure law for surreal su... |
negsval2 27890 | Surreal negation in terms ... |
negsval2d 27891 | Surreal negation in terms ... |
subsid1 27892 | Identity law for subtracti... |
subsid 27893 | Subtraction of a surreal f... |
subadds 27894 | Relationship between addit... |
subaddsd 27895 | Relationship between addit... |
pncans 27896 | Cancellation law for surre... |
pncan3s 27897 | Subtraction and addition o... |
pncan2s 27898 | Cancellation law for surre... |
npcans 27899 | Cancellation law for surre... |
sltsub1 27900 | Subtraction from both side... |
sltsub2 27901 | Subtraction from both side... |
sltsub1d 27902 | Subtraction from both side... |
sltsub2d 27903 | Subtraction from both side... |
negsubsdi2d 27904 | Distribution of negative o... |
addsubsassd 27905 | Associative-type law for s... |
addsubsd 27906 | Law for surreal addition a... |
sltsubsubbd 27907 | Equivalence for the surrea... |
sltsubsub2bd 27908 | Equivalence for the surrea... |
sltsubsub3bd 27909 | Equivalence for the surrea... |
slesubsubbd 27910 | Equivalence for the surrea... |
slesubsub2bd 27911 | Equivalence for the surrea... |
slesubsub3bd 27912 | Equivalence for the surrea... |
sltsubaddd 27913 | Surreal less-than relation... |
sltsubadd2d 27914 | Surreal less-than relation... |
sltaddsubd 27915 | Surreal less-than relation... |
sltaddsub2d 27916 | Surreal less-than relation... |
subsubs4d 27917 | Law for double surreal sub... |
subsubs2d 27918 | Law for double surreal sub... |
nncansd 27919 | Cancellation law for surre... |
posdifsd 27920 | Comparison of two surreals... |
sltsubposd 27921 | Subtraction of a positive ... |
mulsfn 27924 | Surreal multiplication is ... |
mulsval 27925 | The value of surreal multi... |
mulsval2lem 27926 | Lemma for ~ mulsval2 . Ch... |
mulsval2 27927 | The value of surreal multi... |
muls01 27928 | Surreal multiplication by ... |
mulsrid 27929 | Surreal one is a right ide... |
mulsridd 27930 | Surreal one is a right ide... |
mulsproplemcbv 27931 | Lemma for surreal multipli... |
mulsproplem1 27932 | Lemma for surreal multipli... |
mulsproplem2 27933 | Lemma for surreal multipli... |
mulsproplem3 27934 | Lemma for surreal multipli... |
mulsproplem4 27935 | Lemma for surreal multipli... |
mulsproplem5 27936 | Lemma for surreal multipli... |
mulsproplem6 27937 | Lemma for surreal multipli... |
mulsproplem7 27938 | Lemma for surreal multipli... |
mulsproplem8 27939 | Lemma for surreal multipli... |
mulsproplem9 27940 | Lemma for surreal multipli... |
mulsproplem10 27941 | Lemma for surreal multipli... |
mulsproplem11 27942 | Lemma for surreal multipli... |
mulsproplem12 27943 | Lemma for surreal multipli... |
mulsproplem13 27944 | Lemma for surreal multipli... |
mulsproplem14 27945 | Lemma for surreal multipli... |
mulsprop 27946 | Surreals are closed under ... |
mulscutlem 27947 | Lemma for ~ mulscut . Sta... |
mulscut 27948 | Show the cut properties of... |
mulscut2 27949 | Show that the cut involved... |
mulscl 27950 | The surreals are closed un... |
mulscld 27951 | The surreals are closed un... |
sltmul 27952 | An ordering relationship f... |
sltmuld 27953 | An ordering relationship f... |
slemuld 27954 | An ordering relationship f... |
mulscom 27955 | Surreal multiplication com... |
mulscomd 27956 | Surreal multiplication com... |
muls02 27957 | Surreal multiplication by ... |
mulslid 27958 | Surreal one is a left iden... |
mulslidd 27959 | Surreal one is a left iden... |
mulsgt0 27960 | The product of two positiv... |
mulsgt0d 27961 | The product of two positiv... |
mulsge0d 27962 | The product of two non-neg... |
ssltmul1 27963 | One surreal set less-than ... |
ssltmul2 27964 | One surreal set less-than ... |
mulsuniflem 27965 | Lemma for ~ mulsunif . St... |
mulsunif 27966 | Surreal multiplication has... |
addsdilem1 27967 | Lemma for surreal distribu... |
addsdilem2 27968 | Lemma for surreal distribu... |
addsdilem3 27969 | Lemma for ~ addsdi . Show... |
addsdilem4 27970 | Lemma for ~ addsdi . Show... |
addsdi 27971 | Distributive law for surre... |
addsdid 27972 | Distributive law for surre... |
addsdird 27973 | Distributive law for surre... |
subsdid 27974 | Distribution of surreal mu... |
subsdird 27975 | Distribution of surreal mu... |
mulnegs1d 27976 | Product with negative is n... |
mulnegs2d 27977 | Product with negative is n... |
mul2negsd 27978 | Surreal product of two neg... |
mulsasslem1 27979 | Lemma for ~ mulsass . Exp... |
mulsasslem2 27980 | Lemma for ~ mulsass . Exp... |
mulsasslem3 27981 | Lemma for ~ mulsass . Dem... |
mulsass 27982 | Associative law for surrea... |
mulsassd 27983 | Associative law for surrea... |
muls4d 27984 | Rearrangement of four surr... |
mulsunif2lem 27985 | Lemma for ~ mulsunif2 . S... |
mulsunif2 27986 | Alternate expression for s... |
sltmul2 27987 | Multiplication of both sid... |
sltmul2d 27988 | Multiplication of both sid... |
sltmul1d 27989 | Multiplication of both sid... |
slemul2d 27990 | Multiplication of both sid... |
slemul1d 27991 | Multiplication of both sid... |
sltmulneg1d 27992 | Multiplication of both sid... |
sltmulneg2d 27993 | Multiplication of both sid... |
mulscan2dlem 27994 | Lemma for ~ mulscan2d . C... |
mulscan2d 27995 | Cancellation of surreal mu... |
mulscan1d 27996 | Cancellation of surreal mu... |
muls12d 27997 | Commutative/associative la... |
slemul1ad 27998 | Multiplication of both sid... |
sltmul12ad 27999 | Comparison of the product ... |
divsmo 28000 | Uniqueness of surreal inve... |
muls0ord 28001 | If a surreal product is ze... |
mulsne0bd 28002 | The product of two non-zer... |
divsval 28005 | The value of surreal divis... |
norecdiv 28006 | If a surreal has a recipro... |
noreceuw 28007 | If a surreal has a recipro... |
divsmulw 28008 | Relationship between surre... |
divsmulwd 28009 | Relationship between surre... |
divsclw 28010 | Weak division closure law.... |
divsclwd 28011 | Weak division closure law.... |
divscan2wd 28012 | A weak cancellation law fo... |
divscan1wd 28013 | A weak cancellation law fo... |
sltdivmulwd 28014 | Surreal less-than relation... |
sltdivmul2wd 28015 | Surreal less-than relation... |
sltmuldivwd 28016 | Surreal less-than relation... |
sltmuldiv2wd 28017 | Surreal less-than relation... |
divsasswd 28018 | An associative law for sur... |
divs1 28019 | A surreal divided by one i... |
precsexlemcbv 28020 | Lemma for surreal reciproc... |
precsexlem1 28021 | Lemma for surreal reciproc... |
precsexlem2 28022 | Lemma for surreal reciproc... |
precsexlem3 28023 | Lemma for surreal reciproc... |
precsexlem4 28024 | Lemma for surreal reciproc... |
precsexlem5 28025 | Lemma for surreal reciproc... |
precsexlem6 28026 | Lemma for surreal reciproc... |
precsexlem7 28027 | Lemma for surreal reciproc... |
precsexlem8 28028 | Lemma for surreal reciproc... |
precsexlem9 28029 | Lemma for surreal reciproc... |
precsexlem10 28030 | Lemma for surreal reciproc... |
precsexlem11 28031 | Lemma for surreal reciproc... |
precsex 28032 | Every positive surreal has... |
recsex 28033 | A non-zero surreal has a r... |
recsexd 28034 | A non-zero surreal has a r... |
divsmul 28035 | Relationship between surre... |
divsmuld 28036 | Relationship between surre... |
divscl 28037 | Surreal division closure l... |
divscld 28038 | Surreal division closure l... |
divscan2d 28039 | A cancellation law for sur... |
divscan1d 28040 | A cancellation law for sur... |
sltdivmuld 28041 | Surreal less-than relation... |
sltdivmul2d 28042 | Surreal less-than relation... |
sltmuldivd 28043 | Surreal less-than relation... |
sltmuldiv2d 28044 | Surreal less-than relation... |
divsassd 28045 | An associative law for sur... |
divmuldivsd 28046 | Multiplication of two surr... |
abssval 28049 | The value of surreal absol... |
absscl 28050 | Closure law for surreal ab... |
abssid 28051 | The absolute value of a no... |
abs0s 28052 | The absolute value of surr... |
abssnid 28053 | For a negative surreal, it... |
absmuls 28054 | Surreal absolute value dis... |
abssge0 28055 | The absolute value of a su... |
abssor 28056 | The absolute value of a su... |
abssneg 28057 | Surreal absolute value of ... |
sleabs 28058 | A surreal is less than or ... |
absslt 28059 | Surreal absolute value and... |
elons 28062 | Membership in the class of... |
onssno 28063 | The surreal ordinals are a... |
onsno 28064 | A surreal ordinal is a sur... |
0ons 28065 | Surreal zero is a surreal ... |
1ons 28066 | Surreal one is a surreal o... |
elons2 28067 | A surreal is ordinal iff i... |
elons2d 28068 | The cut of any set of surr... |
sltonold 28069 | The class of ordinals less... |
sltonex 28070 | The class of ordinals less... |
onscutleft 28071 | A surreal ordinal is equal... |
seqsex 28074 | Existence of the surreal s... |
seqseq123d 28075 | Equality deduction for the... |
nfseqs 28076 | Hypothesis builder for the... |
seqsval 28077 | The value of the surreal s... |
noseqex 28078 | The next several theorems ... |
noseq0 28079 | The surreal ` A ` is a mem... |
noseqp1 28080 | One plus an element of ` Z... |
noseqind 28081 | Peano's inductive postulat... |
noseqinds 28082 | Induction schema for surre... |
noseqssno 28083 | A surreal sequence is a su... |
noseqno 28084 | An element of a surreal se... |
om2noseq0 28085 | The mapping ` G ` is a one... |
om2noseqsuc 28086 | The value of ` G ` at a su... |
om2noseqfo 28087 | Function statement for ` G... |
om2noseqlt 28088 | Surreal less-than relation... |
om2noseqlt2 28089 | The mapping ` G ` preserve... |
om2noseqf1o 28090 | ` G ` is a bijection. (Co... |
om2noseqiso 28091 | ` G ` is an isomorphism fr... |
om2noseqoi 28092 | An alternative definition ... |
om2noseqrdg 28093 | A helper lemma for the val... |
noseqrdglem 28094 | A helper lemma for the val... |
noseqrdgfn 28095 | The recursive definition g... |
noseqrdg0 28096 | Initial value of a recursi... |
noseqrdgsuc 28097 | Successor value of a recur... |
seqsfn 28098 | The surreal sequence build... |
seqs1 28099 | The value of the surreal s... |
seqsp1 28100 | The value of the surreal s... |
n0sex 28105 | The set of all non-negativ... |
nnsex 28106 | The set of all positive su... |
peano5n0s 28107 | Peano's inductive postulat... |
n0ssno 28108 | The non-negative surreal i... |
nnssn0s 28109 | The positive surreal integ... |
nnssno 28110 | The positive surreal integ... |
n0sno 28111 | A non-negative surreal int... |
nnsno 28112 | A positive surreal integer... |
n0snod 28113 | A non-negative surreal int... |
nnsnod 28114 | A positive surreal integer... |
0n0s 28115 | Peano postulate: ` 0s ` is... |
peano2n0s 28116 | Peano postulate: the succe... |
dfn0s2 28117 | Alternate definition of th... |
n0sind 28118 | Principle of Mathematical ... |
n0scut 28119 | A cut form for surreal nat... |
n0ons 28120 | A surreal natural is a sur... |
nnne0s 28121 | A surreal positive integer... |
n0sge0 28122 | A non-negative integer is ... |
nnsgt0 28123 | A positive integer is grea... |
elnns 28124 | Membership in the positive... |
elnns2 28125 | A positive surreal integer... |
n0addscl 28126 | The non-negative surreal i... |
n0mulscl 28127 | The non-negative surreal i... |
nnaddscl 28128 | The positive surreal integ... |
nnmulscl 28129 | The positive surreal integ... |
1n0s 28130 | Surreal one is a non-negat... |
1nns 28131 | Surreal one is a positive ... |
peano2nns 28132 | Peano postulate for positi... |
n0sbday 28133 | A non-negative surreal int... |
n0ssold 28134 | The non-negative surreal i... |
nnsrecgt0d 28135 | The reciprocal of a positi... |
seqn0sfn 28136 | The surreal sequence build... |
elreno 28139 | Membership in the set of s... |
recut 28140 | The cut involved in defini... |
0reno 28141 | Surreal zero is a surreal ... |
renegscl 28142 | The surreal reals are clos... |
readdscl 28143 | The surreal reals are clos... |
remulscllem1 28144 | Lemma for ~ remulscl . Sp... |
remulscllem2 28145 | Lemma for ~ remulscl . Bo... |
remulscl 28146 | The surreal reals are clos... |
itvndx 28157 | Index value of the Interva... |
lngndx 28158 | Index value of the "line" ... |
itvid 28159 | Utility theorem: index-ind... |
lngid 28160 | Utility theorem: index-ind... |
slotsinbpsd 28161 | The slots ` Base ` , ` +g ... |
slotslnbpsd 28162 | The slots ` Base ` , ` +g ... |
lngndxnitvndx 28163 | The slot for the line is n... |
trkgstr 28164 | Functionality of a Tarski ... |
trkgbas 28165 | The base set of a Tarski g... |
trkgdist 28166 | The measure of a distance ... |
trkgitv 28167 | The congruence relation in... |
istrkgc 28174 | Property of being a Tarski... |
istrkgb 28175 | Property of being a Tarski... |
istrkgcb 28176 | Property of being a Tarski... |
istrkge 28177 | Property of fulfilling Euc... |
istrkgl 28178 | Building lines from the se... |
istrkgld 28179 | Property of fulfilling the... |
istrkg2ld 28180 | Property of fulfilling the... |
istrkg3ld 28181 | Property of fulfilling the... |
axtgcgrrflx 28182 | Axiom of reflexivity of co... |
axtgcgrid 28183 | Axiom of identity of congr... |
axtgsegcon 28184 | Axiom of segment construct... |
axtg5seg 28185 | Five segments axiom, Axiom... |
axtgbtwnid 28186 | Identity of Betweenness. ... |
axtgpasch 28187 | Axiom of (Inner) Pasch, Ax... |
axtgcont1 28188 | Axiom of Continuity. Axio... |
axtgcont 28189 | Axiom of Continuity. Axio... |
axtglowdim2 28190 | Lower dimension axiom for ... |
axtgupdim2 28191 | Upper dimension axiom for ... |
axtgeucl 28192 | Euclid's Axiom. Axiom A10... |
tgjustf 28193 | Given any function ` F ` ,... |
tgjustr 28194 | Given any equivalence rela... |
tgjustc1 28195 | A justification for using ... |
tgjustc2 28196 | A justification for using ... |
tgcgrcomimp 28197 | Congruence commutes on the... |
tgcgrcomr 28198 | Congruence commutes on the... |
tgcgrcoml 28199 | Congruence commutes on the... |
tgcgrcomlr 28200 | Congruence commutes on bot... |
tgcgreqb 28201 | Congruence and equality. ... |
tgcgreq 28202 | Congruence and equality. ... |
tgcgrneq 28203 | Congruence and equality. ... |
tgcgrtriv 28204 | Degenerate segments are co... |
tgcgrextend 28205 | Link congruence over a pai... |
tgsegconeq 28206 | Two points that satisfy th... |
tgbtwntriv2 28207 | Betweenness always holds f... |
tgbtwncom 28208 | Betweenness commutes. The... |
tgbtwncomb 28209 | Betweenness commutes, bico... |
tgbtwnne 28210 | Betweenness and inequality... |
tgbtwntriv1 28211 | Betweenness always holds f... |
tgbtwnswapid 28212 | If you can swap the first ... |
tgbtwnintr 28213 | Inner transitivity law for... |
tgbtwnexch3 28214 | Exchange the first endpoin... |
tgbtwnouttr2 28215 | Outer transitivity law for... |
tgbtwnexch2 28216 | Exchange the outer point o... |
tgbtwnouttr 28217 | Outer transitivity law for... |
tgbtwnexch 28218 | Outer transitivity law for... |
tgtrisegint 28219 | A line segment between two... |
tglowdim1 28220 | Lower dimension axiom for ... |
tglowdim1i 28221 | Lower dimension axiom for ... |
tgldimor 28222 | Excluded-middle like state... |
tgldim0eq 28223 | In dimension zero, any two... |
tgldim0itv 28224 | In dimension zero, any two... |
tgldim0cgr 28225 | In dimension zero, any two... |
tgbtwndiff 28226 | There is always a ` c ` di... |
tgdim01 28227 | In geometries of dimension... |
tgifscgr 28228 | Inner five segment congrue... |
tgcgrsub 28229 | Removing identical parts f... |
iscgrg 28232 | The congruence property fo... |
iscgrgd 28233 | The property for two seque... |
iscgrglt 28234 | The property for two seque... |
trgcgrg 28235 | The property for two trian... |
trgcgr 28236 | Triangle congruence. (Con... |
ercgrg 28237 | The shape congruence relat... |
tgcgrxfr 28238 | A line segment can be divi... |
cgr3id 28239 | Reflexivity law for three-... |
cgr3simp1 28240 | Deduce segment congruence ... |
cgr3simp2 28241 | Deduce segment congruence ... |
cgr3simp3 28242 | Deduce segment congruence ... |
cgr3swap12 28243 | Permutation law for three-... |
cgr3swap23 28244 | Permutation law for three-... |
cgr3swap13 28245 | Permutation law for three-... |
cgr3rotr 28246 | Permutation law for three-... |
cgr3rotl 28247 | Permutation law for three-... |
trgcgrcom 28248 | Commutative law for three-... |
cgr3tr 28249 | Transitivity law for three... |
tgbtwnxfr 28250 | A condition for extending ... |
tgcgr4 28251 | Two quadrilaterals to be c... |
isismt 28254 | Property of being an isome... |
ismot 28255 | Property of being an isome... |
motcgr 28256 | Property of a motion: dist... |
idmot 28257 | The identity is a motion. ... |
motf1o 28258 | Motions are bijections. (... |
motcl 28259 | Closure of motions. (Cont... |
motco 28260 | The composition of two mot... |
cnvmot 28261 | The converse of a motion i... |
motplusg 28262 | The operation for motions ... |
motgrp 28263 | The motions of a geometry ... |
motcgrg 28264 | Property of a motion: dist... |
motcgr3 28265 | Property of a motion: dist... |
tglng 28266 | Lines of a Tarski Geometry... |
tglnfn 28267 | Lines as functions. (Cont... |
tglnunirn 28268 | Lines are sets of points. ... |
tglnpt 28269 | Lines are sets of points. ... |
tglngne 28270 | It takes two different poi... |
tglngval 28271 | The line going through poi... |
tglnssp 28272 | Lines are subset of the ge... |
tgellng 28273 | Property of lying on the l... |
tgcolg 28274 | We choose the notation ` (... |
btwncolg1 28275 | Betweenness implies coline... |
btwncolg2 28276 | Betweenness implies coline... |
btwncolg3 28277 | Betweenness implies coline... |
colcom 28278 | Swapping the points defini... |
colrot1 28279 | Rotating the points defini... |
colrot2 28280 | Rotating the points defini... |
ncolcom 28281 | Swapping non-colinear poin... |
ncolrot1 28282 | Rotating non-colinear poin... |
ncolrot2 28283 | Rotating non-colinear poin... |
tgdim01ln 28284 | In geometries of dimension... |
ncoltgdim2 28285 | If there are three non-col... |
lnxfr 28286 | Transfer law for colineari... |
lnext 28287 | Extend a line with a missi... |
tgfscgr 28288 | Congruence law for the gen... |
lncgr 28289 | Congruence rule for lines.... |
lnid 28290 | Identity law for points on... |
tgidinside 28291 | Law for finding a point in... |
tgbtwnconn1lem1 28292 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1lem2 28293 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1lem3 28294 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1 28295 | Connectivity law for betwe... |
tgbtwnconn2 28296 | Another connectivity law f... |
tgbtwnconn3 28297 | Inner connectivity law for... |
tgbtwnconnln3 28298 | Derive colinearity from be... |
tgbtwnconn22 28299 | Double connectivity law fo... |
tgbtwnconnln1 28300 | Derive colinearity from be... |
tgbtwnconnln2 28301 | Derive colinearity from be... |
legval 28304 | Value of the less-than rel... |
legov 28305 | Value of the less-than rel... |
legov2 28306 | An equivalent definition o... |
legid 28307 | Reflexivity of the less-th... |
btwnleg 28308 | Betweenness implies less-t... |
legtrd 28309 | Transitivity of the less-t... |
legtri3 28310 | Equality from the less-tha... |
legtrid 28311 | Trichotomy law for the les... |
leg0 28312 | Degenerated (zero-length) ... |
legeq 28313 | Deduce equality from "less... |
legbtwn 28314 | Deduce betweenness from "l... |
tgcgrsub2 28315 | Removing identical parts f... |
ltgseg 28316 | The set ` E ` denotes the ... |
ltgov 28317 | Strict "shorter than" geom... |
legov3 28318 | An equivalent definition o... |
legso 28319 | The "shorter than" relatio... |
ishlg 28322 | Rays : Definition 6.1 of ... |
hlcomb 28323 | The half-line relation com... |
hlcomd 28324 | The half-line relation com... |
hlne1 28325 | The half-line relation imp... |
hlne2 28326 | The half-line relation imp... |
hlln 28327 | The half-line relation imp... |
hleqnid 28328 | The endpoint does not belo... |
hlid 28329 | The half-line relation is ... |
hltr 28330 | The half-line relation is ... |
hlbtwn 28331 | Betweenness is a sufficien... |
btwnhl1 28332 | Deduce half-line from betw... |
btwnhl2 28333 | Deduce half-line from betw... |
btwnhl 28334 | Swap betweenness for a hal... |
lnhl 28335 | Either a point ` C ` on th... |
hlcgrex 28336 | Construct a point on a hal... |
hlcgreulem 28337 | Lemma for ~ hlcgreu . (Co... |
hlcgreu 28338 | The point constructed in ~... |
btwnlng1 28339 | Betweenness implies coline... |
btwnlng2 28340 | Betweenness implies coline... |
btwnlng3 28341 | Betweenness implies coline... |
lncom 28342 | Swapping the points defini... |
lnrot1 28343 | Rotating the points defini... |
lnrot2 28344 | Rotating the points defini... |
ncolne1 28345 | Non-colinear points are di... |
ncolne2 28346 | Non-colinear points are di... |
tgisline 28347 | The property of being a pr... |
tglnne 28348 | It takes two different poi... |
tglndim0 28349 | There are no lines in dime... |
tgelrnln 28350 | The property of being a pr... |
tglineeltr 28351 | Transitivity law for lines... |
tglineelsb2 28352 | If ` S ` lies on PQ , then... |
tglinerflx1 28353 | Reflexivity law for line m... |
tglinerflx2 28354 | Reflexivity law for line m... |
tglinecom 28355 | Commutativity law for line... |
tglinethru 28356 | If ` A ` is a line contain... |
tghilberti1 28357 | There is a line through an... |
tghilberti2 28358 | There is at most one line ... |
tglinethrueu 28359 | There is a unique line goi... |
tglnne0 28360 | A line ` A ` has at least ... |
tglnpt2 28361 | Find a second point on a l... |
tglineintmo 28362 | Two distinct lines interse... |
tglineineq 28363 | Two distinct lines interse... |
tglineneq 28364 | Given three non-colinear p... |
tglineinteq 28365 | Two distinct lines interse... |
ncolncol 28366 | Deduce non-colinearity fro... |
coltr 28367 | A transitivity law for col... |
coltr3 28368 | A transitivity law for col... |
colline 28369 | Three points are colinear ... |
tglowdim2l 28370 | Reformulation of the lower... |
tglowdim2ln 28371 | There is always one point ... |
mirreu3 28374 | Existential uniqueness of ... |
mirval 28375 | Value of the point inversi... |
mirfv 28376 | Value of the point inversi... |
mircgr 28377 | Property of the image by t... |
mirbtwn 28378 | Property of the image by t... |
ismir 28379 | Property of the image by t... |
mirf 28380 | Point inversion as functio... |
mircl 28381 | Closure of the point inver... |
mirmir 28382 | The point inversion functi... |
mircom 28383 | Variation on ~ mirmir . (... |
mirreu 28384 | Any point has a unique ant... |
mireq 28385 | Equality deduction for poi... |
mirinv 28386 | The only invariant point o... |
mirne 28387 | Mirror of non-center point... |
mircinv 28388 | The center point is invari... |
mirf1o 28389 | The point inversion functi... |
miriso 28390 | The point inversion functi... |
mirbtwni 28391 | Point inversion preserves ... |
mirbtwnb 28392 | Point inversion preserves ... |
mircgrs 28393 | Point inversion preserves ... |
mirmir2 28394 | Point inversion of a point... |
mirmot 28395 | Point investion is a motio... |
mirln 28396 | If two points are on the s... |
mirln2 28397 | If a point and its mirror ... |
mirconn 28398 | Point inversion of connect... |
mirhl 28399 | If two points ` X ` and ` ... |
mirbtwnhl 28400 | If the center of the point... |
mirhl2 28401 | Deduce half-line relation ... |
mircgrextend 28402 | Link congruence over a pai... |
mirtrcgr 28403 | Point inversion of one poi... |
mirauto 28404 | Point inversion preserves ... |
miduniq 28405 | Uniqueness of the middle p... |
miduniq1 28406 | Uniqueness of the middle p... |
miduniq2 28407 | If two point inversions co... |
colmid 28408 | Colinearity and equidistan... |
symquadlem 28409 | Lemma of the symetrial qua... |
krippenlem 28410 | Lemma for ~ krippen . We ... |
krippen 28411 | Krippenlemma (German for c... |
midexlem 28412 | Lemma for the existence of... |
israg 28417 | Property for 3 points A, B... |
ragcom 28418 | Commutative rule for right... |
ragcol 28419 | The right angle property i... |
ragmir 28420 | Right angle property is pr... |
mirrag 28421 | Right angle is conserved b... |
ragtrivb 28422 | Trivial right angle. Theo... |
ragflat2 28423 | Deduce equality from two r... |
ragflat 28424 | Deduce equality from two r... |
ragtriva 28425 | Trivial right angle. Theo... |
ragflat3 28426 | Right angle and colinearit... |
ragcgr 28427 | Right angle and colinearit... |
motrag 28428 | Right angles are preserved... |
ragncol 28429 | Right angle implies non-co... |
perpln1 28430 | Derive a line from perpend... |
perpln2 28431 | Derive a line from perpend... |
isperp 28432 | Property for 2 lines A, B ... |
perpcom 28433 | The "perpendicular" relati... |
perpneq 28434 | Two perpendicular lines ar... |
isperp2 28435 | Property for 2 lines A, B,... |
isperp2d 28436 | One direction of ~ isperp2... |
ragperp 28437 | Deduce that two lines are ... |
footexALT 28438 | Alternative version of ~ f... |
footexlem1 28439 | Lemma for ~ footex . (Con... |
footexlem2 28440 | Lemma for ~ footex . (Con... |
footex 28441 | From a point ` C ` outside... |
foot 28442 | From a point ` C ` outside... |
footne 28443 | Uniqueness of the foot poi... |
footeq 28444 | Uniqueness of the foot poi... |
hlperpnel 28445 | A point on a half-line whi... |
perprag 28446 | Deduce a right angle from ... |
perpdragALT 28447 | Deduce a right angle from ... |
perpdrag 28448 | Deduce a right angle from ... |
colperp 28449 | Deduce a perpendicularity ... |
colperpexlem1 28450 | Lemma for ~ colperp . Fir... |
colperpexlem2 28451 | Lemma for ~ colperpex . S... |
colperpexlem3 28452 | Lemma for ~ colperpex . C... |
colperpex 28453 | In dimension 2 and above, ... |
mideulem2 28454 | Lemma for ~ opphllem , whi... |
opphllem 28455 | Lemma 8.24 of [Schwabhause... |
mideulem 28456 | Lemma for ~ mideu . We ca... |
midex 28457 | Existence of the midpoint,... |
mideu 28458 | Existence and uniqueness o... |
islnopp 28459 | The property for two point... |
islnoppd 28460 | Deduce that ` A ` and ` B ... |
oppne1 28461 | Points lying on opposite s... |
oppne2 28462 | Points lying on opposite s... |
oppne3 28463 | Points lying on opposite s... |
oppcom 28464 | Commutativity rule for "op... |
opptgdim2 28465 | If two points opposite to ... |
oppnid 28466 | The "opposite to a line" r... |
opphllem1 28467 | Lemma for ~ opphl . (Cont... |
opphllem2 28468 | Lemma for ~ opphl . Lemma... |
opphllem3 28469 | Lemma for ~ opphl : We as... |
opphllem4 28470 | Lemma for ~ opphl . (Cont... |
opphllem5 28471 | Second part of Lemma 9.4 o... |
opphllem6 28472 | First part of Lemma 9.4 of... |
oppperpex 28473 | Restating ~ colperpex usin... |
opphl 28474 | If two points ` A ` and ` ... |
outpasch 28475 | Axiom of Pasch, outer form... |
hlpasch 28476 | An application of the axio... |
ishpg 28479 | Value of the half-plane re... |
hpgbr 28480 | Half-planes : property for... |
hpgne1 28481 | Points on the open half pl... |
hpgne2 28482 | Points on the open half pl... |
lnopp2hpgb 28483 | Theorem 9.8 of [Schwabhaus... |
lnoppnhpg 28484 | If two points lie on the o... |
hpgerlem 28485 | Lemma for the proof that t... |
hpgid 28486 | The half-plane relation is... |
hpgcom 28487 | The half-plane relation co... |
hpgtr 28488 | The half-plane relation is... |
colopp 28489 | Opposite sides of a line f... |
colhp 28490 | Half-plane relation for co... |
hphl 28491 | If two points are on the s... |
midf 28496 | Midpoint as a function. (... |
midcl 28497 | Closure of the midpoint. ... |
ismidb 28498 | Property of the midpoint. ... |
midbtwn 28499 | Betweenness of midpoint. ... |
midcgr 28500 | Congruence of midpoint. (... |
midid 28501 | Midpoint of a null segment... |
midcom 28502 | Commutativity rule for the... |
mirmid 28503 | Point inversion preserves ... |
lmieu 28504 | Uniqueness of the line mir... |
lmif 28505 | Line mirror as a function.... |
lmicl 28506 | Closure of the line mirror... |
islmib 28507 | Property of the line mirro... |
lmicom 28508 | The line mirroring functio... |
lmilmi 28509 | Line mirroring is an invol... |
lmireu 28510 | Any point has a unique ant... |
lmieq 28511 | Equality deduction for lin... |
lmiinv 28512 | The invariants of the line... |
lmicinv 28513 | The mirroring line is an i... |
lmimid 28514 | If we have a right angle, ... |
lmif1o 28515 | The line mirroring functio... |
lmiisolem 28516 | Lemma for ~ lmiiso . (Con... |
lmiiso 28517 | The line mirroring functio... |
lmimot 28518 | Line mirroring is a motion... |
hypcgrlem1 28519 | Lemma for ~ hypcgr , case ... |
hypcgrlem2 28520 | Lemma for ~ hypcgr , case ... |
hypcgr 28521 | If the catheti of two righ... |
lmiopp 28522 | Line mirroring produces po... |
lnperpex 28523 | Existence of a perpendicul... |
trgcopy 28524 | Triangle construction: a c... |
trgcopyeulem 28525 | Lemma for ~ trgcopyeu . (... |
trgcopyeu 28526 | Triangle construction: a c... |
iscgra 28529 | Property for two angles AB... |
iscgra1 28530 | A special version of ~ isc... |
iscgrad 28531 | Sufficient conditions for ... |
cgrane1 28532 | Angles imply inequality. ... |
cgrane2 28533 | Angles imply inequality. ... |
cgrane3 28534 | Angles imply inequality. ... |
cgrane4 28535 | Angles imply inequality. ... |
cgrahl1 28536 | Angle congruence is indepe... |
cgrahl2 28537 | Angle congruence is indepe... |
cgracgr 28538 | First direction of proposi... |
cgraid 28539 | Angle congruence is reflex... |
cgraswap 28540 | Swap rays in a congruence ... |
cgrcgra 28541 | Triangle congruence implie... |
cgracom 28542 | Angle congruence commutes.... |
cgratr 28543 | Angle congruence is transi... |
flatcgra 28544 | Flat angles are congruent.... |
cgraswaplr 28545 | Swap both side of angle co... |
cgrabtwn 28546 | Angle congruence preserves... |
cgrahl 28547 | Angle congruence preserves... |
cgracol 28548 | Angle congruence preserves... |
cgrancol 28549 | Angle congruence preserves... |
dfcgra2 28550 | This is the full statement... |
sacgr 28551 | Supplementary angles of co... |
oacgr 28552 | Vertical angle theorem. V... |
acopy 28553 | Angle construction. Theor... |
acopyeu 28554 | Angle construction. Theor... |
isinag 28558 | Property for point ` X ` t... |
isinagd 28559 | Sufficient conditions for ... |
inagflat 28560 | Any point lies in a flat a... |
inagswap 28561 | Swap the order of the half... |
inagne1 28562 | Deduce inequality from the... |
inagne2 28563 | Deduce inequality from the... |
inagne3 28564 | Deduce inequality from the... |
inaghl 28565 | The "point lie in angle" r... |
isleag 28567 | Geometrical "less than" pr... |
isleagd 28568 | Sufficient condition for "... |
leagne1 28569 | Deduce inequality from the... |
leagne2 28570 | Deduce inequality from the... |
leagne3 28571 | Deduce inequality from the... |
leagne4 28572 | Deduce inequality from the... |
cgrg3col4 28573 | Lemma 11.28 of [Schwabhaus... |
tgsas1 28574 | First congruence theorem: ... |
tgsas 28575 | First congruence theorem: ... |
tgsas2 28576 | First congruence theorem: ... |
tgsas3 28577 | First congruence theorem: ... |
tgasa1 28578 | Second congruence theorem:... |
tgasa 28579 | Second congruence theorem:... |
tgsss1 28580 | Third congruence theorem: ... |
tgsss2 28581 | Third congruence theorem: ... |
tgsss3 28582 | Third congruence theorem: ... |
dfcgrg2 28583 | Congruence for two triangl... |
isoas 28584 | Congruence theorem for iso... |
iseqlg 28587 | Property of a triangle bei... |
iseqlgd 28588 | Condition for a triangle t... |
f1otrgds 28589 | Convenient lemma for ~ f1o... |
f1otrgitv 28590 | Convenient lemma for ~ f1o... |
f1otrg 28591 | A bijection between bases ... |
f1otrge 28592 | A bijection between bases ... |
ttgval 28595 | Define a function to augme... |
ttgvalOLD 28596 | Obsolete proof of ~ ttgval... |
ttglem 28597 | Lemma for ~ ttgbas , ~ ttg... |
ttglemOLD 28598 | Obsolete version of ~ ttgl... |
ttgbas 28599 | The base set of a subcompl... |
ttgbasOLD 28600 | Obsolete proof of ~ ttgbas... |
ttgplusg 28601 | The addition operation of ... |
ttgplusgOLD 28602 | Obsolete proof of ~ ttgplu... |
ttgsub 28603 | The subtraction operation ... |
ttgvsca 28604 | The scalar product of a su... |
ttgvscaOLD 28605 | Obsolete proof of ~ ttgvsc... |
ttgds 28606 | The metric of a subcomplex... |
ttgdsOLD 28607 | Obsolete proof of ~ ttgds ... |
ttgitvval 28608 | Betweenness for a subcompl... |
ttgelitv 28609 | Betweenness for a subcompl... |
ttgbtwnid 28610 | Any subcomplex module equi... |
ttgcontlem1 28611 | Lemma for % ttgcont . (Co... |
xmstrkgc 28612 | Any metric space fulfills ... |
cchhllem 28613 | Lemma for chlbas and chlvs... |
cchhllemOLD 28614 | Obsolete version of ~ cchh... |
elee 28621 | Membership in a Euclidean ... |
mptelee 28622 | A condition for a mapping ... |
eleenn 28623 | If ` A ` is in ` ( EE `` N... |
eleei 28624 | The forward direction of ~... |
eedimeq 28625 | A point belongs to at most... |
brbtwn 28626 | The binary relation form o... |
brcgr 28627 | The binary relation form o... |
fveere 28628 | The function value of a po... |
fveecn 28629 | The function value of a po... |
eqeefv 28630 | Two points are equal iff t... |
eqeelen 28631 | Two points are equal iff t... |
brbtwn2 28632 | Alternate characterization... |
colinearalglem1 28633 | Lemma for ~ colinearalg . ... |
colinearalglem2 28634 | Lemma for ~ colinearalg . ... |
colinearalglem3 28635 | Lemma for ~ colinearalg . ... |
colinearalglem4 28636 | Lemma for ~ colinearalg . ... |
colinearalg 28637 | An algebraic characterizat... |
eleesub 28638 | Membership of a subtractio... |
eleesubd 28639 | Membership of a subtractio... |
axdimuniq 28640 | The unique dimension axiom... |
axcgrrflx 28641 | ` A ` is as far from ` B `... |
axcgrtr 28642 | Congruence is transitive. ... |
axcgrid 28643 | If there is no distance be... |
axsegconlem1 28644 | Lemma for ~ axsegcon . Ha... |
axsegconlem2 28645 | Lemma for ~ axsegcon . Sh... |
axsegconlem3 28646 | Lemma for ~ axsegcon . Sh... |
axsegconlem4 28647 | Lemma for ~ axsegcon . Sh... |
axsegconlem5 28648 | Lemma for ~ axsegcon . Sh... |
axsegconlem6 28649 | Lemma for ~ axsegcon . Sh... |
axsegconlem7 28650 | Lemma for ~ axsegcon . Sh... |
axsegconlem8 28651 | Lemma for ~ axsegcon . Sh... |
axsegconlem9 28652 | Lemma for ~ axsegcon . Sh... |
axsegconlem10 28653 | Lemma for ~ axsegcon . Sh... |
axsegcon 28654 | Any segment ` A B ` can be... |
ax5seglem1 28655 | Lemma for ~ ax5seg . Rexp... |
ax5seglem2 28656 | Lemma for ~ ax5seg . Rexp... |
ax5seglem3a 28657 | Lemma for ~ ax5seg . (Con... |
ax5seglem3 28658 | Lemma for ~ ax5seg . Comb... |
ax5seglem4 28659 | Lemma for ~ ax5seg . Give... |
ax5seglem5 28660 | Lemma for ~ ax5seg . If `... |
ax5seglem6 28661 | Lemma for ~ ax5seg . Give... |
ax5seglem7 28662 | Lemma for ~ ax5seg . An a... |
ax5seglem8 28663 | Lemma for ~ ax5seg . Use ... |
ax5seglem9 28664 | Lemma for ~ ax5seg . Take... |
ax5seg 28665 | The five segment axiom. T... |
axbtwnid 28666 | Points are indivisible. T... |
axpaschlem 28667 | Lemma for ~ axpasch . Set... |
axpasch 28668 | The inner Pasch axiom. Ta... |
axlowdimlem1 28669 | Lemma for ~ axlowdim . Es... |
axlowdimlem2 28670 | Lemma for ~ axlowdim . Sh... |
axlowdimlem3 28671 | Lemma for ~ axlowdim . Se... |
axlowdimlem4 28672 | Lemma for ~ axlowdim . Se... |
axlowdimlem5 28673 | Lemma for ~ axlowdim . Sh... |
axlowdimlem6 28674 | Lemma for ~ axlowdim . Sh... |
axlowdimlem7 28675 | Lemma for ~ axlowdim . Se... |
axlowdimlem8 28676 | Lemma for ~ axlowdim . Ca... |
axlowdimlem9 28677 | Lemma for ~ axlowdim . Ca... |
axlowdimlem10 28678 | Lemma for ~ axlowdim . Se... |
axlowdimlem11 28679 | Lemma for ~ axlowdim . Ca... |
axlowdimlem12 28680 | Lemma for ~ axlowdim . Ca... |
axlowdimlem13 28681 | Lemma for ~ axlowdim . Es... |
axlowdimlem14 28682 | Lemma for ~ axlowdim . Ta... |
axlowdimlem15 28683 | Lemma for ~ axlowdim . Se... |
axlowdimlem16 28684 | Lemma for ~ axlowdim . Se... |
axlowdimlem17 28685 | Lemma for ~ axlowdim . Es... |
axlowdim1 28686 | The lower dimension axiom ... |
axlowdim2 28687 | The lower two-dimensional ... |
axlowdim 28688 | The general lower dimensio... |
axeuclidlem 28689 | Lemma for ~ axeuclid . Ha... |
axeuclid 28690 | Euclid's axiom. Take an a... |
axcontlem1 28691 | Lemma for ~ axcont . Chan... |
axcontlem2 28692 | Lemma for ~ axcont . The ... |
axcontlem3 28693 | Lemma for ~ axcont . Give... |
axcontlem4 28694 | Lemma for ~ axcont . Give... |
axcontlem5 28695 | Lemma for ~ axcont . Comp... |
axcontlem6 28696 | Lemma for ~ axcont . Stat... |
axcontlem7 28697 | Lemma for ~ axcont . Give... |
axcontlem8 28698 | Lemma for ~ axcont . A po... |
axcontlem9 28699 | Lemma for ~ axcont . Give... |
axcontlem10 28700 | Lemma for ~ axcont . Give... |
axcontlem11 28701 | Lemma for ~ axcont . Elim... |
axcontlem12 28702 | Lemma for ~ axcont . Elim... |
axcont 28703 | The axiom of continuity. ... |
eengv 28706 | The value of the Euclidean... |
eengstr 28707 | The Euclidean geometry as ... |
eengbas 28708 | The Base of the Euclidean ... |
ebtwntg 28709 | The betweenness relation u... |
ecgrtg 28710 | The congruence relation us... |
elntg 28711 | The line definition in the... |
elntg2 28712 | The line definition in the... |
eengtrkg 28713 | The geometry structure for... |
eengtrkge 28714 | The geometry structure for... |
edgfid 28717 | Utility theorem: index-ind... |
edgfndx 28718 | Index value of the ~ df-ed... |
edgfndxnn 28719 | The index value of the edg... |
edgfndxid 28720 | The value of the edge func... |
edgfndxidOLD 28721 | Obsolete version of ~ edgf... |
basendxltedgfndx 28722 | The index value of the ` B... |
baseltedgfOLD 28723 | Obsolete proof of ~ basend... |
basendxnedgfndx 28724 | The slots ` Base ` and ` .... |
vtxval 28729 | The set of vertices of a g... |
iedgval 28730 | The set of indexed edges o... |
1vgrex 28731 | A graph with at least one ... |
opvtxval 28732 | The set of vertices of a g... |
opvtxfv 28733 | The set of vertices of a g... |
opvtxov 28734 | The set of vertices of a g... |
opiedgval 28735 | The set of indexed edges o... |
opiedgfv 28736 | The set of indexed edges o... |
opiedgov 28737 | The set of indexed edges o... |
opvtxfvi 28738 | The set of vertices of a g... |
opiedgfvi 28739 | The set of indexed edges o... |
funvtxdmge2val 28740 | The set of vertices of an ... |
funiedgdmge2val 28741 | The set of indexed edges o... |
funvtxdm2val 28742 | The set of vertices of an ... |
funiedgdm2val 28743 | The set of indexed edges o... |
funvtxval0 28744 | The set of vertices of an ... |
basvtxval 28745 | The set of vertices of a g... |
edgfiedgval 28746 | The set of indexed edges o... |
funvtxval 28747 | The set of vertices of a g... |
funiedgval 28748 | The set of indexed edges o... |
structvtxvallem 28749 | Lemma for ~ structvtxval a... |
structvtxval 28750 | The set of vertices of an ... |
structiedg0val 28751 | The set of indexed edges o... |
structgrssvtxlem 28752 | Lemma for ~ structgrssvtx ... |
structgrssvtx 28753 | The set of vertices of a g... |
structgrssiedg 28754 | The set of indexed edges o... |
struct2grstr 28755 | A graph represented as an ... |
struct2grvtx 28756 | The set of vertices of a g... |
struct2griedg 28757 | The set of indexed edges o... |
graop 28758 | Any representation of a gr... |
grastruct 28759 | Any representation of a gr... |
gropd 28760 | If any representation of a... |
grstructd 28761 | If any representation of a... |
gropeld 28762 | If any representation of a... |
grstructeld 28763 | If any representation of a... |
setsvtx 28764 | The vertices of a structur... |
setsiedg 28765 | The (indexed) edges of a s... |
snstrvtxval 28766 | The set of vertices of a g... |
snstriedgval 28767 | The set of indexed edges o... |
vtxval0 28768 | Degenerated case 1 for ver... |
iedgval0 28769 | Degenerated case 1 for edg... |
vtxvalsnop 28770 | Degenerated case 2 for ver... |
iedgvalsnop 28771 | Degenerated case 2 for edg... |
vtxval3sn 28772 | Degenerated case 3 for ver... |
iedgval3sn 28773 | Degenerated case 3 for edg... |
vtxvalprc 28774 | Degenerated case 4 for ver... |
iedgvalprc 28775 | Degenerated case 4 for edg... |
edgval 28778 | The edges of a graph. (Co... |
iedgedg 28779 | An indexed edge is an edge... |
edgopval 28780 | The edges of a graph repre... |
edgov 28781 | The edges of a graph repre... |
edgstruct 28782 | The edges of a graph repre... |
edgiedgb 28783 | A set is an edge iff it is... |
edg0iedg0 28784 | There is no edge in a grap... |
isuhgr 28789 | The predicate "is an undir... |
isushgr 28790 | The predicate "is an undir... |
uhgrf 28791 | The edge function of an un... |
ushgrf 28792 | The edge function of an un... |
uhgrss 28793 | An edge is a subset of ver... |
uhgreq12g 28794 | If two sets have the same ... |
uhgrfun 28795 | The edge function of an un... |
uhgrn0 28796 | An edge is a nonempty subs... |
lpvtx 28797 | The endpoints of a loop (w... |
ushgruhgr 28798 | An undirected simple hyper... |
isuhgrop 28799 | The property of being an u... |
uhgr0e 28800 | The empty graph, with vert... |
uhgr0vb 28801 | The null graph, with no ve... |
uhgr0 28802 | The null graph represented... |
uhgrun 28803 | The union ` U ` of two (un... |
uhgrunop 28804 | The union of two (undirect... |
ushgrun 28805 | The union ` U ` of two (un... |
ushgrunop 28806 | The union of two (undirect... |
uhgrstrrepe 28807 | Replacing (or adding) the ... |
incistruhgr 28808 | An _incidence structure_ `... |
isupgr 28813 | The property of being an u... |
wrdupgr 28814 | The property of being an u... |
upgrf 28815 | The edge function of an un... |
upgrfn 28816 | The edge function of an un... |
upgrss 28817 | An edge is a subset of ver... |
upgrn0 28818 | An edge is a nonempty subs... |
upgrle 28819 | An edge of an undirected p... |
upgrfi 28820 | An edge is a finite subset... |
upgrex 28821 | An edge is an unordered pa... |
upgrbi 28822 | Show that an unordered pai... |
upgrop 28823 | A pseudograph represented ... |
isumgr 28824 | The property of being an u... |
isumgrs 28825 | The simplified property of... |
wrdumgr 28826 | The property of being an u... |
umgrf 28827 | The edge function of an un... |
umgrfn 28828 | The edge function of an un... |
umgredg2 28829 | An edge of a multigraph ha... |
umgrbi 28830 | Show that an unordered pai... |
upgruhgr 28831 | An undirected pseudograph ... |
umgrupgr 28832 | An undirected multigraph i... |
umgruhgr 28833 | An undirected multigraph i... |
upgrle2 28834 | An edge of an undirected p... |
umgrnloopv 28835 | In a multigraph, there is ... |
umgredgprv 28836 | In a multigraph, an edge i... |
umgrnloop 28837 | In a multigraph, there is ... |
umgrnloop0 28838 | A multigraph has no loops.... |
umgr0e 28839 | The empty graph, with vert... |
upgr0e 28840 | The empty graph, with vert... |
upgr1elem 28841 | Lemma for ~ upgr1e and ~ u... |
upgr1e 28842 | A pseudograph with one edg... |
upgr0eop 28843 | The empty graph, with vert... |
upgr1eop 28844 | A pseudograph with one edg... |
upgr0eopALT 28845 | Alternate proof of ~ upgr0... |
upgr1eopALT 28846 | Alternate proof of ~ upgr1... |
upgrun 28847 | The union ` U ` of two pse... |
upgrunop 28848 | The union of two pseudogra... |
umgrun 28849 | The union ` U ` of two mul... |
umgrunop 28850 | The union of two multigrap... |
umgrislfupgrlem 28851 | Lemma for ~ umgrislfupgr a... |
umgrislfupgr 28852 | A multigraph is a loop-fre... |
lfgredgge2 28853 | An edge of a loop-free gra... |
lfgrnloop 28854 | A loop-free graph has no l... |
uhgredgiedgb 28855 | In a hypergraph, a set is ... |
uhgriedg0edg0 28856 | A hypergraph has no edges ... |
uhgredgn0 28857 | An edge of a hypergraph is... |
edguhgr 28858 | An edge of a hypergraph is... |
uhgredgrnv 28859 | An edge of a hypergraph co... |
uhgredgss 28860 | The set of edges of a hype... |
upgredgss 28861 | The set of edges of a pseu... |
umgredgss 28862 | The set of edges of a mult... |
edgupgr 28863 | Properties of an edge of a... |
edgumgr 28864 | Properties of an edge of a... |
uhgrvtxedgiedgb 28865 | In a hypergraph, a vertex ... |
upgredg 28866 | For each edge in a pseudog... |
umgredg 28867 | For each edge in a multigr... |
upgrpredgv 28868 | An edge of a pseudograph a... |
umgrpredgv 28869 | An edge of a multigraph al... |
upgredg2vtx 28870 | For a vertex incident to a... |
upgredgpr 28871 | If a proper pair (of verti... |
edglnl 28872 | The edges incident with a ... |
numedglnl 28873 | The number of edges incide... |
umgredgne 28874 | An edge of a multigraph al... |
umgrnloop2 28875 | A multigraph has no loops.... |
umgredgnlp 28876 | An edge of a multigraph is... |
isuspgr 28881 | The property of being a si... |
isusgr 28882 | The property of being a si... |
uspgrf 28883 | The edge function of a sim... |
usgrf 28884 | The edge function of a sim... |
isusgrs 28885 | The property of being a si... |
usgrfs 28886 | The edge function of a sim... |
usgrfun 28887 | The edge function of a sim... |
usgredgss 28888 | The set of edges of a simp... |
edgusgr 28889 | An edge of a simple graph ... |
isuspgrop 28890 | The property of being an u... |
isusgrop 28891 | The property of being an u... |
usgrop 28892 | A simple graph represented... |
isausgr 28893 | The property of an unorder... |
ausgrusgrb 28894 | The equivalence of the def... |
usgrausgri 28895 | A simple graph represented... |
ausgrumgri 28896 | If an alternatively define... |
ausgrusgri 28897 | The equivalence of the def... |
usgrausgrb 28898 | The equivalence of the def... |
usgredgop 28899 | An edge of a simple graph ... |
usgrf1o 28900 | The edge function of a sim... |
usgrf1 28901 | The edge function of a sim... |
uspgrf1oedg 28902 | The edge function of a sim... |
usgrss 28903 | An edge is a subset of ver... |
uspgrushgr 28904 | A simple pseudograph is an... |
uspgrupgr 28905 | A simple pseudograph is an... |
uspgrupgrushgr 28906 | A graph is a simple pseudo... |
usgruspgr 28907 | A simple graph is a simple... |
usgrumgr 28908 | A simple graph is an undir... |
usgrumgruspgr 28909 | A graph is a simple graph ... |
usgruspgrb 28910 | A class is a simple graph ... |
usgrupgr 28911 | A simple graph is an undir... |
usgruhgr 28912 | A simple graph is an undir... |
usgrislfuspgr 28913 | A simple graph is a loop-f... |
uspgrun 28914 | The union ` U ` of two sim... |
uspgrunop 28915 | The union of two simple ps... |
usgrun 28916 | The union ` U ` of two sim... |
usgrunop 28917 | The union of two simple gr... |
usgredg2 28918 | The value of the "edge fun... |
usgredg2ALT 28919 | Alternate proof of ~ usgre... |
usgredgprv 28920 | In a simple graph, an edge... |
usgredgprvALT 28921 | Alternate proof of ~ usgre... |
usgredgppr 28922 | An edge of a simple graph ... |
usgrpredgv 28923 | An edge of a simple graph ... |
edgssv2 28924 | An edge of a simple graph ... |
usgredg 28925 | For each edge in a simple ... |
usgrnloopv 28926 | In a simple graph, there i... |
usgrnloopvALT 28927 | Alternate proof of ~ usgrn... |
usgrnloop 28928 | In a simple graph, there i... |
usgrnloopALT 28929 | Alternate proof of ~ usgrn... |
usgrnloop0 28930 | A simple graph has no loop... |
usgrnloop0ALT 28931 | Alternate proof of ~ usgrn... |
usgredgne 28932 | An edge of a simple graph ... |
usgrf1oedg 28933 | The edge function of a sim... |
uhgr2edg 28934 | If a vertex is adjacent to... |
umgr2edg 28935 | If a vertex is adjacent to... |
usgr2edg 28936 | If a vertex is adjacent to... |
umgr2edg1 28937 | If a vertex is adjacent to... |
usgr2edg1 28938 | If a vertex is adjacent to... |
umgrvad2edg 28939 | If a vertex is adjacent to... |
umgr2edgneu 28940 | If a vertex is adjacent to... |
usgrsizedg 28941 | In a simple graph, the siz... |
usgredg3 28942 | The value of the "edge fun... |
usgredg4 28943 | For a vertex incident to a... |
usgredgreu 28944 | For a vertex incident to a... |
usgredg2vtx 28945 | For a vertex incident to a... |
uspgredg2vtxeu 28946 | For a vertex incident to a... |
usgredg2vtxeu 28947 | For a vertex incident to a... |
usgredg2vtxeuALT 28948 | Alternate proof of ~ usgre... |
uspgredg2vlem 28949 | Lemma for ~ uspgredg2v . ... |
uspgredg2v 28950 | In a simple pseudograph, t... |
usgredg2vlem1 28951 | Lemma 1 for ~ usgredg2v . ... |
usgredg2vlem2 28952 | Lemma 2 for ~ usgredg2v . ... |
usgredg2v 28953 | In a simple graph, the map... |
usgriedgleord 28954 | Alternate version of ~ usg... |
ushgredgedg 28955 | In a simple hypergraph the... |
usgredgedg 28956 | In a simple graph there is... |
ushgredgedgloop 28957 | In a simple hypergraph the... |
uspgredgleord 28958 | In a simple pseudograph th... |
usgredgleord 28959 | In a simple graph the numb... |
usgredgleordALT 28960 | Alternate proof for ~ usgr... |
usgrstrrepe 28961 | Replacing (or adding) the ... |
usgr0e 28962 | The empty graph, with vert... |
usgr0vb 28963 | The null graph, with no ve... |
uhgr0v0e 28964 | The null graph, with no ve... |
uhgr0vsize0 28965 | The size of a hypergraph w... |
uhgr0edgfi 28966 | A graph of order 0 (i.e. w... |
usgr0v 28967 | The null graph, with no ve... |
uhgr0vusgr 28968 | The null graph, with no ve... |
usgr0 28969 | The null graph represented... |
uspgr1e 28970 | A simple pseudograph with ... |
usgr1e 28971 | A simple graph with one ed... |
usgr0eop 28972 | The empty graph, with vert... |
uspgr1eop 28973 | A simple pseudograph with ... |
uspgr1ewop 28974 | A simple pseudograph with ... |
uspgr1v1eop 28975 | A simple pseudograph with ... |
usgr1eop 28976 | A simple graph with (at le... |
uspgr2v1e2w 28977 | A simple pseudograph with ... |
usgr2v1e2w 28978 | A simple graph with two ve... |
edg0usgr 28979 | A class without edges is a... |
lfuhgr1v0e 28980 | A loop-free hypergraph wit... |
usgr1vr 28981 | A simple graph with one ve... |
usgr1v 28982 | A class with one (or no) v... |
usgr1v0edg 28983 | A class with one (or no) v... |
usgrexmpldifpr 28984 | Lemma for ~ usgrexmpledg :... |
usgrexmplef 28985 | Lemma for ~ usgrexmpl . (... |
usgrexmpllem 28986 | Lemma for ~ usgrexmpl . (... |
usgrexmplvtx 28987 | The vertices ` 0 , 1 , 2 ,... |
usgrexmpledg 28988 | The edges ` { 0 , 1 } , { ... |
usgrexmpl 28989 | ` G ` is a simple graph of... |
griedg0prc 28990 | The class of empty graphs ... |
griedg0ssusgr 28991 | The class of all simple gr... |
usgrprc 28992 | The class of simple graphs... |
relsubgr 28995 | The class of the subgraph ... |
subgrv 28996 | If a class is a subgraph o... |
issubgr 28997 | The property of a set to b... |
issubgr2 28998 | The property of a set to b... |
subgrprop 28999 | The properties of a subgra... |
subgrprop2 29000 | The properties of a subgra... |
uhgrissubgr 29001 | The property of a hypergra... |
subgrprop3 29002 | The properties of a subgra... |
egrsubgr 29003 | An empty graph consisting ... |
0grsubgr 29004 | The null graph (represente... |
0uhgrsubgr 29005 | The null graph (as hypergr... |
uhgrsubgrself 29006 | A hypergraph is a subgraph... |
subgrfun 29007 | The edge function of a sub... |
subgruhgrfun 29008 | The edge function of a sub... |
subgreldmiedg 29009 | An element of the domain o... |
subgruhgredgd 29010 | An edge of a subgraph of a... |
subumgredg2 29011 | An edge of a subgraph of a... |
subuhgr 29012 | A subgraph of a hypergraph... |
subupgr 29013 | A subgraph of a pseudograp... |
subumgr 29014 | A subgraph of a multigraph... |
subusgr 29015 | A subgraph of a simple gra... |
uhgrspansubgrlem 29016 | Lemma for ~ uhgrspansubgr ... |
uhgrspansubgr 29017 | A spanning subgraph ` S ` ... |
uhgrspan 29018 | A spanning subgraph ` S ` ... |
upgrspan 29019 | A spanning subgraph ` S ` ... |
umgrspan 29020 | A spanning subgraph ` S ` ... |
usgrspan 29021 | A spanning subgraph ` S ` ... |
uhgrspanop 29022 | A spanning subgraph of a h... |
upgrspanop 29023 | A spanning subgraph of a p... |
umgrspanop 29024 | A spanning subgraph of a m... |
usgrspanop 29025 | A spanning subgraph of a s... |
uhgrspan1lem1 29026 | Lemma 1 for ~ uhgrspan1 . ... |
uhgrspan1lem2 29027 | Lemma 2 for ~ uhgrspan1 . ... |
uhgrspan1lem3 29028 | Lemma 3 for ~ uhgrspan1 . ... |
uhgrspan1 29029 | The induced subgraph ` S `... |
upgrreslem 29030 | Lemma for ~ upgrres . (Co... |
umgrreslem 29031 | Lemma for ~ umgrres and ~ ... |
upgrres 29032 | A subgraph obtained by rem... |
umgrres 29033 | A subgraph obtained by rem... |
usgrres 29034 | A subgraph obtained by rem... |
upgrres1lem1 29035 | Lemma 1 for ~ upgrres1 . ... |
umgrres1lem 29036 | Lemma for ~ umgrres1 . (C... |
upgrres1lem2 29037 | Lemma 2 for ~ upgrres1 . ... |
upgrres1lem3 29038 | Lemma 3 for ~ upgrres1 . ... |
upgrres1 29039 | A pseudograph obtained by ... |
umgrres1 29040 | A multigraph obtained by r... |
usgrres1 29041 | Restricting a simple graph... |
isfusgr 29044 | The property of being a fi... |
fusgrvtxfi 29045 | A finite simple graph has ... |
isfusgrf1 29046 | The property of being a fi... |
isfusgrcl 29047 | The property of being a fi... |
fusgrusgr 29048 | A finite simple graph is a... |
opfusgr 29049 | A finite simple graph repr... |
usgredgffibi 29050 | The number of edges in a s... |
fusgredgfi 29051 | In a finite simple graph t... |
usgr1v0e 29052 | The size of a (finite) sim... |
usgrfilem 29053 | In a finite simple graph, ... |
fusgrfisbase 29054 | Induction base for ~ fusgr... |
fusgrfisstep 29055 | Induction step in ~ fusgrf... |
fusgrfis 29056 | A finite simple graph is o... |
fusgrfupgrfs 29057 | A finite simple graph is a... |
nbgrprc0 29060 | The set of neighbors is em... |
nbgrcl 29061 | If a class ` X ` has at le... |
nbgrval 29062 | The set of neighbors of a ... |
dfnbgr2 29063 | Alternate definition of th... |
dfnbgr3 29064 | Alternate definition of th... |
nbgrnvtx0 29065 | If a class ` X ` is not a ... |
nbgrel 29066 | Characterization of a neig... |
nbgrisvtx 29067 | Every neighbor ` N ` of a ... |
nbgrssvtx 29068 | The neighbors of a vertex ... |
nbuhgr 29069 | The set of neighbors of a ... |
nbupgr 29070 | The set of neighbors of a ... |
nbupgrel 29071 | A neighbor of a vertex in ... |
nbumgrvtx 29072 | The set of neighbors of a ... |
nbumgr 29073 | The set of neighbors of an... |
nbusgrvtx 29074 | The set of neighbors of a ... |
nbusgr 29075 | The set of neighbors of an... |
nbgr2vtx1edg 29076 | If a graph has two vertice... |
nbuhgr2vtx1edgblem 29077 | Lemma for ~ nbuhgr2vtx1edg... |
nbuhgr2vtx1edgb 29078 | If a hypergraph has two ve... |
nbusgreledg 29079 | A class/vertex is a neighb... |
uhgrnbgr0nb 29080 | A vertex which is not endp... |
nbgr0vtxlem 29081 | Lemma for ~ nbgr0vtx and ~... |
nbgr0vtx 29082 | In a null graph (with no v... |
nbgr0edg 29083 | In an empty graph (with no... |
nbgr1vtx 29084 | In a graph with one vertex... |
nbgrnself 29085 | A vertex in a graph is not... |
nbgrnself2 29086 | A class ` X ` is not a nei... |
nbgrssovtx 29087 | The neighbors of a vertex ... |
nbgrssvwo2 29088 | The neighbors of a vertex ... |
nbgrsym 29089 | In a graph, the neighborho... |
nbupgrres 29090 | The neighborhood of a vert... |
usgrnbcnvfv 29091 | Applying the edge function... |
nbusgredgeu 29092 | For each neighbor of a ver... |
edgnbusgreu 29093 | For each edge incident to ... |
nbusgredgeu0 29094 | For each neighbor of a ver... |
nbusgrf1o0 29095 | The mapping of neighbors o... |
nbusgrf1o1 29096 | The set of neighbors of a ... |
nbusgrf1o 29097 | The set of neighbors of a ... |
nbedgusgr 29098 | The number of neighbors of... |
edgusgrnbfin 29099 | The number of neighbors of... |
nbusgrfi 29100 | The class of neighbors of ... |
nbfiusgrfi 29101 | The class of neighbors of ... |
hashnbusgrnn0 29102 | The number of neighbors of... |
nbfusgrlevtxm1 29103 | The number of neighbors of... |
nbfusgrlevtxm2 29104 | If there is a vertex which... |
nbusgrvtxm1 29105 | If the number of neighbors... |
nb3grprlem1 29106 | Lemma 1 for ~ nb3grpr . (... |
nb3grprlem2 29107 | Lemma 2 for ~ nb3grpr . (... |
nb3grpr 29108 | The neighbors of a vertex ... |
nb3grpr2 29109 | The neighbors of a vertex ... |
nb3gr2nb 29110 | If the neighbors of two ve... |
uvtxval 29113 | The set of all universal v... |
uvtxel 29114 | A universal vertex, i.e. a... |
uvtxisvtx 29115 | A universal vertex is a ve... |
uvtxssvtx 29116 | The set of the universal v... |
vtxnbuvtx 29117 | A universal vertex has all... |
uvtxnbgrss 29118 | A universal vertex has all... |
uvtxnbgrvtx 29119 | A universal vertex is neig... |
uvtx0 29120 | There is no universal vert... |
isuvtx 29121 | The set of all universal v... |
uvtxel1 29122 | Characterization of a univ... |
uvtx01vtx 29123 | If a graph/class has no ed... |
uvtx2vtx1edg 29124 | If a graph has two vertice... |
uvtx2vtx1edgb 29125 | If a hypergraph has two ve... |
uvtxnbgr 29126 | A universal vertex has all... |
uvtxnbgrb 29127 | A vertex is universal iff ... |
uvtxusgr 29128 | The set of all universal v... |
uvtxusgrel 29129 | A universal vertex, i.e. a... |
uvtxnm1nbgr 29130 | A universal vertex has ` n... |
nbusgrvtxm1uvtx 29131 | If the number of neighbors... |
uvtxnbvtxm1 29132 | A universal vertex has ` n... |
nbupgruvtxres 29133 | The neighborhood of a univ... |
uvtxupgrres 29134 | A universal vertex is univ... |
cplgruvtxb 29139 | A graph ` G ` is complete ... |
prcliscplgr 29140 | A proper class (representi... |
iscplgr 29141 | The property of being a co... |
iscplgrnb 29142 | A graph is complete iff al... |
iscplgredg 29143 | A graph ` G ` is complete ... |
iscusgr 29144 | The property of being a co... |
cusgrusgr 29145 | A complete simple graph is... |
cusgrcplgr 29146 | A complete simple graph is... |
iscusgrvtx 29147 | A simple graph is complete... |
cusgruvtxb 29148 | A simple graph is complete... |
iscusgredg 29149 | A simple graph is complete... |
cusgredg 29150 | In a complete simple graph... |
cplgr0 29151 | The null graph (with no ve... |
cusgr0 29152 | The null graph (with no ve... |
cplgr0v 29153 | A null graph (with no vert... |
cusgr0v 29154 | A graph with no vertices a... |
cplgr1vlem 29155 | Lemma for ~ cplgr1v and ~ ... |
cplgr1v 29156 | A graph with one vertex is... |
cusgr1v 29157 | A graph with one vertex an... |
cplgr2v 29158 | An undirected hypergraph w... |
cplgr2vpr 29159 | An undirected hypergraph w... |
nbcplgr 29160 | In a complete graph, each ... |
cplgr3v 29161 | A pseudograph with three (... |
cusgr3vnbpr 29162 | The neighbors of a vertex ... |
cplgrop 29163 | A complete graph represent... |
cusgrop 29164 | A complete simple graph re... |
cusgrexilem1 29165 | Lemma 1 for ~ cusgrexi . ... |
usgrexilem 29166 | Lemma for ~ usgrexi . (Co... |
usgrexi 29167 | An arbitrary set regarded ... |
cusgrexilem2 29168 | Lemma 2 for ~ cusgrexi . ... |
cusgrexi 29169 | An arbitrary set ` V ` reg... |
cusgrexg 29170 | For each set there is a se... |
structtousgr 29171 | Any (extensible) structure... |
structtocusgr 29172 | Any (extensible) structure... |
cffldtocusgr 29173 | The field of complex numbe... |
cusgrres 29174 | Restricting a complete sim... |
cusgrsizeindb0 29175 | Base case of the induction... |
cusgrsizeindb1 29176 | Base case of the induction... |
cusgrsizeindslem 29177 | Lemma for ~ cusgrsizeinds ... |
cusgrsizeinds 29178 | Part 1 of induction step i... |
cusgrsize2inds 29179 | Induction step in ~ cusgrs... |
cusgrsize 29180 | The size of a finite compl... |
cusgrfilem1 29181 | Lemma 1 for ~ cusgrfi . (... |
cusgrfilem2 29182 | Lemma 2 for ~ cusgrfi . (... |
cusgrfilem3 29183 | Lemma 3 for ~ cusgrfi . (... |
cusgrfi 29184 | If the size of a complete ... |
usgredgsscusgredg 29185 | A simple graph is a subgra... |
usgrsscusgr 29186 | A simple graph is a subgra... |
sizusglecusglem1 29187 | Lemma 1 for ~ sizusglecusg... |
sizusglecusglem2 29188 | Lemma 2 for ~ sizusglecusg... |
sizusglecusg 29189 | The size of a simple graph... |
fusgrmaxsize 29190 | The maximum size of a fini... |
vtxdgfval 29193 | The value of the vertex de... |
vtxdgval 29194 | The degree of a vertex. (... |
vtxdgfival 29195 | The degree of a vertex for... |
vtxdgop 29196 | The vertex degree expresse... |
vtxdgf 29197 | The vertex degree function... |
vtxdgelxnn0 29198 | The degree of a vertex is ... |
vtxdg0v 29199 | The degree of a vertex in ... |
vtxdg0e 29200 | The degree of a vertex in ... |
vtxdgfisnn0 29201 | The degree of a vertex in ... |
vtxdgfisf 29202 | The vertex degree function... |
vtxdeqd 29203 | Equality theorem for the v... |
vtxduhgr0e 29204 | The degree of a vertex in ... |
vtxdlfuhgr1v 29205 | The degree of the vertex i... |
vdumgr0 29206 | A vertex in a multigraph h... |
vtxdun 29207 | The degree of a vertex in ... |
vtxdfiun 29208 | The degree of a vertex in ... |
vtxduhgrun 29209 | The degree of a vertex in ... |
vtxduhgrfiun 29210 | The degree of a vertex in ... |
vtxdlfgrval 29211 | The value of the vertex de... |
vtxdumgrval 29212 | The value of the vertex de... |
vtxdusgrval 29213 | The value of the vertex de... |
vtxd0nedgb 29214 | A vertex has degree 0 iff ... |
vtxdushgrfvedglem 29215 | Lemma for ~ vtxdushgrfvedg... |
vtxdushgrfvedg 29216 | The value of the vertex de... |
vtxdusgrfvedg 29217 | The value of the vertex de... |
vtxduhgr0nedg 29218 | If a vertex in a hypergrap... |
vtxdumgr0nedg 29219 | If a vertex in a multigrap... |
vtxduhgr0edgnel 29220 | A vertex in a hypergraph h... |
vtxdusgr0edgnel 29221 | A vertex in a simple graph... |
vtxdusgr0edgnelALT 29222 | Alternate proof of ~ vtxdu... |
vtxdgfusgrf 29223 | The vertex degree function... |
vtxdgfusgr 29224 | In a finite simple graph, ... |
fusgrn0degnn0 29225 | In a nonempty, finite grap... |
1loopgruspgr 29226 | A graph with one edge whic... |
1loopgredg 29227 | The set of edges in a grap... |
1loopgrnb0 29228 | In a graph (simple pseudog... |
1loopgrvd2 29229 | The vertex degree of a one... |
1loopgrvd0 29230 | The vertex degree of a one... |
1hevtxdg0 29231 | The vertex degree of verte... |
1hevtxdg1 29232 | The vertex degree of verte... |
1hegrvtxdg1 29233 | The vertex degree of a gra... |
1hegrvtxdg1r 29234 | The vertex degree of a gra... |
1egrvtxdg1 29235 | The vertex degree of a one... |
1egrvtxdg1r 29236 | The vertex degree of a one... |
1egrvtxdg0 29237 | The vertex degree of a one... |
p1evtxdeqlem 29238 | Lemma for ~ p1evtxdeq and ... |
p1evtxdeq 29239 | If an edge ` E ` which doe... |
p1evtxdp1 29240 | If an edge ` E ` (not bein... |
uspgrloopvtx 29241 | The set of vertices in a g... |
uspgrloopvtxel 29242 | A vertex in a graph (simpl... |
uspgrloopiedg 29243 | The set of edges in a grap... |
uspgrloopedg 29244 | The set of edges in a grap... |
uspgrloopnb0 29245 | In a graph (simple pseudog... |
uspgrloopvd2 29246 | The vertex degree of a one... |
umgr2v2evtx 29247 | The set of vertices in a m... |
umgr2v2evtxel 29248 | A vertex in a multigraph w... |
umgr2v2eiedg 29249 | The edge function in a mul... |
umgr2v2eedg 29250 | The set of edges in a mult... |
umgr2v2e 29251 | A multigraph with two edge... |
umgr2v2enb1 29252 | In a multigraph with two e... |
umgr2v2evd2 29253 | In a multigraph with two e... |
hashnbusgrvd 29254 | In a simple graph, the num... |
usgruvtxvdb 29255 | In a finite simple graph w... |
vdiscusgrb 29256 | A finite simple graph with... |
vdiscusgr 29257 | In a finite complete simpl... |
vtxdusgradjvtx 29258 | The degree of a vertex in ... |
usgrvd0nedg 29259 | If a vertex in a simple gr... |
uhgrvd00 29260 | If every vertex in a hyper... |
usgrvd00 29261 | If every vertex in a simpl... |
vdegp1ai 29262 | The induction step for a v... |
vdegp1bi 29263 | The induction step for a v... |
vdegp1ci 29264 | The induction step for a v... |
vtxdginducedm1lem1 29265 | Lemma 1 for ~ vtxdginduced... |
vtxdginducedm1lem2 29266 | Lemma 2 for ~ vtxdginduced... |
vtxdginducedm1lem3 29267 | Lemma 3 for ~ vtxdginduced... |
vtxdginducedm1lem4 29268 | Lemma 4 for ~ vtxdginduced... |
vtxdginducedm1 29269 | The degree of a vertex ` v... |
vtxdginducedm1fi 29270 | The degree of a vertex ` v... |
finsumvtxdg2ssteplem1 29271 | Lemma for ~ finsumvtxdg2ss... |
finsumvtxdg2ssteplem2 29272 | Lemma for ~ finsumvtxdg2ss... |
finsumvtxdg2ssteplem3 29273 | Lemma for ~ finsumvtxdg2ss... |
finsumvtxdg2ssteplem4 29274 | Lemma for ~ finsumvtxdg2ss... |
finsumvtxdg2sstep 29275 | Induction step of ~ finsum... |
finsumvtxdg2size 29276 | The sum of the degrees of ... |
fusgr1th 29277 | The sum of the degrees of ... |
finsumvtxdgeven 29278 | The sum of the degrees of ... |
vtxdgoddnumeven 29279 | The number of vertices of ... |
fusgrvtxdgonume 29280 | The number of vertices of ... |
isrgr 29285 | The property of a class be... |
rgrprop 29286 | The properties of a k-regu... |
isrusgr 29287 | The property of being a k-... |
rusgrprop 29288 | The properties of a k-regu... |
rusgrrgr 29289 | A k-regular simple graph i... |
rusgrusgr 29290 | A k-regular simple graph i... |
finrusgrfusgr 29291 | A finite regular simple gr... |
isrusgr0 29292 | The property of being a k-... |
rusgrprop0 29293 | The properties of a k-regu... |
usgreqdrusgr 29294 | If all vertices in a simpl... |
fusgrregdegfi 29295 | In a nonempty finite simpl... |
fusgrn0eqdrusgr 29296 | If all vertices in a nonem... |
frusgrnn0 29297 | In a nonempty finite k-reg... |
0edg0rgr 29298 | A graph is 0-regular if it... |
uhgr0edg0rgr 29299 | A hypergraph is 0-regular ... |
uhgr0edg0rgrb 29300 | A hypergraph is 0-regular ... |
usgr0edg0rusgr 29301 | A simple graph is 0-regula... |
0vtxrgr 29302 | A null graph (with no vert... |
0vtxrusgr 29303 | A graph with no vertices a... |
0uhgrrusgr 29304 | The null graph as hypergra... |
0grrusgr 29305 | The null graph represented... |
0grrgr 29306 | The null graph represented... |
cusgrrusgr 29307 | A complete simple graph wi... |
cusgrm1rusgr 29308 | A finite simple graph with... |
rusgrpropnb 29309 | The properties of a k-regu... |
rusgrpropedg 29310 | The properties of a k-regu... |
rusgrpropadjvtx 29311 | The properties of a k-regu... |
rusgrnumwrdl2 29312 | In a k-regular simple grap... |
rusgr1vtxlem 29313 | Lemma for ~ rusgr1vtx . (... |
rusgr1vtx 29314 | If a k-regular simple grap... |
rgrusgrprc 29315 | The class of 0-regular sim... |
rusgrprc 29316 | The class of 0-regular sim... |
rgrprc 29317 | The class of 0-regular gra... |
rgrprcx 29318 | The class of 0-regular gra... |
rgrx0ndm 29319 | 0 is not in the domain of ... |
rgrx0nd 29320 | The potentially alternativ... |
ewlksfval 29327 | The set of s-walks of edge... |
isewlk 29328 | Conditions for a function ... |
ewlkprop 29329 | Properties of an s-walk of... |
ewlkinedg 29330 | The intersection (common v... |
ewlkle 29331 | An s-walk of edges is also... |
upgrewlkle2 29332 | In a pseudograph, there is... |
wkslem1 29333 | Lemma 1 for walks to subst... |
wkslem2 29334 | Lemma 2 for walks to subst... |
wksfval 29335 | The set of walks (in an un... |
iswlk 29336 | Properties of a pair of fu... |
wlkprop 29337 | Properties of a walk. (Co... |
wlkv 29338 | The classes involved in a ... |
iswlkg 29339 | Generalization of ~ iswlk ... |
wlkf 29340 | The mapping enumerating th... |
wlkcl 29341 | A walk has length ` # ( F ... |
wlkp 29342 | The mapping enumerating th... |
wlkpwrd 29343 | The sequence of vertices o... |
wlklenvp1 29344 | The number of vertices of ... |
wksv 29345 | The class of walks is a se... |
wksvOLD 29346 | Obsolete version of ~ wksv... |
wlkn0 29347 | The sequence of vertices o... |
wlklenvm1 29348 | The number of edges of a w... |
ifpsnprss 29349 | Lemma for ~ wlkvtxeledg : ... |
wlkvtxeledg 29350 | Each pair of adjacent vert... |
wlkvtxiedg 29351 | The vertices of a walk are... |
relwlk 29352 | The set ` ( Walks `` G ) `... |
wlkvv 29353 | If there is at least one w... |
wlkop 29354 | A walk is an ordered pair.... |
wlkcpr 29355 | A walk as class with two c... |
wlk2f 29356 | If there is a walk ` W ` t... |
wlkcomp 29357 | A walk expressed by proper... |
wlkcompim 29358 | Implications for the prope... |
wlkelwrd 29359 | The components of a walk a... |
wlkeq 29360 | Conditions for two walks (... |
edginwlk 29361 | The value of the edge func... |
upgredginwlk 29362 | The value of the edge func... |
iedginwlk 29363 | The value of the edge func... |
wlkl1loop 29364 | A walk of length 1 from a ... |
wlk1walk 29365 | A walk is a 1-walk "on the... |
wlk1ewlk 29366 | A walk is an s-walk "on th... |
upgriswlk 29367 | Properties of a pair of fu... |
upgrwlkedg 29368 | The edges of a walk in a p... |
upgrwlkcompim 29369 | Implications for the prope... |
wlkvtxedg 29370 | The vertices of a walk are... |
upgrwlkvtxedg 29371 | The pairs of connected ver... |
uspgr2wlkeq 29372 | Conditions for two walks w... |
uspgr2wlkeq2 29373 | Conditions for two walks w... |
uspgr2wlkeqi 29374 | Conditions for two walks w... |
umgrwlknloop 29375 | In a multigraph, each walk... |
wlkResOLD 29376 | Obsolete version of ~ opab... |
wlkv0 29377 | If there is a walk in the ... |
g0wlk0 29378 | There is no walk in a null... |
0wlk0 29379 | There is no walk for the e... |
wlk0prc 29380 | There is no walk in a null... |
wlklenvclwlk 29381 | The number of vertices in ... |
wlkson 29382 | The set of walks between t... |
iswlkon 29383 | Properties of a pair of fu... |
wlkonprop 29384 | Properties of a walk betwe... |
wlkpvtx 29385 | A walk connects vertices. ... |
wlkepvtx 29386 | The endpoints of a walk ar... |
wlkoniswlk 29387 | A walk between two vertice... |
wlkonwlk 29388 | A walk is a walk between i... |
wlkonwlk1l 29389 | A walk is a walk from its ... |
wlksoneq1eq2 29390 | Two walks with identical s... |
wlkonl1iedg 29391 | If there is a walk between... |
wlkon2n0 29392 | The length of a walk betwe... |
2wlklem 29393 | Lemma for theorems for wal... |
upgr2wlk 29394 | Properties of a pair of fu... |
wlkreslem 29395 | Lemma for ~ wlkres . (Con... |
wlkres 29396 | The restriction ` <. H , Q... |
redwlklem 29397 | Lemma for ~ redwlk . (Con... |
redwlk 29398 | A walk ending at the last ... |
wlkp1lem1 29399 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem2 29400 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem3 29401 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem4 29402 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem5 29403 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem6 29404 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem7 29405 | Lemma for ~ wlkp1 . (Cont... |
wlkp1lem8 29406 | Lemma for ~ wlkp1 . (Cont... |
wlkp1 29407 | Append one path segment (e... |
wlkdlem1 29408 | Lemma 1 for ~ wlkd . (Con... |
wlkdlem2 29409 | Lemma 2 for ~ wlkd . (Con... |
wlkdlem3 29410 | Lemma 3 for ~ wlkd . (Con... |
wlkdlem4 29411 | Lemma 4 for ~ wlkd . (Con... |
wlkd 29412 | Two words representing a w... |
lfgrwlkprop 29413 | Two adjacent vertices in a... |
lfgriswlk 29414 | Conditions for a pair of f... |
lfgrwlknloop 29415 | In a loop-free graph, each... |
reltrls 29420 | The set ` ( Trails `` G ) ... |
trlsfval 29421 | The set of trails (in an u... |
istrl 29422 | Conditions for a pair of c... |
trliswlk 29423 | A trail is a walk. (Contr... |
trlf1 29424 | The enumeration ` F ` of a... |
trlreslem 29425 | Lemma for ~ trlres . Form... |
trlres 29426 | The restriction ` <. H , Q... |
upgrtrls 29427 | The set of trails in a pse... |
upgristrl 29428 | Properties of a pair of fu... |
upgrf1istrl 29429 | Properties of a pair of a ... |
wksonproplem 29430 | Lemma for theorems for pro... |
wksonproplemOLD 29431 | Obsolete version of ~ wkso... |
trlsonfval 29432 | The set of trails between ... |
istrlson 29433 | Properties of a pair of fu... |
trlsonprop 29434 | Properties of a trail betw... |
trlsonistrl 29435 | A trail between two vertic... |
trlsonwlkon 29436 | A trail between two vertic... |
trlontrl 29437 | A trail is a trail between... |
relpths 29446 | The set ` ( Paths `` G ) `... |
pthsfval 29447 | The set of paths (in an un... |
spthsfval 29448 | The set of simple paths (i... |
ispth 29449 | Conditions for a pair of c... |
isspth 29450 | Conditions for a pair of c... |
pthistrl 29451 | A path is a trail (in an u... |
spthispth 29452 | A simple path is a path (i... |
pthiswlk 29453 | A path is a walk (in an un... |
spthiswlk 29454 | A simple path is a walk (i... |
pthdivtx 29455 | The inner vertices of a pa... |
pthdadjvtx 29456 | The adjacent vertices of a... |
2pthnloop 29457 | A path of length at least ... |
upgr2pthnlp 29458 | A path of length at least ... |
spthdifv 29459 | The vertices of a simple p... |
spthdep 29460 | A simple path (at least of... |
pthdepisspth 29461 | A path with different star... |
upgrwlkdvdelem 29462 | Lemma for ~ upgrwlkdvde . ... |
upgrwlkdvde 29463 | In a pseudograph, all edge... |
upgrspthswlk 29464 | The set of simple paths in... |
upgrwlkdvspth 29465 | A walk consisting of diffe... |
pthsonfval 29466 | The set of paths between t... |
spthson 29467 | The set of simple paths be... |
ispthson 29468 | Properties of a pair of fu... |
isspthson 29469 | Properties of a pair of fu... |
pthsonprop 29470 | Properties of a path betwe... |
spthonprop 29471 | Properties of a simple pat... |
pthonispth 29472 | A path between two vertice... |
pthontrlon 29473 | A path between two vertice... |
pthonpth 29474 | A path is a path between i... |
isspthonpth 29475 | A pair of functions is a s... |
spthonisspth 29476 | A simple path between to v... |
spthonpthon 29477 | A simple path between two ... |
spthonepeq 29478 | The endpoints of a simple ... |
uhgrwkspthlem1 29479 | Lemma 1 for ~ uhgrwkspth .... |
uhgrwkspthlem2 29480 | Lemma 2 for ~ uhgrwkspth .... |
uhgrwkspth 29481 | Any walk of length 1 betwe... |
usgr2wlkneq 29482 | The vertices and edges are... |
usgr2wlkspthlem1 29483 | Lemma 1 for ~ usgr2wlkspth... |
usgr2wlkspthlem2 29484 | Lemma 2 for ~ usgr2wlkspth... |
usgr2wlkspth 29485 | In a simple graph, any wal... |
usgr2trlncl 29486 | In a simple graph, any tra... |
usgr2trlspth 29487 | In a simple graph, any tra... |
usgr2pthspth 29488 | In a simple graph, any pat... |
usgr2pthlem 29489 | Lemma for ~ usgr2pth . (C... |
usgr2pth 29490 | In a simple graph, there i... |
usgr2pth0 29491 | In a simply graph, there i... |
pthdlem1 29492 | Lemma 1 for ~ pthd . (Con... |
pthdlem2lem 29493 | Lemma for ~ pthdlem2 . (C... |
pthdlem2 29494 | Lemma 2 for ~ pthd . (Con... |
pthd 29495 | Two words representing a t... |
clwlks 29498 | The set of closed walks (i... |
isclwlk 29499 | A pair of functions repres... |
clwlkiswlk 29500 | A closed walk is a walk (i... |
clwlkwlk 29501 | Closed walks are walks (in... |
clwlkswks 29502 | Closed walks are walks (in... |
isclwlke 29503 | Properties of a pair of fu... |
isclwlkupgr 29504 | Properties of a pair of fu... |
clwlkcomp 29505 | A closed walk expressed by... |
clwlkcompim 29506 | Implications for the prope... |
upgrclwlkcompim 29507 | Implications for the prope... |
clwlkcompbp 29508 | Basic properties of the co... |
clwlkl1loop 29509 | A closed walk of length 1 ... |
crcts 29514 | The set of circuits (in an... |
cycls 29515 | The set of cycles (in an u... |
iscrct 29516 | Sufficient and necessary c... |
iscycl 29517 | Sufficient and necessary c... |
crctprop 29518 | The properties of a circui... |
cyclprop 29519 | The properties of a cycle:... |
crctisclwlk 29520 | A circuit is a closed walk... |
crctistrl 29521 | A circuit is a trail. (Co... |
crctiswlk 29522 | A circuit is a walk. (Con... |
cyclispth 29523 | A cycle is a path. (Contr... |
cycliswlk 29524 | A cycle is a walk. (Contr... |
cycliscrct 29525 | A cycle is a circuit. (Co... |
cyclnspth 29526 | A (non-trivial) cycle is n... |
cyclispthon 29527 | A cycle is a path starting... |
lfgrn1cycl 29528 | In a loop-free graph there... |
usgr2trlncrct 29529 | In a simple graph, any tra... |
umgrn1cycl 29530 | In a multigraph graph (wit... |
uspgrn2crct 29531 | In a simple pseudograph th... |
usgrn2cycl 29532 | In a simple graph there ar... |
crctcshwlkn0lem1 29533 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem2 29534 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem3 29535 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem4 29536 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem5 29537 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem6 29538 | Lemma for ~ crctcshwlkn0 .... |
crctcshwlkn0lem7 29539 | Lemma for ~ crctcshwlkn0 .... |
crctcshlem1 29540 | Lemma for ~ crctcsh . (Co... |
crctcshlem2 29541 | Lemma for ~ crctcsh . (Co... |
crctcshlem3 29542 | Lemma for ~ crctcsh . (Co... |
crctcshlem4 29543 | Lemma for ~ crctcsh . (Co... |
crctcshwlkn0 29544 | Cyclically shifting the in... |
crctcshwlk 29545 | Cyclically shifting the in... |
crctcshtrl 29546 | Cyclically shifting the in... |
crctcsh 29547 | Cyclically shifting the in... |
wwlks 29558 | The set of walks (in an un... |
iswwlks 29559 | A word over the set of ver... |
wwlksn 29560 | The set of walks (in an un... |
iswwlksn 29561 | A word over the set of ver... |
wwlksnprcl 29562 | Derivation of the length o... |
iswwlksnx 29563 | Properties of a word to re... |
wwlkbp 29564 | Basic properties of a walk... |
wwlknbp 29565 | Basic properties of a walk... |
wwlknp 29566 | Properties of a set being ... |
wwlknbp1 29567 | Other basic properties of ... |
wwlknvtx 29568 | The symbols of a word ` W ... |
wwlknllvtx 29569 | If a word ` W ` represents... |
wwlknlsw 29570 | If a word represents a wal... |
wspthsn 29571 | The set of simple paths of... |
iswspthn 29572 | An element of the set of s... |
wspthnp 29573 | Properties of a set being ... |
wwlksnon 29574 | The set of walks of a fixe... |
wspthsnon 29575 | The set of simple paths of... |
iswwlksnon 29576 | The set of walks of a fixe... |
wwlksnon0 29577 | Sufficient conditions for ... |
wwlksonvtx 29578 | If a word ` W ` represents... |
iswspthsnon 29579 | The set of simple paths of... |
wwlknon 29580 | An element of the set of w... |
wspthnon 29581 | An element of the set of s... |
wspthnonp 29582 | Properties of a set being ... |
wspthneq1eq2 29583 | Two simple paths with iden... |
wwlksn0s 29584 | The set of all walks as wo... |
wwlkssswrd 29585 | Walks (represented by word... |
wwlksn0 29586 | A walk of length 0 is repr... |
0enwwlksnge1 29587 | In graphs without edges, t... |
wwlkswwlksn 29588 | A walk of a fixed length a... |
wwlkssswwlksn 29589 | The walks of a fixed lengt... |
wlkiswwlks1 29590 | The sequence of vertices i... |
wlklnwwlkln1 29591 | The sequence of vertices i... |
wlkiswwlks2lem1 29592 | Lemma 1 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem2 29593 | Lemma 2 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem3 29594 | Lemma 3 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem4 29595 | Lemma 4 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem5 29596 | Lemma 5 for ~ wlkiswwlks2 ... |
wlkiswwlks2lem6 29597 | Lemma 6 for ~ wlkiswwlks2 ... |
wlkiswwlks2 29598 | A walk as word corresponds... |
wlkiswwlks 29599 | A walk as word corresponds... |
wlkiswwlksupgr2 29600 | A walk as word corresponds... |
wlkiswwlkupgr 29601 | A walk as word corresponds... |
wlkswwlksf1o 29602 | The mapping of (ordinary) ... |
wlkswwlksen 29603 | The set of walks as words ... |
wwlksm1edg 29604 | Removing the trailing edge... |
wlklnwwlkln2lem 29605 | Lemma for ~ wlklnwwlkln2 a... |
wlklnwwlkln2 29606 | A walk of length ` N ` as ... |
wlklnwwlkn 29607 | A walk of length ` N ` as ... |
wlklnwwlklnupgr2 29608 | A walk of length ` N ` as ... |
wlklnwwlknupgr 29609 | A walk of length ` N ` as ... |
wlknewwlksn 29610 | If a walk in a pseudograph... |
wlknwwlksnbij 29611 | The mapping ` ( t e. T |->... |
wlknwwlksnen 29612 | In a simple pseudograph, t... |
wlknwwlksneqs 29613 | The set of walks of a fixe... |
wwlkseq 29614 | Equality of two walks (as ... |
wwlksnred 29615 | Reduction of a walk (as wo... |
wwlksnext 29616 | Extension of a walk (as wo... |
wwlksnextbi 29617 | Extension of a walk (as wo... |
wwlksnredwwlkn 29618 | For each walk (as word) of... |
wwlksnredwwlkn0 29619 | For each walk (as word) of... |
wwlksnextwrd 29620 | Lemma for ~ wwlksnextbij .... |
wwlksnextfun 29621 | Lemma for ~ wwlksnextbij .... |
wwlksnextinj 29622 | Lemma for ~ wwlksnextbij .... |
wwlksnextsurj 29623 | Lemma for ~ wwlksnextbij .... |
wwlksnextbij0 29624 | Lemma for ~ wwlksnextbij .... |
wwlksnextbij 29625 | There is a bijection betwe... |
wwlksnexthasheq 29626 | The number of the extensio... |
disjxwwlksn 29627 | Sets of walks (as words) e... |
wwlksnndef 29628 | Conditions for ` WWalksN `... |
wwlksnfi 29629 | The number of walks repres... |
wlksnfi 29630 | The number of walks of fix... |
wlksnwwlknvbij 29631 | There is a bijection betwe... |
wwlksnextproplem1 29632 | Lemma 1 for ~ wwlksnextpro... |
wwlksnextproplem2 29633 | Lemma 2 for ~ wwlksnextpro... |
wwlksnextproplem3 29634 | Lemma 3 for ~ wwlksnextpro... |
wwlksnextprop 29635 | Adding additional properti... |
disjxwwlkn 29636 | Sets of walks (as words) e... |
hashwwlksnext 29637 | Number of walks (as words)... |
wwlksnwwlksnon 29638 | A walk of fixed length is ... |
wspthsnwspthsnon 29639 | A simple path of fixed len... |
wspthsnonn0vne 29640 | If the set of simple paths... |
wspthsswwlkn 29641 | The set of simple paths of... |
wspthnfi 29642 | In a finite graph, the set... |
wwlksnonfi 29643 | In a finite graph, the set... |
wspthsswwlknon 29644 | The set of simple paths of... |
wspthnonfi 29645 | In a finite graph, the set... |
wspniunwspnon 29646 | The set of nonempty simple... |
wspn0 29647 | If there are no vertices, ... |
2wlkdlem1 29648 | Lemma 1 for ~ 2wlkd . (Co... |
2wlkdlem2 29649 | Lemma 2 for ~ 2wlkd . (Co... |
2wlkdlem3 29650 | Lemma 3 for ~ 2wlkd . (Co... |
2wlkdlem4 29651 | Lemma 4 for ~ 2wlkd . (Co... |
2wlkdlem5 29652 | Lemma 5 for ~ 2wlkd . (Co... |
2pthdlem1 29653 | Lemma 1 for ~ 2pthd . (Co... |
2wlkdlem6 29654 | Lemma 6 for ~ 2wlkd . (Co... |
2wlkdlem7 29655 | Lemma 7 for ~ 2wlkd . (Co... |
2wlkdlem8 29656 | Lemma 8 for ~ 2wlkd . (Co... |
2wlkdlem9 29657 | Lemma 9 for ~ 2wlkd . (Co... |
2wlkdlem10 29658 | Lemma 10 for ~ 3wlkd . (C... |
2wlkd 29659 | Construction of a walk fro... |
2wlkond 29660 | A walk of length 2 from on... |
2trld 29661 | Construction of a trail fr... |
2trlond 29662 | A trail of length 2 from o... |
2pthd 29663 | A path of length 2 from on... |
2spthd 29664 | A simple path of length 2 ... |
2pthond 29665 | A simple path of length 2 ... |
2pthon3v 29666 | For a vertex adjacent to t... |
umgr2adedgwlklem 29667 | Lemma for ~ umgr2adedgwlk ... |
umgr2adedgwlk 29668 | In a multigraph, two adjac... |
umgr2adedgwlkon 29669 | In a multigraph, two adjac... |
umgr2adedgwlkonALT 29670 | Alternate proof for ~ umgr... |
umgr2adedgspth 29671 | In a multigraph, two adjac... |
umgr2wlk 29672 | In a multigraph, there is ... |
umgr2wlkon 29673 | For each pair of adjacent ... |
elwwlks2s3 29674 | A walk of length 2 as word... |
midwwlks2s3 29675 | There is a vertex between ... |
wwlks2onv 29676 | If a length 3 string repre... |
elwwlks2ons3im 29677 | A walk as word of length 2... |
elwwlks2ons3 29678 | For each walk of length 2 ... |
s3wwlks2on 29679 | A length 3 string which re... |
umgrwwlks2on 29680 | A walk of length 2 between... |
wwlks2onsym 29681 | There is a walk of length ... |
elwwlks2on 29682 | A walk of length 2 between... |
elwspths2on 29683 | A simple path of length 2 ... |
wpthswwlks2on 29684 | For two different vertices... |
2wspdisj 29685 | All simple paths of length... |
2wspiundisj 29686 | All simple paths of length... |
usgr2wspthons3 29687 | A simple path of length 2 ... |
usgr2wspthon 29688 | A simple path of length 2 ... |
elwwlks2 29689 | A walk of length 2 between... |
elwspths2spth 29690 | A simple path of length 2 ... |
rusgrnumwwlkl1 29691 | In a k-regular graph, ther... |
rusgrnumwwlkslem 29692 | Lemma for ~ rusgrnumwwlks ... |
rusgrnumwwlklem 29693 | Lemma for ~ rusgrnumwwlk e... |
rusgrnumwwlkb0 29694 | Induction base 0 for ~ rus... |
rusgrnumwwlkb1 29695 | Induction base 1 for ~ rus... |
rusgr0edg 29696 | Special case for graphs wi... |
rusgrnumwwlks 29697 | Induction step for ~ rusgr... |
rusgrnumwwlk 29698 | In a ` K `-regular graph, ... |
rusgrnumwwlkg 29699 | In a ` K `-regular graph, ... |
rusgrnumwlkg 29700 | In a k-regular graph, the ... |
clwwlknclwwlkdif 29701 | The set ` A ` of walks of ... |
clwwlknclwwlkdifnum 29702 | In a ` K `-regular graph, ... |
clwwlk 29705 | The set of closed walks (i... |
isclwwlk 29706 | Properties of a word to re... |
clwwlkbp 29707 | Basic properties of a clos... |
clwwlkgt0 29708 | There is no empty closed w... |
clwwlksswrd 29709 | Closed walks (represented ... |
clwwlk1loop 29710 | A closed walk of length 1 ... |
clwwlkccatlem 29711 | Lemma for ~ clwwlkccat : i... |
clwwlkccat 29712 | The concatenation of two w... |
umgrclwwlkge2 29713 | A closed walk in a multigr... |
clwlkclwwlklem2a1 29714 | Lemma 1 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a2 29715 | Lemma 2 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a3 29716 | Lemma 3 for ~ clwlkclwwlkl... |
clwlkclwwlklem2fv1 29717 | Lemma 4a for ~ clwlkclwwlk... |
clwlkclwwlklem2fv2 29718 | Lemma 4b for ~ clwlkclwwlk... |
clwlkclwwlklem2a4 29719 | Lemma 4 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a 29720 | Lemma for ~ clwlkclwwlklem... |
clwlkclwwlklem1 29721 | Lemma 1 for ~ clwlkclwwlk ... |
clwlkclwwlklem2 29722 | Lemma 2 for ~ clwlkclwwlk ... |
clwlkclwwlklem3 29723 | Lemma 3 for ~ clwlkclwwlk ... |
clwlkclwwlk 29724 | A closed walk as word of l... |
clwlkclwwlk2 29725 | A closed walk corresponds ... |
clwlkclwwlkflem 29726 | Lemma for ~ clwlkclwwlkf .... |
clwlkclwwlkf1lem2 29727 | Lemma 2 for ~ clwlkclwwlkf... |
clwlkclwwlkf1lem3 29728 | Lemma 3 for ~ clwlkclwwlkf... |
clwlkclwwlkfolem 29729 | Lemma for ~ clwlkclwwlkfo ... |
clwlkclwwlkf 29730 | ` F ` is a function from t... |
clwlkclwwlkfo 29731 | ` F ` is a function from t... |
clwlkclwwlkf1 29732 | ` F ` is a one-to-one func... |
clwlkclwwlkf1o 29733 | ` F ` is a bijection betwe... |
clwlkclwwlken 29734 | The set of the nonempty cl... |
clwwisshclwwslemlem 29735 | Lemma for ~ clwwisshclwwsl... |
clwwisshclwwslem 29736 | Lemma for ~ clwwisshclwws ... |
clwwisshclwws 29737 | Cyclically shifting a clos... |
clwwisshclwwsn 29738 | Cyclically shifting a clos... |
erclwwlkrel 29739 | ` .~ ` is a relation. (Co... |
erclwwlkeq 29740 | Two classes are equivalent... |
erclwwlkeqlen 29741 | If two classes are equival... |
erclwwlkref 29742 | ` .~ ` is a reflexive rela... |
erclwwlksym 29743 | ` .~ ` is a symmetric rela... |
erclwwlktr 29744 | ` .~ ` is a transitive rel... |
erclwwlk 29745 | ` .~ ` is an equivalence r... |
clwwlkn 29748 | The set of closed walks of... |
isclwwlkn 29749 | A word over the set of ver... |
clwwlkn0 29750 | There is no closed walk of... |
clwwlkneq0 29751 | Sufficient conditions for ... |
clwwlkclwwlkn 29752 | A closed walk of a fixed l... |
clwwlksclwwlkn 29753 | The closed walks of a fixe... |
clwwlknlen 29754 | The length of a word repre... |
clwwlknnn 29755 | The length of a closed wal... |
clwwlknwrd 29756 | A closed walk of a fixed l... |
clwwlknbp 29757 | Basic properties of a clos... |
isclwwlknx 29758 | Characterization of a word... |
clwwlknp 29759 | Properties of a set being ... |
clwwlknwwlksn 29760 | A word representing a clos... |
clwwlknlbonbgr1 29761 | The last but one vertex in... |
clwwlkinwwlk 29762 | If the initial vertex of a... |
clwwlkn1 29763 | A closed walk of length 1 ... |
loopclwwlkn1b 29764 | The singleton word consist... |
clwwlkn1loopb 29765 | A word represents a closed... |
clwwlkn2 29766 | A closed walk of length 2 ... |
clwwlknfi 29767 | If there is only a finite ... |
clwwlkel 29768 | Obtaining a closed walk (a... |
clwwlkf 29769 | Lemma 1 for ~ clwwlkf1o : ... |
clwwlkfv 29770 | Lemma 2 for ~ clwwlkf1o : ... |
clwwlkf1 29771 | Lemma 3 for ~ clwwlkf1o : ... |
clwwlkfo 29772 | Lemma 4 for ~ clwwlkf1o : ... |
clwwlkf1o 29773 | F is a 1-1 onto function, ... |
clwwlken 29774 | The set of closed walks of... |
clwwlknwwlkncl 29775 | Obtaining a closed walk (a... |
clwwlkwwlksb 29776 | A nonempty word over verti... |
clwwlknwwlksnb 29777 | A word over vertices repre... |
clwwlkext2edg 29778 | If a word concatenated wit... |
wwlksext2clwwlk 29779 | If a word represents a wal... |
wwlksubclwwlk 29780 | Any prefix of a word repre... |
clwwnisshclwwsn 29781 | Cyclically shifting a clos... |
eleclclwwlknlem1 29782 | Lemma 1 for ~ eleclclwwlkn... |
eleclclwwlknlem2 29783 | Lemma 2 for ~ eleclclwwlkn... |
clwwlknscsh 29784 | The set of cyclical shifts... |
clwwlknccat 29785 | The concatenation of two w... |
umgr2cwwk2dif 29786 | If a word represents a clo... |
umgr2cwwkdifex 29787 | If a word represents a clo... |
erclwwlknrel 29788 | ` .~ ` is a relation. (Co... |
erclwwlkneq 29789 | Two classes are equivalent... |
erclwwlkneqlen 29790 | If two classes are equival... |
erclwwlknref 29791 | ` .~ ` is a reflexive rela... |
erclwwlknsym 29792 | ` .~ ` is a symmetric rela... |
erclwwlkntr 29793 | ` .~ ` is a transitive rel... |
erclwwlkn 29794 | ` .~ ` is an equivalence r... |
qerclwwlknfi 29795 | The quotient set of the se... |
hashclwwlkn0 29796 | The number of closed walks... |
eclclwwlkn1 29797 | An equivalence class accor... |
eleclclwwlkn 29798 | A member of an equivalence... |
hashecclwwlkn1 29799 | The size of every equivale... |
umgrhashecclwwlk 29800 | The size of every equivale... |
fusgrhashclwwlkn 29801 | The size of the set of clo... |
clwwlkndivn 29802 | The size of the set of clo... |
clwlknf1oclwwlknlem1 29803 | Lemma 1 for ~ clwlknf1oclw... |
clwlknf1oclwwlknlem2 29804 | Lemma 2 for ~ clwlknf1oclw... |
clwlknf1oclwwlknlem3 29805 | Lemma 3 for ~ clwlknf1oclw... |
clwlknf1oclwwlkn 29806 | There is a one-to-one onto... |
clwlkssizeeq 29807 | The size of the set of clo... |
clwlksndivn 29808 | The size of the set of clo... |
clwwlknonmpo 29811 | ` ( ClWWalksNOn `` G ) ` i... |
clwwlknon 29812 | The set of closed walks on... |
isclwwlknon 29813 | A word over the set of ver... |
clwwlk0on0 29814 | There is no word over the ... |
clwwlknon0 29815 | Sufficient conditions for ... |
clwwlknonfin 29816 | In a finite graph ` G ` , ... |
clwwlknonel 29817 | Characterization of a word... |
clwwlknonccat 29818 | The concatenation of two w... |
clwwlknon1 29819 | The set of closed walks on... |
clwwlknon1loop 29820 | If there is a loop at vert... |
clwwlknon1nloop 29821 | If there is no loop at ver... |
clwwlknon1sn 29822 | The set of (closed) walks ... |
clwwlknon1le1 29823 | There is at most one (clos... |
clwwlknon2 29824 | The set of closed walks on... |
clwwlknon2x 29825 | The set of closed walks on... |
s2elclwwlknon2 29826 | Sufficient conditions of a... |
clwwlknon2num 29827 | In a ` K `-regular graph `... |
clwwlknonwwlknonb 29828 | A word over vertices repre... |
clwwlknonex2lem1 29829 | Lemma 1 for ~ clwwlknonex2... |
clwwlknonex2lem2 29830 | Lemma 2 for ~ clwwlknonex2... |
clwwlknonex2 29831 | Extending a closed walk ` ... |
clwwlknonex2e 29832 | Extending a closed walk ` ... |
clwwlknondisj 29833 | The sets of closed walks o... |
clwwlknun 29834 | The set of closed walks of... |
clwwlkvbij 29835 | There is a bijection betwe... |
0ewlk 29836 | The empty set (empty seque... |
1ewlk 29837 | A sequence of 1 edge is an... |
0wlk 29838 | A pair of an empty set (of... |
is0wlk 29839 | A pair of an empty set (of... |
0wlkonlem1 29840 | Lemma 1 for ~ 0wlkon and ~... |
0wlkonlem2 29841 | Lemma 2 for ~ 0wlkon and ~... |
0wlkon 29842 | A walk of length 0 from a ... |
0wlkons1 29843 | A walk of length 0 from a ... |
0trl 29844 | A pair of an empty set (of... |
is0trl 29845 | A pair of an empty set (of... |
0trlon 29846 | A trail of length 0 from a... |
0pth 29847 | A pair of an empty set (of... |
0spth 29848 | A pair of an empty set (of... |
0pthon 29849 | A path of length 0 from a ... |
0pthon1 29850 | A path of length 0 from a ... |
0pthonv 29851 | For each vertex there is a... |
0clwlk 29852 | A pair of an empty set (of... |
0clwlkv 29853 | Any vertex (more precisely... |
0clwlk0 29854 | There is no closed walk in... |
0crct 29855 | A pair of an empty set (of... |
0cycl 29856 | A pair of an empty set (of... |
1pthdlem1 29857 | Lemma 1 for ~ 1pthd . (Co... |
1pthdlem2 29858 | Lemma 2 for ~ 1pthd . (Co... |
1wlkdlem1 29859 | Lemma 1 for ~ 1wlkd . (Co... |
1wlkdlem2 29860 | Lemma 2 for ~ 1wlkd . (Co... |
1wlkdlem3 29861 | Lemma 3 for ~ 1wlkd . (Co... |
1wlkdlem4 29862 | Lemma 4 for ~ 1wlkd . (Co... |
1wlkd 29863 | In a graph with two vertic... |
1trld 29864 | In a graph with two vertic... |
1pthd 29865 | In a graph with two vertic... |
1pthond 29866 | In a graph with two vertic... |
upgr1wlkdlem1 29867 | Lemma 1 for ~ upgr1wlkd . ... |
upgr1wlkdlem2 29868 | Lemma 2 for ~ upgr1wlkd . ... |
upgr1wlkd 29869 | In a pseudograph with two ... |
upgr1trld 29870 | In a pseudograph with two ... |
upgr1pthd 29871 | In a pseudograph with two ... |
upgr1pthond 29872 | In a pseudograph with two ... |
lppthon 29873 | A loop (which is an edge a... |
lp1cycl 29874 | A loop (which is an edge a... |
1pthon2v 29875 | For each pair of adjacent ... |
1pthon2ve 29876 | For each pair of adjacent ... |
wlk2v2elem1 29877 | Lemma 1 for ~ wlk2v2e : ` ... |
wlk2v2elem2 29878 | Lemma 2 for ~ wlk2v2e : T... |
wlk2v2e 29879 | In a graph with two vertic... |
ntrl2v2e 29880 | A walk which is not a trai... |
3wlkdlem1 29881 | Lemma 1 for ~ 3wlkd . (Co... |
3wlkdlem2 29882 | Lemma 2 for ~ 3wlkd . (Co... |
3wlkdlem3 29883 | Lemma 3 for ~ 3wlkd . (Co... |
3wlkdlem4 29884 | Lemma 4 for ~ 3wlkd . (Co... |
3wlkdlem5 29885 | Lemma 5 for ~ 3wlkd . (Co... |
3pthdlem1 29886 | Lemma 1 for ~ 3pthd . (Co... |
3wlkdlem6 29887 | Lemma 6 for ~ 3wlkd . (Co... |
3wlkdlem7 29888 | Lemma 7 for ~ 3wlkd . (Co... |
3wlkdlem8 29889 | Lemma 8 for ~ 3wlkd . (Co... |
3wlkdlem9 29890 | Lemma 9 for ~ 3wlkd . (Co... |
3wlkdlem10 29891 | Lemma 10 for ~ 3wlkd . (C... |
3wlkd 29892 | Construction of a walk fro... |
3wlkond 29893 | A walk of length 3 from on... |
3trld 29894 | Construction of a trail fr... |
3trlond 29895 | A trail of length 3 from o... |
3pthd 29896 | A path of length 3 from on... |
3pthond 29897 | A path of length 3 from on... |
3spthd 29898 | A simple path of length 3 ... |
3spthond 29899 | A simple path of length 3 ... |
3cycld 29900 | Construction of a 3-cycle ... |
3cyclpd 29901 | Construction of a 3-cycle ... |
upgr3v3e3cycl 29902 | If there is a cycle of len... |
uhgr3cyclexlem 29903 | Lemma for ~ uhgr3cyclex . ... |
uhgr3cyclex 29904 | If there are three differe... |
umgr3cyclex 29905 | If there are three (differ... |
umgr3v3e3cycl 29906 | If and only if there is a ... |
upgr4cycl4dv4e 29907 | If there is a cycle of len... |
dfconngr1 29910 | Alternative definition of ... |
isconngr 29911 | The property of being a co... |
isconngr1 29912 | The property of being a co... |
cusconngr 29913 | A complete hypergraph is c... |
0conngr 29914 | A graph without vertices i... |
0vconngr 29915 | A graph without vertices i... |
1conngr 29916 | A graph with (at most) one... |
conngrv2edg 29917 | A vertex in a connected gr... |
vdn0conngrumgrv2 29918 | A vertex in a connected mu... |
releupth 29921 | The set ` ( EulerPaths `` ... |
eupths 29922 | The Eulerian paths on the ... |
iseupth 29923 | The property " ` <. F , P ... |
iseupthf1o 29924 | The property " ` <. F , P ... |
eupthi 29925 | Properties of an Eulerian ... |
eupthf1o 29926 | The ` F ` function in an E... |
eupthfi 29927 | Any graph with an Eulerian... |
eupthseg 29928 | The ` N ` -th edge in an e... |
upgriseupth 29929 | The property " ` <. F , P ... |
upgreupthi 29930 | Properties of an Eulerian ... |
upgreupthseg 29931 | The ` N ` -th edge in an e... |
eupthcl 29932 | An Eulerian path has lengt... |
eupthistrl 29933 | An Eulerian path is a trai... |
eupthiswlk 29934 | An Eulerian path is a walk... |
eupthpf 29935 | The ` P ` function in an E... |
eupth0 29936 | There is an Eulerian path ... |
eupthres 29937 | The restriction ` <. H , Q... |
eupthp1 29938 | Append one path segment to... |
eupth2eucrct 29939 | Append one path segment to... |
eupth2lem1 29940 | Lemma for ~ eupth2 . (Con... |
eupth2lem2 29941 | Lemma for ~ eupth2 . (Con... |
trlsegvdeglem1 29942 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem2 29943 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem3 29944 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem4 29945 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem5 29946 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem6 29947 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem7 29948 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeg 29949 | Formerly part of proof of ... |
eupth2lem3lem1 29950 | Lemma for ~ eupth2lem3 . ... |
eupth2lem3lem2 29951 | Lemma for ~ eupth2lem3 . ... |
eupth2lem3lem3 29952 | Lemma for ~ eupth2lem3 , f... |
eupth2lem3lem4 29953 | Lemma for ~ eupth2lem3 , f... |
eupth2lem3lem5 29954 | Lemma for ~ eupth2 . (Con... |
eupth2lem3lem6 29955 | Formerly part of proof of ... |
eupth2lem3lem7 29956 | Lemma for ~ eupth2lem3 : ... |
eupthvdres 29957 | Formerly part of proof of ... |
eupth2lem3 29958 | Lemma for ~ eupth2 . (Con... |
eupth2lemb 29959 | Lemma for ~ eupth2 (induct... |
eupth2lems 29960 | Lemma for ~ eupth2 (induct... |
eupth2 29961 | The only vertices of odd d... |
eulerpathpr 29962 | A graph with an Eulerian p... |
eulerpath 29963 | A pseudograph with an Eule... |
eulercrct 29964 | A pseudograph with an Eule... |
eucrctshift 29965 | Cyclically shifting the in... |
eucrct2eupth1 29966 | Removing one edge ` ( I ``... |
eucrct2eupth 29967 | Removing one edge ` ( I ``... |
konigsbergvtx 29968 | The set of vertices of the... |
konigsbergiedg 29969 | The indexed edges of the K... |
konigsbergiedgw 29970 | The indexed edges of the K... |
konigsbergssiedgwpr 29971 | Each subset of the indexed... |
konigsbergssiedgw 29972 | Each subset of the indexed... |
konigsbergumgr 29973 | The Königsberg graph ... |
konigsberglem1 29974 | Lemma 1 for ~ konigsberg :... |
konigsberglem2 29975 | Lemma 2 for ~ konigsberg :... |
konigsberglem3 29976 | Lemma 3 for ~ konigsberg :... |
konigsberglem4 29977 | Lemma 4 for ~ konigsberg :... |
konigsberglem5 29978 | Lemma 5 for ~ konigsberg :... |
konigsberg 29979 | The Königsberg Bridge... |
isfrgr 29982 | The property of being a fr... |
frgrusgr 29983 | A friendship graph is a si... |
frgr0v 29984 | Any null graph (set with n... |
frgr0vb 29985 | Any null graph (without ve... |
frgruhgr0v 29986 | Any null graph (without ve... |
frgr0 29987 | The null graph (graph with... |
frcond1 29988 | The friendship condition: ... |
frcond2 29989 | The friendship condition: ... |
frgreu 29990 | Variant of ~ frcond2 : An... |
frcond3 29991 | The friendship condition, ... |
frcond4 29992 | The friendship condition, ... |
frgr1v 29993 | Any graph with (at most) o... |
nfrgr2v 29994 | Any graph with two (differ... |
frgr3vlem1 29995 | Lemma 1 for ~ frgr3v . (C... |
frgr3vlem2 29996 | Lemma 2 for ~ frgr3v . (C... |
frgr3v 29997 | Any graph with three verti... |
1vwmgr 29998 | Every graph with one verte... |
3vfriswmgrlem 29999 | Lemma for ~ 3vfriswmgr . ... |
3vfriswmgr 30000 | Every friendship graph wit... |
1to2vfriswmgr 30001 | Every friendship graph wit... |
1to3vfriswmgr 30002 | Every friendship graph wit... |
1to3vfriendship 30003 | The friendship theorem for... |
2pthfrgrrn 30004 | Between any two (different... |
2pthfrgrrn2 30005 | Between any two (different... |
2pthfrgr 30006 | Between any two (different... |
3cyclfrgrrn1 30007 | Every vertex in a friendsh... |
3cyclfrgrrn 30008 | Every vertex in a friendsh... |
3cyclfrgrrn2 30009 | Every vertex in a friendsh... |
3cyclfrgr 30010 | Every vertex in a friendsh... |
4cycl2v2nb 30011 | In a (maybe degenerate) 4-... |
4cycl2vnunb 30012 | In a 4-cycle, two distinct... |
n4cyclfrgr 30013 | There is no 4-cycle in a f... |
4cyclusnfrgr 30014 | A graph with a 4-cycle is ... |
frgrnbnb 30015 | If two neighbors ` U ` and... |
frgrconngr 30016 | A friendship graph is conn... |
vdgn0frgrv2 30017 | A vertex in a friendship g... |
vdgn1frgrv2 30018 | Any vertex in a friendship... |
vdgn1frgrv3 30019 | Any vertex in a friendship... |
vdgfrgrgt2 30020 | Any vertex in a friendship... |
frgrncvvdeqlem1 30021 | Lemma 1 for ~ frgrncvvdeq ... |
frgrncvvdeqlem2 30022 | Lemma 2 for ~ frgrncvvdeq ... |
frgrncvvdeqlem3 30023 | Lemma 3 for ~ frgrncvvdeq ... |
frgrncvvdeqlem4 30024 | Lemma 4 for ~ frgrncvvdeq ... |
frgrncvvdeqlem5 30025 | Lemma 5 for ~ frgrncvvdeq ... |
frgrncvvdeqlem6 30026 | Lemma 6 for ~ frgrncvvdeq ... |
frgrncvvdeqlem7 30027 | Lemma 7 for ~ frgrncvvdeq ... |
frgrncvvdeqlem8 30028 | Lemma 8 for ~ frgrncvvdeq ... |
frgrncvvdeqlem9 30029 | Lemma 9 for ~ frgrncvvdeq ... |
frgrncvvdeqlem10 30030 | Lemma 10 for ~ frgrncvvdeq... |
frgrncvvdeq 30031 | In a friendship graph, two... |
frgrwopreglem4a 30032 | In a friendship graph any ... |
frgrwopreglem5a 30033 | If a friendship graph has ... |
frgrwopreglem1 30034 | Lemma 1 for ~ frgrwopreg :... |
frgrwopreglem2 30035 | Lemma 2 for ~ frgrwopreg .... |
frgrwopreglem3 30036 | Lemma 3 for ~ frgrwopreg .... |
frgrwopreglem4 30037 | Lemma 4 for ~ frgrwopreg .... |
frgrwopregasn 30038 | According to statement 5 i... |
frgrwopregbsn 30039 | According to statement 5 i... |
frgrwopreg1 30040 | According to statement 5 i... |
frgrwopreg2 30041 | According to statement 5 i... |
frgrwopreglem5lem 30042 | Lemma for ~ frgrwopreglem5... |
frgrwopreglem5 30043 | Lemma 5 for ~ frgrwopreg .... |
frgrwopreglem5ALT 30044 | Alternate direct proof of ... |
frgrwopreg 30045 | In a friendship graph ther... |
frgrregorufr0 30046 | In a friendship graph ther... |
frgrregorufr 30047 | If there is a vertex havin... |
frgrregorufrg 30048 | If there is a vertex havin... |
frgr2wwlkeu 30049 | For two different vertices... |
frgr2wwlkn0 30050 | In a friendship graph, the... |
frgr2wwlk1 30051 | In a friendship graph, the... |
frgr2wsp1 30052 | In a friendship graph, the... |
frgr2wwlkeqm 30053 | If there is a (simple) pat... |
frgrhash2wsp 30054 | The number of simple paths... |
fusgreg2wsplem 30055 | Lemma for ~ fusgreg2wsp an... |
fusgr2wsp2nb 30056 | The set of paths of length... |
fusgreghash2wspv 30057 | According to statement 7 i... |
fusgreg2wsp 30058 | In a finite simple graph, ... |
2wspmdisj 30059 | The sets of paths of lengt... |
fusgreghash2wsp 30060 | In a finite k-regular grap... |
frrusgrord0lem 30061 | Lemma for ~ frrusgrord0 . ... |
frrusgrord0 30062 | If a nonempty finite frien... |
frrusgrord 30063 | If a nonempty finite frien... |
numclwwlk2lem1lem 30064 | Lemma for ~ numclwwlk2lem1... |
2clwwlklem 30065 | Lemma for ~ clwwnonrepclww... |
clwwnrepclwwn 30066 | If the initial vertex of a... |
clwwnonrepclwwnon 30067 | If the initial vertex of a... |
2clwwlk2clwwlklem 30068 | Lemma for ~ 2clwwlk2clwwlk... |
2clwwlk 30069 | Value of operation ` C ` ,... |
2clwwlk2 30070 | The set ` ( X C 2 ) ` of d... |
2clwwlkel 30071 | Characterization of an ele... |
2clwwlk2clwwlk 30072 | An element of the value of... |
numclwwlk1lem2foalem 30073 | Lemma for ~ numclwwlk1lem2... |
extwwlkfab 30074 | The set ` ( X C N ) ` of d... |
extwwlkfabel 30075 | Characterization of an ele... |
numclwwlk1lem2foa 30076 | Going forth and back from ... |
numclwwlk1lem2f 30077 | ` T ` is a function, mappi... |
numclwwlk1lem2fv 30078 | Value of the function ` T ... |
numclwwlk1lem2f1 30079 | ` T ` is a 1-1 function. ... |
numclwwlk1lem2fo 30080 | ` T ` is an onto function.... |
numclwwlk1lem2f1o 30081 | ` T ` is a 1-1 onto functi... |
numclwwlk1lem2 30082 | The set of double loops of... |
numclwwlk1 30083 | Statement 9 in [Huneke] p.... |
clwwlknonclwlknonf1o 30084 | ` F ` is a bijection betwe... |
clwwlknonclwlknonen 30085 | The sets of the two repres... |
dlwwlknondlwlknonf1olem1 30086 | Lemma 1 for ~ dlwwlknondlw... |
dlwwlknondlwlknonf1o 30087 | ` F ` is a bijection betwe... |
dlwwlknondlwlknonen 30088 | The sets of the two repres... |
wlkl0 30089 | There is exactly one walk ... |
clwlknon2num 30090 | There are k walks of lengt... |
numclwlk1lem1 30091 | Lemma 1 for ~ numclwlk1 (S... |
numclwlk1lem2 30092 | Lemma 2 for ~ numclwlk1 (S... |
numclwlk1 30093 | Statement 9 in [Huneke] p.... |
numclwwlkovh0 30094 | Value of operation ` H ` ,... |
numclwwlkovh 30095 | Value of operation ` H ` ,... |
numclwwlkovq 30096 | Value of operation ` Q ` ,... |
numclwwlkqhash 30097 | In a ` K `-regular graph, ... |
numclwwlk2lem1 30098 | In a friendship graph, for... |
numclwlk2lem2f 30099 | ` R ` is a function mappin... |
numclwlk2lem2fv 30100 | Value of the function ` R ... |
numclwlk2lem2f1o 30101 | ` R ` is a 1-1 onto functi... |
numclwwlk2lem3 30102 | In a friendship graph, the... |
numclwwlk2 30103 | Statement 10 in [Huneke] p... |
numclwwlk3lem1 30104 | Lemma 2 for ~ numclwwlk3 .... |
numclwwlk3lem2lem 30105 | Lemma for ~ numclwwlk3lem2... |
numclwwlk3lem2 30106 | Lemma 1 for ~ numclwwlk3 :... |
numclwwlk3 30107 | Statement 12 in [Huneke] p... |
numclwwlk4 30108 | The total number of closed... |
numclwwlk5lem 30109 | Lemma for ~ numclwwlk5 . ... |
numclwwlk5 30110 | Statement 13 in [Huneke] p... |
numclwwlk7lem 30111 | Lemma for ~ numclwwlk7 , ~... |
numclwwlk6 30112 | For a prime divisor ` P ` ... |
numclwwlk7 30113 | Statement 14 in [Huneke] p... |
numclwwlk8 30114 | The size of the set of clo... |
frgrreggt1 30115 | If a finite nonempty frien... |
frgrreg 30116 | If a finite nonempty frien... |
frgrregord013 30117 | If a finite friendship gra... |
frgrregord13 30118 | If a nonempty finite frien... |
frgrogt3nreg 30119 | If a finite friendship gra... |
friendshipgt3 30120 | The friendship theorem for... |
friendship 30121 | The friendship theorem: I... |
conventions 30122 |
H... |
conventions-labels 30123 |
... |
conventions-comments 30124 |
... |
natded 30125 | Here are typical n... |
ex-natded5.2 30126 | Theorem 5.2 of [Clemente] ... |
ex-natded5.2-2 30127 | A more efficient proof of ... |
ex-natded5.2i 30128 | The same as ~ ex-natded5.2... |
ex-natded5.3 30129 | Theorem 5.3 of [Clemente] ... |
ex-natded5.3-2 30130 | A more efficient proof of ... |
ex-natded5.3i 30131 | The same as ~ ex-natded5.3... |
ex-natded5.5 30132 | Theorem 5.5 of [Clemente] ... |
ex-natded5.7 30133 | Theorem 5.7 of [Clemente] ... |
ex-natded5.7-2 30134 | A more efficient proof of ... |
ex-natded5.8 30135 | Theorem 5.8 of [Clemente] ... |
ex-natded5.8-2 30136 | A more efficient proof of ... |
ex-natded5.13 30137 | Theorem 5.13 of [Clemente]... |
ex-natded5.13-2 30138 | A more efficient proof of ... |
ex-natded9.20 30139 | Theorem 9.20 of [Clemente]... |
ex-natded9.20-2 30140 | A more efficient proof of ... |
ex-natded9.26 30141 | Theorem 9.26 of [Clemente]... |
ex-natded9.26-2 30142 | A more efficient proof of ... |
ex-or 30143 | Example for ~ df-or . Exa... |
ex-an 30144 | Example for ~ df-an . Exa... |
ex-dif 30145 | Example for ~ df-dif . Ex... |
ex-un 30146 | Example for ~ df-un . Exa... |
ex-in 30147 | Example for ~ df-in . Exa... |
ex-uni 30148 | Example for ~ df-uni . Ex... |
ex-ss 30149 | Example for ~ df-ss . Exa... |
ex-pss 30150 | Example for ~ df-pss . Ex... |
ex-pw 30151 | Example for ~ df-pw . Exa... |
ex-pr 30152 | Example for ~ df-pr . (Co... |
ex-br 30153 | Example for ~ df-br . Exa... |
ex-opab 30154 | Example for ~ df-opab . E... |
ex-eprel 30155 | Example for ~ df-eprel . ... |
ex-id 30156 | Example for ~ df-id . Exa... |
ex-po 30157 | Example for ~ df-po . Exa... |
ex-xp 30158 | Example for ~ df-xp . Exa... |
ex-cnv 30159 | Example for ~ df-cnv . Ex... |
ex-co 30160 | Example for ~ df-co . Exa... |
ex-dm 30161 | Example for ~ df-dm . Exa... |
ex-rn 30162 | Example for ~ df-rn . Exa... |
ex-res 30163 | Example for ~ df-res . Ex... |
ex-ima 30164 | Example for ~ df-ima . Ex... |
ex-fv 30165 | Example for ~ df-fv . Exa... |
ex-1st 30166 | Example for ~ df-1st . Ex... |
ex-2nd 30167 | Example for ~ df-2nd . Ex... |
1kp2ke3k 30168 | Example for ~ df-dec , 100... |
ex-fl 30169 | Example for ~ df-fl . Exa... |
ex-ceil 30170 | Example for ~ df-ceil . (... |
ex-mod 30171 | Example for ~ df-mod . (C... |
ex-exp 30172 | Example for ~ df-exp . (C... |
ex-fac 30173 | Example for ~ df-fac . (C... |
ex-bc 30174 | Example for ~ df-bc . (Co... |
ex-hash 30175 | Example for ~ df-hash . (... |
ex-sqrt 30176 | Example for ~ df-sqrt . (... |
ex-abs 30177 | Example for ~ df-abs . (C... |
ex-dvds 30178 | Example for ~ df-dvds : 3 ... |
ex-gcd 30179 | Example for ~ df-gcd . (C... |
ex-lcm 30180 | Example for ~ df-lcm . (C... |
ex-prmo 30181 | Example for ~ df-prmo : ` ... |
aevdemo 30182 | Proof illustrating the com... |
ex-ind-dvds 30183 | Example of a proof by indu... |
ex-fpar 30184 | Formalized example provide... |
avril1 30185 | Poisson d'Avril's Theorem.... |
2bornot2b 30186 | The law of excluded middle... |
helloworld 30187 | The classic "Hello world" ... |
1p1e2apr1 30188 | One plus one equals two. ... |
eqid1 30189 | Law of identity (reflexivi... |
1div0apr 30190 | Division by zero is forbid... |
topnfbey 30191 | Nothing seems to be imposs... |
9p10ne21 30192 | 9 + 10 is not equal to 21.... |
9p10ne21fool 30193 | 9 + 10 equals 21. This as... |
nrt2irr 30195 | The ` N ` -th root of 2 is... |
isplig 30198 | The predicate "is a planar... |
ispligb 30199 | The predicate "is a planar... |
tncp 30200 | In any planar incidence ge... |
l2p 30201 | For any line in a planar i... |
lpni 30202 | For any line in a planar i... |
nsnlplig 30203 | There is no "one-point lin... |
nsnlpligALT 30204 | Alternate version of ~ nsn... |
n0lplig 30205 | There is no "empty line" i... |
n0lpligALT 30206 | Alternate version of ~ n0l... |
eulplig 30207 | Through two distinct point... |
pliguhgr 30208 | Any planar incidence geome... |
dummylink 30209 | Alias for ~ a1ii that may ... |
id1 30210 | Alias for ~ idALT that may... |
isgrpo 30219 | The predicate "is a group ... |
isgrpoi 30220 | Properties that determine ... |
grpofo 30221 | A group operation maps ont... |
grpocl 30222 | Closure law for a group op... |
grpolidinv 30223 | A group has a left identit... |
grpon0 30224 | The base set of a group is... |
grpoass 30225 | A group operation is assoc... |
grpoidinvlem1 30226 | Lemma for ~ grpoidinv . (... |
grpoidinvlem2 30227 | Lemma for ~ grpoidinv . (... |
grpoidinvlem3 30228 | Lemma for ~ grpoidinv . (... |
grpoidinvlem4 30229 | Lemma for ~ grpoidinv . (... |
grpoidinv 30230 | A group has a left and rig... |
grpoideu 30231 | The left identity element ... |
grporndm 30232 | A group's range in terms o... |
0ngrp 30233 | The empty set is not a gro... |
gidval 30234 | The value of the identity ... |
grpoidval 30235 | Lemma for ~ grpoidcl and o... |
grpoidcl 30236 | The identity element of a ... |
grpoidinv2 30237 | A group's properties using... |
grpolid 30238 | The identity element of a ... |
grporid 30239 | The identity element of a ... |
grporcan 30240 | Right cancellation law for... |
grpoinveu 30241 | The left inverse element o... |
grpoid 30242 | Two ways of saying that an... |
grporn 30243 | The range of a group opera... |
grpoinvfval 30244 | The inverse function of a ... |
grpoinvval 30245 | The inverse of a group ele... |
grpoinvcl 30246 | A group element's inverse ... |
grpoinv 30247 | The properties of a group ... |
grpolinv 30248 | The left inverse of a grou... |
grporinv 30249 | The right inverse of a gro... |
grpoinvid1 30250 | The inverse of a group ele... |
grpoinvid2 30251 | The inverse of a group ele... |
grpolcan 30252 | Left cancellation law for ... |
grpo2inv 30253 | Double inverse law for gro... |
grpoinvf 30254 | Mapping of the inverse fun... |
grpoinvop 30255 | The inverse of the group o... |
grpodivfval 30256 | Group division (or subtrac... |
grpodivval 30257 | Group division (or subtrac... |
grpodivinv 30258 | Group division by an inver... |
grpoinvdiv 30259 | Inverse of a group divisio... |
grpodivf 30260 | Mapping for group division... |
grpodivcl 30261 | Closure of group division ... |
grpodivdiv 30262 | Double group division. (C... |
grpomuldivass 30263 | Associative-type law for m... |
grpodivid 30264 | Division of a group member... |
grponpcan 30265 | Cancellation law for group... |
isablo 30268 | The predicate "is an Abeli... |
ablogrpo 30269 | An Abelian group operation... |
ablocom 30270 | An Abelian group operation... |
ablo32 30271 | Commutative/associative la... |
ablo4 30272 | Commutative/associative la... |
isabloi 30273 | Properties that determine ... |
ablomuldiv 30274 | Law for group multiplicati... |
ablodivdiv 30275 | Law for double group divis... |
ablodivdiv4 30276 | Law for double group divis... |
ablodiv32 30277 | Swap the second and third ... |
ablonncan 30278 | Cancellation law for group... |
ablonnncan1 30279 | Cancellation law for group... |
vcrel 30282 | The class of all complex v... |
vciOLD 30283 | Obsolete version of ~ cvsi... |
vcsm 30284 | Functionality of th scalar... |
vccl 30285 | Closure of the scalar prod... |
vcidOLD 30286 | Identity element for the s... |
vcdi 30287 | Distributive law for the s... |
vcdir 30288 | Distributive law for the s... |
vcass 30289 | Associative law for the sc... |
vc2OLD 30290 | A vector plus itself is tw... |
vcablo 30291 | Vector addition is an Abel... |
vcgrp 30292 | Vector addition is a group... |
vclcan 30293 | Left cancellation law for ... |
vczcl 30294 | The zero vector is a vecto... |
vc0rid 30295 | The zero vector is a right... |
vc0 30296 | Zero times a vector is the... |
vcz 30297 | Anything times the zero ve... |
vcm 30298 | Minus 1 times a vector is ... |
isvclem 30299 | Lemma for ~ isvcOLD . (Co... |
vcex 30300 | The components of a comple... |
isvcOLD 30301 | The predicate "is a comple... |
isvciOLD 30302 | Properties that determine ... |
cnaddabloOLD 30303 | Obsolete version of ~ cnad... |
cnidOLD 30304 | Obsolete version of ~ cnad... |
cncvcOLD 30305 | Obsolete version of ~ cncv... |
nvss 30315 | Structure of the class of ... |
nvvcop 30316 | A normed complex vector sp... |
nvrel 30324 | The class of all normed co... |
vafval 30325 | Value of the function for ... |
bafval 30326 | Value of the function for ... |
smfval 30327 | Value of the function for ... |
0vfval 30328 | Value of the function for ... |
nmcvfval 30329 | Value of the norm function... |
nvop2 30330 | A normed complex vector sp... |
nvvop 30331 | The vector space component... |
isnvlem 30332 | Lemma for ~ isnv . (Contr... |
nvex 30333 | The components of a normed... |
isnv 30334 | The predicate "is a normed... |
isnvi 30335 | Properties that determine ... |
nvi 30336 | The properties of a normed... |
nvvc 30337 | The vector space component... |
nvablo 30338 | The vector addition operat... |
nvgrp 30339 | The vector addition operat... |
nvgf 30340 | Mapping for the vector add... |
nvsf 30341 | Mapping for the scalar mul... |
nvgcl 30342 | Closure law for the vector... |
nvcom 30343 | The vector addition (group... |
nvass 30344 | The vector addition (group... |
nvadd32 30345 | Commutative/associative la... |
nvrcan 30346 | Right cancellation law for... |
nvadd4 30347 | Rearrangement of 4 terms i... |
nvscl 30348 | Closure law for the scalar... |
nvsid 30349 | Identity element for the s... |
nvsass 30350 | Associative law for the sc... |
nvscom 30351 | Commutative law for the sc... |
nvdi 30352 | Distributive law for the s... |
nvdir 30353 | Distributive law for the s... |
nv2 30354 | A vector plus itself is tw... |
vsfval 30355 | Value of the function for ... |
nvzcl 30356 | Closure law for the zero v... |
nv0rid 30357 | The zero vector is a right... |
nv0lid 30358 | The zero vector is a left ... |
nv0 30359 | Zero times a vector is the... |
nvsz 30360 | Anything times the zero ve... |
nvinv 30361 | Minus 1 times a vector is ... |
nvinvfval 30362 | Function for the negative ... |
nvm 30363 | Vector subtraction in term... |
nvmval 30364 | Value of vector subtractio... |
nvmval2 30365 | Value of vector subtractio... |
nvmfval 30366 | Value of the function for ... |
nvmf 30367 | Mapping for the vector sub... |
nvmcl 30368 | Closure law for the vector... |
nvnnncan1 30369 | Cancellation law for vecto... |
nvmdi 30370 | Distributive law for scala... |
nvnegneg 30371 | Double negative of a vecto... |
nvmul0or 30372 | If a scalar product is zer... |
nvrinv 30373 | A vector minus itself. (C... |
nvlinv 30374 | Minus a vector plus itself... |
nvpncan2 30375 | Cancellation law for vecto... |
nvpncan 30376 | Cancellation law for vecto... |
nvaddsub 30377 | Commutative/associative la... |
nvnpcan 30378 | Cancellation law for a nor... |
nvaddsub4 30379 | Rearrangement of 4 terms i... |
nvmeq0 30380 | The difference between two... |
nvmid 30381 | A vector minus itself is t... |
nvf 30382 | Mapping for the norm funct... |
nvcl 30383 | The norm of a normed compl... |
nvcli 30384 | The norm of a normed compl... |
nvs 30385 | Proportionality property o... |
nvsge0 30386 | The norm of a scalar produ... |
nvm1 30387 | The norm of the negative o... |
nvdif 30388 | The norm of the difference... |
nvpi 30389 | The norm of a vector plus ... |
nvz0 30390 | The norm of a zero vector ... |
nvz 30391 | The norm of a vector is ze... |
nvtri 30392 | Triangle inequality for th... |
nvmtri 30393 | Triangle inequality for th... |
nvabs 30394 | Norm difference property o... |
nvge0 30395 | The norm of a normed compl... |
nvgt0 30396 | A nonzero norm is positive... |
nv1 30397 | From any nonzero vector, c... |
nvop 30398 | A complex inner product sp... |
cnnv 30399 | The set of complex numbers... |
cnnvg 30400 | The vector addition (group... |
cnnvba 30401 | The base set of the normed... |
cnnvs 30402 | The scalar product operati... |
cnnvnm 30403 | The norm operation of the ... |
cnnvm 30404 | The vector subtraction ope... |
elimnv 30405 | Hypothesis elimination lem... |
elimnvu 30406 | Hypothesis elimination lem... |
imsval 30407 | Value of the induced metri... |
imsdval 30408 | Value of the induced metri... |
imsdval2 30409 | Value of the distance func... |
nvnd 30410 | The norm of a normed compl... |
imsdf 30411 | Mapping for the induced me... |
imsmetlem 30412 | Lemma for ~ imsmet . (Con... |
imsmet 30413 | The induced metric of a no... |
imsxmet 30414 | The induced metric of a no... |
cnims 30415 | The metric induced on the ... |
vacn 30416 | Vector addition is jointly... |
nmcvcn 30417 | The norm of a normed compl... |
nmcnc 30418 | The norm of a normed compl... |
smcnlem 30419 | Lemma for ~ smcn . (Contr... |
smcn 30420 | Scalar multiplication is j... |
vmcn 30421 | Vector subtraction is join... |
dipfval 30424 | The inner product function... |
ipval 30425 | Value of the inner product... |
ipval2lem2 30426 | Lemma for ~ ipval3 . (Con... |
ipval2lem3 30427 | Lemma for ~ ipval3 . (Con... |
ipval2lem4 30428 | Lemma for ~ ipval3 . (Con... |
ipval2 30429 | Expansion of the inner pro... |
4ipval2 30430 | Four times the inner produ... |
ipval3 30431 | Expansion of the inner pro... |
ipidsq 30432 | The inner product of a vec... |
ipnm 30433 | Norm expressed in terms of... |
dipcl 30434 | An inner product is a comp... |
ipf 30435 | Mapping for the inner prod... |
dipcj 30436 | The complex conjugate of a... |
ipipcj 30437 | An inner product times its... |
diporthcom 30438 | Orthogonality (meaning inn... |
dip0r 30439 | Inner product with a zero ... |
dip0l 30440 | Inner product with a zero ... |
ipz 30441 | The inner product of a vec... |
dipcn 30442 | Inner product is jointly c... |
sspval 30445 | The set of all subspaces o... |
isssp 30446 | The predicate "is a subspa... |
sspid 30447 | A normed complex vector sp... |
sspnv 30448 | A subspace is a normed com... |
sspba 30449 | The base set of a subspace... |
sspg 30450 | Vector addition on a subsp... |
sspgval 30451 | Vector addition on a subsp... |
ssps 30452 | Scalar multiplication on a... |
sspsval 30453 | Scalar multiplication on a... |
sspmlem 30454 | Lemma for ~ sspm and other... |
sspmval 30455 | Vector addition on a subsp... |
sspm 30456 | Vector subtraction on a su... |
sspz 30457 | The zero vector of a subsp... |
sspn 30458 | The norm on a subspace is ... |
sspnval 30459 | The norm on a subspace in ... |
sspimsval 30460 | The induced metric on a su... |
sspims 30461 | The induced metric on a su... |
lnoval 30474 | The set of linear operator... |
islno 30475 | The predicate "is a linear... |
lnolin 30476 | Basic linearity property o... |
lnof 30477 | A linear operator is a map... |
lno0 30478 | The value of a linear oper... |
lnocoi 30479 | The composition of two lin... |
lnoadd 30480 | Addition property of a lin... |
lnosub 30481 | Subtraction property of a ... |
lnomul 30482 | Scalar multiplication prop... |
nvo00 30483 | Two ways to express a zero... |
nmoofval 30484 | The operator norm function... |
nmooval 30485 | The operator norm function... |
nmosetre 30486 | The set in the supremum of... |
nmosetn0 30487 | The set in the supremum of... |
nmoxr 30488 | The norm of an operator is... |
nmooge0 30489 | The norm of an operator is... |
nmorepnf 30490 | The norm of an operator is... |
nmoreltpnf 30491 | The norm of any operator i... |
nmogtmnf 30492 | The norm of an operator is... |
nmoolb 30493 | A lower bound for an opera... |
nmoubi 30494 | An upper bound for an oper... |
nmoub3i 30495 | An upper bound for an oper... |
nmoub2i 30496 | An upper bound for an oper... |
nmobndi 30497 | Two ways to express that a... |
nmounbi 30498 | Two ways two express that ... |
nmounbseqi 30499 | An unbounded operator dete... |
nmounbseqiALT 30500 | Alternate shorter proof of... |
nmobndseqi 30501 | A bounded sequence determi... |
nmobndseqiALT 30502 | Alternate shorter proof of... |
bloval 30503 | The class of bounded linea... |
isblo 30504 | The predicate "is a bounde... |
isblo2 30505 | The predicate "is a bounde... |
bloln 30506 | A bounded operator is a li... |
blof 30507 | A bounded operator is an o... |
nmblore 30508 | The norm of a bounded oper... |
0ofval 30509 | The zero operator between ... |
0oval 30510 | Value of the zero operator... |
0oo 30511 | The zero operator is an op... |
0lno 30512 | The zero operator is linea... |
nmoo0 30513 | The operator norm of the z... |
0blo 30514 | The zero operator is a bou... |
nmlno0lem 30515 | Lemma for ~ nmlno0i . (Co... |
nmlno0i 30516 | The norm of a linear opera... |
nmlno0 30517 | The norm of a linear opera... |
nmlnoubi 30518 | An upper bound for the ope... |
nmlnogt0 30519 | The norm of a nonzero line... |
lnon0 30520 | The domain of a nonzero li... |
nmblolbii 30521 | A lower bound for the norm... |
nmblolbi 30522 | A lower bound for the norm... |
isblo3i 30523 | The predicate "is a bounde... |
blo3i 30524 | Properties that determine ... |
blometi 30525 | Upper bound for the distan... |
blocnilem 30526 | Lemma for ~ blocni and ~ l... |
blocni 30527 | A linear operator is conti... |
lnocni 30528 | If a linear operator is co... |
blocn 30529 | A linear operator is conti... |
blocn2 30530 | A bounded linear operator ... |
ajfval 30531 | The adjoint function. (Co... |
hmoval 30532 | The set of Hermitian (self... |
ishmo 30533 | The predicate "is a hermit... |
phnv 30536 | Every complex inner produc... |
phrel 30537 | The class of all complex i... |
phnvi 30538 | Every complex inner produc... |
isphg 30539 | The predicate "is a comple... |
phop 30540 | A complex inner product sp... |
cncph 30541 | The set of complex numbers... |
elimph 30542 | Hypothesis elimination lem... |
elimphu 30543 | Hypothesis elimination lem... |
isph 30544 | The predicate "is an inner... |
phpar2 30545 | The parallelogram law for ... |
phpar 30546 | The parallelogram law for ... |
ip0i 30547 | A slight variant of Equati... |
ip1ilem 30548 | Lemma for ~ ip1i . (Contr... |
ip1i 30549 | Equation 6.47 of [Ponnusam... |
ip2i 30550 | Equation 6.48 of [Ponnusam... |
ipdirilem 30551 | Lemma for ~ ipdiri . (Con... |
ipdiri 30552 | Distributive law for inner... |
ipasslem1 30553 | Lemma for ~ ipassi . Show... |
ipasslem2 30554 | Lemma for ~ ipassi . Show... |
ipasslem3 30555 | Lemma for ~ ipassi . Show... |
ipasslem4 30556 | Lemma for ~ ipassi . Show... |
ipasslem5 30557 | Lemma for ~ ipassi . Show... |
ipasslem7 30558 | Lemma for ~ ipassi . Show... |
ipasslem8 30559 | Lemma for ~ ipassi . By ~... |
ipasslem9 30560 | Lemma for ~ ipassi . Conc... |
ipasslem10 30561 | Lemma for ~ ipassi . Show... |
ipasslem11 30562 | Lemma for ~ ipassi . Show... |
ipassi 30563 | Associative law for inner ... |
dipdir 30564 | Distributive law for inner... |
dipdi 30565 | Distributive law for inner... |
ip2dii 30566 | Inner product of two sums.... |
dipass 30567 | Associative law for inner ... |
dipassr 30568 | "Associative" law for seco... |
dipassr2 30569 | "Associative" law for inne... |
dipsubdir 30570 | Distributive law for inner... |
dipsubdi 30571 | Distributive law for inner... |
pythi 30572 | The Pythagorean theorem fo... |
siilem1 30573 | Lemma for ~ sii . (Contri... |
siilem2 30574 | Lemma for ~ sii . (Contri... |
siii 30575 | Inference from ~ sii . (C... |
sii 30576 | Obsolete version of ~ ipca... |
ipblnfi 30577 | A function ` F ` generated... |
ip2eqi 30578 | Two vectors are equal iff ... |
phoeqi 30579 | A condition implying that ... |
ajmoi 30580 | Every operator has at most... |
ajfuni 30581 | The adjoint function is a ... |
ajfun 30582 | The adjoint function is a ... |
ajval 30583 | Value of the adjoint funct... |
iscbn 30586 | A complex Banach space is ... |
cbncms 30587 | The induced metric on comp... |
bnnv 30588 | Every complex Banach space... |
bnrel 30589 | The class of all complex B... |
bnsscmcl 30590 | A subspace of a Banach spa... |
cnbn 30591 | The set of complex numbers... |
ubthlem1 30592 | Lemma for ~ ubth . The fu... |
ubthlem2 30593 | Lemma for ~ ubth . Given ... |
ubthlem3 30594 | Lemma for ~ ubth . Prove ... |
ubth 30595 | Uniform Boundedness Theore... |
minvecolem1 30596 | Lemma for ~ minveco . The... |
minvecolem2 30597 | Lemma for ~ minveco . Any... |
minvecolem3 30598 | Lemma for ~ minveco . The... |
minvecolem4a 30599 | Lemma for ~ minveco . ` F ... |
minvecolem4b 30600 | Lemma for ~ minveco . The... |
minvecolem4c 30601 | Lemma for ~ minveco . The... |
minvecolem4 30602 | Lemma for ~ minveco . The... |
minvecolem5 30603 | Lemma for ~ minveco . Dis... |
minvecolem6 30604 | Lemma for ~ minveco . Any... |
minvecolem7 30605 | Lemma for ~ minveco . Sin... |
minveco 30606 | Minimizing vector theorem,... |
ishlo 30609 | The predicate "is a comple... |
hlobn 30610 | Every complex Hilbert spac... |
hlph 30611 | Every complex Hilbert spac... |
hlrel 30612 | The class of all complex H... |
hlnv 30613 | Every complex Hilbert spac... |
hlnvi 30614 | Every complex Hilbert spac... |
hlvc 30615 | Every complex Hilbert spac... |
hlcmet 30616 | The induced metric on a co... |
hlmet 30617 | The induced metric on a co... |
hlpar2 30618 | The parallelogram law sati... |
hlpar 30619 | The parallelogram law sati... |
hlex 30620 | The base set of a Hilbert ... |
hladdf 30621 | Mapping for Hilbert space ... |
hlcom 30622 | Hilbert space vector addit... |
hlass 30623 | Hilbert space vector addit... |
hl0cl 30624 | The Hilbert space zero vec... |
hladdid 30625 | Hilbert space addition wit... |
hlmulf 30626 | Mapping for Hilbert space ... |
hlmulid 30627 | Hilbert space scalar multi... |
hlmulass 30628 | Hilbert space scalar multi... |
hldi 30629 | Hilbert space scalar multi... |
hldir 30630 | Hilbert space scalar multi... |
hlmul0 30631 | Hilbert space scalar multi... |
hlipf 30632 | Mapping for Hilbert space ... |
hlipcj 30633 | Conjugate law for Hilbert ... |
hlipdir 30634 | Distributive law for Hilbe... |
hlipass 30635 | Associative law for Hilber... |
hlipgt0 30636 | The inner product of a Hil... |
hlcompl 30637 | Completeness of a Hilbert ... |
cnchl 30638 | The set of complex numbers... |
htthlem 30639 | Lemma for ~ htth . The co... |
htth 30640 | Hellinger-Toeplitz Theorem... |
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
h2hva 30696 | The group (addition) opera... |
h2hsm 30697 | The scalar product operati... |
h2hnm 30698 | The norm function of Hilbe... |
h2hvs 30699 | The vector subtraction ope... |
h2hmetdval 30700 | Value of the distance func... |
h2hcau 30701 | The Cauchy sequences of Hi... |
h2hlm 30702 | The limit sequences of Hil... |
axhilex-zf 30703 | Derive Axiom ~ ax-hilex fr... |
axhfvadd-zf 30704 | Derive Axiom ~ ax-hfvadd f... |
axhvcom-zf 30705 | Derive Axiom ~ ax-hvcom fr... |
axhvass-zf 30706 | Derive Axiom ~ ax-hvass fr... |
axhv0cl-zf 30707 | Derive Axiom ~ ax-hv0cl fr... |
axhvaddid-zf 30708 | Derive Axiom ~ ax-hvaddid ... |
axhfvmul-zf 30709 | Derive Axiom ~ ax-hfvmul f... |
axhvmulid-zf 30710 | Derive Axiom ~ ax-hvmulid ... |
axhvmulass-zf 30711 | Derive Axiom ~ ax-hvmulass... |
axhvdistr1-zf 30712 | Derive Axiom ~ ax-hvdistr1... |
axhvdistr2-zf 30713 | Derive Axiom ~ ax-hvdistr2... |
axhvmul0-zf 30714 | Derive Axiom ~ ax-hvmul0 f... |
axhfi-zf 30715 | Derive Axiom ~ ax-hfi from... |
axhis1-zf 30716 | Derive Axiom ~ ax-his1 fro... |
axhis2-zf 30717 | Derive Axiom ~ ax-his2 fro... |
axhis3-zf 30718 | Derive Axiom ~ ax-his3 fro... |
axhis4-zf 30719 | Derive Axiom ~ ax-his4 fro... |
axhcompl-zf 30720 | Derive Axiom ~ ax-hcompl f... |
hvmulex 30733 | The Hilbert space scalar p... |
hvaddcl 30734 | Closure of vector addition... |
hvmulcl 30735 | Closure of scalar multipli... |
hvmulcli 30736 | Closure inference for scal... |
hvsubf 30737 | Mapping domain and codomai... |
hvsubval 30738 | Value of vector subtractio... |
hvsubcl 30739 | Closure of vector subtract... |
hvaddcli 30740 | Closure of vector addition... |
hvcomi 30741 | Commutation of vector addi... |
hvsubvali 30742 | Value of vector subtractio... |
hvsubcli 30743 | Closure of vector subtract... |
ifhvhv0 30744 | Prove ` if ( A e. ~H , A ,... |
hvaddlid 30745 | Addition with the zero vec... |
hvmul0 30746 | Scalar multiplication with... |
hvmul0or 30747 | If a scalar product is zer... |
hvsubid 30748 | Subtraction of a vector fr... |
hvnegid 30749 | Addition of negative of a ... |
hv2neg 30750 | Two ways to express the ne... |
hvaddlidi 30751 | Addition with the zero vec... |
hvnegidi 30752 | Addition of negative of a ... |
hv2negi 30753 | Two ways to express the ne... |
hvm1neg 30754 | Convert minus one times a ... |
hvaddsubval 30755 | Value of vector addition i... |
hvadd32 30756 | Commutative/associative la... |
hvadd12 30757 | Commutative/associative la... |
hvadd4 30758 | Hilbert vector space addit... |
hvsub4 30759 | Hilbert vector space addit... |
hvaddsub12 30760 | Commutative/associative la... |
hvpncan 30761 | Addition/subtraction cance... |
hvpncan2 30762 | Addition/subtraction cance... |
hvaddsubass 30763 | Associativity of sum and d... |
hvpncan3 30764 | Subtraction and addition o... |
hvmulcom 30765 | Scalar multiplication comm... |
hvsubass 30766 | Hilbert vector space assoc... |
hvsub32 30767 | Hilbert vector space commu... |
hvmulassi 30768 | Scalar multiplication asso... |
hvmulcomi 30769 | Scalar multiplication comm... |
hvmul2negi 30770 | Double negative in scalar ... |
hvsubdistr1 30771 | Scalar multiplication dist... |
hvsubdistr2 30772 | Scalar multiplication dist... |
hvdistr1i 30773 | Scalar multiplication dist... |
hvsubdistr1i 30774 | Scalar multiplication dist... |
hvassi 30775 | Hilbert vector space assoc... |
hvadd32i 30776 | Hilbert vector space commu... |
hvsubassi 30777 | Hilbert vector space assoc... |
hvsub32i 30778 | Hilbert vector space commu... |
hvadd12i 30779 | Hilbert vector space commu... |
hvadd4i 30780 | Hilbert vector space addit... |
hvsubsub4i 30781 | Hilbert vector space addit... |
hvsubsub4 30782 | Hilbert vector space addit... |
hv2times 30783 | Two times a vector. (Cont... |
hvnegdii 30784 | Distribution of negative o... |
hvsubeq0i 30785 | If the difference between ... |
hvsubcan2i 30786 | Vector cancellation law. ... |
hvaddcani 30787 | Cancellation law for vecto... |
hvsubaddi 30788 | Relationship between vecto... |
hvnegdi 30789 | Distribution of negative o... |
hvsubeq0 30790 | If the difference between ... |
hvaddeq0 30791 | If the sum of two vectors ... |
hvaddcan 30792 | Cancellation law for vecto... |
hvaddcan2 30793 | Cancellation law for vecto... |
hvmulcan 30794 | Cancellation law for scala... |
hvmulcan2 30795 | Cancellation law for scala... |
hvsubcan 30796 | Cancellation law for vecto... |
hvsubcan2 30797 | Cancellation law for vecto... |
hvsub0 30798 | Subtraction of a zero vect... |
hvsubadd 30799 | Relationship between vecto... |
hvaddsub4 30800 | Hilbert vector space addit... |
hicl 30802 | Closure of inner product. ... |
hicli 30803 | Closure inference for inne... |
his5 30808 | Associative law for inner ... |
his52 30809 | Associative law for inner ... |
his35 30810 | Move scalar multiplication... |
his35i 30811 | Move scalar multiplication... |
his7 30812 | Distributive law for inner... |
hiassdi 30813 | Distributive/associative l... |
his2sub 30814 | Distributive law for inner... |
his2sub2 30815 | Distributive law for inner... |
hire 30816 | A necessary and sufficient... |
hiidrcl 30817 | Real closure of inner prod... |
hi01 30818 | Inner product with the 0 v... |
hi02 30819 | Inner product with the 0 v... |
hiidge0 30820 | Inner product with self is... |
his6 30821 | Zero inner product with se... |
his1i 30822 | Conjugate law for inner pr... |
abshicom 30823 | Commuted inner products ha... |
hial0 30824 | A vector whose inner produ... |
hial02 30825 | A vector whose inner produ... |
hisubcomi 30826 | Two vector subtractions si... |
hi2eq 30827 | Lemma used to prove equali... |
hial2eq 30828 | Two vectors whose inner pr... |
hial2eq2 30829 | Two vectors whose inner pr... |
orthcom 30830 | Orthogonality commutes. (... |
normlem0 30831 | Lemma used to derive prope... |
normlem1 30832 | Lemma used to derive prope... |
normlem2 30833 | Lemma used to derive prope... |
normlem3 30834 | Lemma used to derive prope... |
normlem4 30835 | Lemma used to derive prope... |
normlem5 30836 | Lemma used to derive prope... |
normlem6 30837 | Lemma used to derive prope... |
normlem7 30838 | Lemma used to derive prope... |
normlem8 30839 | Lemma used to derive prope... |
normlem9 30840 | Lemma used to derive prope... |
normlem7tALT 30841 | Lemma used to derive prope... |
bcseqi 30842 | Equality case of Bunjakova... |
normlem9at 30843 | Lemma used to derive prope... |
dfhnorm2 30844 | Alternate definition of th... |
normf 30845 | The norm function maps fro... |
normval 30846 | The value of the norm of a... |
normcl 30847 | Real closure of the norm o... |
normge0 30848 | The norm of a vector is no... |
normgt0 30849 | The norm of nonzero vector... |
norm0 30850 | The norm of a zero vector.... |
norm-i 30851 | Theorem 3.3(i) of [Beran] ... |
normne0 30852 | A norm is nonzero iff its ... |
normcli 30853 | Real closure of the norm o... |
normsqi 30854 | The square of a norm. (Co... |
norm-i-i 30855 | Theorem 3.3(i) of [Beran] ... |
normsq 30856 | The square of a norm. (Co... |
normsub0i 30857 | Two vectors are equal iff ... |
normsub0 30858 | Two vectors are equal iff ... |
norm-ii-i 30859 | Triangle inequality for no... |
norm-ii 30860 | Triangle inequality for no... |
norm-iii-i 30861 | Theorem 3.3(iii) of [Beran... |
norm-iii 30862 | Theorem 3.3(iii) of [Beran... |
normsubi 30863 | Negative doesn't change th... |
normpythi 30864 | Analogy to Pythagorean the... |
normsub 30865 | Swapping order of subtract... |
normneg 30866 | The norm of a vector equal... |
normpyth 30867 | Analogy to Pythagorean the... |
normpyc 30868 | Corollary to Pythagorean t... |
norm3difi 30869 | Norm of differences around... |
norm3adifii 30870 | Norm of differences around... |
norm3lem 30871 | Lemma involving norm of di... |
norm3dif 30872 | Norm of differences around... |
norm3dif2 30873 | Norm of differences around... |
norm3lemt 30874 | Lemma involving norm of di... |
norm3adifi 30875 | Norm of differences around... |
normpari 30876 | Parallelogram law for norm... |
normpar 30877 | Parallelogram law for norm... |
normpar2i 30878 | Corollary of parallelogram... |
polid2i 30879 | Generalized polarization i... |
polidi 30880 | Polarization identity. Re... |
polid 30881 | Polarization identity. Re... |
hilablo 30882 | Hilbert space vector addit... |
hilid 30883 | The group identity element... |
hilvc 30884 | Hilbert space is a complex... |
hilnormi 30885 | Hilbert space norm in term... |
hilhhi 30886 | Deduce the structure of Hi... |
hhnv 30887 | Hilbert space is a normed ... |
hhva 30888 | The group (addition) opera... |
hhba 30889 | The base set of Hilbert sp... |
hh0v 30890 | The zero vector of Hilbert... |
hhsm 30891 | The scalar product operati... |
hhvs 30892 | The vector subtraction ope... |
hhnm 30893 | The norm function of Hilbe... |
hhims 30894 | The induced metric of Hilb... |
hhims2 30895 | Hilbert space distance met... |
hhmet 30896 | The induced metric of Hilb... |
hhxmet 30897 | The induced metric of Hilb... |
hhmetdval 30898 | Value of the distance func... |
hhip 30899 | The inner product operatio... |
hhph 30900 | The Hilbert space of the H... |
bcsiALT 30901 | Bunjakovaskij-Cauchy-Schwa... |
bcsiHIL 30902 | Bunjakovaskij-Cauchy-Schwa... |
bcs 30903 | Bunjakovaskij-Cauchy-Schwa... |
bcs2 30904 | Corollary of the Bunjakova... |
bcs3 30905 | Corollary of the Bunjakova... |
hcau 30906 | Member of the set of Cauch... |
hcauseq 30907 | A Cauchy sequences on a Hi... |
hcaucvg 30908 | A Cauchy sequence on a Hil... |
seq1hcau 30909 | A sequence on a Hilbert sp... |
hlimi 30910 | Express the predicate: Th... |
hlimseqi 30911 | A sequence with a limit on... |
hlimveci 30912 | Closure of the limit of a ... |
hlimconvi 30913 | Convergence of a sequence ... |
hlim2 30914 | The limit of a sequence on... |
hlimadd 30915 | Limit of the sum of two se... |
hilmet 30916 | The Hilbert space norm det... |
hilxmet 30917 | The Hilbert space norm det... |
hilmetdval 30918 | Value of the distance func... |
hilims 30919 | Hilbert space distance met... |
hhcau 30920 | The Cauchy sequences of Hi... |
hhlm 30921 | The limit sequences of Hil... |
hhcmpl 30922 | Lemma used for derivation ... |
hilcompl 30923 | Lemma used for derivation ... |
hhcms 30925 | The Hilbert space induced ... |
hhhl 30926 | The Hilbert space structur... |
hilcms 30927 | The Hilbert space norm det... |
hilhl 30928 | The Hilbert space of the H... |
issh 30930 | Subspace ` H ` of a Hilber... |
issh2 30931 | Subspace ` H ` of a Hilber... |
shss 30932 | A subspace is a subset of ... |
shel 30933 | A member of a subspace of ... |
shex 30934 | The set of subspaces of a ... |
shssii 30935 | A closed subspace of a Hil... |
sheli 30936 | A member of a subspace of ... |
shelii 30937 | A member of a subspace of ... |
sh0 30938 | The zero vector belongs to... |
shaddcl 30939 | Closure of vector addition... |
shmulcl 30940 | Closure of vector scalar m... |
issh3 30941 | Subspace ` H ` of a Hilber... |
shsubcl 30942 | Closure of vector subtract... |
isch 30944 | Closed subspace ` H ` of a... |
isch2 30945 | Closed subspace ` H ` of a... |
chsh 30946 | A closed subspace is a sub... |
chsssh 30947 | Closed subspaces are subsp... |
chex 30948 | The set of closed subspace... |
chshii 30949 | A closed subspace is a sub... |
ch0 30950 | The zero vector belongs to... |
chss 30951 | A closed subspace of a Hil... |
chel 30952 | A member of a closed subsp... |
chssii 30953 | A closed subspace of a Hil... |
cheli 30954 | A member of a closed subsp... |
chelii 30955 | A member of a closed subsp... |
chlimi 30956 | The limit property of a cl... |
hlim0 30957 | The zero sequence in Hilbe... |
hlimcaui 30958 | If a sequence in Hilbert s... |
hlimf 30959 | Function-like behavior of ... |
hlimuni 30960 | A Hilbert space sequence c... |
hlimreui 30961 | The limit of a Hilbert spa... |
hlimeui 30962 | The limit of a Hilbert spa... |
isch3 30963 | A Hilbert subspace is clos... |
chcompl 30964 | Completeness of a closed s... |
helch 30965 | The Hilbert lattice one (w... |
ifchhv 30966 | Prove ` if ( A e. CH , A ,... |
helsh 30967 | Hilbert space is a subspac... |
shsspwh 30968 | Subspaces are subsets of H... |
chsspwh 30969 | Closed subspaces are subse... |
hsn0elch 30970 | The zero subspace belongs ... |
norm1 30971 | From any nonzero Hilbert s... |
norm1exi 30972 | A normalized vector exists... |
norm1hex 30973 | A normalized vector can ex... |
elch0 30976 | Membership in zero for clo... |
h0elch 30977 | The zero subspace is a clo... |
h0elsh 30978 | The zero subspace is a sub... |
hhssva 30979 | The vector addition operat... |
hhsssm 30980 | The scalar multiplication ... |
hhssnm 30981 | The norm operation on a su... |
issubgoilem 30982 | Lemma for ~ hhssabloilem .... |
hhssabloilem 30983 | Lemma for ~ hhssabloi . F... |
hhssabloi 30984 | Abelian group property of ... |
hhssablo 30985 | Abelian group property of ... |
hhssnv 30986 | Normed complex vector spac... |
hhssnvt 30987 | Normed complex vector spac... |
hhsst 30988 | A member of ` SH ` is a su... |
hhshsslem1 30989 | Lemma for ~ hhsssh . (Con... |
hhshsslem2 30990 | Lemma for ~ hhsssh . (Con... |
hhsssh 30991 | The predicate " ` H ` is a... |
hhsssh2 30992 | The predicate " ` H ` is a... |
hhssba 30993 | The base set of a subspace... |
hhssvs 30994 | The vector subtraction ope... |
hhssvsf 30995 | Mapping of the vector subt... |
hhssims 30996 | Induced metric of a subspa... |
hhssims2 30997 | Induced metric of a subspa... |
hhssmet 30998 | Induced metric of a subspa... |
hhssmetdval 30999 | Value of the distance func... |
hhsscms 31000 | The induced metric of a cl... |
hhssbnOLD 31001 | Obsolete version of ~ cssb... |
ocval 31002 | Value of orthogonal comple... |
ocel 31003 | Membership in orthogonal c... |
shocel 31004 | Membership in orthogonal c... |
ocsh 31005 | The orthogonal complement ... |
shocsh 31006 | The orthogonal complement ... |
ocss 31007 | An orthogonal complement i... |
shocss 31008 | An orthogonal complement i... |
occon 31009 | Contraposition law for ort... |
occon2 31010 | Double contraposition for ... |
occon2i 31011 | Double contraposition for ... |
oc0 31012 | The zero vector belongs to... |
ocorth 31013 | Members of a subset and it... |
shocorth 31014 | Members of a subspace and ... |
ococss 31015 | Inclusion in complement of... |
shococss 31016 | Inclusion in complement of... |
shorth 31017 | Members of orthogonal subs... |
ocin 31018 | Intersection of a Hilbert ... |
occon3 31019 | Hilbert lattice contraposi... |
ocnel 31020 | A nonzero vector in the co... |
chocvali 31021 | Value of the orthogonal co... |
shuni 31022 | Two subspaces with trivial... |
chocunii 31023 | Lemma for uniqueness part ... |
pjhthmo 31024 | Projection Theorem, unique... |
occllem 31025 | Lemma for ~ occl . (Contr... |
occl 31026 | Closure of complement of H... |
shoccl 31027 | Closure of complement of H... |
choccl 31028 | Closure of complement of H... |
choccli 31029 | Closure of ` CH ` orthocom... |
shsval 31034 | Value of subspace sum of t... |
shsss 31035 | The subspace sum is a subs... |
shsel 31036 | Membership in the subspace... |
shsel3 31037 | Membership in the subspace... |
shseli 31038 | Membership in subspace sum... |
shscli 31039 | Closure of subspace sum. ... |
shscl 31040 | Closure of subspace sum. ... |
shscom 31041 | Commutative law for subspa... |
shsva 31042 | Vector sum belongs to subs... |
shsel1 31043 | A subspace sum contains a ... |
shsel2 31044 | A subspace sum contains a ... |
shsvs 31045 | Vector subtraction belongs... |
shsub1 31046 | Subspace sum is an upper b... |
shsub2 31047 | Subspace sum is an upper b... |
choc0 31048 | The orthocomplement of the... |
choc1 31049 | The orthocomplement of the... |
chocnul 31050 | Orthogonal complement of t... |
shintcli 31051 | Closure of intersection of... |
shintcl 31052 | The intersection of a none... |
chintcli 31053 | The intersection of a none... |
chintcl 31054 | The intersection (infimum)... |
spanval 31055 | Value of the linear span o... |
hsupval 31056 | Value of supremum of set o... |
chsupval 31057 | The value of the supremum ... |
spancl 31058 | The span of a subset of Hi... |
elspancl 31059 | A member of a span is a ve... |
shsupcl 31060 | Closure of the subspace su... |
hsupcl 31061 | Closure of supremum of set... |
chsupcl 31062 | Closure of supremum of sub... |
hsupss 31063 | Subset relation for suprem... |
chsupss 31064 | Subset relation for suprem... |
hsupunss 31065 | The union of a set of Hilb... |
chsupunss 31066 | The union of a set of clos... |
spanss2 31067 | A subset of Hilbert space ... |
shsupunss 31068 | The union of a set of subs... |
spanid 31069 | A subspace of Hilbert spac... |
spanss 31070 | Ordering relationship for ... |
spanssoc 31071 | The span of a subset of Hi... |
sshjval 31072 | Value of join for subsets ... |
shjval 31073 | Value of join in ` SH ` . ... |
chjval 31074 | Value of join in ` CH ` . ... |
chjvali 31075 | Value of join in ` CH ` . ... |
sshjval3 31076 | Value of join for subsets ... |
sshjcl 31077 | Closure of join for subset... |
shjcl 31078 | Closure of join in ` SH ` ... |
chjcl 31079 | Closure of join in ` CH ` ... |
shjcom 31080 | Commutative law for Hilber... |
shless 31081 | Subset implies subset of s... |
shlej1 31082 | Add disjunct to both sides... |
shlej2 31083 | Add disjunct to both sides... |
shincli 31084 | Closure of intersection of... |
shscomi 31085 | Commutative law for subspa... |
shsvai 31086 | Vector sum belongs to subs... |
shsel1i 31087 | A subspace sum contains a ... |
shsel2i 31088 | A subspace sum contains a ... |
shsvsi 31089 | Vector subtraction belongs... |
shunssi 31090 | Union is smaller than subs... |
shunssji 31091 | Union is smaller than Hilb... |
shsleji 31092 | Subspace sum is smaller th... |
shjcomi 31093 | Commutative law for join i... |
shsub1i 31094 | Subspace sum is an upper b... |
shsub2i 31095 | Subspace sum is an upper b... |
shub1i 31096 | Hilbert lattice join is an... |
shjcli 31097 | Closure of ` CH ` join. (... |
shjshcli 31098 | ` SH ` closure of join. (... |
shlessi 31099 | Subset implies subset of s... |
shlej1i 31100 | Add disjunct to both sides... |
shlej2i 31101 | Add disjunct to both sides... |
shslej 31102 | Subspace sum is smaller th... |
shincl 31103 | Closure of intersection of... |
shub1 31104 | Hilbert lattice join is an... |
shub2 31105 | A subspace is a subset of ... |
shsidmi 31106 | Idempotent law for Hilbert... |
shslubi 31107 | The least upper bound law ... |
shlesb1i 31108 | Hilbert lattice ordering i... |
shsval2i 31109 | An alternate way to expres... |
shsval3i 31110 | An alternate way to expres... |
shmodsi 31111 | The modular law holds for ... |
shmodi 31112 | The modular law is implied... |
pjhthlem1 31113 | Lemma for ~ pjhth . (Cont... |
pjhthlem2 31114 | Lemma for ~ pjhth . (Cont... |
pjhth 31115 | Projection Theorem: Any H... |
pjhtheu 31116 | Projection Theorem: Any H... |
pjhfval 31118 | The value of the projectio... |
pjhval 31119 | Value of a projection. (C... |
pjpreeq 31120 | Equality with a projection... |
pjeq 31121 | Equality with a projection... |
axpjcl 31122 | Closure of a projection in... |
pjhcl 31123 | Closure of a projection in... |
omlsilem 31124 | Lemma for orthomodular law... |
omlsii 31125 | Subspace inference form of... |
omlsi 31126 | Subspace form of orthomodu... |
ococi 31127 | Complement of complement o... |
ococ 31128 | Complement of complement o... |
dfch2 31129 | Alternate definition of th... |
ococin 31130 | The double complement is t... |
hsupval2 31131 | Alternate definition of su... |
chsupval2 31132 | The value of the supremum ... |
sshjval2 31133 | Value of join in the set o... |
chsupid 31134 | A subspace is the supremum... |
chsupsn 31135 | Value of supremum of subse... |
shlub 31136 | Hilbert lattice join is th... |
shlubi 31137 | Hilbert lattice join is th... |
pjhtheu2 31138 | Uniqueness of ` y ` for th... |
pjcli 31139 | Closure of a projection in... |
pjhcli 31140 | Closure of a projection in... |
pjpjpre 31141 | Decomposition of a vector ... |
axpjpj 31142 | Decomposition of a vector ... |
pjclii 31143 | Closure of a projection in... |
pjhclii 31144 | Closure of a projection in... |
pjpj0i 31145 | Decomposition of a vector ... |
pjpji 31146 | Decomposition of a vector ... |
pjpjhth 31147 | Projection Theorem: Any H... |
pjpjhthi 31148 | Projection Theorem: Any H... |
pjop 31149 | Orthocomplement projection... |
pjpo 31150 | Projection in terms of ort... |
pjopi 31151 | Orthocomplement projection... |
pjpoi 31152 | Projection in terms of ort... |
pjoc1i 31153 | Projection of a vector in ... |
pjchi 31154 | Projection of a vector in ... |
pjoccl 31155 | The part of a vector that ... |
pjoc1 31156 | Projection of a vector in ... |
pjomli 31157 | Subspace form of orthomodu... |
pjoml 31158 | Subspace form of orthomodu... |
pjococi 31159 | Proof of orthocomplement t... |
pjoc2i 31160 | Projection of a vector in ... |
pjoc2 31161 | Projection of a vector in ... |
sh0le 31162 | The zero subspace is the s... |
ch0le 31163 | The zero subspace is the s... |
shle0 31164 | No subspace is smaller tha... |
chle0 31165 | No Hilbert lattice element... |
chnlen0 31166 | A Hilbert lattice element ... |
ch0pss 31167 | The zero subspace is a pro... |
orthin 31168 | The intersection of orthog... |
ssjo 31169 | The lattice join of a subs... |
shne0i 31170 | A nonzero subspace has a n... |
shs0i 31171 | Hilbert subspace sum with ... |
shs00i 31172 | Two subspaces are zero iff... |
ch0lei 31173 | The closed subspace zero i... |
chle0i 31174 | No Hilbert closed subspace... |
chne0i 31175 | A nonzero closed subspace ... |
chocini 31176 | Intersection of a closed s... |
chj0i 31177 | Join with lattice zero in ... |
chm1i 31178 | Meet with lattice one in `... |
chjcli 31179 | Closure of ` CH ` join. (... |
chsleji 31180 | Subspace sum is smaller th... |
chseli 31181 | Membership in subspace sum... |
chincli 31182 | Closure of Hilbert lattice... |
chsscon3i 31183 | Hilbert lattice contraposi... |
chsscon1i 31184 | Hilbert lattice contraposi... |
chsscon2i 31185 | Hilbert lattice contraposi... |
chcon2i 31186 | Hilbert lattice contraposi... |
chcon1i 31187 | Hilbert lattice contraposi... |
chcon3i 31188 | Hilbert lattice contraposi... |
chunssji 31189 | Union is smaller than ` CH... |
chjcomi 31190 | Commutative law for join i... |
chub1i 31191 | ` CH ` join is an upper bo... |
chub2i 31192 | ` CH ` join is an upper bo... |
chlubi 31193 | Hilbert lattice join is th... |
chlubii 31194 | Hilbert lattice join is th... |
chlej1i 31195 | Add join to both sides of ... |
chlej2i 31196 | Add join to both sides of ... |
chlej12i 31197 | Add join to both sides of ... |
chlejb1i 31198 | Hilbert lattice ordering i... |
chdmm1i 31199 | De Morgan's law for meet i... |
chdmm2i 31200 | De Morgan's law for meet i... |
chdmm3i 31201 | De Morgan's law for meet i... |
chdmm4i 31202 | De Morgan's law for meet i... |
chdmj1i 31203 | De Morgan's law for join i... |
chdmj2i 31204 | De Morgan's law for join i... |
chdmj3i 31205 | De Morgan's law for join i... |
chdmj4i 31206 | De Morgan's law for join i... |
chnlei 31207 | Equivalent expressions for... |
chjassi 31208 | Associative law for Hilber... |
chj00i 31209 | Two Hilbert lattice elemen... |
chjoi 31210 | The join of a closed subsp... |
chj1i 31211 | Join with Hilbert lattice ... |
chm0i 31212 | Meet with Hilbert lattice ... |
chm0 31213 | Meet with Hilbert lattice ... |
shjshsi 31214 | Hilbert lattice join equal... |
shjshseli 31215 | A closed subspace sum equa... |
chne0 31216 | A nonzero closed subspace ... |
chocin 31217 | Intersection of a closed s... |
chssoc 31218 | A closed subspace less tha... |
chj0 31219 | Join with Hilbert lattice ... |
chslej 31220 | Subspace sum is smaller th... |
chincl 31221 | Closure of Hilbert lattice... |
chsscon3 31222 | Hilbert lattice contraposi... |
chsscon1 31223 | Hilbert lattice contraposi... |
chsscon2 31224 | Hilbert lattice contraposi... |
chpsscon3 31225 | Hilbert lattice contraposi... |
chpsscon1 31226 | Hilbert lattice contraposi... |
chpsscon2 31227 | Hilbert lattice contraposi... |
chjcom 31228 | Commutative law for Hilber... |
chub1 31229 | Hilbert lattice join is gr... |
chub2 31230 | Hilbert lattice join is gr... |
chlub 31231 | Hilbert lattice join is th... |
chlej1 31232 | Add join to both sides of ... |
chlej2 31233 | Add join to both sides of ... |
chlejb1 31234 | Hilbert lattice ordering i... |
chlejb2 31235 | Hilbert lattice ordering i... |
chnle 31236 | Equivalent expressions for... |
chjo 31237 | The join of a closed subsp... |
chabs1 31238 | Hilbert lattice absorption... |
chabs2 31239 | Hilbert lattice absorption... |
chabs1i 31240 | Hilbert lattice absorption... |
chabs2i 31241 | Hilbert lattice absorption... |
chjidm 31242 | Idempotent law for Hilbert... |
chjidmi 31243 | Idempotent law for Hilbert... |
chj12i 31244 | A rearrangement of Hilbert... |
chj4i 31245 | Rearrangement of the join ... |
chjjdiri 31246 | Hilbert lattice join distr... |
chdmm1 31247 | De Morgan's law for meet i... |
chdmm2 31248 | De Morgan's law for meet i... |
chdmm3 31249 | De Morgan's law for meet i... |
chdmm4 31250 | De Morgan's law for meet i... |
chdmj1 31251 | De Morgan's law for join i... |
chdmj2 31252 | De Morgan's law for join i... |
chdmj3 31253 | De Morgan's law for join i... |
chdmj4 31254 | De Morgan's law for join i... |
chjass 31255 | Associative law for Hilber... |
chj12 31256 | A rearrangement of Hilbert... |
chj4 31257 | Rearrangement of the join ... |
ledii 31258 | An ortholattice is distrib... |
lediri 31259 | An ortholattice is distrib... |
lejdii 31260 | An ortholattice is distrib... |
lejdiri 31261 | An ortholattice is distrib... |
ledi 31262 | An ortholattice is distrib... |
spansn0 31263 | The span of the singleton ... |
span0 31264 | The span of the empty set ... |
elspani 31265 | Membership in the span of ... |
spanuni 31266 | The span of a union is the... |
spanun 31267 | The span of a union is the... |
sshhococi 31268 | The join of two Hilbert sp... |
hne0 31269 | Hilbert space has a nonzer... |
chsup0 31270 | The supremum of the empty ... |
h1deoi 31271 | Membership in orthocomplem... |
h1dei 31272 | Membership in 1-dimensiona... |
h1did 31273 | A generating vector belong... |
h1dn0 31274 | A nonzero vector generates... |
h1de2i 31275 | Membership in 1-dimensiona... |
h1de2bi 31276 | Membership in 1-dimensiona... |
h1de2ctlem 31277 | Lemma for ~ h1de2ci . (Co... |
h1de2ci 31278 | Membership in 1-dimensiona... |
spansni 31279 | The span of a singleton in... |
elspansni 31280 | Membership in the span of ... |
spansn 31281 | The span of a singleton in... |
spansnch 31282 | The span of a Hilbert spac... |
spansnsh 31283 | The span of a Hilbert spac... |
spansnchi 31284 | The span of a singleton in... |
spansnid 31285 | A vector belongs to the sp... |
spansnmul 31286 | A scalar product with a ve... |
elspansncl 31287 | A member of a span of a si... |
elspansn 31288 | Membership in the span of ... |
elspansn2 31289 | Membership in the span of ... |
spansncol 31290 | The singletons of collinea... |
spansneleqi 31291 | Membership relation implie... |
spansneleq 31292 | Membership relation that i... |
spansnss 31293 | The span of the singleton ... |
elspansn3 31294 | A member of the span of th... |
elspansn4 31295 | A span membership conditio... |
elspansn5 31296 | A vector belonging to both... |
spansnss2 31297 | The span of the singleton ... |
normcan 31298 | Cancellation-type law that... |
pjspansn 31299 | A projection on the span o... |
spansnpji 31300 | A subset of Hilbert space ... |
spanunsni 31301 | The span of the union of a... |
spanpr 31302 | The span of a pair of vect... |
h1datomi 31303 | A 1-dimensional subspace i... |
h1datom 31304 | A 1-dimensional subspace i... |
cmbr 31306 | Binary relation expressing... |
pjoml2i 31307 | Variation of orthomodular ... |
pjoml3i 31308 | Variation of orthomodular ... |
pjoml4i 31309 | Variation of orthomodular ... |
pjoml5i 31310 | The orthomodular law. Rem... |
pjoml6i 31311 | An equivalent of the ortho... |
cmbri 31312 | Binary relation expressing... |
cmcmlem 31313 | Commutation is symmetric. ... |
cmcmi 31314 | Commutation is symmetric. ... |
cmcm2i 31315 | Commutation with orthocomp... |
cmcm3i 31316 | Commutation with orthocomp... |
cmcm4i 31317 | Commutation with orthocomp... |
cmbr2i 31318 | Alternate definition of th... |
cmcmii 31319 | Commutation is symmetric. ... |
cmcm2ii 31320 | Commutation with orthocomp... |
cmcm3ii 31321 | Commutation with orthocomp... |
cmbr3i 31322 | Alternate definition for t... |
cmbr4i 31323 | Alternate definition for t... |
lecmi 31324 | Comparable Hilbert lattice... |
lecmii 31325 | Comparable Hilbert lattice... |
cmj1i 31326 | A Hilbert lattice element ... |
cmj2i 31327 | A Hilbert lattice element ... |
cmm1i 31328 | A Hilbert lattice element ... |
cmm2i 31329 | A Hilbert lattice element ... |
cmbr3 31330 | Alternate definition for t... |
cm0 31331 | The zero Hilbert lattice e... |
cmidi 31332 | The commutes relation is r... |
pjoml2 31333 | Variation of orthomodular ... |
pjoml3 31334 | Variation of orthomodular ... |
pjoml5 31335 | The orthomodular law. Rem... |
cmcm 31336 | Commutation is symmetric. ... |
cmcm3 31337 | Commutation with orthocomp... |
cmcm2 31338 | Commutation with orthocomp... |
lecm 31339 | Comparable Hilbert lattice... |
fh1 31340 | Foulis-Holland Theorem. I... |
fh2 31341 | Foulis-Holland Theorem. I... |
cm2j 31342 | A lattice element that com... |
fh1i 31343 | Foulis-Holland Theorem. I... |
fh2i 31344 | Foulis-Holland Theorem. I... |
fh3i 31345 | Variation of the Foulis-Ho... |
fh4i 31346 | Variation of the Foulis-Ho... |
cm2ji 31347 | A lattice element that com... |
cm2mi 31348 | A lattice element that com... |
qlax1i 31349 | One of the equations showi... |
qlax2i 31350 | One of the equations showi... |
qlax3i 31351 | One of the equations showi... |
qlax4i 31352 | One of the equations showi... |
qlax5i 31353 | One of the equations showi... |
qlaxr1i 31354 | One of the conditions show... |
qlaxr2i 31355 | One of the conditions show... |
qlaxr4i 31356 | One of the conditions show... |
qlaxr5i 31357 | One of the conditions show... |
qlaxr3i 31358 | A variation of the orthomo... |
chscllem1 31359 | Lemma for ~ chscl . (Cont... |
chscllem2 31360 | Lemma for ~ chscl . (Cont... |
chscllem3 31361 | Lemma for ~ chscl . (Cont... |
chscllem4 31362 | Lemma for ~ chscl . (Cont... |
chscl 31363 | The subspace sum of two cl... |
osumi 31364 | If two closed subspaces of... |
osumcori 31365 | Corollary of ~ osumi . (C... |
osumcor2i 31366 | Corollary of ~ osumi , sho... |
osum 31367 | If two closed subspaces of... |
spansnji 31368 | The subspace sum of a clos... |
spansnj 31369 | The subspace sum of a clos... |
spansnscl 31370 | The subspace sum of a clos... |
sumspansn 31371 | The sum of two vectors bel... |
spansnm0i 31372 | The meet of different one-... |
nonbooli 31373 | A Hilbert lattice with two... |
spansncvi 31374 | Hilbert space has the cove... |
spansncv 31375 | Hilbert space has the cove... |
5oalem1 31376 | Lemma for orthoarguesian l... |
5oalem2 31377 | Lemma for orthoarguesian l... |
5oalem3 31378 | Lemma for orthoarguesian l... |
5oalem4 31379 | Lemma for orthoarguesian l... |
5oalem5 31380 | Lemma for orthoarguesian l... |
5oalem6 31381 | Lemma for orthoarguesian l... |
5oalem7 31382 | Lemma for orthoarguesian l... |
5oai 31383 | Orthoarguesian law 5OA. Th... |
3oalem1 31384 | Lemma for 3OA (weak) ortho... |
3oalem2 31385 | Lemma for 3OA (weak) ortho... |
3oalem3 31386 | Lemma for 3OA (weak) ortho... |
3oalem4 31387 | Lemma for 3OA (weak) ortho... |
3oalem5 31388 | Lemma for 3OA (weak) ortho... |
3oalem6 31389 | Lemma for 3OA (weak) ortho... |
3oai 31390 | 3OA (weak) orthoarguesian ... |
pjorthi 31391 | Projection components on o... |
pjch1 31392 | Property of identity proje... |
pjo 31393 | The orthogonal projection.... |
pjcompi 31394 | Component of a projection.... |
pjidmi 31395 | A projection is idempotent... |
pjadjii 31396 | A projection is self-adjoi... |
pjaddii 31397 | Projection of vector sum i... |
pjinormii 31398 | The inner product of a pro... |
pjmulii 31399 | Projection of (scalar) pro... |
pjsubii 31400 | Projection of vector diffe... |
pjsslem 31401 | Lemma for subset relations... |
pjss2i 31402 | Subset relationship for pr... |
pjssmii 31403 | Projection meet property. ... |
pjssge0ii 31404 | Theorem 4.5(iv)->(v) of [B... |
pjdifnormii 31405 | Theorem 4.5(v)<->(vi) of [... |
pjcji 31406 | The projection on a subspa... |
pjadji 31407 | A projection is self-adjoi... |
pjaddi 31408 | Projection of vector sum i... |
pjinormi 31409 | The inner product of a pro... |
pjsubi 31410 | Projection of vector diffe... |
pjmuli 31411 | Projection of scalar produ... |
pjige0i 31412 | The inner product of a pro... |
pjige0 31413 | The inner product of a pro... |
pjcjt2 31414 | The projection on a subspa... |
pj0i 31415 | The projection of the zero... |
pjch 31416 | Projection of a vector in ... |
pjid 31417 | The projection of a vector... |
pjvec 31418 | The set of vectors belongi... |
pjocvec 31419 | The set of vectors belongi... |
pjocini 31420 | Membership of projection i... |
pjini 31421 | Membership of projection i... |
pjjsi 31422 | A sufficient condition for... |
pjfni 31423 | Functionality of a project... |
pjrni 31424 | The range of a projection.... |
pjfoi 31425 | A projection maps onto its... |
pjfi 31426 | The mapping of a projectio... |
pjvi 31427 | The value of a projection ... |
pjhfo 31428 | A projection maps onto its... |
pjrn 31429 | The range of a projection.... |
pjhf 31430 | The mapping of a projectio... |
pjfn 31431 | Functionality of a project... |
pjsumi 31432 | The projection on a subspa... |
pj11i 31433 | One-to-one correspondence ... |
pjdsi 31434 | Vector decomposition into ... |
pjds3i 31435 | Vector decomposition into ... |
pj11 31436 | One-to-one correspondence ... |
pjmfn 31437 | Functionality of the proje... |
pjmf1 31438 | The projector function map... |
pjoi0 31439 | The inner product of proje... |
pjoi0i 31440 | The inner product of proje... |
pjopythi 31441 | Pythagorean theorem for pr... |
pjopyth 31442 | Pythagorean theorem for pr... |
pjnormi 31443 | The norm of the projection... |
pjpythi 31444 | Pythagorean theorem for pr... |
pjneli 31445 | If a vector does not belon... |
pjnorm 31446 | The norm of the projection... |
pjpyth 31447 | Pythagorean theorem for pr... |
pjnel 31448 | If a vector does not belon... |
pjnorm2 31449 | A vector belongs to the su... |
mayete3i 31450 | Mayet's equation E_3. Par... |
mayetes3i 31451 | Mayet's equation E^*_3, de... |
hosmval 31457 | Value of the sum of two Hi... |
hommval 31458 | Value of the scalar produc... |
hodmval 31459 | Value of the difference of... |
hfsmval 31460 | Value of the sum of two Hi... |
hfmmval 31461 | Value of the scalar produc... |
hosval 31462 | Value of the sum of two Hi... |
homval 31463 | Value of the scalar produc... |
hodval 31464 | Value of the difference of... |
hfsval 31465 | Value of the sum of two Hi... |
hfmval 31466 | Value of the scalar produc... |
hoscl 31467 | Closure of the sum of two ... |
homcl 31468 | Closure of the scalar prod... |
hodcl 31469 | Closure of the difference ... |
ho0val 31472 | Value of the zero Hilbert ... |
ho0f 31473 | Functionality of the zero ... |
df0op2 31474 | Alternate definition of Hi... |
dfiop2 31475 | Alternate definition of Hi... |
hoif 31476 | Functionality of the Hilbe... |
hoival 31477 | The value of the Hilbert s... |
hoico1 31478 | Composition with the Hilbe... |
hoico2 31479 | Composition with the Hilbe... |
hoaddcl 31480 | The sum of Hilbert space o... |
homulcl 31481 | The scalar product of a Hi... |
hoeq 31482 | Equality of Hilbert space ... |
hoeqi 31483 | Equality of Hilbert space ... |
hoscli 31484 | Closure of Hilbert space o... |
hodcli 31485 | Closure of Hilbert space o... |
hocoi 31486 | Composition of Hilbert spa... |
hococli 31487 | Closure of composition of ... |
hocofi 31488 | Mapping of composition of ... |
hocofni 31489 | Functionality of compositi... |
hoaddcli 31490 | Mapping of sum of Hilbert ... |
hosubcli 31491 | Mapping of difference of H... |
hoaddfni 31492 | Functionality of sum of Hi... |
hosubfni 31493 | Functionality of differenc... |
hoaddcomi 31494 | Commutativity of sum of Hi... |
hosubcl 31495 | Mapping of difference of H... |
hoaddcom 31496 | Commutativity of sum of Hi... |
hodsi 31497 | Relationship between Hilbe... |
hoaddassi 31498 | Associativity of sum of Hi... |
hoadd12i 31499 | Commutative/associative la... |
hoadd32i 31500 | Commutative/associative la... |
hocadddiri 31501 | Distributive law for Hilbe... |
hocsubdiri 31502 | Distributive law for Hilbe... |
ho2coi 31503 | Double composition of Hilb... |
hoaddass 31504 | Associativity of sum of Hi... |
hoadd32 31505 | Commutative/associative la... |
hoadd4 31506 | Rearrangement of 4 terms i... |
hocsubdir 31507 | Distributive law for Hilbe... |
hoaddridi 31508 | Sum of a Hilbert space ope... |
hodidi 31509 | Difference of a Hilbert sp... |
ho0coi 31510 | Composition of the zero op... |
hoid1i 31511 | Composition of Hilbert spa... |
hoid1ri 31512 | Composition of Hilbert spa... |
hoaddrid 31513 | Sum of a Hilbert space ope... |
hodid 31514 | Difference of a Hilbert sp... |
hon0 31515 | A Hilbert space operator i... |
hodseqi 31516 | Subtraction and addition o... |
ho0subi 31517 | Subtraction of Hilbert spa... |
honegsubi 31518 | Relationship between Hilbe... |
ho0sub 31519 | Subtraction of Hilbert spa... |
hosubid1 31520 | The zero operator subtract... |
honegsub 31521 | Relationship between Hilbe... |
homullid 31522 | An operator equals its sca... |
homco1 31523 | Associative law for scalar... |
homulass 31524 | Scalar product associative... |
hoadddi 31525 | Scalar product distributiv... |
hoadddir 31526 | Scalar product reverse dis... |
homul12 31527 | Swap first and second fact... |
honegneg 31528 | Double negative of a Hilbe... |
hosubneg 31529 | Relationship between opera... |
hosubdi 31530 | Scalar product distributiv... |
honegdi 31531 | Distribution of negative o... |
honegsubdi 31532 | Distribution of negative o... |
honegsubdi2 31533 | Distribution of negative o... |
hosubsub2 31534 | Law for double subtraction... |
hosub4 31535 | Rearrangement of 4 terms i... |
hosubadd4 31536 | Rearrangement of 4 terms i... |
hoaddsubass 31537 | Associative-type law for a... |
hoaddsub 31538 | Law for operator addition ... |
hosubsub 31539 | Law for double subtraction... |
hosubsub4 31540 | Law for double subtraction... |
ho2times 31541 | Two times a Hilbert space ... |
hoaddsubassi 31542 | Associativity of sum and d... |
hoaddsubi 31543 | Law for sum and difference... |
hosd1i 31544 | Hilbert space operator sum... |
hosd2i 31545 | Hilbert space operator sum... |
hopncani 31546 | Hilbert space operator can... |
honpcani 31547 | Hilbert space operator can... |
hosubeq0i 31548 | If the difference between ... |
honpncani 31549 | Hilbert space operator can... |
ho01i 31550 | A condition implying that ... |
ho02i 31551 | A condition implying that ... |
hoeq1 31552 | A condition implying that ... |
hoeq2 31553 | A condition implying that ... |
adjmo 31554 | Every Hilbert space operat... |
adjsym 31555 | Symmetry property of an ad... |
eigrei 31556 | A necessary and sufficient... |
eigre 31557 | A necessary and sufficient... |
eigposi 31558 | A sufficient condition (fi... |
eigorthi 31559 | A necessary and sufficient... |
eigorth 31560 | A necessary and sufficient... |
nmopval 31578 | Value of the norm of a Hil... |
elcnop 31579 | Property defining a contin... |
ellnop 31580 | Property defining a linear... |
lnopf 31581 | A linear Hilbert space ope... |
elbdop 31582 | Property defining a bounde... |
bdopln 31583 | A bounded linear Hilbert s... |
bdopf 31584 | A bounded linear Hilbert s... |
nmopsetretALT 31585 | The set in the supremum of... |
nmopsetretHIL 31586 | The set in the supremum of... |
nmopsetn0 31587 | The set in the supremum of... |
nmopxr 31588 | The norm of a Hilbert spac... |
nmoprepnf 31589 | The norm of a Hilbert spac... |
nmopgtmnf 31590 | The norm of a Hilbert spac... |
nmopreltpnf 31591 | The norm of a Hilbert spac... |
nmopre 31592 | The norm of a bounded oper... |
elbdop2 31593 | Property defining a bounde... |
elunop 31594 | Property defining a unitar... |
elhmop 31595 | Property defining a Hermit... |
hmopf 31596 | A Hermitian operator is a ... |
hmopex 31597 | The class of Hermitian ope... |
nmfnval 31598 | Value of the norm of a Hil... |
nmfnsetre 31599 | The set in the supremum of... |
nmfnsetn0 31600 | The set in the supremum of... |
nmfnxr 31601 | The norm of any Hilbert sp... |
nmfnrepnf 31602 | The norm of a Hilbert spac... |
nlfnval 31603 | Value of the null space of... |
elcnfn 31604 | Property defining a contin... |
ellnfn 31605 | Property defining a linear... |
lnfnf 31606 | A linear Hilbert space fun... |
dfadj2 31607 | Alternate definition of th... |
funadj 31608 | Functionality of the adjoi... |
dmadjss 31609 | The domain of the adjoint ... |
dmadjop 31610 | A member of the domain of ... |
adjeu 31611 | Elementhood in the domain ... |
adjval 31612 | Value of the adjoint funct... |
adjval2 31613 | Value of the adjoint funct... |
cnvadj 31614 | The adjoint function equal... |
funcnvadj 31615 | The converse of the adjoin... |
adj1o 31616 | The adjoint function maps ... |
dmadjrn 31617 | The adjoint of an operator... |
eigvecval 31618 | The set of eigenvectors of... |
eigvalfval 31619 | The eigenvalues of eigenve... |
specval 31620 | The value of the spectrum ... |
speccl 31621 | The spectrum of an operato... |
hhlnoi 31622 | The linear operators of Hi... |
hhnmoi 31623 | The norm of an operator in... |
hhbloi 31624 | A bounded linear operator ... |
hh0oi 31625 | The zero operator in Hilbe... |
hhcno 31626 | The continuous operators o... |
hhcnf 31627 | The continuous functionals... |
dmadjrnb 31628 | The adjoint of an operator... |
nmoplb 31629 | A lower bound for an opera... |
nmopub 31630 | An upper bound for an oper... |
nmopub2tALT 31631 | An upper bound for an oper... |
nmopub2tHIL 31632 | An upper bound for an oper... |
nmopge0 31633 | The norm of any Hilbert sp... |
nmopgt0 31634 | A linear Hilbert space ope... |
cnopc 31635 | Basic continuity property ... |
lnopl 31636 | Basic linearity property o... |
unop 31637 | Basic inner product proper... |
unopf1o 31638 | A unitary operator in Hilb... |
unopnorm 31639 | A unitary operator is idem... |
cnvunop 31640 | The inverse (converse) of ... |
unopadj 31641 | The inverse (converse) of ... |
unoplin 31642 | A unitary operator is line... |
counop 31643 | The composition of two uni... |
hmop 31644 | Basic inner product proper... |
hmopre 31645 | The inner product of the v... |
nmfnlb 31646 | A lower bound for a functi... |
nmfnleub 31647 | An upper bound for the nor... |
nmfnleub2 31648 | An upper bound for the nor... |
nmfnge0 31649 | The norm of any Hilbert sp... |
elnlfn 31650 | Membership in the null spa... |
elnlfn2 31651 | Membership in the null spa... |
cnfnc 31652 | Basic continuity property ... |
lnfnl 31653 | Basic linearity property o... |
adjcl 31654 | Closure of the adjoint of ... |
adj1 31655 | Property of an adjoint Hil... |
adj2 31656 | Property of an adjoint Hil... |
adjeq 31657 | A property that determines... |
adjadj 31658 | Double adjoint. Theorem 3... |
adjvalval 31659 | Value of the value of the ... |
unopadj2 31660 | The adjoint of a unitary o... |
hmopadj 31661 | A Hermitian operator is se... |
hmdmadj 31662 | Every Hermitian operator h... |
hmopadj2 31663 | An operator is Hermitian i... |
hmoplin 31664 | A Hermitian operator is li... |
brafval 31665 | The bra of a vector, expre... |
braval 31666 | A bra-ket juxtaposition, e... |
braadd 31667 | Linearity property of bra ... |
bramul 31668 | Linearity property of bra ... |
brafn 31669 | The bra function is a func... |
bralnfn 31670 | The Dirac bra function is ... |
bracl 31671 | Closure of the bra functio... |
bra0 31672 | The Dirac bra of the zero ... |
brafnmul 31673 | Anti-linearity property of... |
kbfval 31674 | The outer product of two v... |
kbop 31675 | The outer product of two v... |
kbval 31676 | The value of the operator ... |
kbmul 31677 | Multiplication property of... |
kbpj 31678 | If a vector ` A ` has norm... |
eleigvec 31679 | Membership in the set of e... |
eleigvec2 31680 | Membership in the set of e... |
eleigveccl 31681 | Closure of an eigenvector ... |
eigvalval 31682 | The eigenvalue of an eigen... |
eigvalcl 31683 | An eigenvalue is a complex... |
eigvec1 31684 | Property of an eigenvector... |
eighmre 31685 | The eigenvalues of a Hermi... |
eighmorth 31686 | Eigenvectors of a Hermitia... |
nmopnegi 31687 | Value of the norm of the n... |
lnop0 31688 | The value of a linear Hilb... |
lnopmul 31689 | Multiplicative property of... |
lnopli 31690 | Basic scalar product prope... |
lnopfi 31691 | A linear Hilbert space ope... |
lnop0i 31692 | The value of a linear Hilb... |
lnopaddi 31693 | Additive property of a lin... |
lnopmuli 31694 | Multiplicative property of... |
lnopaddmuli 31695 | Sum/product property of a ... |
lnopsubi 31696 | Subtraction property for a... |
lnopsubmuli 31697 | Subtraction/product proper... |
lnopmulsubi 31698 | Product/subtraction proper... |
homco2 31699 | Move a scalar product out ... |
idunop 31700 | The identity function (res... |
0cnop 31701 | The identically zero funct... |
0cnfn 31702 | The identically zero funct... |
idcnop 31703 | The identity function (res... |
idhmop 31704 | The Hilbert space identity... |
0hmop 31705 | The identically zero funct... |
0lnop 31706 | The identically zero funct... |
0lnfn 31707 | The identically zero funct... |
nmop0 31708 | The norm of the zero opera... |
nmfn0 31709 | The norm of the identicall... |
hmopbdoptHIL 31710 | A Hermitian operator is a ... |
hoddii 31711 | Distributive law for Hilbe... |
hoddi 31712 | Distributive law for Hilbe... |
nmop0h 31713 | The norm of any operator o... |
idlnop 31714 | The identity function (res... |
0bdop 31715 | The identically zero opera... |
adj0 31716 | Adjoint of the zero operat... |
nmlnop0iALT 31717 | A linear operator with a z... |
nmlnop0iHIL 31718 | A linear operator with a z... |
nmlnopgt0i 31719 | A linear Hilbert space ope... |
nmlnop0 31720 | A linear operator with a z... |
nmlnopne0 31721 | A linear operator with a n... |
lnopmi 31722 | The scalar product of a li... |
lnophsi 31723 | The sum of two linear oper... |
lnophdi 31724 | The difference of two line... |
lnopcoi 31725 | The composition of two lin... |
lnopco0i 31726 | The composition of a linea... |
lnopeq0lem1 31727 | Lemma for ~ lnopeq0i . Ap... |
lnopeq0lem2 31728 | Lemma for ~ lnopeq0i . (C... |
lnopeq0i 31729 | A condition implying that ... |
lnopeqi 31730 | Two linear Hilbert space o... |
lnopeq 31731 | Two linear Hilbert space o... |
lnopunilem1 31732 | Lemma for ~ lnopunii . (C... |
lnopunilem2 31733 | Lemma for ~ lnopunii . (C... |
lnopunii 31734 | If a linear operator (whos... |
elunop2 31735 | An operator is unitary iff... |
nmopun 31736 | Norm of a unitary Hilbert ... |
unopbd 31737 | A unitary operator is a bo... |
lnophmlem1 31738 | Lemma for ~ lnophmi . (Co... |
lnophmlem2 31739 | Lemma for ~ lnophmi . (Co... |
lnophmi 31740 | A linear operator is Hermi... |
lnophm 31741 | A linear operator is Hermi... |
hmops 31742 | The sum of two Hermitian o... |
hmopm 31743 | The scalar product of a He... |
hmopd 31744 | The difference of two Herm... |
hmopco 31745 | The composition of two com... |
nmbdoplbi 31746 | A lower bound for the norm... |
nmbdoplb 31747 | A lower bound for the norm... |
nmcexi 31748 | Lemma for ~ nmcopexi and ~... |
nmcopexi 31749 | The norm of a continuous l... |
nmcoplbi 31750 | A lower bound for the norm... |
nmcopex 31751 | The norm of a continuous l... |
nmcoplb 31752 | A lower bound for the norm... |
nmophmi 31753 | The norm of the scalar pro... |
bdophmi 31754 | The scalar product of a bo... |
lnconi 31755 | Lemma for ~ lnopconi and ~... |
lnopconi 31756 | A condition equivalent to ... |
lnopcon 31757 | A condition equivalent to ... |
lnopcnbd 31758 | A linear operator is conti... |
lncnopbd 31759 | A continuous linear operat... |
lncnbd 31760 | A continuous linear operat... |
lnopcnre 31761 | A linear operator is conti... |
lnfnli 31762 | Basic property of a linear... |
lnfnfi 31763 | A linear Hilbert space fun... |
lnfn0i 31764 | The value of a linear Hilb... |
lnfnaddi 31765 | Additive property of a lin... |
lnfnmuli 31766 | Multiplicative property of... |
lnfnaddmuli 31767 | Sum/product property of a ... |
lnfnsubi 31768 | Subtraction property for a... |
lnfn0 31769 | The value of a linear Hilb... |
lnfnmul 31770 | Multiplicative property of... |
nmbdfnlbi 31771 | A lower bound for the norm... |
nmbdfnlb 31772 | A lower bound for the norm... |
nmcfnexi 31773 | The norm of a continuous l... |
nmcfnlbi 31774 | A lower bound for the norm... |
nmcfnex 31775 | The norm of a continuous l... |
nmcfnlb 31776 | A lower bound of the norm ... |
lnfnconi 31777 | A condition equivalent to ... |
lnfncon 31778 | A condition equivalent to ... |
lnfncnbd 31779 | A linear functional is con... |
imaelshi 31780 | The image of a subspace un... |
rnelshi 31781 | The range of a linear oper... |
nlelshi 31782 | The null space of a linear... |
nlelchi 31783 | The null space of a contin... |
riesz3i 31784 | A continuous linear functi... |
riesz4i 31785 | A continuous linear functi... |
riesz4 31786 | A continuous linear functi... |
riesz1 31787 | Part 1 of the Riesz repres... |
riesz2 31788 | Part 2 of the Riesz repres... |
cnlnadjlem1 31789 | Lemma for ~ cnlnadji (Theo... |
cnlnadjlem2 31790 | Lemma for ~ cnlnadji . ` G... |
cnlnadjlem3 31791 | Lemma for ~ cnlnadji . By... |
cnlnadjlem4 31792 | Lemma for ~ cnlnadji . Th... |
cnlnadjlem5 31793 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem6 31794 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem7 31795 | Lemma for ~ cnlnadji . He... |
cnlnadjlem8 31796 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem9 31797 | Lemma for ~ cnlnadji . ` F... |
cnlnadji 31798 | Every continuous linear op... |
cnlnadjeui 31799 | Every continuous linear op... |
cnlnadjeu 31800 | Every continuous linear op... |
cnlnadj 31801 | Every continuous linear op... |
cnlnssadj 31802 | Every continuous linear Hi... |
bdopssadj 31803 | Every bounded linear Hilbe... |
bdopadj 31804 | Every bounded linear Hilbe... |
adjbdln 31805 | The adjoint of a bounded l... |
adjbdlnb 31806 | An operator is bounded and... |
adjbd1o 31807 | The mapping of adjoints of... |
adjlnop 31808 | The adjoint of an operator... |
adjsslnop 31809 | Every operator with an adj... |
nmopadjlei 31810 | Property of the norm of an... |
nmopadjlem 31811 | Lemma for ~ nmopadji . (C... |
nmopadji 31812 | Property of the norm of an... |
adjeq0 31813 | An operator is zero iff it... |
adjmul 31814 | The adjoint of the scalar ... |
adjadd 31815 | The adjoint of the sum of ... |
nmoptrii 31816 | Triangle inequality for th... |
nmopcoi 31817 | Upper bound for the norm o... |
bdophsi 31818 | The sum of two bounded lin... |
bdophdi 31819 | The difference between two... |
bdopcoi 31820 | The composition of two bou... |
nmoptri2i 31821 | Triangle-type inequality f... |
adjcoi 31822 | The adjoint of a compositi... |
nmopcoadji 31823 | The norm of an operator co... |
nmopcoadj2i 31824 | The norm of an operator co... |
nmopcoadj0i 31825 | An operator composed with ... |
unierri 31826 | If we approximate a chain ... |
branmfn 31827 | The norm of the bra functi... |
brabn 31828 | The bra of a vector is a b... |
rnbra 31829 | The set of bras equals the... |
bra11 31830 | The bra function maps vect... |
bracnln 31831 | A bra is a continuous line... |
cnvbraval 31832 | Value of the converse of t... |
cnvbracl 31833 | Closure of the converse of... |
cnvbrabra 31834 | The converse bra of the br... |
bracnvbra 31835 | The bra of the converse br... |
bracnlnval 31836 | The vector that a continuo... |
cnvbramul 31837 | Multiplication property of... |
kbass1 31838 | Dirac bra-ket associative ... |
kbass2 31839 | Dirac bra-ket associative ... |
kbass3 31840 | Dirac bra-ket associative ... |
kbass4 31841 | Dirac bra-ket associative ... |
kbass5 31842 | Dirac bra-ket associative ... |
kbass6 31843 | Dirac bra-ket associative ... |
leopg 31844 | Ordering relation for posi... |
leop 31845 | Ordering relation for oper... |
leop2 31846 | Ordering relation for oper... |
leop3 31847 | Operator ordering in terms... |
leoppos 31848 | Binary relation defining a... |
leoprf2 31849 | The ordering relation for ... |
leoprf 31850 | The ordering relation for ... |
leopsq 31851 | The square of a Hermitian ... |
0leop 31852 | The zero operator is a pos... |
idleop 31853 | The identity operator is a... |
leopadd 31854 | The sum of two positive op... |
leopmuli 31855 | The scalar product of a no... |
leopmul 31856 | The scalar product of a po... |
leopmul2i 31857 | Scalar product applied to ... |
leoptri 31858 | The positive operator orde... |
leoptr 31859 | The positive operator orde... |
leopnmid 31860 | A bounded Hermitian operat... |
nmopleid 31861 | A nonzero, bounded Hermiti... |
opsqrlem1 31862 | Lemma for opsqri . (Contr... |
opsqrlem2 31863 | Lemma for opsqri . ` F `` ... |
opsqrlem3 31864 | Lemma for opsqri . (Contr... |
opsqrlem4 31865 | Lemma for opsqri . (Contr... |
opsqrlem5 31866 | Lemma for opsqri . (Contr... |
opsqrlem6 31867 | Lemma for opsqri . (Contr... |
pjhmopi 31868 | A projector is a Hermitian... |
pjlnopi 31869 | A projector is a linear op... |
pjnmopi 31870 | The operator norm of a pro... |
pjbdlni 31871 | A projector is a bounded l... |
pjhmop 31872 | A projection is a Hermitia... |
hmopidmchi 31873 | An idempotent Hermitian op... |
hmopidmpji 31874 | An idempotent Hermitian op... |
hmopidmch 31875 | An idempotent Hermitian op... |
hmopidmpj 31876 | An idempotent Hermitian op... |
pjsdii 31877 | Distributive law for Hilbe... |
pjddii 31878 | Distributive law for Hilbe... |
pjsdi2i 31879 | Chained distributive law f... |
pjcoi 31880 | Composition of projections... |
pjcocli 31881 | Closure of composition of ... |
pjcohcli 31882 | Closure of composition of ... |
pjadjcoi 31883 | Adjoint of composition of ... |
pjcofni 31884 | Functionality of compositi... |
pjss1coi 31885 | Subset relationship for pr... |
pjss2coi 31886 | Subset relationship for pr... |
pjssmi 31887 | Projection meet property. ... |
pjssge0i 31888 | Theorem 4.5(iv)->(v) of [B... |
pjdifnormi 31889 | Theorem 4.5(v)<->(vi) of [... |
pjnormssi 31890 | Theorem 4.5(i)<->(vi) of [... |
pjorthcoi 31891 | Composition of projections... |
pjscji 31892 | The projection of orthogon... |
pjssumi 31893 | The projection on a subspa... |
pjssposi 31894 | Projector ordering can be ... |
pjordi 31895 | The definition of projecto... |
pjssdif2i 31896 | The projection subspace of... |
pjssdif1i 31897 | A necessary and sufficient... |
pjimai 31898 | The image of a projection.... |
pjidmcoi 31899 | A projection is idempotent... |
pjoccoi 31900 | Composition of projections... |
pjtoi 31901 | Subspace sum of projection... |
pjoci 31902 | Projection of orthocomplem... |
pjidmco 31903 | A projection operator is i... |
dfpjop 31904 | Definition of projection o... |
pjhmopidm 31905 | Two ways to express the se... |
elpjidm 31906 | A projection operator is i... |
elpjhmop 31907 | A projection operator is H... |
0leopj 31908 | A projector is a positive ... |
pjadj2 31909 | A projector is self-adjoin... |
pjadj3 31910 | A projector is self-adjoin... |
elpjch 31911 | Reconstruction of the subs... |
elpjrn 31912 | Reconstruction of the subs... |
pjinvari 31913 | A closed subspace ` H ` wi... |
pjin1i 31914 | Lemma for Theorem 1.22 of ... |
pjin2i 31915 | Lemma for Theorem 1.22 of ... |
pjin3i 31916 | Lemma for Theorem 1.22 of ... |
pjclem1 31917 | Lemma for projection commu... |
pjclem2 31918 | Lemma for projection commu... |
pjclem3 31919 | Lemma for projection commu... |
pjclem4a 31920 | Lemma for projection commu... |
pjclem4 31921 | Lemma for projection commu... |
pjci 31922 | Two subspaces commute iff ... |
pjcmul1i 31923 | A necessary and sufficient... |
pjcmul2i 31924 | The projection subspace of... |
pjcohocli 31925 | Closure of composition of ... |
pjadj2coi 31926 | Adjoint of double composit... |
pj2cocli 31927 | Closure of double composit... |
pj3lem1 31928 | Lemma for projection tripl... |
pj3si 31929 | Stronger projection triple... |
pj3i 31930 | Projection triplet theorem... |
pj3cor1i 31931 | Projection triplet corolla... |
pjs14i 31932 | Theorem S-14 of Watanabe, ... |
isst 31935 | Property of a state. (Con... |
ishst 31936 | Property of a complex Hilb... |
sticl 31937 | ` [ 0 , 1 ] ` closure of t... |
stcl 31938 | Real closure of the value ... |
hstcl 31939 | Closure of the value of a ... |
hst1a 31940 | Unit value of a Hilbert-sp... |
hstel2 31941 | Properties of a Hilbert-sp... |
hstorth 31942 | Orthogonality property of ... |
hstosum 31943 | Orthogonal sum property of... |
hstoc 31944 | Sum of a Hilbert-space-val... |
hstnmoc 31945 | Sum of norms of a Hilbert-... |
stge0 31946 | The value of a state is no... |
stle1 31947 | The value of a state is le... |
hstle1 31948 | The norm of the value of a... |
hst1h 31949 | The norm of a Hilbert-spac... |
hst0h 31950 | The norm of a Hilbert-spac... |
hstpyth 31951 | Pythagorean property of a ... |
hstle 31952 | Ordering property of a Hil... |
hstles 31953 | Ordering property of a Hil... |
hstoh 31954 | A Hilbert-space-valued sta... |
hst0 31955 | A Hilbert-space-valued sta... |
sthil 31956 | The value of a state at th... |
stj 31957 | The value of a state on a ... |
sto1i 31958 | The state of a subspace pl... |
sto2i 31959 | The state of the orthocomp... |
stge1i 31960 | If a state is greater than... |
stle0i 31961 | If a state is less than or... |
stlei 31962 | Ordering law for states. ... |
stlesi 31963 | Ordering law for states. ... |
stji1i 31964 | Join of components of Sasa... |
stm1i 31965 | State of component of unit... |
stm1ri 31966 | State of component of unit... |
stm1addi 31967 | Sum of states whose meet i... |
staddi 31968 | If the sum of 2 states is ... |
stm1add3i 31969 | Sum of states whose meet i... |
stadd3i 31970 | If the sum of 3 states is ... |
st0 31971 | The state of the zero subs... |
strlem1 31972 | Lemma for strong state the... |
strlem2 31973 | Lemma for strong state the... |
strlem3a 31974 | Lemma for strong state the... |
strlem3 31975 | Lemma for strong state the... |
strlem4 31976 | Lemma for strong state the... |
strlem5 31977 | Lemma for strong state the... |
strlem6 31978 | Lemma for strong state the... |
stri 31979 | Strong state theorem. The... |
strb 31980 | Strong state theorem (bidi... |
hstrlem2 31981 | Lemma for strong set of CH... |
hstrlem3a 31982 | Lemma for strong set of CH... |
hstrlem3 31983 | Lemma for strong set of CH... |
hstrlem4 31984 | Lemma for strong set of CH... |
hstrlem5 31985 | Lemma for strong set of CH... |
hstrlem6 31986 | Lemma for strong set of CH... |
hstri 31987 | Hilbert space admits a str... |
hstrbi 31988 | Strong CH-state theorem (b... |
largei 31989 | A Hilbert lattice admits a... |
jplem1 31990 | Lemma for Jauch-Piron theo... |
jplem2 31991 | Lemma for Jauch-Piron theo... |
jpi 31992 | The function ` S ` , that ... |
golem1 31993 | Lemma for Godowski's equat... |
golem2 31994 | Lemma for Godowski's equat... |
goeqi 31995 | Godowski's equation, shown... |
stcltr1i 31996 | Property of a strong class... |
stcltr2i 31997 | Property of a strong class... |
stcltrlem1 31998 | Lemma for strong classical... |
stcltrlem2 31999 | Lemma for strong classical... |
stcltrthi 32000 | Theorem for classically st... |
cvbr 32004 | Binary relation expressing... |
cvbr2 32005 | Binary relation expressing... |
cvcon3 32006 | Contraposition law for the... |
cvpss 32007 | The covers relation implie... |
cvnbtwn 32008 | The covers relation implie... |
cvnbtwn2 32009 | The covers relation implie... |
cvnbtwn3 32010 | The covers relation implie... |
cvnbtwn4 32011 | The covers relation implie... |
cvnsym 32012 | The covers relation is not... |
cvnref 32013 | The covers relation is not... |
cvntr 32014 | The covers relation is not... |
spansncv2 32015 | Hilbert space has the cove... |
mdbr 32016 | Binary relation expressing... |
mdi 32017 | Consequence of the modular... |
mdbr2 32018 | Binary relation expressing... |
mdbr3 32019 | Binary relation expressing... |
mdbr4 32020 | Binary relation expressing... |
dmdbr 32021 | Binary relation expressing... |
dmdmd 32022 | The dual modular pair prop... |
mddmd 32023 | The modular pair property ... |
dmdi 32024 | Consequence of the dual mo... |
dmdbr2 32025 | Binary relation expressing... |
dmdi2 32026 | Consequence of the dual mo... |
dmdbr3 32027 | Binary relation expressing... |
dmdbr4 32028 | Binary relation expressing... |
dmdi4 32029 | Consequence of the dual mo... |
dmdbr5 32030 | Binary relation expressing... |
mddmd2 32031 | Relationship between modul... |
mdsl0 32032 | A sublattice condition tha... |
ssmd1 32033 | Ordering implies the modul... |
ssmd2 32034 | Ordering implies the modul... |
ssdmd1 32035 | Ordering implies the dual ... |
ssdmd2 32036 | Ordering implies the dual ... |
dmdsl3 32037 | Sublattice mapping for a d... |
mdsl3 32038 | Sublattice mapping for a m... |
mdslle1i 32039 | Order preservation of the ... |
mdslle2i 32040 | Order preservation of the ... |
mdslj1i 32041 | Join preservation of the o... |
mdslj2i 32042 | Meet preservation of the r... |
mdsl1i 32043 | If the modular pair proper... |
mdsl2i 32044 | If the modular pair proper... |
mdsl2bi 32045 | If the modular pair proper... |
cvmdi 32046 | The covering property impl... |
mdslmd1lem1 32047 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem2 32048 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem3 32049 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem4 32050 | Lemma for ~ mdslmd1i . (C... |
mdslmd1i 32051 | Preservation of the modula... |
mdslmd2i 32052 | Preservation of the modula... |
mdsldmd1i 32053 | Preservation of the dual m... |
mdslmd3i 32054 | Modular pair conditions th... |
mdslmd4i 32055 | Modular pair condition tha... |
csmdsymi 32056 | Cross-symmetry implies M-s... |
mdexchi 32057 | An exchange lemma for modu... |
cvmd 32058 | The covering property impl... |
cvdmd 32059 | The covering property impl... |
ela 32061 | Atoms in a Hilbert lattice... |
elat2 32062 | Expanded membership relati... |
elatcv0 32063 | A Hilbert lattice element ... |
atcv0 32064 | An atom covers the zero su... |
atssch 32065 | Atoms are a subset of the ... |
atelch 32066 | An atom is a Hilbert latti... |
atne0 32067 | An atom is not the Hilbert... |
atss 32068 | A lattice element smaller ... |
atsseq 32069 | Two atoms in a subset rela... |
atcveq0 32070 | A Hilbert lattice element ... |
h1da 32071 | A 1-dimensional subspace i... |
spansna 32072 | The span of the singleton ... |
sh1dle 32073 | A 1-dimensional subspace i... |
ch1dle 32074 | A 1-dimensional subspace i... |
atom1d 32075 | The 1-dimensional subspace... |
superpos 32076 | Superposition Principle. ... |
chcv1 32077 | The Hilbert lattice has th... |
chcv2 32078 | The Hilbert lattice has th... |
chjatom 32079 | The join of a closed subsp... |
shatomici 32080 | The lattice of Hilbert sub... |
hatomici 32081 | The Hilbert lattice is ato... |
hatomic 32082 | A Hilbert lattice is atomi... |
shatomistici 32083 | The lattice of Hilbert sub... |
hatomistici 32084 | ` CH ` is atomistic, i.e. ... |
chpssati 32085 | Two Hilbert lattice elemen... |
chrelati 32086 | The Hilbert lattice is rel... |
chrelat2i 32087 | A consequence of relative ... |
cvati 32088 | If a Hilbert lattice eleme... |
cvbr4i 32089 | An alternate way to expres... |
cvexchlem 32090 | Lemma for ~ cvexchi . (Co... |
cvexchi 32091 | The Hilbert lattice satisf... |
chrelat2 32092 | A consequence of relative ... |
chrelat3 32093 | A consequence of relative ... |
chrelat3i 32094 | A consequence of the relat... |
chrelat4i 32095 | A consequence of relative ... |
cvexch 32096 | The Hilbert lattice satisf... |
cvp 32097 | The Hilbert lattice satisf... |
atnssm0 32098 | The meet of a Hilbert latt... |
atnemeq0 32099 | The meet of distinct atoms... |
atssma 32100 | The meet with an atom's su... |
atcv0eq 32101 | Two atoms covering the zer... |
atcv1 32102 | Two atoms covering the zer... |
atexch 32103 | The Hilbert lattice satisf... |
atomli 32104 | An assertion holding in at... |
atoml2i 32105 | An assertion holding in at... |
atordi 32106 | An ordering law for a Hilb... |
atcvatlem 32107 | Lemma for ~ atcvati . (Co... |
atcvati 32108 | A nonzero Hilbert lattice ... |
atcvat2i 32109 | A Hilbert lattice element ... |
atord 32110 | An ordering law for a Hilb... |
atcvat2 32111 | A Hilbert lattice element ... |
chirredlem1 32112 | Lemma for ~ chirredi . (C... |
chirredlem2 32113 | Lemma for ~ chirredi . (C... |
chirredlem3 32114 | Lemma for ~ chirredi . (C... |
chirredlem4 32115 | Lemma for ~ chirredi . (C... |
chirredi 32116 | The Hilbert lattice is irr... |
chirred 32117 | The Hilbert lattice is irr... |
atcvat3i 32118 | A condition implying that ... |
atcvat4i 32119 | A condition implying exist... |
atdmd 32120 | Two Hilbert lattice elemen... |
atmd 32121 | Two Hilbert lattice elemen... |
atmd2 32122 | Two Hilbert lattice elemen... |
atabsi 32123 | Absorption of an incompara... |
atabs2i 32124 | Absorption of an incompara... |
mdsymlem1 32125 | Lemma for ~ mdsymi . (Con... |
mdsymlem2 32126 | Lemma for ~ mdsymi . (Con... |
mdsymlem3 32127 | Lemma for ~ mdsymi . (Con... |
mdsymlem4 32128 | Lemma for ~ mdsymi . This... |
mdsymlem5 32129 | Lemma for ~ mdsymi . (Con... |
mdsymlem6 32130 | Lemma for ~ mdsymi . This... |
mdsymlem7 32131 | Lemma for ~ mdsymi . Lemm... |
mdsymlem8 32132 | Lemma for ~ mdsymi . Lemm... |
mdsymi 32133 | M-symmetry of the Hilbert ... |
mdsym 32134 | M-symmetry of the Hilbert ... |
dmdsym 32135 | Dual M-symmetry of the Hil... |
atdmd2 32136 | Two Hilbert lattice elemen... |
sumdmdii 32137 | If the subspace sum of two... |
cmmdi 32138 | Commuting subspaces form a... |
cmdmdi 32139 | Commuting subspaces form a... |
sumdmdlem 32140 | Lemma for ~ sumdmdi . The... |
sumdmdlem2 32141 | Lemma for ~ sumdmdi . (Co... |
sumdmdi 32142 | The subspace sum of two Hi... |
dmdbr4ati 32143 | Dual modular pair property... |
dmdbr5ati 32144 | Dual modular pair property... |
dmdbr6ati 32145 | Dual modular pair property... |
dmdbr7ati 32146 | Dual modular pair property... |
mdoc1i 32147 | Orthocomplements form a mo... |
mdoc2i 32148 | Orthocomplements form a mo... |
dmdoc1i 32149 | Orthocomplements form a du... |
dmdoc2i 32150 | Orthocomplements form a du... |
mdcompli 32151 | A condition equivalent to ... |
dmdcompli 32152 | A condition equivalent to ... |
mddmdin0i 32153 | If dual modular implies mo... |
cdjreui 32154 | A member of the sum of dis... |
cdj1i 32155 | Two ways to express " ` A ... |
cdj3lem1 32156 | A property of " ` A ` and ... |
cdj3lem2 32157 | Lemma for ~ cdj3i . Value... |
cdj3lem2a 32158 | Lemma for ~ cdj3i . Closu... |
cdj3lem2b 32159 | Lemma for ~ cdj3i . The f... |
cdj3lem3 32160 | Lemma for ~ cdj3i . Value... |
cdj3lem3a 32161 | Lemma for ~ cdj3i . Closu... |
cdj3lem3b 32162 | Lemma for ~ cdj3i . The s... |
cdj3i 32163 | Two ways to express " ` A ... |
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
mathbox 32164 | (_This theorem is a dummy ... |
sa-abvi 32165 | A theorem about the univer... |
xfree 32166 | A partial converse to ~ 19... |
xfree2 32167 | A partial converse to ~ 19... |
addltmulALT 32168 | A proof readability experi... |
bian1d 32169 | Adding a superfluous conju... |
or3di 32170 | Distributive law for disju... |
or3dir 32171 | Distributive law for disju... |
3o1cs 32172 | Deduction eliminating disj... |
3o2cs 32173 | Deduction eliminating disj... |
3o3cs 32174 | Deduction eliminating disj... |
13an22anass 32175 | Associative law for four c... |
sbc2iedf 32176 | Conversion of implicit sub... |
rspc2daf 32177 | Double restricted speciali... |
ralcom4f 32178 | Commutation of restricted ... |
rexcom4f 32179 | Commutation of restricted ... |
19.9d2rf 32180 | A deduction version of one... |
19.9d2r 32181 | A deduction version of one... |
r19.29ffa 32182 | A commonly used pattern ba... |
eqtrb 32183 | A transposition of equalit... |
eqelbid 32184 | A variable elimination law... |
opsbc2ie 32185 | Conversion of implicit sub... |
opreu2reuALT 32186 | Correspondence between uni... |
2reucom 32189 | Double restricted existent... |
2reu2rex1 32190 | Double restricted existent... |
2reureurex 32191 | Double restricted existent... |
2reu2reu2 32192 | Double restricted existent... |
opreu2reu1 32193 | Equivalent definition of t... |
sq2reunnltb 32194 | There exists a unique deco... |
addsqnot2reu 32195 | For each complex number ` ... |
sbceqbidf 32196 | Equality theorem for class... |
sbcies 32197 | A special version of class... |
mo5f 32198 | Alternate definition of "a... |
nmo 32199 | Negation of "at most one".... |
reuxfrdf 32200 | Transfer existential uniqu... |
rexunirn 32201 | Restricted existential qua... |
rmoxfrd 32202 | Transfer "at most one" res... |
rmoun 32203 | "At most one" restricted e... |
rmounid 32204 | A case where an "at most o... |
riotaeqbidva 32205 | Equivalent wff's yield equ... |
dmrab 32206 | Domain of a restricted cla... |
difrab2 32207 | Difference of two restrict... |
rabexgfGS 32208 | Separation Scheme in terms... |
rabsnel 32209 | Truth implied by equality ... |
eqrrabd 32210 | Deduce equality with a res... |
foresf1o 32211 | From a surjective function... |
rabfodom 32212 | Domination relation for re... |
abrexdomjm 32213 | An indexed set is dominate... |
abrexdom2jm 32214 | An indexed set is dominate... |
abrexexd 32215 | Existence of a class abstr... |
elabreximd 32216 | Class substitution in an i... |
elabreximdv 32217 | Class substitution in an i... |
abrexss 32218 | A necessary condition for ... |
elunsn 32219 | Elementhood to a union wit... |
nelun 32220 | Negated membership for a u... |
snsssng 32221 | If a singleton is a subset... |
inin 32222 | Intersection with an inter... |
inindif 32223 | See ~ inundif . (Contribu... |
difininv 32224 | Condition for the intersec... |
difeq 32225 | Rewriting an equation with... |
eqdif 32226 | If both set differences of... |
indifbi 32227 | Two ways to express equali... |
diffib 32228 | Case where ~ diffi is a bi... |
difxp1ss 32229 | Difference law for Cartesi... |
difxp2ss 32230 | Difference law for Cartesi... |
indifundif 32231 | A remarkable equation with... |
elpwincl1 32232 | Closure of intersection wi... |
elpwdifcl 32233 | Closure of class differenc... |
elpwiuncl 32234 | Closure of indexed union w... |
eqsnd 32235 | Deduce that a set is a sin... |
elpreq 32236 | Equality wihin a pair. (C... |
nelpr 32237 | A set ` A ` not in a pair ... |
inpr0 32238 | Rewrite an empty intersect... |
neldifpr1 32239 | The first element of a pai... |
neldifpr2 32240 | The second element of a pa... |
unidifsnel 32241 | The other element of a pai... |
unidifsnne 32242 | The other element of a pai... |
ifeqeqx 32243 | An equality theorem tailor... |
elimifd 32244 | Elimination of a condition... |
elim2if 32245 | Elimination of two conditi... |
elim2ifim 32246 | Elimination of two conditi... |
ifeq3da 32247 | Given an expression ` C ` ... |
ifnetrue 32248 | Deduce truth from a condit... |
ifnefals 32249 | Deduce falsehood from a co... |
ifnebib 32250 | The converse of ~ ifbi hol... |
uniinn0 32251 | Sufficient and necessary c... |
uniin1 32252 | Union of intersection. Ge... |
uniin2 32253 | Union of intersection. Ge... |
difuncomp 32254 | Express a class difference... |
elpwunicl 32255 | Closure of a set union wit... |
cbviunf 32256 | Rule used to change the bo... |
iuneq12daf 32257 | Equality deduction for ind... |
iunin1f 32258 | Indexed union of intersect... |
ssiun3 32259 | Subset equivalence for an ... |
ssiun2sf 32260 | Subset relationship for an... |
iuninc 32261 | The union of an increasing... |
iundifdifd 32262 | The intersection of a set ... |
iundifdif 32263 | The intersection of a set ... |
iunrdx 32264 | Re-index an indexed union.... |
iunpreima 32265 | Preimage of an indexed uni... |
iunrnmptss 32266 | A subset relation for an i... |
iunxunsn 32267 | Appending a set to an inde... |
iunxunpr 32268 | Appending two sets to an i... |
iinabrex 32269 | Rewriting an indexed inter... |
disjnf 32270 | In case ` x ` is not free ... |
cbvdisjf 32271 | Change bound variables in ... |
disjss1f 32272 | A subset of a disjoint col... |
disjeq1f 32273 | Equality theorem for disjo... |
disjxun0 32274 | Simplify a disjoint union.... |
disjdifprg 32275 | A trivial partition into a... |
disjdifprg2 32276 | A trivial partition of a s... |
disji2f 32277 | Property of a disjoint col... |
disjif 32278 | Property of a disjoint col... |
disjorf 32279 | Two ways to say that a col... |
disjorsf 32280 | Two ways to say that a col... |
disjif2 32281 | Property of a disjoint col... |
disjabrex 32282 | Rewriting a disjoint colle... |
disjabrexf 32283 | Rewriting a disjoint colle... |
disjpreima 32284 | A preimage of a disjoint s... |
disjrnmpt 32285 | Rewriting a disjoint colle... |
disjin 32286 | If a collection is disjoin... |
disjin2 32287 | If a collection is disjoin... |
disjxpin 32288 | Derive a disjunction over ... |
iundisjf 32289 | Rewrite a countable union ... |
iundisj2f 32290 | A disjoint union is disjoi... |
disjrdx 32291 | Re-index a disjunct collec... |
disjex 32292 | Two ways to say that two c... |
disjexc 32293 | A variant of ~ disjex , ap... |
disjunsn 32294 | Append an element to a dis... |
disjun0 32295 | Adding the empty element p... |
disjiunel 32296 | A set of elements B of a d... |
disjuniel 32297 | A set of elements B of a d... |
xpdisjres 32298 | Restriction of a constant ... |
opeldifid 32299 | Ordered pair elementhood o... |
difres 32300 | Case when class difference... |
imadifxp 32301 | Image of the difference wi... |
relfi 32302 | A relation (set) is finite... |
0res 32303 | Restriction of the empty f... |
fcoinver 32304 | Build an equivalence relat... |
fcoinvbr 32305 | Binary relation for the eq... |
brabgaf 32306 | The law of concretion for ... |
brelg 32307 | Two things in a binary rel... |
br8d 32308 | Substitution for an eight-... |
opabdm 32309 | Domain of an ordered-pair ... |
opabrn 32310 | Range of an ordered-pair c... |
opabssi 32311 | Sufficient condition for a... |
opabid2ss 32312 | One direction of ~ opabid2... |
ssrelf 32313 | A subclass relationship de... |
eqrelrd2 32314 | A version of ~ eqrelrdv2 w... |
erbr3b 32315 | Biconditional for equivale... |
iunsnima 32316 | Image of a singleton by an... |
iunsnima2 32317 | Version of ~ iunsnima with... |
ac6sf2 32318 | Alternate version of ~ ac6... |
fnresin 32319 | Restriction of a function ... |
f1o3d 32320 | Describe an implicit one-t... |
eldmne0 32321 | A function of nonempty dom... |
f1rnen 32322 | Equinumerosity of the rang... |
rinvf1o 32323 | Sufficient conditions for ... |
fresf1o 32324 | Conditions for a restricti... |
nfpconfp 32325 | The set of fixed points of... |
fmptco1f1o 32326 | The action of composing (t... |
cofmpt2 32327 | Express composition of a m... |
f1mptrn 32328 | Express injection for a ma... |
dfimafnf 32329 | Alternate definition of th... |
funimass4f 32330 | Membership relation for th... |
elimampt 32331 | Membership in the image of... |
suppss2f 32332 | Show that the support of a... |
ofrn 32333 | The range of the function ... |
ofrn2 32334 | The range of the function ... |
off2 32335 | The function operation pro... |
ofresid 32336 | Applying an operation rest... |
fimarab 32337 | Expressing the image of a ... |
unipreima 32338 | Preimage of a class union.... |
opfv 32339 | Value of a function produc... |
xppreima 32340 | The preimage of a Cartesia... |
2ndimaxp 32341 | Image of a cartesian produ... |
djussxp2 32342 | Stronger version of ~ djus... |
2ndresdju 32343 | The ` 2nd ` function restr... |
2ndresdjuf1o 32344 | The ` 2nd ` function restr... |
xppreima2 32345 | The preimage of a Cartesia... |
abfmpunirn 32346 | Membership in a union of a... |
rabfmpunirn 32347 | Membership in a union of a... |
abfmpeld 32348 | Membership in an element o... |
abfmpel 32349 | Membership in an element o... |
fmptdF 32350 | Domain and codomain of the... |
fmptcof2 32351 | Composition of two functio... |
fcomptf 32352 | Express composition of two... |
acunirnmpt 32353 | Axiom of choice for the un... |
acunirnmpt2 32354 | Axiom of choice for the un... |
acunirnmpt2f 32355 | Axiom of choice for the un... |
aciunf1lem 32356 | Choice in an index union. ... |
aciunf1 32357 | Choice in an index union. ... |
ofoprabco 32358 | Function operation as a co... |
ofpreima 32359 | Express the preimage of a ... |
ofpreima2 32360 | Express the preimage of a ... |
funcnvmpt 32361 | Condition for a function i... |
funcnv5mpt 32362 | Two ways to say that a fun... |
funcnv4mpt 32363 | Two ways to say that a fun... |
preimane 32364 | Different elements have di... |
fnpreimac 32365 | Choose a set ` x ` contain... |
fgreu 32366 | Exactly one point of a fun... |
fcnvgreu 32367 | If the converse of a relat... |
rnmposs 32368 | The range of an operation ... |
mptssALT 32369 | Deduce subset relation of ... |
dfcnv2 32370 | Alternative definition of ... |
fnimatp 32371 | The image of an unordered ... |
rnexd 32372 | The range of a set is a se... |
imaexd 32373 | The image of a set is a se... |
mpomptxf 32374 | Express a two-argument fun... |
suppovss 32375 | A bound for the support of... |
fvdifsupp 32376 | Function value is zero out... |
suppiniseg 32377 | Relation between the suppo... |
fsuppinisegfi 32378 | The initial segment ` ( ``... |
fressupp 32379 | The restriction of a funct... |
fdifsuppconst 32380 | A function is a zero const... |
ressupprn 32381 | The range of a function re... |
supppreima 32382 | Express the support of a f... |
fsupprnfi 32383 | Finite support implies fin... |
mptiffisupp 32384 | Conditions for a mapping f... |
cosnopne 32385 | Composition of two ordered... |
cosnop 32386 | Composition of two ordered... |
cnvprop 32387 | Converse of a pair of orde... |
brprop 32388 | Binary relation for a pair... |
mptprop 32389 | Rewrite pairs of ordered p... |
coprprop 32390 | Composition of two pairs o... |
gtiso 32391 | Two ways to write a strict... |
isoun 32392 | Infer an isomorphism from ... |
disjdsct 32393 | A disjoint collection is d... |
df1stres 32394 | Definition for a restricti... |
df2ndres 32395 | Definition for a restricti... |
1stpreimas 32396 | The preimage of a singleto... |
1stpreima 32397 | The preimage by ` 1st ` is... |
2ndpreima 32398 | The preimage by ` 2nd ` is... |
curry2ima 32399 | The image of a curried fun... |
preiman0 32400 | The preimage of a nonempty... |
intimafv 32401 | The intersection of an ima... |
ecref 32402 | All elements are in their ... |
supssd 32403 | Inequality deduction for s... |
infssd 32404 | Inequality deduction for i... |
imafi2 32405 | The image by a finite set ... |
unifi3 32406 | If a union is finite, then... |
snct 32407 | A singleton is countable. ... |
prct 32408 | An unordered pair is count... |
mpocti 32409 | An operation is countable ... |
abrexct 32410 | An image set of a countabl... |
mptctf 32411 | A countable mapping set is... |
abrexctf 32412 | An image set of a countabl... |
padct 32413 | Index a countable set with... |
cnvoprabOLD 32414 | The converse of a class ab... |
f1od2 32415 | Sufficient condition for a... |
fcobij 32416 | Composing functions with a... |
fcobijfs 32417 | Composing finitely support... |
suppss3 32418 | Deduce a function's suppor... |
fsuppcurry1 32419 | Finite support of a currie... |
fsuppcurry2 32420 | Finite support of a currie... |
offinsupp1 32421 | Finite support for a funct... |
ffs2 32422 | Rewrite a function's suppo... |
ffsrn 32423 | The range of a finitely su... |
resf1o 32424 | Restriction of functions t... |
maprnin 32425 | Restricting the range of t... |
fpwrelmapffslem 32426 | Lemma for ~ fpwrelmapffs .... |
fpwrelmap 32427 | Define a canonical mapping... |
fpwrelmapffs 32428 | Define a canonical mapping... |
creq0 32429 | The real representation of... |
1nei 32430 | The imaginary unit ` _i ` ... |
1neg1t1neg1 32431 | An integer unit times itse... |
nnmulge 32432 | Multiplying by a positive ... |
lt2addrd 32433 | If the right-hand side of ... |
xrlelttric 32434 | Trichotomy law for extende... |
xaddeq0 32435 | Two extended reals which a... |
xrinfm 32436 | The extended real numbers ... |
le2halvesd 32437 | A sum is less than the who... |
xraddge02 32438 | A number is less than or e... |
xrge0addge 32439 | A number is less than or e... |
xlt2addrd 32440 | If the right-hand side of ... |
xrsupssd 32441 | Inequality deduction for s... |
xrge0infss 32442 | Any subset of nonnegative ... |
xrge0infssd 32443 | Inequality deduction for i... |
xrge0addcld 32444 | Nonnegative extended reals... |
xrge0subcld 32445 | Condition for closure of n... |
infxrge0lb 32446 | A member of a set of nonne... |
infxrge0glb 32447 | The infimum of a set of no... |
infxrge0gelb 32448 | The infimum of a set of no... |
xrofsup 32449 | The supremum is preserved ... |
supxrnemnf 32450 | The supremum of a nonempty... |
xnn0gt0 32451 | Nonzero extended nonnegati... |
xnn01gt 32452 | An extended nonnegative in... |
nn0xmulclb 32453 | Finite multiplication in t... |
joiniooico 32454 | Disjoint joining an open i... |
ubico 32455 | A right-open interval does... |
xeqlelt 32456 | Equality in terms of 'less... |
eliccelico 32457 | Relate elementhood to a cl... |
elicoelioo 32458 | Relate elementhood to a cl... |
iocinioc2 32459 | Intersection between two o... |
xrdifh 32460 | Class difference of a half... |
iocinif 32461 | Relate intersection of two... |
difioo 32462 | The difference between two... |
difico 32463 | The difference between two... |
uzssico 32464 | Upper integer sets are a s... |
fz2ssnn0 32465 | A finite set of sequential... |
nndiffz1 32466 | Upper set of the positive ... |
ssnnssfz 32467 | For any finite subset of `... |
fzne1 32468 | Elementhood in a finite se... |
fzm1ne1 32469 | Elementhood of an integer ... |
fzspl 32470 | Split the last element of ... |
fzdif2 32471 | Split the last element of ... |
fzodif2 32472 | Split the last element of ... |
fzodif1 32473 | Set difference of two half... |
fzsplit3 32474 | Split a finite interval of... |
bcm1n 32475 | The proportion of one bino... |
iundisjfi 32476 | Rewrite a countable union ... |
iundisj2fi 32477 | A disjoint union is disjoi... |
iundisjcnt 32478 | Rewrite a countable union ... |
iundisj2cnt 32479 | A countable disjoint union... |
fzone1 32480 | Elementhood in a half-open... |
fzom1ne1 32481 | Elementhood in a half-open... |
f1ocnt 32482 | Given a countable set ` A ... |
fz1nnct 32483 | NN and integer ranges star... |
fz1nntr 32484 | NN and integer ranges star... |
nn0difffzod 32485 | A nonnegative integer that... |
suppssnn0 32486 | Show that the support of a... |
hashunif 32487 | The cardinality of a disjo... |
hashxpe 32488 | The size of the Cartesian ... |
hashgt1 32489 | Restate "set contains at l... |
numdenneg 32490 | Numerator and denominator ... |
divnumden2 32491 | Calculate the reduced form... |
nnindf 32492 | Principle of Mathematical ... |
nn0min 32493 | Extracting the minimum pos... |
subne0nn 32494 | A nonnegative difference i... |
ltesubnnd 32495 | Subtracting an integer num... |
fprodeq02 32496 | If one of the factors is z... |
pr01ssre 32497 | The range of the indicator... |
fprodex01 32498 | A product of factors equal... |
prodpr 32499 | A product over a pair is t... |
prodtp 32500 | A product over a triple is... |
fsumub 32501 | An upper bound for a term ... |
fsumiunle 32502 | Upper bound for a sum of n... |
dfdec100 32503 | Split the hundreds from a ... |
dp2eq1 32506 | Equality theorem for the d... |
dp2eq2 32507 | Equality theorem for the d... |
dp2eq1i 32508 | Equality theorem for the d... |
dp2eq2i 32509 | Equality theorem for the d... |
dp2eq12i 32510 | Equality theorem for the d... |
dp20u 32511 | Add a zero in the tenths (... |
dp20h 32512 | Add a zero in the unit pla... |
dp2cl 32513 | Closure for the decimal fr... |
dp2clq 32514 | Closure for a decimal frac... |
rpdp2cl 32515 | Closure for a decimal frac... |
rpdp2cl2 32516 | Closure for a decimal frac... |
dp2lt10 32517 | Decimal fraction builds re... |
dp2lt 32518 | Comparing two decimal frac... |
dp2ltsuc 32519 | Comparing a decimal fracti... |
dp2ltc 32520 | Comparing two decimal expa... |
dpval 32523 | Define the value of the de... |
dpcl 32524 | Prove that the closure of ... |
dpfrac1 32525 | Prove a simple equivalence... |
dpval2 32526 | Value of the decimal point... |
dpval3 32527 | Value of the decimal point... |
dpmul10 32528 | Multiply by 10 a decimal e... |
decdiv10 32529 | Divide a decimal number by... |
dpmul100 32530 | Multiply by 100 a decimal ... |
dp3mul10 32531 | Multiply by 10 a decimal e... |
dpmul1000 32532 | Multiply by 1000 a decimal... |
dpval3rp 32533 | Value of the decimal point... |
dp0u 32534 | Add a zero in the tenths p... |
dp0h 32535 | Remove a zero in the units... |
rpdpcl 32536 | Closure of the decimal poi... |
dplt 32537 | Comparing two decimal expa... |
dplti 32538 | Comparing a decimal expans... |
dpgti 32539 | Comparing a decimal expans... |
dpltc 32540 | Comparing two decimal inte... |
dpexpp1 32541 | Add one zero to the mantis... |
0dp2dp 32542 | Multiply by 10 a decimal e... |
dpadd2 32543 | Addition with one decimal,... |
dpadd 32544 | Addition with one decimal.... |
dpadd3 32545 | Addition with two decimals... |
dpmul 32546 | Multiplication with one de... |
dpmul4 32547 | An upper bound to multipli... |
threehalves 32548 | Example theorem demonstrat... |
1mhdrd 32549 | Example theorem demonstrat... |
xdivval 32552 | Value of division: the (un... |
xrecex 32553 | Existence of reciprocal of... |
xmulcand 32554 | Cancellation law for exten... |
xreceu 32555 | Existential uniqueness of ... |
xdivcld 32556 | Closure law for the extend... |
xdivcl 32557 | Closure law for the extend... |
xdivmul 32558 | Relationship between divis... |
rexdiv 32559 | The extended real division... |
xdivrec 32560 | Relationship between divis... |
xdivid 32561 | A number divided by itself... |
xdiv0 32562 | Division into zero is zero... |
xdiv0rp 32563 | Division into zero is zero... |
eliccioo 32564 | Membership in a closed int... |
elxrge02 32565 | Elementhood in the set of ... |
xdivpnfrp 32566 | Plus infinity divided by a... |
rpxdivcld 32567 | Closure law for extended d... |
xrpxdivcld 32568 | Closure law for extended d... |
wrdfd 32569 | A word is a zero-based seq... |
wrdres 32570 | Condition for the restrict... |
wrdsplex 32571 | Existence of a split of a ... |
pfx1s2 32572 | The prefix of length 1 of ... |
pfxrn2 32573 | The range of a prefix of a... |
pfxrn3 32574 | Express the range of a pre... |
pfxf1 32575 | Condition for a prefix to ... |
s1f1 32576 | Conditions for a length 1 ... |
s2rn 32577 | Range of a length 2 string... |
s2f1 32578 | Conditions for a length 2 ... |
s3rn 32579 | Range of a length 3 string... |
s3f1 32580 | Conditions for a length 3 ... |
s3clhash 32581 | Closure of the words of le... |
ccatf1 32582 | Conditions for a concatena... |
pfxlsw2ccat 32583 | Reconstruct a word from it... |
wrdt2ind 32584 | Perform an induction over ... |
swrdrn2 32585 | The range of a subword is ... |
swrdrn3 32586 | Express the range of a sub... |
swrdf1 32587 | Condition for a subword to... |
swrdrndisj 32588 | Condition for the range of... |
splfv3 32589 | Symbols to the right of a ... |
1cshid 32590 | Cyclically shifting a sing... |
cshw1s2 32591 | Cyclically shifting a leng... |
cshwrnid 32592 | Cyclically shifting a word... |
cshf1o 32593 | Condition for the cyclic s... |
ressplusf 32594 | The group operation functi... |
ressnm 32595 | The norm in a restricted s... |
abvpropd2 32596 | Weaker version of ~ abvpro... |
oppgle 32597 | less-than relation of an o... |
oppgleOLD 32598 | Obsolete version of ~ oppg... |
oppglt 32599 | less-than relation of an o... |
ressprs 32600 | The restriction of a prose... |
oduprs 32601 | Being a proset is a self-d... |
posrasymb 32602 | A poset ordering is asymet... |
resspos 32603 | The restriction of a Poset... |
resstos 32604 | The restriction of a Toset... |
odutos 32605 | Being a toset is a self-du... |
tlt2 32606 | In a Toset, two elements m... |
tlt3 32607 | In a Toset, two elements m... |
trleile 32608 | In a Toset, two elements m... |
toslublem 32609 | Lemma for ~ toslub and ~ x... |
toslub 32610 | In a toset, the lowest upp... |
tosglblem 32611 | Lemma for ~ tosglb and ~ x... |
tosglb 32612 | Same theorem as ~ toslub ,... |
clatp0cl 32613 | The poset zero of a comple... |
clatp1cl 32614 | The poset one of a complet... |
mntoval 32619 | Operation value of the mon... |
ismnt 32620 | Express the statement " ` ... |
ismntd 32621 | Property of being a monoto... |
mntf 32622 | A monotone function is a f... |
mgcoval 32623 | Operation value of the mon... |
mgcval 32624 | Monotone Galois connection... |
mgcf1 32625 | The lower adjoint ` F ` of... |
mgcf2 32626 | The upper adjoint ` G ` of... |
mgccole1 32627 | An inequality for the kern... |
mgccole2 32628 | Inequality for the closure... |
mgcmnt1 32629 | The lower adjoint ` F ` of... |
mgcmnt2 32630 | The upper adjoint ` G ` of... |
mgcmntco 32631 | A Galois connection like s... |
dfmgc2lem 32632 | Lemma for dfmgc2, backward... |
dfmgc2 32633 | Alternate definition of th... |
mgcmnt1d 32634 | Galois connection implies ... |
mgcmnt2d 32635 | Galois connection implies ... |
mgccnv 32636 | The inverse Galois connect... |
pwrssmgc 32637 | Given a function ` F ` , e... |
mgcf1olem1 32638 | Property of a Galois conne... |
mgcf1olem2 32639 | Property of a Galois conne... |
mgcf1o 32640 | Given a Galois connection,... |
xrs0 32643 | The zero of the extended r... |
xrslt 32644 | The "strictly less than" r... |
xrsinvgval 32645 | The inversion operation in... |
xrsmulgzz 32646 | The "multiple" function in... |
xrstos 32647 | The extended real numbers ... |
xrsclat 32648 | The extended real numbers ... |
xrsp0 32649 | The poset 0 of the extende... |
xrsp1 32650 | The poset 1 of the extende... |
xrge0base 32651 | The base of the extended n... |
xrge00 32652 | The zero of the extended n... |
xrge0plusg 32653 | The additive law of the ex... |
xrge0le 32654 | The "less than or equal to... |
xrge0mulgnn0 32655 | The group multiple functio... |
xrge0addass 32656 | Associativity of extended ... |
xrge0addgt0 32657 | The sum of nonnegative and... |
xrge0adddir 32658 | Right-distributivity of ex... |
xrge0adddi 32659 | Left-distributivity of ext... |
xrge0npcan 32660 | Extended nonnegative real ... |
fsumrp0cl 32661 | Closure of a finite sum of... |
abliso 32662 | The image of an Abelian gr... |
lmhmghmd 32663 | A module homomorphism is a... |
mhmimasplusg 32664 | Value of the operation of ... |
lmhmimasvsca 32665 | Value of the scalar produc... |
gsumsubg 32666 | The group sum in a subgrou... |
gsumsra 32667 | The group sum in a subring... |
gsummpt2co 32668 | Split a finite sum into a ... |
gsummpt2d 32669 | Express a finite sum over ... |
lmodvslmhm 32670 | Scalar multiplication in a... |
gsumvsmul1 32671 | Pull a scalar multiplicati... |
gsummptres 32672 | Extend a finite group sum ... |
gsummptres2 32673 | Extend a finite group sum ... |
gsumzresunsn 32674 | Append an element to a fin... |
gsumpart 32675 | Express a group sum as a d... |
gsumhashmul 32676 | Express a group sum by gro... |
xrge0tsmsd 32677 | Any finite or infinite sum... |
xrge0tsmsbi 32678 | Any limit of a finite or i... |
xrge0tsmseq 32679 | Any limit of a finite or i... |
cntzun 32680 | The centralizer of a union... |
cntzsnid 32681 | The centralizer of the ide... |
cntrcrng 32682 | The center of a ring is a ... |
isomnd 32687 | A (left) ordered monoid is... |
isogrp 32688 | A (left-)ordered group is ... |
ogrpgrp 32689 | A left-ordered group is a ... |
omndmnd 32690 | A left-ordered monoid is a... |
omndtos 32691 | A left-ordered monoid is a... |
omndadd 32692 | In an ordered monoid, the ... |
omndaddr 32693 | In a right ordered monoid,... |
omndadd2d 32694 | In a commutative left orde... |
omndadd2rd 32695 | In a left- and right- orde... |
submomnd 32696 | A submonoid of an ordered ... |
xrge0omnd 32697 | The nonnegative extended r... |
omndmul2 32698 | In an ordered monoid, the ... |
omndmul3 32699 | In an ordered monoid, the ... |
omndmul 32700 | In a commutative ordered m... |
ogrpinv0le 32701 | In an ordered group, the o... |
ogrpsub 32702 | In an ordered group, the o... |
ogrpaddlt 32703 | In an ordered group, stric... |
ogrpaddltbi 32704 | In a right ordered group, ... |
ogrpaddltrd 32705 | In a right ordered group, ... |
ogrpaddltrbid 32706 | In a right ordered group, ... |
ogrpsublt 32707 | In an ordered group, stric... |
ogrpinv0lt 32708 | In an ordered group, the o... |
ogrpinvlt 32709 | In an ordered group, the o... |
gsumle 32710 | A finite sum in an ordered... |
symgfcoeu 32711 | Uniqueness property of per... |
symgcom 32712 | Two permutations ` X ` and... |
symgcom2 32713 | Two permutations ` X ` and... |
symgcntz 32714 | All elements of a (finite)... |
odpmco 32715 | The composition of two odd... |
symgsubg 32716 | The value of the group sub... |
pmtrprfv2 32717 | In a transposition of two ... |
pmtrcnel 32718 | Composing a permutation ` ... |
pmtrcnel2 32719 | Variation on ~ pmtrcnel . ... |
pmtrcnelor 32720 | Composing a permutation ` ... |
pmtridf1o 32721 | Transpositions of ` X ` an... |
pmtridfv1 32722 | Value at X of the transpos... |
pmtridfv2 32723 | Value at Y of the transpos... |
psgnid 32724 | Permutation sign of the id... |
psgndmfi 32725 | For a finite base set, the... |
pmtrto1cl 32726 | Useful lemma for the follo... |
psgnfzto1stlem 32727 | Lemma for ~ psgnfzto1st . ... |
fzto1stfv1 32728 | Value of our permutation `... |
fzto1st1 32729 | Special case where the per... |
fzto1st 32730 | The function moving one el... |
fzto1stinvn 32731 | Value of the inverse of ou... |
psgnfzto1st 32732 | The permutation sign for m... |
tocycval 32735 | Value of the cycle builder... |
tocycfv 32736 | Function value of a permut... |
tocycfvres1 32737 | A cyclic permutation is a ... |
tocycfvres2 32738 | A cyclic permutation is th... |
cycpmfvlem 32739 | Lemma for ~ cycpmfv1 and ~... |
cycpmfv1 32740 | Value of a cycle function ... |
cycpmfv2 32741 | Value of a cycle function ... |
cycpmfv3 32742 | Values outside of the orbi... |
cycpmcl 32743 | Cyclic permutations are pe... |
tocycf 32744 | The permutation cycle buil... |
tocyc01 32745 | Permutation cycles built f... |
cycpm2tr 32746 | A cyclic permutation of 2 ... |
cycpm2cl 32747 | Closure for the 2-cycles. ... |
cyc2fv1 32748 | Function value of a 2-cycl... |
cyc2fv2 32749 | Function value of a 2-cycl... |
trsp2cyc 32750 | Exhibit the word a transpo... |
cycpmco2f1 32751 | The word U used in ~ cycpm... |
cycpmco2rn 32752 | The orbit of the compositi... |
cycpmco2lem1 32753 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem2 32754 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem3 32755 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem4 32756 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem5 32757 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem6 32758 | Lemma for ~ cycpmco2 . (C... |
cycpmco2lem7 32759 | Lemma for ~ cycpmco2 . (C... |
cycpmco2 32760 | The composition of a cycli... |
cyc2fvx 32761 | Function value of a 2-cycl... |
cycpm3cl 32762 | Closure of the 3-cycles in... |
cycpm3cl2 32763 | Closure of the 3-cycles in... |
cyc3fv1 32764 | Function value of a 3-cycl... |
cyc3fv2 32765 | Function value of a 3-cycl... |
cyc3fv3 32766 | Function value of a 3-cycl... |
cyc3co2 32767 | Represent a 3-cycle as a c... |
cycpmconjvlem 32768 | Lemma for ~ cycpmconjv . ... |
cycpmconjv 32769 | A formula for computing co... |
cycpmrn 32770 | The range of the word used... |
tocyccntz 32771 | All elements of a (finite)... |
evpmval 32772 | Value of the set of even p... |
cnmsgn0g 32773 | The neutral element of the... |
evpmsubg 32774 | The alternating group is a... |
evpmid 32775 | The identity is an even pe... |
altgnsg 32776 | The alternating group ` ( ... |
cyc3evpm 32777 | 3-Cycles are even permutat... |
cyc3genpmlem 32778 | Lemma for ~ cyc3genpm . (... |
cyc3genpm 32779 | The alternating group ` A ... |
cycpmgcl 32780 | Cyclic permutations are pe... |
cycpmconjslem1 32781 | Lemma for ~ cycpmconjs . ... |
cycpmconjslem2 32782 | Lemma for ~ cycpmconjs . ... |
cycpmconjs 32783 | All cycles of the same len... |
cyc3conja 32784 | All 3-cycles are conjugate... |
sgnsv 32787 | The sign mapping. (Contri... |
sgnsval 32788 | The sign value. (Contribu... |
sgnsf 32789 | The sign function. (Contr... |
inftmrel 32794 | The infinitesimal relation... |
isinftm 32795 | Express ` x ` is infinites... |
isarchi 32796 | Express the predicate " ` ... |
pnfinf 32797 | Plus infinity is an infini... |
xrnarchi 32798 | The completed real line is... |
isarchi2 32799 | Alternative way to express... |
submarchi 32800 | A submonoid is archimedean... |
isarchi3 32801 | This is the usual definiti... |
archirng 32802 | Property of Archimedean or... |
archirngz 32803 | Property of Archimedean le... |
archiexdiv 32804 | In an Archimedean group, g... |
archiabllem1a 32805 | Lemma for ~ archiabl : In... |
archiabllem1b 32806 | Lemma for ~ archiabl . (C... |
archiabllem1 32807 | Archimedean ordered groups... |
archiabllem2a 32808 | Lemma for ~ archiabl , whi... |
archiabllem2c 32809 | Lemma for ~ archiabl . (C... |
archiabllem2b 32810 | Lemma for ~ archiabl . (C... |
archiabllem2 32811 | Archimedean ordered groups... |
archiabl 32812 | Archimedean left- and righ... |
isslmd 32815 | The predicate "is a semimo... |
slmdlema 32816 | Lemma for properties of a ... |
lmodslmd 32817 | Left semimodules generaliz... |
slmdcmn 32818 | A semimodule is a commutat... |
slmdmnd 32819 | A semimodule is a monoid. ... |
slmdsrg 32820 | The scalar component of a ... |
slmdbn0 32821 | The base set of a semimodu... |
slmdacl 32822 | Closure of ring addition f... |
slmdmcl 32823 | Closure of ring multiplica... |
slmdsn0 32824 | The set of scalars in a se... |
slmdvacl 32825 | Closure of vector addition... |
slmdass 32826 | Semiring left module vecto... |
slmdvscl 32827 | Closure of scalar product ... |
slmdvsdi 32828 | Distributive law for scala... |
slmdvsdir 32829 | Distributive law for scala... |
slmdvsass 32830 | Associative law for scalar... |
slmd0cl 32831 | The ring zero in a semimod... |
slmd1cl 32832 | The ring unity in a semiri... |
slmdvs1 32833 | Scalar product with ring u... |
slmd0vcl 32834 | The zero vector is a vecto... |
slmd0vlid 32835 | Left identity law for the ... |
slmd0vrid 32836 | Right identity law for the... |
slmd0vs 32837 | Zero times a vector is the... |
slmdvs0 32838 | Anything times the zero ve... |
gsumvsca1 32839 | Scalar product of a finite... |
gsumvsca2 32840 | Scalar product of a finite... |
prmsimpcyc 32841 | A group of prime order is ... |
idomdomd 32842 | An integral domain is a do... |
idomringd 32843 | An integral domain is a ri... |
domnlcan 32844 | Left-cancellation law for ... |
idomrcan 32845 | Right-cancellation law for... |
urpropd 32846 | Sufficient condition for r... |
0ringsubrg 32847 | A subring of a zero ring i... |
frobrhm 32848 | In a commutative ring with... |
ress1r 32849 | ` 1r ` is unaffected by re... |
ringinvval 32850 | The ring inverse expressed... |
dvrcan5 32851 | Cancellation law for commo... |
subrgchr 32852 | If ` A ` is a subring of `... |
rmfsupp2 32853 | A mapping of a multiplicat... |
eufndx 32856 | Index value of the Euclide... |
eufid 32857 | Utility theorem: index-ind... |
ringinveu 32860 | If a ring unit element ` X... |
isdrng4 32861 | A division ring is a ring ... |
rndrhmcl 32862 | The image of a division ri... |
sdrgdvcl 32863 | A sub-division-ring is clo... |
sdrginvcl 32864 | A sub-division-ring is clo... |
primefldchr 32865 | The characteristic of a pr... |
fldgenval 32868 | Value of the field generat... |
fldgenssid 32869 | The field generated by a s... |
fldgensdrg 32870 | A generated subfield is a ... |
fldgenssv 32871 | A generated subfield is a ... |
fldgenss 32872 | Generated subfields preser... |
fldgenidfld 32873 | The subfield generated by ... |
fldgenssp 32874 | The field generated by a s... |
fldgenid 32875 | The subfield of a field ` ... |
fldgenfld 32876 | A generated subfield is a ... |
primefldgen1 32877 | The prime field of a divis... |
1fldgenq 32878 | The field of rational numb... |
isorng 32883 | An ordered ring is a ring ... |
orngring 32884 | An ordered ring is a ring.... |
orngogrp 32885 | An ordered ring is an orde... |
isofld 32886 | An ordered field is a fiel... |
orngmul 32887 | In an ordered ring, the or... |
orngsqr 32888 | In an ordered ring, all sq... |
ornglmulle 32889 | In an ordered ring, multip... |
orngrmulle 32890 | In an ordered ring, multip... |
ornglmullt 32891 | In an ordered ring, multip... |
orngrmullt 32892 | In an ordered ring, multip... |
orngmullt 32893 | In an ordered ring, the st... |
ofldfld 32894 | An ordered field is a fiel... |
ofldtos 32895 | An ordered field is a tota... |
orng0le1 32896 | In an ordered ring, the ri... |
ofldlt1 32897 | In an ordered field, the r... |
ofldchr 32898 | The characteristic of an o... |
suborng 32899 | Every subring of an ordere... |
subofld 32900 | Every subfield of an order... |
isarchiofld 32901 | Axiom of Archimedes : a ch... |
rhmdvd 32902 | A ring homomorphism preser... |
kerunit 32903 | If a unit element lies in ... |
reldmresv 32906 | The scalar restriction is ... |
resvval 32907 | Value of structure restric... |
resvid2 32908 | General behavior of trivia... |
resvval2 32909 | Value of nontrivial struct... |
resvsca 32910 | Base set of a structure re... |
resvlem 32911 | Other elements of a scalar... |
resvlemOLD 32912 | Obsolete version of ~ resv... |
resvbas 32913 | ` Base ` is unaffected by ... |
resvbasOLD 32914 | Obsolete proof of ~ resvba... |
resvplusg 32915 | ` +g ` is unaffected by sc... |
resvplusgOLD 32916 | Obsolete proof of ~ resvpl... |
resvvsca 32917 | ` .s ` is unaffected by sc... |
resvvscaOLD 32918 | Obsolete proof of ~ resvvs... |
resvmulr 32919 | ` .r ` is unaffected by sc... |
resvmulrOLD 32920 | Obsolete proof of ~ resvmu... |
resv0g 32921 | ` 0g ` is unaffected by sc... |
resv1r 32922 | ` 1r ` is unaffected by sc... |
resvcmn 32923 | Scalar restriction preserv... |
gzcrng 32924 | The gaussian integers form... |
reofld 32925 | The real numbers form an o... |
nn0omnd 32926 | The nonnegative integers f... |
rearchi 32927 | The field of the real numb... |
nn0archi 32928 | The monoid of the nonnegat... |
xrge0slmod 32929 | The extended nonnegative r... |
qusker 32930 | The kernel of a quotient m... |
eqgvscpbl 32931 | The left coset equivalence... |
qusvscpbl 32932 | The quotient map distribut... |
qusvsval 32933 | Value of the scalar multip... |
imaslmod 32934 | The image structure of a l... |
imasmhm 32935 | Given a function ` F ` wit... |
imasghm 32936 | Given a function ` F ` wit... |
imasrhm 32937 | Given a function ` F ` wit... |
imaslmhm 32938 | Given a function ` F ` wit... |
quslmod 32939 | If ` G ` is a submodule in... |
quslmhm 32940 | If ` G ` is a submodule of... |
quslvec 32941 | If ` S ` is a vector subsp... |
ecxpid 32942 | The equivalence class of a... |
eqg0el 32943 | Equivalence class of a quo... |
qsxpid 32944 | The quotient set of a cart... |
qusxpid 32945 | The Group quotient equival... |
qustriv 32946 | The quotient of a group ` ... |
qustrivr 32947 | Converse of ~ qustriv . (... |
znfermltl 32948 | Fermat's little theorem in... |
islinds5 32949 | A set is linearly independ... |
ellspds 32950 | Variation on ~ ellspd . (... |
0ellsp 32951 | Zero is in all spans. (Co... |
0nellinds 32952 | The group identity cannot ... |
rspsnel 32953 | Membership in a principal ... |
rspsnid 32954 | A principal ideal contains... |
elrsp 32955 | Write the elements of a ri... |
rspidlid 32956 | The ideal span of an ideal... |
pidlnz 32957 | A principal ideal generate... |
dvdsruassoi 32958 | If two elements ` X ` and ... |
dvdsruasso 32959 | Two elements ` X ` and ` Y... |
dvdsrspss 32960 | In a ring, an element ` X ... |
rspsnasso 32961 | Two elements ` X ` and ` Y... |
lbslsp 32962 | Any element of a left modu... |
lindssn 32963 | Any singleton of a nonzero... |
lindflbs 32964 | Conditions for an independ... |
islbs5 32965 | An equivalent formulation ... |
linds2eq 32966 | Deduce equality of element... |
lindfpropd 32967 | Property deduction for lin... |
lindspropd 32968 | Property deduction for lin... |
elgrplsmsn 32969 | Membership in a sumset wit... |
lsmsnorb 32970 | The sumset of a group with... |
lsmsnorb2 32971 | The sumset of a single ele... |
elringlsm 32972 | Membership in a product of... |
elringlsmd 32973 | Membership in a product of... |
ringlsmss 32974 | Closure of the product of ... |
ringlsmss1 32975 | The product of an ideal ` ... |
ringlsmss2 32976 | The product with an ideal ... |
lsmsnpridl 32977 | The product of the ring wi... |
lsmsnidl 32978 | The product of the ring wi... |
lsmidllsp 32979 | The sum of two ideals is t... |
lsmidl 32980 | The sum of two ideals is a... |
lsmssass 32981 | Group sum is associative, ... |
grplsm0l 32982 | Sumset with the identity s... |
grplsmid 32983 | The direct sum of an eleme... |
qusmul 32984 | Value of the ring operatio... |
quslsm 32985 | Express the image by the q... |
qusbas2 32986 | Alternate definition of th... |
qus0g 32987 | The identity element of a ... |
qusima 32988 | The image of a subgroup by... |
qusrn 32989 | The natural map from eleme... |
nsgqus0 32990 | A normal subgroup ` N ` is... |
nsgmgclem 32991 | Lemma for ~ nsgmgc . (Con... |
nsgmgc 32992 | There is a monotone Galois... |
nsgqusf1olem1 32993 | Lemma for ~ nsgqusf1o . (... |
nsgqusf1olem2 32994 | Lemma for ~ nsgqusf1o . (... |
nsgqusf1olem3 32995 | Lemma for ~ nsgqusf1o . (... |
nsgqusf1o 32996 | The canonical projection h... |
ghmquskerlem1 32997 | Lemma for ~ ghmqusker (Con... |
ghmquskerco 32998 | In the case of theorem ~ g... |
ghmquskerlem2 32999 | Lemma for ~ ghmqusker . (... |
ghmquskerlem3 33000 | The mapping ` H ` induced ... |
ghmqusker 33001 | A surjective group homomor... |
gicqusker 33002 | The image ` H ` of a group... |
lmhmqusker 33003 | A surjective module homomo... |
lmicqusker 33004 | The image ` H ` of a modul... |
intlidl 33005 | The intersection of a none... |
rhmpreimaidl 33006 | The preimage of an ideal b... |
kerlidl 33007 | The kernel of a ring homom... |
0ringidl 33008 | The zero ideal is the only... |
pidlnzb 33009 | A principal ideal is nonze... |
lidlunitel 33010 | If an ideal ` I ` contains... |
unitpidl1 33011 | The ideal ` I ` generated ... |
rhmquskerlem 33012 | The mapping ` J ` induced ... |
rhmqusker 33013 | A surjective ring homomorp... |
ricqusker 33014 | The image ` H ` of a ring ... |
elrspunidl 33015 | Elementhood in the span of... |
elrspunsn 33016 | Membership to the span of ... |
lidlincl 33017 | Ideals are closed under in... |
idlinsubrg 33018 | The intersection between a... |
rhmimaidl 33019 | The image of an ideal ` I ... |
drngidl 33020 | A nonzero ring is a divisi... |
drngidlhash 33021 | A ring is a division ring ... |
prmidlval 33024 | The class of prime ideals ... |
isprmidl 33025 | The predicate "is a prime ... |
prmidlnr 33026 | A prime ideal is a proper ... |
prmidl 33027 | The main property of a pri... |
prmidl2 33028 | A condition that shows an ... |
idlmulssprm 33029 | Let ` P ` be a prime ideal... |
pridln1 33030 | A proper ideal cannot cont... |
prmidlidl 33031 | A prime ideal is an ideal.... |
prmidlssidl 33032 | Prime ideals as a subset o... |
lidlnsg 33033 | An ideal is a normal subgr... |
cringm4 33034 | Commutative/associative la... |
isprmidlc 33035 | The predicate "is prime id... |
prmidlc 33036 | Property of a prime ideal ... |
0ringprmidl 33037 | The trivial ring does not ... |
prmidl0 33038 | The zero ideal of a commut... |
rhmpreimaprmidl 33039 | The preimage of a prime id... |
qsidomlem1 33040 | If the quotient ring of a ... |
qsidomlem2 33041 | A quotient by a prime idea... |
qsidom 33042 | An ideal ` I ` in the comm... |
qsnzr 33043 | A quotient of a non-zero r... |
mxidlval 33046 | The set of maximal ideals ... |
ismxidl 33047 | The predicate "is a maxima... |
mxidlidl 33048 | A maximal ideal is an idea... |
mxidlnr 33049 | A maximal ideal is proper.... |
mxidlmax 33050 | A maximal ideal is a maxim... |
mxidln1 33051 | One is not contained in an... |
mxidlnzr 33052 | A ring with a maximal idea... |
mxidlmaxv 33053 | An ideal ` I ` strictly co... |
crngmxidl 33054 | In a commutative ring, max... |
mxidlprm 33055 | Every maximal ideal is pri... |
mxidlirredi 33056 | In an integral domain, the... |
mxidlirred 33057 | In a principal ideal domai... |
ssmxidllem 33058 | The set ` P ` used in the ... |
ssmxidl 33059 | Let ` R ` be a ring, and l... |
drnglidl1ne0 33060 | In a nonzero ring, the zer... |
drng0mxidl 33061 | In a division ring, the ze... |
drngmxidl 33062 | The zero ideal is the only... |
krull 33063 | Krull's theorem: Any nonz... |
mxidlnzrb 33064 | A ring is nonzero if and o... |
opprabs 33065 | The opposite ring of the o... |
oppreqg 33066 | Group coset equivalence re... |
opprnsg 33067 | Normal subgroups of the op... |
opprlidlabs 33068 | The ideals of the opposite... |
oppr2idl 33069 | Two sided ideal of the opp... |
opprmxidlabs 33070 | The maximal ideal of the o... |
opprqusbas 33071 | The base of the quotient o... |
opprqusplusg 33072 | The group operation of the... |
opprqus0g 33073 | The group identity element... |
opprqusmulr 33074 | The multiplication operati... |
opprqus1r 33075 | The ring unity of the quot... |
opprqusdrng 33076 | The quotient of the opposi... |
qsdrngilem 33077 | Lemma for ~ qsdrngi . (Co... |
qsdrngi 33078 | A quotient by a maximal le... |
qsdrnglem2 33079 | Lemma for ~ qsdrng . (Con... |
qsdrng 33080 | An ideal ` M ` is both lef... |
qsfld 33081 | An ideal ` M ` in the comm... |
mxidlprmALT 33082 | Every maximal ideal is pri... |
idlsrgstr 33085 | A constructed semiring of ... |
idlsrgval 33086 | Lemma for ~ idlsrgbas thro... |
idlsrgbas 33087 | Base of the ideals of a ri... |
idlsrgplusg 33088 | Additive operation of the ... |
idlsrg0g 33089 | The zero ideal is the addi... |
idlsrgmulr 33090 | Multiplicative operation o... |
idlsrgtset 33091 | Topology component of the ... |
idlsrgmulrval 33092 | Value of the ring multipli... |
idlsrgmulrcl 33093 | Ideals of a ring ` R ` are... |
idlsrgmulrss1 33094 | In a commutative ring, the... |
idlsrgmulrss2 33095 | The product of two ideals ... |
idlsrgmulrssin 33096 | In a commutative ring, the... |
idlsrgmnd 33097 | The ideals of a ring form ... |
idlsrgcmnd 33098 | The ideals of a ring form ... |
isufd 33101 | The property of being a Un... |
rprmval 33102 | The prime elements of a ri... |
isrprm 33103 | Property for ` P ` to be a... |
0ringmon1p 33104 | There are no monic polynom... |
fply1 33105 | Conditions for a function ... |
ply1lvec 33106 | In a division ring, the un... |
evls1fn 33107 | Functionality of the subri... |
evls1dm 33108 | The domain of the subring ... |
evls1fvf 33109 | The subring evaluation fun... |
evls1scafv 33110 | Value of the univariate po... |
evls1expd 33111 | Univariate polynomial eval... |
evls1varpwval 33112 | Univariate polynomial eval... |
evls1fpws 33113 | Evaluation of a univariate... |
ressply1evl 33114 | Evaluation of a univariate... |
evls1addd 33115 | Univariate polynomial eval... |
evls1muld 33116 | Univariate polynomial eval... |
evls1vsca 33117 | Univariate polynomial eval... |
ressdeg1 33118 | The degree of a univariate... |
ply1ascl0 33119 | The zero scalar as a polyn... |
ply1ascl1 33120 | The multiplicative unit sc... |
ply1asclunit 33121 | A non-zero scalar polynomi... |
deg1le0eq0 33122 | A polynomial with nonposit... |
ressply10g 33123 | A restricted polynomial al... |
ressply1mon1p 33124 | The monic polynomials of a... |
ressply1invg 33125 | An element of a restricted... |
ressply1sub 33126 | A restricted polynomial al... |
asclply1subcl 33127 | Closure of the algebra sca... |
ply1fermltl 33128 | Fermat's little theorem fo... |
coe1mon 33129 | Coefficient vector of a mo... |
ply1moneq 33130 | Two monomials are equal if... |
ply1degltel 33131 | Characterize elementhood i... |
ply1degleel 33132 | Characterize elementhood i... |
ply1degltlss 33133 | The space ` S ` of the uni... |
gsummoncoe1fzo 33134 | A coefficient of the polyn... |
ply1gsumz 33135 | If a polynomial given as a... |
deg1addlt 33136 | If both factors have degre... |
ig1pnunit 33137 | The polynomial ideal gener... |
ig1pmindeg 33138 | The polynomial ideal gener... |
q1pdir 33139 | Distribution of univariate... |
q1pvsca 33140 | Scalar multiplication prop... |
r1pvsca 33141 | Scalar multiplication prop... |
r1p0 33142 | Polynomial remainder opera... |
r1pcyc 33143 | The polynomial remainder o... |
r1padd1 33144 | Addition property of the p... |
r1pid2 33145 | Identity law for polynomia... |
r1plmhm 33146 | The univariate polynomial ... |
r1pquslmic 33147 | The univariate polynomial ... |
sra1r 33148 | The unity element of a sub... |
sradrng 33149 | Condition for a subring al... |
srasubrg 33150 | A subring of the original ... |
sralvec 33151 | Given a sub division ring ... |
srafldlvec 33152 | Given a subfield ` F ` of ... |
resssra 33153 | The subring algebra of a r... |
lsssra 33154 | A subring is a subspace of... |
drgext0g 33155 | The additive neutral eleme... |
drgextvsca 33156 | The scalar multiplication ... |
drgext0gsca 33157 | The additive neutral eleme... |
drgextsubrg 33158 | The scalar field is a subr... |
drgextlsp 33159 | The scalar field is a subs... |
drgextgsum 33160 | Group sum in a division ri... |
lvecdimfi 33161 | Finite version of ~ lvecdi... |
dimval 33164 | The dimension of a vector ... |
dimvalfi 33165 | The dimension of a vector ... |
dimcl 33166 | Closure of the vector spac... |
lmimdim 33167 | Module isomorphisms preser... |
lmicdim 33168 | Module isomorphisms preser... |
lvecdim0i 33169 | A vector space of dimensio... |
lvecdim0 33170 | A vector space of dimensio... |
lssdimle 33171 | The dimension of a linear ... |
dimpropd 33172 | If two structures have the... |
rlmdim 33173 | The left vector space indu... |
rgmoddimOLD 33174 | Obsolete version of ~ rlmd... |
frlmdim 33175 | Dimension of a free left m... |
tnglvec 33176 | Augmenting a structure wit... |
tngdim 33177 | Dimension of a left vector... |
rrxdim 33178 | Dimension of the generaliz... |
matdim 33179 | Dimension of the space of ... |
lbslsat 33180 | A nonzero vector ` X ` is ... |
lsatdim 33181 | A line, spanned by a nonze... |
drngdimgt0 33182 | The dimension of a vector ... |
lmhmlvec2 33183 | A homomorphism of left vec... |
kerlmhm 33184 | The kernel of a vector spa... |
imlmhm 33185 | The image of a vector spac... |
ply1degltdimlem 33186 | Lemma for ~ ply1degltdim .... |
ply1degltdim 33187 | The space ` S ` of the uni... |
lindsunlem 33188 | Lemma for ~ lindsun . (Co... |
lindsun 33189 | Condition for the union of... |
lbsdiflsp0 33190 | The linear spans of two di... |
dimkerim 33191 | Given a linear map ` F ` b... |
qusdimsum 33192 | Let ` W ` be a vector spac... |
fedgmullem1 33193 | Lemma for ~ fedgmul . (Co... |
fedgmullem2 33194 | Lemma for ~ fedgmul . (Co... |
fedgmul 33195 | The multiplicativity formu... |
relfldext 33204 | The field extension is a r... |
brfldext 33205 | The field extension relati... |
ccfldextrr 33206 | The field of the complex n... |
fldextfld1 33207 | A field extension is only ... |
fldextfld2 33208 | A field extension is only ... |
fldextsubrg 33209 | Field extension implies a ... |
fldextress 33210 | Field extension implies a ... |
brfinext 33211 | The finite field extension... |
extdgval 33212 | Value of the field extensi... |
fldextsralvec 33213 | The subring algebra associ... |
extdgcl 33214 | Closure of the field exten... |
extdggt0 33215 | Degrees of field extension... |
fldexttr 33216 | Field extension is a trans... |
fldextid 33217 | The field extension relati... |
extdgid 33218 | A trivial field extension ... |
extdgmul 33219 | The multiplicativity formu... |
finexttrb 33220 | The extension ` E ` of ` K... |
extdg1id 33221 | If the degree of the exten... |
extdg1b 33222 | The degree of the extensio... |
fldextchr 33223 | The characteristic of a su... |
evls1fldgencl 33224 | Closure of the subring pol... |
ccfldsrarelvec 33225 | The subring algebra of the... |
ccfldextdgrr 33226 | The degree of the field ex... |
irngval 33229 | The elements of a field ` ... |
elirng 33230 | Property for an element ` ... |
irngss 33231 | All elements of a subring ... |
irngssv 33232 | An integral element is an ... |
0ringirng 33233 | A zero ring ` R ` has no i... |
irngnzply1lem 33234 | In the case of a field ` E... |
irngnzply1 33235 | In the case of a field ` E... |
evls1fvcl 33238 | Variant of ~ fveval1fvcl f... |
evls1maprhm 33239 | The function ` F ` mapping... |
evls1maplmhm 33240 | The function ` F ` mapping... |
evls1maprnss 33241 | The function ` F ` mapping... |
ply1annidllem 33242 | Write the set ` Q ` of pol... |
ply1annidl 33243 | The set ` Q ` of polynomia... |
ply1annnr 33244 | The set ` Q ` of polynomia... |
ply1annig1p 33245 | The ideal ` Q ` of polynom... |
minplyval 33246 | Expand the value of the mi... |
minplycl 33247 | The minimal polynomial is ... |
ply1annprmidl 33248 | The set ` Q ` of polynomia... |
minplyirredlem 33249 | Lemma for ~ minplyirred . ... |
minplyirred 33250 | A nonzero minimal polynomi... |
irngnminplynz 33251 | Integral elements have non... |
minplym1p 33252 | A minimal polynomial is mo... |
algextdeglem1 33253 | Lemma for ~ algextdeg . (... |
algextdeglem2 33254 | Lemma for ~ algextdeg . (... |
algextdeglem3 33255 | Lemma for ~ algextdeg . (... |
algextdeglem4 33256 | Lemma for ~ algextdeg . (... |
algextdeglem5 33257 | Lemma for ~ algextdeg . (... |
algextdeglem6 33258 | Lemma for ~ algextdeg . (... |
algextdeglem7 33259 | Lemma for ~ algextdeg . (... |
algextdeglem8 33260 | Lemma for ~ algextdeg . (... |
algextdeg 33261 | The degree of an algebraic... |
smatfval 33264 | Value of the submatrix. (... |
smatrcl 33265 | Closure of the rectangular... |
smatlem 33266 | Lemma for the next theorem... |
smattl 33267 | Entries of a submatrix, to... |
smattr 33268 | Entries of a submatrix, to... |
smatbl 33269 | Entries of a submatrix, bo... |
smatbr 33270 | Entries of a submatrix, bo... |
smatcl 33271 | Closure of the square subm... |
matmpo 33272 | Write a square matrix as a... |
1smat1 33273 | The submatrix of the ident... |
submat1n 33274 | One case where the submatr... |
submatres 33275 | Special case where the sub... |
submateqlem1 33276 | Lemma for ~ submateq . (C... |
submateqlem2 33277 | Lemma for ~ submateq . (C... |
submateq 33278 | Sufficient condition for t... |
submatminr1 33279 | If we take a submatrix by ... |
lmatval 33282 | Value of the literal matri... |
lmatfval 33283 | Entries of a literal matri... |
lmatfvlem 33284 | Useful lemma to extract li... |
lmatcl 33285 | Closure of the literal mat... |
lmat22lem 33286 | Lemma for ~ lmat22e11 and ... |
lmat22e11 33287 | Entry of a 2x2 literal mat... |
lmat22e12 33288 | Entry of a 2x2 literal mat... |
lmat22e21 33289 | Entry of a 2x2 literal mat... |
lmat22e22 33290 | Entry of a 2x2 literal mat... |
lmat22det 33291 | The determinant of a liter... |
mdetpmtr1 33292 | The determinant of a matri... |
mdetpmtr2 33293 | The determinant of a matri... |
mdetpmtr12 33294 | The determinant of a matri... |
mdetlap1 33295 | A Laplace expansion of the... |
madjusmdetlem1 33296 | Lemma for ~ madjusmdet . ... |
madjusmdetlem2 33297 | Lemma for ~ madjusmdet . ... |
madjusmdetlem3 33298 | Lemma for ~ madjusmdet . ... |
madjusmdetlem4 33299 | Lemma for ~ madjusmdet . ... |
madjusmdet 33300 | Express the cofactor of th... |
mdetlap 33301 | Laplace expansion of the d... |
ist0cld 33302 | The predicate "is a T_0 sp... |
txomap 33303 | Given two open maps ` F ` ... |
qtopt1 33304 | If every equivalence class... |
qtophaus 33305 | If an open map's graph in ... |
circtopn 33306 | The topology of the unit c... |
circcn 33307 | The function gluing the re... |
reff 33308 | For any cover refinement, ... |
locfinreflem 33309 | A locally finite refinemen... |
locfinref 33310 | A locally finite refinemen... |
iscref 33313 | The property that every op... |
crefeq 33314 | Equality theorem for the "... |
creftop 33315 | A space where every open c... |
crefi 33316 | The property that every op... |
crefdf 33317 | A formulation of ~ crefi e... |
crefss 33318 | The "every open cover has ... |
cmpcref 33319 | Equivalent definition of c... |
cmpfiref 33320 | Every open cover of a Comp... |
ldlfcntref 33323 | Every open cover of a Lind... |
ispcmp 33326 | The predicate "is a paraco... |
cmppcmp 33327 | Every compact space is par... |
dispcmp 33328 | Every discrete space is pa... |
pcmplfin 33329 | Given a paracompact topolo... |
pcmplfinf 33330 | Given a paracompact topolo... |
rspecval 33333 | Value of the spectrum of t... |
rspecbas 33334 | The prime ideals form the ... |
rspectset 33335 | Topology component of the ... |
rspectopn 33336 | The topology component of ... |
zarcls0 33337 | The closure of the identit... |
zarcls1 33338 | The unit ideal ` B ` is th... |
zarclsun 33339 | The union of two closed se... |
zarclsiin 33340 | In a Zariski topology, the... |
zarclsint 33341 | The intersection of a fami... |
zarclssn 33342 | The closed points of Zaris... |
zarcls 33343 | The open sets of the Zaris... |
zartopn 33344 | The Zariski topology is a ... |
zartop 33345 | The Zariski topology is a ... |
zartopon 33346 | The points of the Zariski ... |
zar0ring 33347 | The Zariski Topology of th... |
zart0 33348 | The Zariski topology is T_... |
zarmxt1 33349 | The Zariski topology restr... |
zarcmplem 33350 | Lemma for ~ zarcmp . (Con... |
zarcmp 33351 | The Zariski topology is co... |
rspectps 33352 | The spectrum of a ring ` R... |
rhmpreimacnlem 33353 | Lemma for ~ rhmpreimacn . ... |
rhmpreimacn 33354 | The function mapping a pri... |
metidval 33359 | Value of the metric identi... |
metidss 33360 | As a relation, the metric ... |
metidv 33361 | ` A ` and ` B ` identify b... |
metideq 33362 | Basic property of the metr... |
metider 33363 | The metric identification ... |
pstmval 33364 | Value of the metric induce... |
pstmfval 33365 | Function value of the metr... |
pstmxmet 33366 | The metric induced by a ps... |
hauseqcn 33367 | In a Hausdorff topology, t... |
elunitge0 33368 | An element of the closed u... |
unitssxrge0 33369 | The closed unit interval i... |
unitdivcld 33370 | Necessary conditions for a... |
iistmd 33371 | The closed unit interval f... |
unicls 33372 | The union of the closed se... |
tpr2tp 33373 | The usual topology on ` ( ... |
tpr2uni 33374 | The usual topology on ` ( ... |
xpinpreima 33375 | Rewrite the cartesian prod... |
xpinpreima2 33376 | Rewrite the cartesian prod... |
sqsscirc1 33377 | The complex square of side... |
sqsscirc2 33378 | The complex square of side... |
cnre2csqlem 33379 | Lemma for ~ cnre2csqima . ... |
cnre2csqima 33380 | Image of a centered square... |
tpr2rico 33381 | For any point of an open s... |
cnvordtrestixx 33382 | The restriction of the 'gr... |
prsdm 33383 | Domain of the relation of ... |
prsrn 33384 | Range of the relation of a... |
prsss 33385 | Relation of a subproset. ... |
prsssdm 33386 | Domain of a subproset rela... |
ordtprsval 33387 | Value of the order topolog... |
ordtprsuni 33388 | Value of the order topolog... |
ordtcnvNEW 33389 | The order dual generates t... |
ordtrestNEW 33390 | The subspace topology of a... |
ordtrest2NEWlem 33391 | Lemma for ~ ordtrest2NEW .... |
ordtrest2NEW 33392 | An interval-closed set ` A... |
ordtconnlem1 33393 | Connectedness in the order... |
ordtconn 33394 | Connectedness in the order... |
mndpluscn 33395 | A mapping that is both a h... |
mhmhmeotmd 33396 | Deduce a Topological Monoi... |
rmulccn 33397 | Multiplication by a real c... |
raddcn 33398 | Addition in the real numbe... |
xrmulc1cn 33399 | The operation multiplying ... |
fmcncfil 33400 | The image of a Cauchy filt... |
xrge0hmph 33401 | The extended nonnegative r... |
xrge0iifcnv 33402 | Define a bijection from ` ... |
xrge0iifcv 33403 | The defined function's val... |
xrge0iifiso 33404 | The defined bijection from... |
xrge0iifhmeo 33405 | Expose a homeomorphism fro... |
xrge0iifhom 33406 | The defined function from ... |
xrge0iif1 33407 | Condition for the defined ... |
xrge0iifmhm 33408 | The defined function from ... |
xrge0pluscn 33409 | The addition operation of ... |
xrge0mulc1cn 33410 | The operation multiplying ... |
xrge0tps 33411 | The extended nonnegative r... |
xrge0topn 33412 | The topology of the extend... |
xrge0haus 33413 | The topology of the extend... |
xrge0tmd 33414 | The extended nonnegative r... |
xrge0tmdALT 33415 | Alternate proof of ~ xrge0... |
lmlim 33416 | Relate a limit in a given ... |
lmlimxrge0 33417 | Relate a limit in the nonn... |
rge0scvg 33418 | Implication of convergence... |
fsumcvg4 33419 | A serie with finite suppor... |
pnfneige0 33420 | A neighborhood of ` +oo ` ... |
lmxrge0 33421 | Express "sequence ` F ` co... |
lmdvg 33422 | If a monotonic sequence of... |
lmdvglim 33423 | If a monotonic real number... |
pl1cn 33424 | A univariate polynomial is... |
zringnm 33427 | The norm (function) for a ... |
zzsnm 33428 | The norm of the ring of th... |
zlm0 33429 | Zero of a ` ZZ ` -module. ... |
zlm1 33430 | Unity element of a ` ZZ ` ... |
zlmds 33431 | Distance in a ` ZZ ` -modu... |
zlmdsOLD 33432 | Obsolete proof of ~ zlmds ... |
zlmtset 33433 | Topology in a ` ZZ ` -modu... |
zlmtsetOLD 33434 | Obsolete proof of ~ zlmtse... |
zlmnm 33435 | Norm of a ` ZZ ` -module (... |
zhmnrg 33436 | The ` ZZ ` -module built f... |
nmmulg 33437 | The norm of a group produc... |
zrhnm 33438 | The norm of the image by `... |
cnzh 33439 | The ` ZZ ` -module of ` CC... |
rezh 33440 | The ` ZZ ` -module of ` RR... |
qqhval 33443 | Value of the canonical hom... |
zrhf1ker 33444 | The kernel of the homomorp... |
zrhchr 33445 | The kernel of the homomorp... |
zrhker 33446 | The kernel of the homomorp... |
zrhunitpreima 33447 | The preimage by ` ZRHom ` ... |
elzrhunit 33448 | Condition for the image by... |
elzdif0 33449 | Lemma for ~ qqhval2 . (Co... |
qqhval2lem 33450 | Lemma for ~ qqhval2 . (Co... |
qqhval2 33451 | Value of the canonical hom... |
qqhvval 33452 | Value of the canonical hom... |
qqh0 33453 | The image of ` 0 ` by the ... |
qqh1 33454 | The image of ` 1 ` by the ... |
qqhf 33455 | ` QQHom ` as a function. ... |
qqhvq 33456 | The image of a quotient by... |
qqhghm 33457 | The ` QQHom ` homomorphism... |
qqhrhm 33458 | The ` QQHom ` homomorphism... |
qqhnm 33459 | The norm of the image by `... |
qqhcn 33460 | The ` QQHom ` homomorphism... |
qqhucn 33461 | The ` QQHom ` homomorphism... |
rrhval 33465 | Value of the canonical hom... |
rrhcn 33466 | If the topology of ` R ` i... |
rrhf 33467 | If the topology of ` R ` i... |
isrrext 33469 | Express the property " ` R... |
rrextnrg 33470 | An extension of ` RR ` is ... |
rrextdrg 33471 | An extension of ` RR ` is ... |
rrextnlm 33472 | The norm of an extension o... |
rrextchr 33473 | The ring characteristic of... |
rrextcusp 33474 | An extension of ` RR ` is ... |
rrexttps 33475 | An extension of ` RR ` is ... |
rrexthaus 33476 | The topology of an extensi... |
rrextust 33477 | The uniformity of an exten... |
rerrext 33478 | The field of the real numb... |
cnrrext 33479 | The field of the complex n... |
qqtopn 33480 | The topology of the field ... |
rrhfe 33481 | If ` R ` is an extension o... |
rrhcne 33482 | If ` R ` is an extension o... |
rrhqima 33483 | The ` RRHom ` homomorphism... |
rrh0 33484 | The image of ` 0 ` by the ... |
xrhval 33487 | The value of the embedding... |
zrhre 33488 | The ` ZRHom ` homomorphism... |
qqhre 33489 | The ` QQHom ` homomorphism... |
rrhre 33490 | The ` RRHom ` homomorphism... |
relmntop 33493 | Manifold is a relation. (... |
ismntoplly 33494 | Property of being a manifo... |
ismntop 33495 | Property of being a manifo... |
nexple 33496 | A lower bound for an expon... |
indv 33499 | Value of the indicator fun... |
indval 33500 | Value of the indicator fun... |
indval2 33501 | Alternate value of the ind... |
indf 33502 | An indicator function as a... |
indfval 33503 | Value of the indicator fun... |
ind1 33504 | Value of the indicator fun... |
ind0 33505 | Value of the indicator fun... |
ind1a 33506 | Value of the indicator fun... |
indpi1 33507 | Preimage of the singleton ... |
indsum 33508 | Finite sum of a product wi... |
indsumin 33509 | Finite sum of a product wi... |
prodindf 33510 | The product of indicators ... |
indf1o 33511 | The bijection between a po... |
indpreima 33512 | A function with range ` { ... |
indf1ofs 33513 | The bijection between fini... |
esumex 33516 | An extended sum is a set b... |
esumcl 33517 | Closure for extended sum i... |
esumeq12dvaf 33518 | Equality deduction for ext... |
esumeq12dva 33519 | Equality deduction for ext... |
esumeq12d 33520 | Equality deduction for ext... |
esumeq1 33521 | Equality theorem for an ex... |
esumeq1d 33522 | Equality theorem for an ex... |
esumeq2 33523 | Equality theorem for exten... |
esumeq2d 33524 | Equality deduction for ext... |
esumeq2dv 33525 | Equality deduction for ext... |
esumeq2sdv 33526 | Equality deduction for ext... |
nfesum1 33527 | Bound-variable hypothesis ... |
nfesum2 33528 | Bound-variable hypothesis ... |
cbvesum 33529 | Change bound variable in a... |
cbvesumv 33530 | Change bound variable in a... |
esumid 33531 | Identify the extended sum ... |
esumgsum 33532 | A finite extended sum is t... |
esumval 33533 | Develop the value of the e... |
esumel 33534 | The extended sum is a limi... |
esumnul 33535 | Extended sum over the empt... |
esum0 33536 | Extended sum of zero. (Co... |
esumf1o 33537 | Re-index an extended sum u... |
esumc 33538 | Convert from the collectio... |
esumrnmpt 33539 | Rewrite an extended sum in... |
esumsplit 33540 | Split an extended sum into... |
esummono 33541 | Extended sum is monotonic.... |
esumpad 33542 | Extend an extended sum by ... |
esumpad2 33543 | Remove zeroes from an exte... |
esumadd 33544 | Addition of infinite sums.... |
esumle 33545 | If all of the terms of an ... |
gsumesum 33546 | Relate a group sum on ` ( ... |
esumlub 33547 | The extended sum is the lo... |
esumaddf 33548 | Addition of infinite sums.... |
esumlef 33549 | If all of the terms of an ... |
esumcst 33550 | The extended sum of a cons... |
esumsnf 33551 | The extended sum of a sing... |
esumsn 33552 | The extended sum of a sing... |
esumpr 33553 | Extended sum over a pair. ... |
esumpr2 33554 | Extended sum over a pair, ... |
esumrnmpt2 33555 | Rewrite an extended sum in... |
esumfzf 33556 | Formulating a partial exte... |
esumfsup 33557 | Formulating an extended su... |
esumfsupre 33558 | Formulating an extended su... |
esumss 33559 | Change the index set to a ... |
esumpinfval 33560 | The value of the extended ... |
esumpfinvallem 33561 | Lemma for ~ esumpfinval . ... |
esumpfinval 33562 | The value of the extended ... |
esumpfinvalf 33563 | Same as ~ esumpfinval , mi... |
esumpinfsum 33564 | The value of the extended ... |
esumpcvgval 33565 | The value of the extended ... |
esumpmono 33566 | The partial sums in an ext... |
esumcocn 33567 | Lemma for ~ esummulc2 and ... |
esummulc1 33568 | An extended sum multiplied... |
esummulc2 33569 | An extended sum multiplied... |
esumdivc 33570 | An extended sum divided by... |
hashf2 33571 | Lemma for ~ hasheuni . (C... |
hasheuni 33572 | The cardinality of a disjo... |
esumcvg 33573 | The sequence of partial su... |
esumcvg2 33574 | Simpler version of ~ esumc... |
esumcvgsum 33575 | The value of the extended ... |
esumsup 33576 | Express an extended sum as... |
esumgect 33577 | "Send ` n ` to ` +oo ` " i... |
esumcvgre 33578 | All terms of a converging ... |
esum2dlem 33579 | Lemma for ~ esum2d (finite... |
esum2d 33580 | Write a double extended su... |
esumiun 33581 | Sum over a nonnecessarily ... |
ofceq 33584 | Equality theorem for funct... |
ofcfval 33585 | Value of an operation appl... |
ofcval 33586 | Evaluate a function/consta... |
ofcfn 33587 | The function operation pro... |
ofcfeqd2 33588 | Equality theorem for funct... |
ofcfval3 33589 | General value of ` ( F oFC... |
ofcf 33590 | The function/constant oper... |
ofcfval2 33591 | The function operation exp... |
ofcfval4 33592 | The function/constant oper... |
ofcc 33593 | Left operation by a consta... |
ofcof 33594 | Relate function operation ... |
sigaex 33597 | Lemma for ~ issiga and ~ i... |
sigaval 33598 | The set of sigma-algebra w... |
issiga 33599 | An alternative definition ... |
isrnsiga 33600 | The property of being a si... |
0elsiga 33601 | A sigma-algebra contains t... |
baselsiga 33602 | A sigma-algebra contains i... |
sigasspw 33603 | A sigma-algebra is a set o... |
sigaclcu 33604 | A sigma-algebra is closed ... |
sigaclcuni 33605 | A sigma-algebra is closed ... |
sigaclfu 33606 | A sigma-algebra is closed ... |
sigaclcu2 33607 | A sigma-algebra is closed ... |
sigaclfu2 33608 | A sigma-algebra is closed ... |
sigaclcu3 33609 | A sigma-algebra is closed ... |
issgon 33610 | Property of being a sigma-... |
sgon 33611 | A sigma-algebra is a sigma... |
elsigass 33612 | An element of a sigma-alge... |
elrnsiga 33613 | Dropping the base informat... |
isrnsigau 33614 | The property of being a si... |
unielsiga 33615 | A sigma-algebra contains i... |
dmvlsiga 33616 | Lebesgue-measurable subset... |
pwsiga 33617 | Any power set forms a sigm... |
prsiga 33618 | The smallest possible sigm... |
sigaclci 33619 | A sigma-algebra is closed ... |
difelsiga 33620 | A sigma-algebra is closed ... |
unelsiga 33621 | A sigma-algebra is closed ... |
inelsiga 33622 | A sigma-algebra is closed ... |
sigainb 33623 | Building a sigma-algebra f... |
insiga 33624 | The intersection of a coll... |
sigagenval 33627 | Value of the generated sig... |
sigagensiga 33628 | A generated sigma-algebra ... |
sgsiga 33629 | A generated sigma-algebra ... |
unisg 33630 | The sigma-algebra generate... |
dmsigagen 33631 | A sigma-algebra can be gen... |
sssigagen 33632 | A set is a subset of the s... |
sssigagen2 33633 | A subset of the generating... |
elsigagen 33634 | Any element of a set is al... |
elsigagen2 33635 | Any countable union of ele... |
sigagenss 33636 | The generated sigma-algebr... |
sigagenss2 33637 | Sufficient condition for i... |
sigagenid 33638 | The sigma-algebra generate... |
ispisys 33639 | The property of being a pi... |
ispisys2 33640 | The property of being a pi... |
inelpisys 33641 | Pi-systems are closed unde... |
sigapisys 33642 | All sigma-algebras are pi-... |
isldsys 33643 | The property of being a la... |
pwldsys 33644 | The power set of the unive... |
unelldsys 33645 | Lambda-systems are closed ... |
sigaldsys 33646 | All sigma-algebras are lam... |
ldsysgenld 33647 | The intersection of all la... |
sigapildsyslem 33648 | Lemma for ~ sigapildsys . ... |
sigapildsys 33649 | Sigma-algebra are exactly ... |
ldgenpisyslem1 33650 | Lemma for ~ ldgenpisys . ... |
ldgenpisyslem2 33651 | Lemma for ~ ldgenpisys . ... |
ldgenpisyslem3 33652 | Lemma for ~ ldgenpisys . ... |
ldgenpisys 33653 | The lambda system ` E ` ge... |
dynkin 33654 | Dynkin's lambda-pi theorem... |
isros 33655 | The property of being a ri... |
rossspw 33656 | A ring of sets is a collec... |
0elros 33657 | A ring of sets contains th... |
unelros 33658 | A ring of sets is closed u... |
difelros 33659 | A ring of sets is closed u... |
inelros 33660 | A ring of sets is closed u... |
fiunelros 33661 | A ring of sets is closed u... |
issros 33662 | The property of being a se... |
srossspw 33663 | A semiring of sets is a co... |
0elsros 33664 | A semiring of sets contain... |
inelsros 33665 | A semiring of sets is clos... |
diffiunisros 33666 | In semiring of sets, compl... |
rossros 33667 | Rings of sets are semiring... |
brsiga 33670 | The Borel Algebra on real ... |
brsigarn 33671 | The Borel Algebra is a sig... |
brsigasspwrn 33672 | The Borel Algebra is a set... |
unibrsiga 33673 | The union of the Borel Alg... |
cldssbrsiga 33674 | A Borel Algebra contains a... |
sxval 33677 | Value of the product sigma... |
sxsiga 33678 | A product sigma-algebra is... |
sxsigon 33679 | A product sigma-algebra is... |
sxuni 33680 | The base set of a product ... |
elsx 33681 | The cartesian product of t... |
measbase 33684 | The base set of a measure ... |
measval 33685 | The value of the ` measure... |
ismeas 33686 | The property of being a me... |
isrnmeas 33687 | The property of being a me... |
dmmeas 33688 | The domain of a measure is... |
measbasedom 33689 | The base set of a measure ... |
measfrge0 33690 | A measure is a function ov... |
measfn 33691 | A measure is a function on... |
measvxrge0 33692 | The values of a measure ar... |
measvnul 33693 | The measure of the empty s... |
measge0 33694 | A measure is nonnegative. ... |
measle0 33695 | If the measure of a given ... |
measvun 33696 | The measure of a countable... |
measxun2 33697 | The measure the union of t... |
measun 33698 | The measure the union of t... |
measvunilem 33699 | Lemma for ~ measvuni . (C... |
measvunilem0 33700 | Lemma for ~ measvuni . (C... |
measvuni 33701 | The measure of a countable... |
measssd 33702 | A measure is monotone with... |
measunl 33703 | A measure is sub-additive ... |
measiuns 33704 | The measure of the union o... |
measiun 33705 | A measure is sub-additive.... |
meascnbl 33706 | A measure is continuous fr... |
measinblem 33707 | Lemma for ~ measinb . (Co... |
measinb 33708 | Building a measure restric... |
measres 33709 | Building a measure restric... |
measinb2 33710 | Building a measure restric... |
measdivcst 33711 | Division of a measure by a... |
measdivcstALTV 33712 | Alternate version of ~ mea... |
cntmeas 33713 | The Counting measure is a ... |
pwcntmeas 33714 | The counting measure is a ... |
cntnevol 33715 | Counting and Lebesgue meas... |
voliune 33716 | The Lebesgue measure funct... |
volfiniune 33717 | The Lebesgue measure funct... |
volmeas 33718 | The Lebesgue measure is a ... |
ddeval1 33721 | Value of the delta measure... |
ddeval0 33722 | Value of the delta measure... |
ddemeas 33723 | The Dirac delta measure is... |
relae 33727 | 'almost everywhere' is a r... |
brae 33728 | 'almost everywhere' relati... |
braew 33729 | 'almost everywhere' relati... |
truae 33730 | A truth holds almost every... |
aean 33731 | A conjunction holds almost... |
faeval 33733 | Value of the 'almost every... |
relfae 33734 | The 'almost everywhere' bu... |
brfae 33735 | 'almost everywhere' relati... |
ismbfm 33738 | The predicate " ` F ` is a... |
elunirnmbfm 33739 | The property of being a me... |
mbfmfun 33740 | A measurable function is a... |
mbfmf 33741 | A measurable function as a... |
isanmbfmOLD 33742 | Obsolete version of ~ isan... |
mbfmcnvima 33743 | The preimage by a measurab... |
isanmbfm 33744 | The predicate to be a meas... |
mbfmbfmOLD 33745 | A measurable function to a... |
mbfmbfm 33746 | A measurable function to a... |
mbfmcst 33747 | A constant function is mea... |
1stmbfm 33748 | The first projection map i... |
2ndmbfm 33749 | The second projection map ... |
imambfm 33750 | If the sigma-algebra in th... |
cnmbfm 33751 | A continuous function is m... |
mbfmco 33752 | The composition of two mea... |
mbfmco2 33753 | The pair building of two m... |
mbfmvolf 33754 | Measurable functions with ... |
elmbfmvol2 33755 | Measurable functions with ... |
mbfmcnt 33756 | All functions are measurab... |
br2base 33757 | The base set for the gener... |
dya2ub 33758 | An upper bound for a dyadi... |
sxbrsigalem0 33759 | The closed half-spaces of ... |
sxbrsigalem3 33760 | The sigma-algebra generate... |
dya2iocival 33761 | The function ` I ` returns... |
dya2iocress 33762 | Dyadic intervals are subse... |
dya2iocbrsiga 33763 | Dyadic intervals are Borel... |
dya2icobrsiga 33764 | Dyadic intervals are Borel... |
dya2icoseg 33765 | For any point and any clos... |
dya2icoseg2 33766 | For any point and any open... |
dya2iocrfn 33767 | The function returning dya... |
dya2iocct 33768 | The dyadic rectangle set i... |
dya2iocnrect 33769 | For any point of an open r... |
dya2iocnei 33770 | For any point of an open s... |
dya2iocuni 33771 | Every open set of ` ( RR X... |
dya2iocucvr 33772 | The dyadic rectangular set... |
sxbrsigalem1 33773 | The Borel algebra on ` ( R... |
sxbrsigalem2 33774 | The sigma-algebra generate... |
sxbrsigalem4 33775 | The Borel algebra on ` ( R... |
sxbrsigalem5 33776 | First direction for ~ sxbr... |
sxbrsigalem6 33777 | First direction for ~ sxbr... |
sxbrsiga 33778 | The product sigma-algebra ... |
omsval 33781 | Value of the function mapp... |
omsfval 33782 | Value of the outer measure... |
omscl 33783 | A closure lemma for the co... |
omsf 33784 | A constructed outer measur... |
oms0 33785 | A constructed outer measur... |
omsmon 33786 | A constructed outer measur... |
omssubaddlem 33787 | For any small margin ` E `... |
omssubadd 33788 | A constructed outer measur... |
carsgval 33791 | Value of the Caratheodory ... |
carsgcl 33792 | Closure of the Caratheodor... |
elcarsg 33793 | Property of being a Carath... |
baselcarsg 33794 | The universe set, ` O ` , ... |
0elcarsg 33795 | The empty set is Caratheod... |
carsguni 33796 | The union of all Caratheod... |
elcarsgss 33797 | Caratheodory measurable se... |
difelcarsg 33798 | The Caratheodory measurabl... |
inelcarsg 33799 | The Caratheodory measurabl... |
unelcarsg 33800 | The Caratheodory-measurabl... |
difelcarsg2 33801 | The Caratheodory-measurabl... |
carsgmon 33802 | Utility lemma: Apply mono... |
carsgsigalem 33803 | Lemma for the following th... |
fiunelcarsg 33804 | The Caratheodory measurabl... |
carsgclctunlem1 33805 | Lemma for ~ carsgclctun . ... |
carsggect 33806 | The outer measure is count... |
carsgclctunlem2 33807 | Lemma for ~ carsgclctun . ... |
carsgclctunlem3 33808 | Lemma for ~ carsgclctun . ... |
carsgclctun 33809 | The Caratheodory measurabl... |
carsgsiga 33810 | The Caratheodory measurabl... |
omsmeas 33811 | The restriction of a const... |
pmeasmono 33812 | This theorem's hypotheses ... |
pmeasadd 33813 | A premeasure on a ring of ... |
itgeq12dv 33814 | Equality theorem for an in... |
sitgval 33820 | Value of the simple functi... |
issibf 33821 | The predicate " ` F ` is a... |
sibf0 33822 | The constant zero function... |
sibfmbl 33823 | A simple function is measu... |
sibff 33824 | A simple function is a fun... |
sibfrn 33825 | A simple function has fini... |
sibfima 33826 | Any preimage of a singleto... |
sibfinima 33827 | The measure of the interse... |
sibfof 33828 | Applying function operatio... |
sitgfval 33829 | Value of the Bochner integ... |
sitgclg 33830 | Closure of the Bochner int... |
sitgclbn 33831 | Closure of the Bochner int... |
sitgclcn 33832 | Closure of the Bochner int... |
sitgclre 33833 | Closure of the Bochner int... |
sitg0 33834 | The integral of the consta... |
sitgf 33835 | The integral for simple fu... |
sitgaddlemb 33836 | Lemma for * sitgadd . (Co... |
sitmval 33837 | Value of the simple functi... |
sitmfval 33838 | Value of the integral dist... |
sitmcl 33839 | Closure of the integral di... |
sitmf 33840 | The integral metric as a f... |
oddpwdc 33842 | Lemma for ~ eulerpart . T... |
oddpwdcv 33843 | Lemma for ~ eulerpart : va... |
eulerpartlemsv1 33844 | Lemma for ~ eulerpart . V... |
eulerpartlemelr 33845 | Lemma for ~ eulerpart . (... |
eulerpartlemsv2 33846 | Lemma for ~ eulerpart . V... |
eulerpartlemsf 33847 | Lemma for ~ eulerpart . (... |
eulerpartlems 33848 | Lemma for ~ eulerpart . (... |
eulerpartlemsv3 33849 | Lemma for ~ eulerpart . V... |
eulerpartlemgc 33850 | Lemma for ~ eulerpart . (... |
eulerpartleme 33851 | Lemma for ~ eulerpart . (... |
eulerpartlemv 33852 | Lemma for ~ eulerpart . (... |
eulerpartlemo 33853 | Lemma for ~ eulerpart : ` ... |
eulerpartlemd 33854 | Lemma for ~ eulerpart : ` ... |
eulerpartlem1 33855 | Lemma for ~ eulerpart . (... |
eulerpartlemb 33856 | Lemma for ~ eulerpart . T... |
eulerpartlemt0 33857 | Lemma for ~ eulerpart . (... |
eulerpartlemf 33858 | Lemma for ~ eulerpart : O... |
eulerpartlemt 33859 | Lemma for ~ eulerpart . (... |
eulerpartgbij 33860 | Lemma for ~ eulerpart : T... |
eulerpartlemgv 33861 | Lemma for ~ eulerpart : va... |
eulerpartlemr 33862 | Lemma for ~ eulerpart . (... |
eulerpartlemmf 33863 | Lemma for ~ eulerpart . (... |
eulerpartlemgvv 33864 | Lemma for ~ eulerpart : va... |
eulerpartlemgu 33865 | Lemma for ~ eulerpart : R... |
eulerpartlemgh 33866 | Lemma for ~ eulerpart : T... |
eulerpartlemgf 33867 | Lemma for ~ eulerpart : I... |
eulerpartlemgs2 33868 | Lemma for ~ eulerpart : T... |
eulerpartlemn 33869 | Lemma for ~ eulerpart . (... |
eulerpart 33870 | Euler's theorem on partiti... |
subiwrd 33873 | Lemma for ~ sseqp1 . (Con... |
subiwrdlen 33874 | Length of a subword of an ... |
iwrdsplit 33875 | Lemma for ~ sseqp1 . (Con... |
sseqval 33876 | Value of the strong sequen... |
sseqfv1 33877 | Value of the strong sequen... |
sseqfn 33878 | A strong recursive sequenc... |
sseqmw 33879 | Lemma for ~ sseqf amd ~ ss... |
sseqf 33880 | A strong recursive sequenc... |
sseqfres 33881 | The first elements in the ... |
sseqfv2 33882 | Value of the strong sequen... |
sseqp1 33883 | Value of the strong sequen... |
fiblem 33886 | Lemma for ~ fib0 , ~ fib1 ... |
fib0 33887 | Value of the Fibonacci seq... |
fib1 33888 | Value of the Fibonacci seq... |
fibp1 33889 | Value of the Fibonacci seq... |
fib2 33890 | Value of the Fibonacci seq... |
fib3 33891 | Value of the Fibonacci seq... |
fib4 33892 | Value of the Fibonacci seq... |
fib5 33893 | Value of the Fibonacci seq... |
fib6 33894 | Value of the Fibonacci seq... |
elprob 33897 | The property of being a pr... |
domprobmeas 33898 | A probability measure is a... |
domprobsiga 33899 | The domain of a probabilit... |
probtot 33900 | The probability of the uni... |
prob01 33901 | A probability is an elemen... |
probnul 33902 | The probability of the emp... |
unveldomd 33903 | The universe is an element... |
unveldom 33904 | The universe is an element... |
nuleldmp 33905 | The empty set is an elemen... |
probcun 33906 | The probability of the uni... |
probun 33907 | The probability of the uni... |
probdif 33908 | The probability of the dif... |
probinc 33909 | A probability law is incre... |
probdsb 33910 | The probability of the com... |
probmeasd 33911 | A probability measure is a... |
probvalrnd 33912 | The value of a probability... |
probtotrnd 33913 | The probability of the uni... |
totprobd 33914 | Law of total probability, ... |
totprob 33915 | Law of total probability. ... |
probfinmeasb 33916 | Build a probability measur... |
probfinmeasbALTV 33917 | Alternate version of ~ pro... |
probmeasb 33918 | Build a probability from a... |
cndprobval 33921 | The value of the condition... |
cndprobin 33922 | An identity linking condit... |
cndprob01 33923 | The conditional probabilit... |
cndprobtot 33924 | The conditional probabilit... |
cndprobnul 33925 | The conditional probabilit... |
cndprobprob 33926 | The conditional probabilit... |
bayesth 33927 | Bayes Theorem. (Contribut... |
rrvmbfm 33930 | A real-valued random varia... |
isrrvv 33931 | Elementhood to the set of ... |
rrvvf 33932 | A real-valued random varia... |
rrvfn 33933 | A real-valued random varia... |
rrvdm 33934 | The domain of a random var... |
rrvrnss 33935 | The range of a random vari... |
rrvf2 33936 | A real-valued random varia... |
rrvdmss 33937 | The domain of a random var... |
rrvfinvima 33938 | For a real-value random va... |
0rrv 33939 | The constant function equa... |
rrvadd 33940 | The sum of two random vari... |
rrvmulc 33941 | A random variable multipli... |
rrvsum 33942 | An indexed sum of random v... |
orvcval 33945 | Value of the preimage mapp... |
orvcval2 33946 | Another way to express the... |
elorvc 33947 | Elementhood of a preimage.... |
orvcval4 33948 | The value of the preimage ... |
orvcoel 33949 | If the relation produces o... |
orvccel 33950 | If the relation produces c... |
elorrvc 33951 | Elementhood of a preimage ... |
orrvcval4 33952 | The value of the preimage ... |
orrvcoel 33953 | If the relation produces o... |
orrvccel 33954 | If the relation produces c... |
orvcgteel 33955 | Preimage maps produced by ... |
orvcelval 33956 | Preimage maps produced by ... |
orvcelel 33957 | Preimage maps produced by ... |
dstrvval 33958 | The value of the distribut... |
dstrvprob 33959 | The distribution of a rand... |
orvclteel 33960 | Preimage maps produced by ... |
dstfrvel 33961 | Elementhood of preimage ma... |
dstfrvunirn 33962 | The limit of all preimage ... |
orvclteinc 33963 | Preimage maps produced by ... |
dstfrvinc 33964 | A cumulative distribution ... |
dstfrvclim1 33965 | The limit of the cumulativ... |
coinfliplem 33966 | Division in the extended r... |
coinflipprob 33967 | The ` P ` we defined for c... |
coinflipspace 33968 | The space of our coin-flip... |
coinflipuniv 33969 | The universe of our coin-f... |
coinfliprv 33970 | The ` X ` we defined for c... |
coinflippv 33971 | The probability of heads i... |
coinflippvt 33972 | The probability of tails i... |
ballotlemoex 33973 | ` O ` is a set. (Contribu... |
ballotlem1 33974 | The size of the universe i... |
ballotlemelo 33975 | Elementhood in ` O ` . (C... |
ballotlem2 33976 | The probability that the f... |
ballotlemfval 33977 | The value of ` F ` . (Con... |
ballotlemfelz 33978 | ` ( F `` C ) ` has values ... |
ballotlemfp1 33979 | If the ` J ` th ballot is ... |
ballotlemfc0 33980 | ` F ` takes value 0 betwee... |
ballotlemfcc 33981 | ` F ` takes value 0 betwee... |
ballotlemfmpn 33982 | ` ( F `` C ) ` finishes co... |
ballotlemfval0 33983 | ` ( F `` C ) ` always star... |
ballotleme 33984 | Elements of ` E ` . (Cont... |
ballotlemodife 33985 | Elements of ` ( O \ E ) ` ... |
ballotlem4 33986 | If the first pick is a vot... |
ballotlem5 33987 | If A is not ahead througho... |
ballotlemi 33988 | Value of ` I ` for a given... |
ballotlemiex 33989 | Properties of ` ( I `` C )... |
ballotlemi1 33990 | The first tie cannot be re... |
ballotlemii 33991 | The first tie cannot be re... |
ballotlemsup 33992 | The set of zeroes of ` F `... |
ballotlemimin 33993 | ` ( I `` C ) ` is the firs... |
ballotlemic 33994 | If the first vote is for B... |
ballotlem1c 33995 | If the first vote is for A... |
ballotlemsval 33996 | Value of ` S ` . (Contrib... |
ballotlemsv 33997 | Value of ` S ` evaluated a... |
ballotlemsgt1 33998 | ` S ` maps values less tha... |
ballotlemsdom 33999 | Domain of ` S ` for a give... |
ballotlemsel1i 34000 | The range ` ( 1 ... ( I ``... |
ballotlemsf1o 34001 | The defined ` S ` is a bij... |
ballotlemsi 34002 | The image by ` S ` of the ... |
ballotlemsima 34003 | The image by ` S ` of an i... |
ballotlemieq 34004 | If two countings share the... |
ballotlemrval 34005 | Value of ` R ` . (Contrib... |
ballotlemscr 34006 | The image of ` ( R `` C ) ... |
ballotlemrv 34007 | Value of ` R ` evaluated a... |
ballotlemrv1 34008 | Value of ` R ` before the ... |
ballotlemrv2 34009 | Value of ` R ` after the t... |
ballotlemro 34010 | Range of ` R ` is included... |
ballotlemgval 34011 | Expand the value of ` .^ `... |
ballotlemgun 34012 | A property of the defined ... |
ballotlemfg 34013 | Express the value of ` ( F... |
ballotlemfrc 34014 | Express the value of ` ( F... |
ballotlemfrci 34015 | Reverse counting preserves... |
ballotlemfrceq 34016 | Value of ` F ` for a rever... |
ballotlemfrcn0 34017 | Value of ` F ` for a rever... |
ballotlemrc 34018 | Range of ` R ` . (Contrib... |
ballotlemirc 34019 | Applying ` R ` does not ch... |
ballotlemrinv0 34020 | Lemma for ~ ballotlemrinv ... |
ballotlemrinv 34021 | ` R ` is its own inverse :... |
ballotlem1ri 34022 | When the vote on the first... |
ballotlem7 34023 | ` R ` is a bijection betwe... |
ballotlem8 34024 | There are as many counting... |
ballotth 34025 | Bertrand's ballot problem ... |
sgncl 34026 | Closure of the signum. (C... |
sgnclre 34027 | Closure of the signum. (C... |
sgnneg 34028 | Negation of the signum. (... |
sgn3da 34029 | A conditional containing a... |
sgnmul 34030 | Signum of a product. (Con... |
sgnmulrp2 34031 | Multiplication by a positi... |
sgnsub 34032 | Subtraction of a number of... |
sgnnbi 34033 | Negative signum. (Contrib... |
sgnpbi 34034 | Positive signum. (Contrib... |
sgn0bi 34035 | Zero signum. (Contributed... |
sgnsgn 34036 | Signum is idempotent. (Co... |
sgnmulsgn 34037 | If two real numbers are of... |
sgnmulsgp 34038 | If two real numbers are of... |
fzssfzo 34039 | Condition for an integer i... |
gsumncl 34040 | Closure of a group sum in ... |
gsumnunsn 34041 | Closure of a group sum in ... |
ccatmulgnn0dir 34042 | Concatenation of words fol... |
ofcccat 34043 | Letterwise operations on w... |
ofcs1 34044 | Letterwise operations on a... |
ofcs2 34045 | Letterwise operations on a... |
plymul02 34046 | Product of a polynomial wi... |
plymulx0 34047 | Coefficients of a polynomi... |
plymulx 34048 | Coefficients of a polynomi... |
plyrecld 34049 | Closure of a polynomial wi... |
signsplypnf 34050 | The quotient of a polynomi... |
signsply0 34051 | Lemma for the rule of sign... |
signspval 34052 | The value of the skipping ... |
signsw0glem 34053 | Neutral element property o... |
signswbase 34054 | The base of ` W ` is the u... |
signswplusg 34055 | The operation of ` W ` . ... |
signsw0g 34056 | The neutral element of ` W... |
signswmnd 34057 | ` W ` is a monoid structur... |
signswrid 34058 | The zero-skipping operatio... |
signswlid 34059 | The zero-skipping operatio... |
signswn0 34060 | The zero-skipping operatio... |
signswch 34061 | The zero-skipping operatio... |
signslema 34062 | Computational part of ~~? ... |
signstfv 34063 | Value of the zero-skipping... |
signstfval 34064 | Value of the zero-skipping... |
signstcl 34065 | Closure of the zero skippi... |
signstf 34066 | The zero skipping sign wor... |
signstlen 34067 | Length of the zero skippin... |
signstf0 34068 | Sign of a single letter wo... |
signstfvn 34069 | Zero-skipping sign in a wo... |
signsvtn0 34070 | If the last letter is nonz... |
signstfvp 34071 | Zero-skipping sign in a wo... |
signstfvneq0 34072 | In case the first letter i... |
signstfvcl 34073 | Closure of the zero skippi... |
signstfvc 34074 | Zero-skipping sign in a wo... |
signstres 34075 | Restriction of a zero skip... |
signstfveq0a 34076 | Lemma for ~ signstfveq0 . ... |
signstfveq0 34077 | In case the last letter is... |
signsvvfval 34078 | The value of ` V ` , which... |
signsvvf 34079 | ` V ` is a function. (Con... |
signsvf0 34080 | There is no change of sign... |
signsvf1 34081 | In a single-letter word, w... |
signsvfn 34082 | Number of changes in a wor... |
signsvtp 34083 | Adding a letter of the sam... |
signsvtn 34084 | Adding a letter of a diffe... |
signsvfpn 34085 | Adding a letter of the sam... |
signsvfnn 34086 | Adding a letter of a diffe... |
signlem0 34087 | Adding a zero as the highe... |
signshf 34088 | ` H ` , corresponding to t... |
signshwrd 34089 | ` H ` , corresponding to t... |
signshlen 34090 | Length of ` H ` , correspo... |
signshnz 34091 | ` H ` is not the empty wor... |
efcld 34092 | Closure law for the expone... |
iblidicc 34093 | The identity function is i... |
rpsqrtcn 34094 | Continuity of the real pos... |
divsqrtid 34095 | A real number divided by i... |
cxpcncf1 34096 | The power function on comp... |
efmul2picn 34097 | Multiplying by ` ( _i x. (... |
fct2relem 34098 | Lemma for ~ ftc2re . (Con... |
ftc2re 34099 | The Fundamental Theorem of... |
fdvposlt 34100 | Functions with a positive ... |
fdvneggt 34101 | Functions with a negative ... |
fdvposle 34102 | Functions with a nonnegati... |
fdvnegge 34103 | Functions with a nonpositi... |
prodfzo03 34104 | A product of three factors... |
actfunsnf1o 34105 | The action ` F ` of extend... |
actfunsnrndisj 34106 | The action ` F ` of extend... |
itgexpif 34107 | The basis for the circle m... |
fsum2dsub 34108 | Lemma for ~ breprexp - Re-... |
reprval 34111 | Value of the representatio... |
repr0 34112 | There is exactly one repre... |
reprf 34113 | Members of the representat... |
reprsum 34114 | Sums of values of the memb... |
reprle 34115 | Upper bound to the terms i... |
reprsuc 34116 | Express the representation... |
reprfi 34117 | Bounded representations ar... |
reprss 34118 | Representations with terms... |
reprinrn 34119 | Representations with term ... |
reprlt 34120 | There are no representatio... |
hashreprin 34121 | Express a sum of represent... |
reprgt 34122 | There are no representatio... |
reprinfz1 34123 | For the representation of ... |
reprfi2 34124 | Corollary of ~ reprinfz1 .... |
reprfz1 34125 | Corollary of ~ reprinfz1 .... |
hashrepr 34126 | Develop the number of repr... |
reprpmtf1o 34127 | Transposing ` 0 ` and ` X ... |
reprdifc 34128 | Express the representation... |
chpvalz 34129 | Value of the second Chebys... |
chtvalz 34130 | Value of the Chebyshev fun... |
breprexplema 34131 | Lemma for ~ breprexp (indu... |
breprexplemb 34132 | Lemma for ~ breprexp (clos... |
breprexplemc 34133 | Lemma for ~ breprexp (indu... |
breprexp 34134 | Express the ` S ` th power... |
breprexpnat 34135 | Express the ` S ` th power... |
vtsval 34138 | Value of the Vinogradov tr... |
vtscl 34139 | Closure of the Vinogradov ... |
vtsprod 34140 | Express the Vinogradov tri... |
circlemeth 34141 | The Hardy, Littlewood and ... |
circlemethnat 34142 | The Hardy, Littlewood and ... |
circlevma 34143 | The Circle Method, where t... |
circlemethhgt 34144 | The circle method, where t... |
hgt750lemc 34148 | An upper bound to the summ... |
hgt750lemd 34149 | An upper bound to the summ... |
hgt749d 34150 | A deduction version of ~ a... |
logdivsqrle 34151 | Conditions for ` ( ( log `... |
hgt750lem 34152 | Lemma for ~ tgoldbachgtd .... |
hgt750lem2 34153 | Decimal multiplication gal... |
hgt750lemf 34154 | Lemma for the statement 7.... |
hgt750lemg 34155 | Lemma for the statement 7.... |
oddprm2 34156 | Two ways to write the set ... |
hgt750lemb 34157 | An upper bound on the cont... |
hgt750lema 34158 | An upper bound on the cont... |
hgt750leme 34159 | An upper bound on the cont... |
tgoldbachgnn 34160 | Lemma for ~ tgoldbachgtd .... |
tgoldbachgtde 34161 | Lemma for ~ tgoldbachgtd .... |
tgoldbachgtda 34162 | Lemma for ~ tgoldbachgtd .... |
tgoldbachgtd 34163 | Odd integers greater than ... |
tgoldbachgt 34164 | Odd integers greater than ... |
istrkg2d 34167 | Property of fulfilling dim... |
axtglowdim2ALTV 34168 | Alternate version of ~ axt... |
axtgupdim2ALTV 34169 | Alternate version of ~ axt... |
afsval 34172 | Value of the AFS relation ... |
brafs 34173 | Binary relation form of th... |
tg5segofs 34174 | Rephrase ~ axtg5seg using ... |
lpadval 34177 | Value of the ` leftpad ` f... |
lpadlem1 34178 | Lemma for the ` leftpad ` ... |
lpadlem3 34179 | Lemma for ~ lpadlen1 . (C... |
lpadlen1 34180 | Length of a left-padded wo... |
lpadlem2 34181 | Lemma for the ` leftpad ` ... |
lpadlen2 34182 | Length of a left-padded wo... |
lpadmax 34183 | Length of a left-padded wo... |
lpadleft 34184 | The contents of prefix of ... |
lpadright 34185 | The suffix of a left-padde... |
bnj170 34198 | ` /\ ` -manipulation. (Co... |
bnj240 34199 | ` /\ ` -manipulation. (Co... |
bnj248 34200 | ` /\ ` -manipulation. (Co... |
bnj250 34201 | ` /\ ` -manipulation. (Co... |
bnj251 34202 | ` /\ ` -manipulation. (Co... |
bnj252 34203 | ` /\ ` -manipulation. (Co... |
bnj253 34204 | ` /\ ` -manipulation. (Co... |
bnj255 34205 | ` /\ ` -manipulation. (Co... |
bnj256 34206 | ` /\ ` -manipulation. (Co... |
bnj257 34207 | ` /\ ` -manipulation. (Co... |
bnj258 34208 | ` /\ ` -manipulation. (Co... |
bnj268 34209 | ` /\ ` -manipulation. (Co... |
bnj290 34210 | ` /\ ` -manipulation. (Co... |
bnj291 34211 | ` /\ ` -manipulation. (Co... |
bnj312 34212 | ` /\ ` -manipulation. (Co... |
bnj334 34213 | ` /\ ` -manipulation. (Co... |
bnj345 34214 | ` /\ ` -manipulation. (Co... |
bnj422 34215 | ` /\ ` -manipulation. (Co... |
bnj432 34216 | ` /\ ` -manipulation. (Co... |
bnj446 34217 | ` /\ ` -manipulation. (Co... |
bnj23 34218 | First-order logic and set ... |
bnj31 34219 | First-order logic and set ... |
bnj62 34220 | First-order logic and set ... |
bnj89 34221 | First-order logic and set ... |
bnj90 34222 | First-order logic and set ... |
bnj101 34223 | First-order logic and set ... |
bnj105 34224 | First-order logic and set ... |
bnj115 34225 | First-order logic and set ... |
bnj132 34226 | First-order logic and set ... |
bnj133 34227 | First-order logic and set ... |
bnj156 34228 | First-order logic and set ... |
bnj158 34229 | First-order logic and set ... |
bnj168 34230 | First-order logic and set ... |
bnj206 34231 | First-order logic and set ... |
bnj216 34232 | First-order logic and set ... |
bnj219 34233 | First-order logic and set ... |
bnj226 34234 | First-order logic and set ... |
bnj228 34235 | First-order logic and set ... |
bnj519 34236 | First-order logic and set ... |
bnj524 34237 | First-order logic and set ... |
bnj525 34238 | First-order logic and set ... |
bnj534 34239 | First-order logic and set ... |
bnj538 34240 | First-order logic and set ... |
bnj529 34241 | First-order logic and set ... |
bnj551 34242 | First-order logic and set ... |
bnj563 34243 | First-order logic and set ... |
bnj564 34244 | First-order logic and set ... |
bnj593 34245 | First-order logic and set ... |
bnj596 34246 | First-order logic and set ... |
bnj610 34247 | Pass from equality ( ` x =... |
bnj642 34248 | ` /\ ` -manipulation. (Co... |
bnj643 34249 | ` /\ ` -manipulation. (Co... |
bnj645 34250 | ` /\ ` -manipulation. (Co... |
bnj658 34251 | ` /\ ` -manipulation. (Co... |
bnj667 34252 | ` /\ ` -manipulation. (Co... |
bnj705 34253 | ` /\ ` -manipulation. (Co... |
bnj706 34254 | ` /\ ` -manipulation. (Co... |
bnj707 34255 | ` /\ ` -manipulation. (Co... |
bnj708 34256 | ` /\ ` -manipulation. (Co... |
bnj721 34257 | ` /\ ` -manipulation. (Co... |
bnj832 34258 | ` /\ ` -manipulation. (Co... |
bnj835 34259 | ` /\ ` -manipulation. (Co... |
bnj836 34260 | ` /\ ` -manipulation. (Co... |
bnj837 34261 | ` /\ ` -manipulation. (Co... |
bnj769 34262 | ` /\ ` -manipulation. (Co... |
bnj770 34263 | ` /\ ` -manipulation. (Co... |
bnj771 34264 | ` /\ ` -manipulation. (Co... |
bnj887 34265 | ` /\ ` -manipulation. (Co... |
bnj918 34266 | First-order logic and set ... |
bnj919 34267 | First-order logic and set ... |
bnj923 34268 | First-order logic and set ... |
bnj927 34269 | First-order logic and set ... |
bnj931 34270 | First-order logic and set ... |
bnj937 34271 | First-order logic and set ... |
bnj941 34272 | First-order logic and set ... |
bnj945 34273 | Technical lemma for ~ bnj6... |
bnj946 34274 | First-order logic and set ... |
bnj951 34275 | ` /\ ` -manipulation. (Co... |
bnj956 34276 | First-order logic and set ... |
bnj976 34277 | First-order logic and set ... |
bnj982 34278 | First-order logic and set ... |
bnj1019 34279 | First-order logic and set ... |
bnj1023 34280 | First-order logic and set ... |
bnj1095 34281 | First-order logic and set ... |
bnj1096 34282 | First-order logic and set ... |
bnj1098 34283 | First-order logic and set ... |
bnj1101 34284 | First-order logic and set ... |
bnj1113 34285 | First-order logic and set ... |
bnj1109 34286 | First-order logic and set ... |
bnj1131 34287 | First-order logic and set ... |
bnj1138 34288 | First-order logic and set ... |
bnj1142 34289 | First-order logic and set ... |
bnj1143 34290 | First-order logic and set ... |
bnj1146 34291 | First-order logic and set ... |
bnj1149 34292 | First-order logic and set ... |
bnj1185 34293 | First-order logic and set ... |
bnj1196 34294 | First-order logic and set ... |
bnj1198 34295 | First-order logic and set ... |
bnj1209 34296 | First-order logic and set ... |
bnj1211 34297 | First-order logic and set ... |
bnj1213 34298 | First-order logic and set ... |
bnj1212 34299 | First-order logic and set ... |
bnj1219 34300 | First-order logic and set ... |
bnj1224 34301 | First-order logic and set ... |
bnj1230 34302 | First-order logic and set ... |
bnj1232 34303 | First-order logic and set ... |
bnj1235 34304 | First-order logic and set ... |
bnj1239 34305 | First-order logic and set ... |
bnj1238 34306 | First-order logic and set ... |
bnj1241 34307 | First-order logic and set ... |
bnj1247 34308 | First-order logic and set ... |
bnj1254 34309 | First-order logic and set ... |
bnj1262 34310 | First-order logic and set ... |
bnj1266 34311 | First-order logic and set ... |
bnj1265 34312 | First-order logic and set ... |
bnj1275 34313 | First-order logic and set ... |
bnj1276 34314 | First-order logic and set ... |
bnj1292 34315 | First-order logic and set ... |
bnj1293 34316 | First-order logic and set ... |
bnj1294 34317 | First-order logic and set ... |
bnj1299 34318 | First-order logic and set ... |
bnj1304 34319 | First-order logic and set ... |
bnj1316 34320 | First-order logic and set ... |
bnj1317 34321 | First-order logic and set ... |
bnj1322 34322 | First-order logic and set ... |
bnj1340 34323 | First-order logic and set ... |
bnj1345 34324 | First-order logic and set ... |
bnj1350 34325 | First-order logic and set ... |
bnj1351 34326 | First-order logic and set ... |
bnj1352 34327 | First-order logic and set ... |
bnj1361 34328 | First-order logic and set ... |
bnj1366 34329 | First-order logic and set ... |
bnj1379 34330 | First-order logic and set ... |
bnj1383 34331 | First-order logic and set ... |
bnj1385 34332 | First-order logic and set ... |
bnj1386 34333 | First-order logic and set ... |
bnj1397 34334 | First-order logic and set ... |
bnj1400 34335 | First-order logic and set ... |
bnj1405 34336 | First-order logic and set ... |
bnj1422 34337 | First-order logic and set ... |
bnj1424 34338 | First-order logic and set ... |
bnj1436 34339 | First-order logic and set ... |
bnj1441 34340 | First-order logic and set ... |
bnj1441g 34341 | First-order logic and set ... |
bnj1454 34342 | First-order logic and set ... |
bnj1459 34343 | First-order logic and set ... |
bnj1464 34344 | Conversion of implicit sub... |
bnj1465 34345 | First-order logic and set ... |
bnj1468 34346 | Conversion of implicit sub... |
bnj1476 34347 | First-order logic and set ... |
bnj1502 34348 | First-order logic and set ... |
bnj1503 34349 | First-order logic and set ... |
bnj1517 34350 | First-order logic and set ... |
bnj1521 34351 | First-order logic and set ... |
bnj1533 34352 | First-order logic and set ... |
bnj1534 34353 | First-order logic and set ... |
bnj1536 34354 | First-order logic and set ... |
bnj1538 34355 | First-order logic and set ... |
bnj1541 34356 | First-order logic and set ... |
bnj1542 34357 | First-order logic and set ... |
bnj110 34358 | Well-founded induction res... |
bnj157 34359 | Well-founded induction res... |
bnj66 34360 | Technical lemma for ~ bnj6... |
bnj91 34361 | First-order logic and set ... |
bnj92 34362 | First-order logic and set ... |
bnj93 34363 | Technical lemma for ~ bnj9... |
bnj95 34364 | Technical lemma for ~ bnj1... |
bnj96 34365 | Technical lemma for ~ bnj1... |
bnj97 34366 | Technical lemma for ~ bnj1... |
bnj98 34367 | Technical lemma for ~ bnj1... |
bnj106 34368 | First-order logic and set ... |
bnj118 34369 | First-order logic and set ... |
bnj121 34370 | First-order logic and set ... |
bnj124 34371 | Technical lemma for ~ bnj1... |
bnj125 34372 | Technical lemma for ~ bnj1... |
bnj126 34373 | Technical lemma for ~ bnj1... |
bnj130 34374 | Technical lemma for ~ bnj1... |
bnj149 34375 | Technical lemma for ~ bnj1... |
bnj150 34376 | Technical lemma for ~ bnj1... |
bnj151 34377 | Technical lemma for ~ bnj1... |
bnj154 34378 | Technical lemma for ~ bnj1... |
bnj155 34379 | Technical lemma for ~ bnj1... |
bnj153 34380 | Technical lemma for ~ bnj8... |
bnj207 34381 | Technical lemma for ~ bnj8... |
bnj213 34382 | First-order logic and set ... |
bnj222 34383 | Technical lemma for ~ bnj2... |
bnj229 34384 | Technical lemma for ~ bnj5... |
bnj517 34385 | Technical lemma for ~ bnj5... |
bnj518 34386 | Technical lemma for ~ bnj8... |
bnj523 34387 | Technical lemma for ~ bnj8... |
bnj526 34388 | Technical lemma for ~ bnj8... |
bnj528 34389 | Technical lemma for ~ bnj8... |
bnj535 34390 | Technical lemma for ~ bnj8... |
bnj539 34391 | Technical lemma for ~ bnj8... |
bnj540 34392 | Technical lemma for ~ bnj8... |
bnj543 34393 | Technical lemma for ~ bnj8... |
bnj544 34394 | Technical lemma for ~ bnj8... |
bnj545 34395 | Technical lemma for ~ bnj8... |
bnj546 34396 | Technical lemma for ~ bnj8... |
bnj548 34397 | Technical lemma for ~ bnj8... |
bnj553 34398 | Technical lemma for ~ bnj8... |
bnj554 34399 | Technical lemma for ~ bnj8... |
bnj556 34400 | Technical lemma for ~ bnj8... |
bnj557 34401 | Technical lemma for ~ bnj8... |
bnj558 34402 | Technical lemma for ~ bnj8... |
bnj561 34403 | Technical lemma for ~ bnj8... |
bnj562 34404 | Technical lemma for ~ bnj8... |
bnj570 34405 | Technical lemma for ~ bnj8... |
bnj571 34406 | Technical lemma for ~ bnj8... |
bnj605 34407 | Technical lemma. This lem... |
bnj581 34408 | Technical lemma for ~ bnj5... |
bnj589 34409 | Technical lemma for ~ bnj8... |
bnj590 34410 | Technical lemma for ~ bnj8... |
bnj591 34411 | Technical lemma for ~ bnj8... |
bnj594 34412 | Technical lemma for ~ bnj8... |
bnj580 34413 | Technical lemma for ~ bnj5... |
bnj579 34414 | Technical lemma for ~ bnj8... |
bnj602 34415 | Equality theorem for the `... |
bnj607 34416 | Technical lemma for ~ bnj8... |
bnj609 34417 | Technical lemma for ~ bnj8... |
bnj611 34418 | Technical lemma for ~ bnj8... |
bnj600 34419 | Technical lemma for ~ bnj8... |
bnj601 34420 | Technical lemma for ~ bnj8... |
bnj852 34421 | Technical lemma for ~ bnj6... |
bnj864 34422 | Technical lemma for ~ bnj6... |
bnj865 34423 | Technical lemma for ~ bnj6... |
bnj873 34424 | Technical lemma for ~ bnj6... |
bnj849 34425 | Technical lemma for ~ bnj6... |
bnj882 34426 | Definition (using hypothes... |
bnj18eq1 34427 | Equality theorem for trans... |
bnj893 34428 | Property of ` _trCl ` . U... |
bnj900 34429 | Technical lemma for ~ bnj6... |
bnj906 34430 | Property of ` _trCl ` . (... |
bnj908 34431 | Technical lemma for ~ bnj6... |
bnj911 34432 | Technical lemma for ~ bnj6... |
bnj916 34433 | Technical lemma for ~ bnj6... |
bnj917 34434 | Technical lemma for ~ bnj6... |
bnj934 34435 | Technical lemma for ~ bnj6... |
bnj929 34436 | Technical lemma for ~ bnj6... |
bnj938 34437 | Technical lemma for ~ bnj6... |
bnj944 34438 | Technical lemma for ~ bnj6... |
bnj953 34439 | Technical lemma for ~ bnj6... |
bnj958 34440 | Technical lemma for ~ bnj6... |
bnj1000 34441 | Technical lemma for ~ bnj8... |
bnj965 34442 | Technical lemma for ~ bnj8... |
bnj964 34443 | Technical lemma for ~ bnj6... |
bnj966 34444 | Technical lemma for ~ bnj6... |
bnj967 34445 | Technical lemma for ~ bnj6... |
bnj969 34446 | Technical lemma for ~ bnj6... |
bnj970 34447 | Technical lemma for ~ bnj6... |
bnj910 34448 | Technical lemma for ~ bnj6... |
bnj978 34449 | Technical lemma for ~ bnj6... |
bnj981 34450 | Technical lemma for ~ bnj6... |
bnj983 34451 | Technical lemma for ~ bnj6... |
bnj984 34452 | Technical lemma for ~ bnj6... |
bnj985v 34453 | Version of ~ bnj985 with a... |
bnj985 34454 | Technical lemma for ~ bnj6... |
bnj986 34455 | Technical lemma for ~ bnj6... |
bnj996 34456 | Technical lemma for ~ bnj6... |
bnj998 34457 | Technical lemma for ~ bnj6... |
bnj999 34458 | Technical lemma for ~ bnj6... |
bnj1001 34459 | Technical lemma for ~ bnj6... |
bnj1006 34460 | Technical lemma for ~ bnj6... |
bnj1014 34461 | Technical lemma for ~ bnj6... |
bnj1015 34462 | Technical lemma for ~ bnj6... |
bnj1018g 34463 | Version of ~ bnj1018 with ... |
bnj1018 34464 | Technical lemma for ~ bnj6... |
bnj1020 34465 | Technical lemma for ~ bnj6... |
bnj1021 34466 | Technical lemma for ~ bnj6... |
bnj907 34467 | Technical lemma for ~ bnj6... |
bnj1029 34468 | Property of ` _trCl ` . (... |
bnj1033 34469 | Technical lemma for ~ bnj6... |
bnj1034 34470 | Technical lemma for ~ bnj6... |
bnj1039 34471 | Technical lemma for ~ bnj6... |
bnj1040 34472 | Technical lemma for ~ bnj6... |
bnj1047 34473 | Technical lemma for ~ bnj6... |
bnj1049 34474 | Technical lemma for ~ bnj6... |
bnj1052 34475 | Technical lemma for ~ bnj6... |
bnj1053 34476 | Technical lemma for ~ bnj6... |
bnj1071 34477 | Technical lemma for ~ bnj6... |
bnj1083 34478 | Technical lemma for ~ bnj6... |
bnj1090 34479 | Technical lemma for ~ bnj6... |
bnj1093 34480 | Technical lemma for ~ bnj6... |
bnj1097 34481 | Technical lemma for ~ bnj6... |
bnj1110 34482 | Technical lemma for ~ bnj6... |
bnj1112 34483 | Technical lemma for ~ bnj6... |
bnj1118 34484 | Technical lemma for ~ bnj6... |
bnj1121 34485 | Technical lemma for ~ bnj6... |
bnj1123 34486 | Technical lemma for ~ bnj6... |
bnj1030 34487 | Technical lemma for ~ bnj6... |
bnj1124 34488 | Property of ` _trCl ` . (... |
bnj1133 34489 | Technical lemma for ~ bnj6... |
bnj1128 34490 | Technical lemma for ~ bnj6... |
bnj1127 34491 | Property of ` _trCl ` . (... |
bnj1125 34492 | Property of ` _trCl ` . (... |
bnj1145 34493 | Technical lemma for ~ bnj6... |
bnj1147 34494 | Property of ` _trCl ` . (... |
bnj1137 34495 | Property of ` _trCl ` . (... |
bnj1148 34496 | Property of ` _pred ` . (... |
bnj1136 34497 | Technical lemma for ~ bnj6... |
bnj1152 34498 | Technical lemma for ~ bnj6... |
bnj1154 34499 | Property of ` Fr ` . (Con... |
bnj1171 34500 | Technical lemma for ~ bnj6... |
bnj1172 34501 | Technical lemma for ~ bnj6... |
bnj1173 34502 | Technical lemma for ~ bnj6... |
bnj1174 34503 | Technical lemma for ~ bnj6... |
bnj1175 34504 | Technical lemma for ~ bnj6... |
bnj1176 34505 | Technical lemma for ~ bnj6... |
bnj1177 34506 | Technical lemma for ~ bnj6... |
bnj1186 34507 | Technical lemma for ~ bnj6... |
bnj1190 34508 | Technical lemma for ~ bnj6... |
bnj1189 34509 | Technical lemma for ~ bnj6... |
bnj69 34510 | Existence of a minimal ele... |
bnj1228 34511 | Existence of a minimal ele... |
bnj1204 34512 | Well-founded induction. T... |
bnj1234 34513 | Technical lemma for ~ bnj6... |
bnj1245 34514 | Technical lemma for ~ bnj6... |
bnj1256 34515 | Technical lemma for ~ bnj6... |
bnj1259 34516 | Technical lemma for ~ bnj6... |
bnj1253 34517 | Technical lemma for ~ bnj6... |
bnj1279 34518 | Technical lemma for ~ bnj6... |
bnj1286 34519 | Technical lemma for ~ bnj6... |
bnj1280 34520 | Technical lemma for ~ bnj6... |
bnj1296 34521 | Technical lemma for ~ bnj6... |
bnj1309 34522 | Technical lemma for ~ bnj6... |
bnj1307 34523 | Technical lemma for ~ bnj6... |
bnj1311 34524 | Technical lemma for ~ bnj6... |
bnj1318 34525 | Technical lemma for ~ bnj6... |
bnj1326 34526 | Technical lemma for ~ bnj6... |
bnj1321 34527 | Technical lemma for ~ bnj6... |
bnj1364 34528 | Property of ` _FrSe ` . (... |
bnj1371 34529 | Technical lemma for ~ bnj6... |
bnj1373 34530 | Technical lemma for ~ bnj6... |
bnj1374 34531 | Technical lemma for ~ bnj6... |
bnj1384 34532 | Technical lemma for ~ bnj6... |
bnj1388 34533 | Technical lemma for ~ bnj6... |
bnj1398 34534 | Technical lemma for ~ bnj6... |
bnj1413 34535 | Property of ` _trCl ` . (... |
bnj1408 34536 | Technical lemma for ~ bnj1... |
bnj1414 34537 | Property of ` _trCl ` . (... |
bnj1415 34538 | Technical lemma for ~ bnj6... |
bnj1416 34539 | Technical lemma for ~ bnj6... |
bnj1418 34540 | Property of ` _pred ` . (... |
bnj1417 34541 | Technical lemma for ~ bnj6... |
bnj1421 34542 | Technical lemma for ~ bnj6... |
bnj1444 34543 | Technical lemma for ~ bnj6... |
bnj1445 34544 | Technical lemma for ~ bnj6... |
bnj1446 34545 | Technical lemma for ~ bnj6... |
bnj1447 34546 | Technical lemma for ~ bnj6... |
bnj1448 34547 | Technical lemma for ~ bnj6... |
bnj1449 34548 | Technical lemma for ~ bnj6... |
bnj1442 34549 | Technical lemma for ~ bnj6... |
bnj1450 34550 | Technical lemma for ~ bnj6... |
bnj1423 34551 | Technical lemma for ~ bnj6... |
bnj1452 34552 | Technical lemma for ~ bnj6... |
bnj1466 34553 | Technical lemma for ~ bnj6... |
bnj1467 34554 | Technical lemma for ~ bnj6... |
bnj1463 34555 | Technical lemma for ~ bnj6... |
bnj1489 34556 | Technical lemma for ~ bnj6... |
bnj1491 34557 | Technical lemma for ~ bnj6... |
bnj1312 34558 | Technical lemma for ~ bnj6... |
bnj1493 34559 | Technical lemma for ~ bnj6... |
bnj1497 34560 | Technical lemma for ~ bnj6... |
bnj1498 34561 | Technical lemma for ~ bnj6... |
bnj60 34562 | Well-founded recursion, pa... |
bnj1514 34563 | Technical lemma for ~ bnj1... |
bnj1518 34564 | Technical lemma for ~ bnj1... |
bnj1519 34565 | Technical lemma for ~ bnj1... |
bnj1520 34566 | Technical lemma for ~ bnj1... |
bnj1501 34567 | Technical lemma for ~ bnj1... |
bnj1500 34568 | Well-founded recursion, pa... |
bnj1525 34569 | Technical lemma for ~ bnj1... |
bnj1529 34570 | Technical lemma for ~ bnj1... |
bnj1523 34571 | Technical lemma for ~ bnj1... |
bnj1522 34572 | Well-founded recursion, pa... |
exdifsn 34573 | There exists an element in... |
srcmpltd 34574 | If a statement is true for... |
prsrcmpltd 34575 | If a statement is true for... |
dff15 34576 | A one-to-one function in t... |
f1resveqaeq 34577 | If a function restricted t... |
f1resrcmplf1dlem 34578 | Lemma for ~ f1resrcmplf1d ... |
f1resrcmplf1d 34579 | If a function's restrictio... |
funen1cnv 34580 | If a function is equinumer... |
fnrelpredd 34581 | A function that preserves ... |
cardpred 34582 | The cardinality function p... |
nummin 34583 | Every nonempty class of nu... |
fineqvrep 34584 | If the Axiom of Infinity i... |
fineqvpow 34585 | If the Axiom of Infinity i... |
fineqvac 34586 | If the Axiom of Infinity i... |
fineqvacALT 34587 | Shorter proof of ~ fineqva... |
zltp1ne 34588 | Integer ordering relation.... |
nnltp1ne 34589 | Positive integer ordering ... |
nn0ltp1ne 34590 | Nonnegative integer orderi... |
0nn0m1nnn0 34591 | A number is zero if and on... |
f1resfz0f1d 34592 | If a function with a seque... |
fisshasheq 34593 | A finite set is equal to i... |
hashf1dmcdm 34594 | The size of the domain of ... |
revpfxsfxrev 34595 | The reverse of a prefix of... |
swrdrevpfx 34596 | A subword expressed in ter... |
lfuhgr 34597 | A hypergraph is loop-free ... |
lfuhgr2 34598 | A hypergraph is loop-free ... |
lfuhgr3 34599 | A hypergraph is loop-free ... |
cplgredgex 34600 | Any two (distinct) vertice... |
cusgredgex 34601 | Any two (distinct) vertice... |
cusgredgex2 34602 | Any two distinct vertices ... |
pfxwlk 34603 | A prefix of a walk is a wa... |
revwlk 34604 | The reverse of a walk is a... |
revwlkb 34605 | Two words represent a walk... |
swrdwlk 34606 | Two matching subwords of a... |
pthhashvtx 34607 | A graph containing a path ... |
pthisspthorcycl 34608 | A path is either a simple ... |
spthcycl 34609 | A walk is a trivial path i... |
usgrgt2cycl 34610 | A non-trivial cycle in a s... |
usgrcyclgt2v 34611 | A simple graph with a non-... |
subgrwlk 34612 | If a walk exists in a subg... |
subgrtrl 34613 | If a trail exists in a sub... |
subgrpth 34614 | If a path exists in a subg... |
subgrcycl 34615 | If a cycle exists in a sub... |
cusgr3cyclex 34616 | Every complete simple grap... |
loop1cycl 34617 | A hypergraph has a cycle o... |
2cycld 34618 | Construction of a 2-cycle ... |
2cycl2d 34619 | Construction of a 2-cycle ... |
umgr2cycllem 34620 | Lemma for ~ umgr2cycl . (... |
umgr2cycl 34621 | A multigraph with two dist... |
dfacycgr1 34624 | An alternate definition of... |
isacycgr 34625 | The property of being an a... |
isacycgr1 34626 | The property of being an a... |
acycgrcycl 34627 | Any cycle in an acyclic gr... |
acycgr0v 34628 | A null graph (with no vert... |
acycgr1v 34629 | A multigraph with one vert... |
acycgr2v 34630 | A simple graph with two ve... |
prclisacycgr 34631 | A proper class (representi... |
acycgrislfgr 34632 | An acyclic hypergraph is a... |
upgracycumgr 34633 | An acyclic pseudograph is ... |
umgracycusgr 34634 | An acyclic multigraph is a... |
upgracycusgr 34635 | An acyclic pseudograph is ... |
cusgracyclt3v 34636 | A complete simple graph is... |
pthacycspth 34637 | A path in an acyclic graph... |
acycgrsubgr 34638 | The subgraph of an acyclic... |
quartfull 34645 | The quartic equation, writ... |
deranglem 34646 | Lemma for derangements. (... |
derangval 34647 | Define the derangement fun... |
derangf 34648 | The derangement number is ... |
derang0 34649 | The derangement number of ... |
derangsn 34650 | The derangement number of ... |
derangenlem 34651 | One half of ~ derangen . ... |
derangen 34652 | The derangement number is ... |
subfacval 34653 | The subfactorial is define... |
derangen2 34654 | Write the derangement numb... |
subfacf 34655 | The subfactorial is a func... |
subfaclefac 34656 | The subfactorial is less t... |
subfac0 34657 | The subfactorial at zero. ... |
subfac1 34658 | The subfactorial at one. ... |
subfacp1lem1 34659 | Lemma for ~ subfacp1 . Th... |
subfacp1lem2a 34660 | Lemma for ~ subfacp1 . Pr... |
subfacp1lem2b 34661 | Lemma for ~ subfacp1 . Pr... |
subfacp1lem3 34662 | Lemma for ~ subfacp1 . In... |
subfacp1lem4 34663 | Lemma for ~ subfacp1 . Th... |
subfacp1lem5 34664 | Lemma for ~ subfacp1 . In... |
subfacp1lem6 34665 | Lemma for ~ subfacp1 . By... |
subfacp1 34666 | A two-term recurrence for ... |
subfacval2 34667 | A closed-form expression f... |
subfaclim 34668 | The subfactorial converges... |
subfacval3 34669 | Another closed form expres... |
derangfmla 34670 | The derangements formula, ... |
erdszelem1 34671 | Lemma for ~ erdsze . (Con... |
erdszelem2 34672 | Lemma for ~ erdsze . (Con... |
erdszelem3 34673 | Lemma for ~ erdsze . (Con... |
erdszelem4 34674 | Lemma for ~ erdsze . (Con... |
erdszelem5 34675 | Lemma for ~ erdsze . (Con... |
erdszelem6 34676 | Lemma for ~ erdsze . (Con... |
erdszelem7 34677 | Lemma for ~ erdsze . (Con... |
erdszelem8 34678 | Lemma for ~ erdsze . (Con... |
erdszelem9 34679 | Lemma for ~ erdsze . (Con... |
erdszelem10 34680 | Lemma for ~ erdsze . (Con... |
erdszelem11 34681 | Lemma for ~ erdsze . (Con... |
erdsze 34682 | The Erdős-Szekeres th... |
erdsze2lem1 34683 | Lemma for ~ erdsze2 . (Co... |
erdsze2lem2 34684 | Lemma for ~ erdsze2 . (Co... |
erdsze2 34685 | Generalize the statement o... |
kur14lem1 34686 | Lemma for ~ kur14 . (Cont... |
kur14lem2 34687 | Lemma for ~ kur14 . Write... |
kur14lem3 34688 | Lemma for ~ kur14 . A clo... |
kur14lem4 34689 | Lemma for ~ kur14 . Compl... |
kur14lem5 34690 | Lemma for ~ kur14 . Closu... |
kur14lem6 34691 | Lemma for ~ kur14 . If ` ... |
kur14lem7 34692 | Lemma for ~ kur14 : main p... |
kur14lem8 34693 | Lemma for ~ kur14 . Show ... |
kur14lem9 34694 | Lemma for ~ kur14 . Since... |
kur14lem10 34695 | Lemma for ~ kur14 . Disch... |
kur14 34696 | Kuratowski's closure-compl... |
ispconn 34703 | The property of being a pa... |
pconncn 34704 | The property of being a pa... |
pconntop 34705 | A simply connected space i... |
issconn 34706 | The property of being a si... |
sconnpconn 34707 | A simply connected space i... |
sconntop 34708 | A simply connected space i... |
sconnpht 34709 | A closed path in a simply ... |
cnpconn 34710 | An image of a path-connect... |
pconnconn 34711 | A path-connected space is ... |
txpconn 34712 | The topological product of... |
ptpconn 34713 | The topological product of... |
indispconn 34714 | The indiscrete topology (o... |
connpconn 34715 | A connected and locally pa... |
qtoppconn 34716 | A quotient of a path-conne... |
pconnpi1 34717 | All fundamental groups in ... |
sconnpht2 34718 | Any two paths in a simply ... |
sconnpi1 34719 | A path-connected topologic... |
txsconnlem 34720 | Lemma for ~ txsconn . (Co... |
txsconn 34721 | The topological product of... |
cvxpconn 34722 | A convex subset of the com... |
cvxsconn 34723 | A convex subset of the com... |
blsconn 34724 | An open ball in the comple... |
cnllysconn 34725 | The topology of the comple... |
resconn 34726 | A subset of ` RR ` is simp... |
ioosconn 34727 | An open interval is simply... |
iccsconn 34728 | A closed interval is simpl... |
retopsconn 34729 | The real numbers are simpl... |
iccllysconn 34730 | A closed interval is local... |
rellysconn 34731 | The real numbers are local... |
iisconn 34732 | The unit interval is simpl... |
iillysconn 34733 | The unit interval is local... |
iinllyconn 34734 | The unit interval is local... |
fncvm 34737 | Lemma for covering maps. ... |
cvmscbv 34738 | Change bound variables in ... |
iscvm 34739 | The property of being a co... |
cvmtop1 34740 | Reverse closure for a cove... |
cvmtop2 34741 | Reverse closure for a cove... |
cvmcn 34742 | A covering map is a contin... |
cvmcov 34743 | Property of a covering map... |
cvmsrcl 34744 | Reverse closure for an eve... |
cvmsi 34745 | One direction of ~ cvmsval... |
cvmsval 34746 | Elementhood in the set ` S... |
cvmsss 34747 | An even covering is a subs... |
cvmsn0 34748 | An even covering is nonemp... |
cvmsuni 34749 | An even covering of ` U ` ... |
cvmsdisj 34750 | An even covering of ` U ` ... |
cvmshmeo 34751 | Every element of an even c... |
cvmsf1o 34752 | ` F ` , localized to an el... |
cvmscld 34753 | The sets of an even coveri... |
cvmsss2 34754 | An open subset of an evenl... |
cvmcov2 34755 | The covering map property ... |
cvmseu 34756 | Every element in ` U. T ` ... |
cvmsiota 34757 | Identify the unique elemen... |
cvmopnlem 34758 | Lemma for ~ cvmopn . (Con... |
cvmfolem 34759 | Lemma for ~ cvmfo . (Cont... |
cvmopn 34760 | A covering map is an open ... |
cvmliftmolem1 34761 | Lemma for ~ cvmliftmo . (... |
cvmliftmolem2 34762 | Lemma for ~ cvmliftmo . (... |
cvmliftmoi 34763 | A lift of a continuous fun... |
cvmliftmo 34764 | A lift of a continuous fun... |
cvmliftlem1 34765 | Lemma for ~ cvmlift . In ... |
cvmliftlem2 34766 | Lemma for ~ cvmlift . ` W ... |
cvmliftlem3 34767 | Lemma for ~ cvmlift . Sin... |
cvmliftlem4 34768 | Lemma for ~ cvmlift . The... |
cvmliftlem5 34769 | Lemma for ~ cvmlift . Def... |
cvmliftlem6 34770 | Lemma for ~ cvmlift . Ind... |
cvmliftlem7 34771 | Lemma for ~ cvmlift . Pro... |
cvmliftlem8 34772 | Lemma for ~ cvmlift . The... |
cvmliftlem9 34773 | Lemma for ~ cvmlift . The... |
cvmliftlem10 34774 | Lemma for ~ cvmlift . The... |
cvmliftlem11 34775 | Lemma for ~ cvmlift . (Co... |
cvmliftlem13 34776 | Lemma for ~ cvmlift . The... |
cvmliftlem14 34777 | Lemma for ~ cvmlift . Put... |
cvmliftlem15 34778 | Lemma for ~ cvmlift . Dis... |
cvmlift 34779 | One of the important prope... |
cvmfo 34780 | A covering map is an onto ... |
cvmliftiota 34781 | Write out a function ` H `... |
cvmlift2lem1 34782 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem9a 34783 | Lemma for ~ cvmlift2 and ~... |
cvmlift2lem2 34784 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem3 34785 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem4 34786 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem5 34787 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem6 34788 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem7 34789 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem8 34790 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem9 34791 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem10 34792 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem11 34793 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem12 34794 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem13 34795 | Lemma for ~ cvmlift2 . (C... |
cvmlift2 34796 | A two-dimensional version ... |
cvmliftphtlem 34797 | Lemma for ~ cvmliftpht . ... |
cvmliftpht 34798 | If ` G ` and ` H ` are pat... |
cvmlift3lem1 34799 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem2 34800 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem3 34801 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem4 34802 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem5 34803 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem6 34804 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem7 34805 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem8 34806 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem9 34807 | Lemma for ~ cvmlift2 . (C... |
cvmlift3 34808 | A general version of ~ cvm... |
snmlff 34809 | The function ` F ` from ~ ... |
snmlfval 34810 | The function ` F ` from ~ ... |
snmlval 34811 | The property " ` A ` is si... |
snmlflim 34812 | If ` A ` is simply normal,... |
goel 34827 | A "Godel-set of membership... |
goelel3xp 34828 | A "Godel-set of membership... |
goeleq12bg 34829 | Two "Godel-set of membersh... |
gonafv 34830 | The "Godel-set for the She... |
goaleq12d 34831 | Equality of the "Godel-set... |
gonanegoal 34832 | The Godel-set for the Shef... |
satf 34833 | The satisfaction predicate... |
satfsucom 34834 | The satisfaction predicate... |
satfn 34835 | The satisfaction predicate... |
satom 34836 | The satisfaction predicate... |
satfvsucom 34837 | The satisfaction predicate... |
satfv0 34838 | The value of the satisfact... |
satfvsuclem1 34839 | Lemma 1 for ~ satfvsuc . ... |
satfvsuclem2 34840 | Lemma 2 for ~ satfvsuc . ... |
satfvsuc 34841 | The value of the satisfact... |
satfv1lem 34842 | Lemma for ~ satfv1 . (Con... |
satfv1 34843 | The value of the satisfact... |
satfsschain 34844 | The binary relation of a s... |
satfvsucsuc 34845 | The satisfaction predicate... |
satfbrsuc 34846 | The binary relation of a s... |
satfrel 34847 | The value of the satisfact... |
satfdmlem 34848 | Lemma for ~ satfdm . (Con... |
satfdm 34849 | The domain of the satisfac... |
satfrnmapom 34850 | The range of the satisfact... |
satfv0fun 34851 | The value of the satisfact... |
satf0 34852 | The satisfaction predicate... |
satf0sucom 34853 | The satisfaction predicate... |
satf00 34854 | The value of the satisfact... |
satf0suclem 34855 | Lemma for ~ satf0suc , ~ s... |
satf0suc 34856 | The value of the satisfact... |
satf0op 34857 | An element of a value of t... |
satf0n0 34858 | The value of the satisfact... |
sat1el2xp 34859 | The first component of an ... |
fmlafv 34860 | The valid Godel formulas o... |
fmla 34861 | The set of all valid Godel... |
fmla0 34862 | The valid Godel formulas o... |
fmla0xp 34863 | The valid Godel formulas o... |
fmlasuc0 34864 | The valid Godel formulas o... |
fmlafvel 34865 | A class is a valid Godel f... |
fmlasuc 34866 | The valid Godel formulas o... |
fmla1 34867 | The valid Godel formulas o... |
isfmlasuc 34868 | The characterization of a ... |
fmlasssuc 34869 | The Godel formulas of heig... |
fmlaomn0 34870 | The empty set is not a God... |
fmlan0 34871 | The empty set is not a God... |
gonan0 34872 | The "Godel-set of NAND" is... |
goaln0 34873 | The "Godel-set of universa... |
gonarlem 34874 | Lemma for ~ gonar (inducti... |
gonar 34875 | If the "Godel-set of NAND"... |
goalrlem 34876 | Lemma for ~ goalr (inducti... |
goalr 34877 | If the "Godel-set of unive... |
fmla0disjsuc 34878 | The set of valid Godel for... |
fmlasucdisj 34879 | The valid Godel formulas o... |
satfdmfmla 34880 | The domain of the satisfac... |
satffunlem 34881 | Lemma for ~ satffunlem1lem... |
satffunlem1lem1 34882 | Lemma for ~ satffunlem1 . ... |
satffunlem1lem2 34883 | Lemma 2 for ~ satffunlem1 ... |
satffunlem2lem1 34884 | Lemma 1 for ~ satffunlem2 ... |
dmopab3rexdif 34885 | The domain of an ordered p... |
satffunlem2lem2 34886 | Lemma 2 for ~ satffunlem2 ... |
satffunlem1 34887 | Lemma 1 for ~ satffun : in... |
satffunlem2 34888 | Lemma 2 for ~ satffun : in... |
satffun 34889 | The value of the satisfact... |
satff 34890 | The satisfaction predicate... |
satfun 34891 | The satisfaction predicate... |
satfvel 34892 | An element of the value of... |
satfv0fvfmla0 34893 | The value of the satisfact... |
satefv 34894 | The simplified satisfactio... |
sate0 34895 | The simplified satisfactio... |
satef 34896 | The simplified satisfactio... |
sate0fv0 34897 | A simplified satisfaction ... |
satefvfmla0 34898 | The simplified satisfactio... |
sategoelfvb 34899 | Characterization of a valu... |
sategoelfv 34900 | Condition of a valuation `... |
ex-sategoelel 34901 | Example of a valuation of ... |
ex-sategoel 34902 | Instance of ~ sategoelfv f... |
satfv1fvfmla1 34903 | The value of the satisfact... |
2goelgoanfmla1 34904 | Two Godel-sets of membersh... |
satefvfmla1 34905 | The simplified satisfactio... |
ex-sategoelelomsuc 34906 | Example of a valuation of ... |
ex-sategoelel12 34907 | Example of a valuation of ... |
prv 34908 | The "proves" relation on a... |
elnanelprv 34909 | The wff ` ( A e. B -/\ B e... |
prv0 34910 | Every wff encoded as ` U `... |
prv1n 34911 | No wff encoded as a Godel-... |
mvtval 34980 | The set of variable typeco... |
mrexval 34981 | The set of "raw expression... |
mexval 34982 | The set of expressions, wh... |
mexval2 34983 | The set of expressions, wh... |
mdvval 34984 | The set of disjoint variab... |
mvrsval 34985 | The set of variables in an... |
mvrsfpw 34986 | The set of variables in an... |
mrsubffval 34987 | The substitution of some v... |
mrsubfval 34988 | The substitution of some v... |
mrsubval 34989 | The substitution of some v... |
mrsubcv 34990 | The value of a substituted... |
mrsubvr 34991 | The value of a substituted... |
mrsubff 34992 | A substitution is a functi... |
mrsubrn 34993 | Although it is defined for... |
mrsubff1 34994 | When restricted to complet... |
mrsubff1o 34995 | When restricted to complet... |
mrsub0 34996 | The value of the substitut... |
mrsubf 34997 | A substitution is a functi... |
mrsubccat 34998 | Substitution distributes o... |
mrsubcn 34999 | A substitution does not ch... |
elmrsubrn 35000 | Characterization of the su... |
mrsubco 35001 | The composition of two sub... |
mrsubvrs 35002 | The set of variables in a ... |
msubffval 35003 | A substitution applied to ... |
msubfval 35004 | A substitution applied to ... |
msubval 35005 | A substitution applied to ... |
msubrsub 35006 | A substitution applied to ... |
msubty 35007 | The type of a substituted ... |
elmsubrn 35008 | Characterization of substi... |
msubrn 35009 | Although it is defined for... |
msubff 35010 | A substitution is a functi... |
msubco 35011 | The composition of two sub... |
msubf 35012 | A substitution is a functi... |
mvhfval 35013 | Value of the function mapp... |
mvhval 35014 | Value of the function mapp... |
mpstval 35015 | A pre-statement is an orde... |
elmpst 35016 | Property of being a pre-st... |
msrfval 35017 | Value of the reduct of a p... |
msrval 35018 | Value of the reduct of a p... |
mpstssv 35019 | A pre-statement is an orde... |
mpst123 35020 | Decompose a pre-statement ... |
mpstrcl 35021 | The elements of a pre-stat... |
msrf 35022 | The reduct of a pre-statem... |
msrrcl 35023 | If ` X ` and ` Y ` have th... |
mstaval 35024 | Value of the set of statem... |
msrid 35025 | The reduct of a statement ... |
msrfo 35026 | The reduct of a pre-statem... |
mstapst 35027 | A statement is a pre-state... |
elmsta 35028 | Property of being a statem... |
ismfs 35029 | A formal system is a tuple... |
mfsdisj 35030 | The constants and variable... |
mtyf2 35031 | The type function maps var... |
mtyf 35032 | The type function maps var... |
mvtss 35033 | The set of variable typeco... |
maxsta 35034 | An axiom is a statement. ... |
mvtinf 35035 | Each variable typecode has... |
msubff1 35036 | When restricted to complet... |
msubff1o 35037 | When restricted to complet... |
mvhf 35038 | The function mapping varia... |
mvhf1 35039 | The function mapping varia... |
msubvrs 35040 | The set of variables in a ... |
mclsrcl 35041 | Reverse closure for the cl... |
mclsssvlem 35042 | Lemma for ~ mclsssv . (Co... |
mclsval 35043 | The function mapping varia... |
mclsssv 35044 | The closure of a set of ex... |
ssmclslem 35045 | Lemma for ~ ssmcls . (Con... |
vhmcls 35046 | All variable hypotheses ar... |
ssmcls 35047 | The original expressions a... |
ss2mcls 35048 | The closure is monotonic u... |
mclsax 35049 | The closure is closed unde... |
mclsind 35050 | Induction theorem for clos... |
mppspstlem 35051 | Lemma for ~ mppspst . (Co... |
mppsval 35052 | Definition of a provable p... |
elmpps 35053 | Definition of a provable p... |
mppspst 35054 | A provable pre-statement i... |
mthmval 35055 | A theorem is a pre-stateme... |
elmthm 35056 | A theorem is a pre-stateme... |
mthmi 35057 | A statement whose reduct i... |
mthmsta 35058 | A theorem is a pre-stateme... |
mppsthm 35059 | A provable pre-statement i... |
mthmblem 35060 | Lemma for ~ mthmb . (Cont... |
mthmb 35061 | If two statements have the... |
mthmpps 35062 | Given a theorem, there is ... |
mclsppslem 35063 | The closure is closed unde... |
mclspps 35064 | The closure is closed unde... |
problem1 35139 | Practice problem 1. Clues... |
problem2 35140 | Practice problem 2. Clues... |
problem3 35141 | Practice problem 3. Clues... |
problem4 35142 | Practice problem 4. Clues... |
problem5 35143 | Practice problem 5. Clues... |
quad3 35144 | Variant of quadratic equat... |
climuzcnv 35145 | Utility lemma to convert b... |
sinccvglem 35146 | ` ( ( sin `` x ) / x ) ~~>... |
sinccvg 35147 | ` ( ( sin `` x ) / x ) ~~>... |
circum 35148 | The circumference of a cir... |
elfzm12 35149 | Membership in a curtailed ... |
nn0seqcvg 35150 | A strictly-decreasing nonn... |
lediv2aALT 35151 | Division of both sides of ... |
abs2sqlei 35152 | The absolute values of two... |
abs2sqlti 35153 | The absolute values of two... |
abs2sqle 35154 | The absolute values of two... |
abs2sqlt 35155 | The absolute values of two... |
abs2difi 35156 | Difference of absolute val... |
abs2difabsi 35157 | Absolute value of differen... |
currybi 35158 | Biconditional version of C... |
axextprim 35165 | ~ ax-ext without distinct ... |
axrepprim 35166 | ~ ax-rep without distinct ... |
axunprim 35167 | ~ ax-un without distinct v... |
axpowprim 35168 | ~ ax-pow without distinct ... |
axregprim 35169 | ~ ax-reg without distinct ... |
axinfprim 35170 | ~ ax-inf without distinct ... |
axacprim 35171 | ~ ax-ac without distinct v... |
untelirr 35172 | We call a class "untanged"... |
untuni 35173 | The union of a class is un... |
untsucf 35174 | If a class is untangled, t... |
unt0 35175 | The null set is untangled.... |
untint 35176 | If there is an untangled e... |
efrunt 35177 | If ` A ` is well-founded b... |
untangtr 35178 | A transitive class is unta... |
3jaodd 35179 | Double deduction form of ~... |
3orit 35180 | Closed form of ~ 3ori . (... |
biimpexp 35181 | A biconditional in the ant... |
nepss 35182 | Two classes are unequal if... |
3ccased 35183 | Triple disjunction form of... |
dfso3 35184 | Expansion of the definitio... |
brtpid1 35185 | A binary relation involvin... |
brtpid2 35186 | A binary relation involvin... |
brtpid3 35187 | A binary relation involvin... |
iota5f 35188 | A method for computing iot... |
jath 35189 | Closed form of ~ ja . Pro... |
xpab 35190 | Cartesian product of two c... |
nnuni 35191 | The union of a finite ordi... |
sqdivzi 35192 | Distribution of square ove... |
supfz 35193 | The supremum of a finite s... |
inffz 35194 | The infimum of a finite se... |
fz0n 35195 | The sequence ` ( 0 ... ( N... |
shftvalg 35196 | Value of a sequence shifte... |
divcnvlin 35197 | Limit of the ratio of two ... |
climlec3 35198 | Comparison of a constant t... |
logi 35199 | Calculate the logarithm of... |
iexpire 35200 | ` _i ` raised to itself is... |
bcneg1 35201 | The binomial coefficent ov... |
bcm1nt 35202 | The proportion of one bion... |
bcprod 35203 | A product identity for bin... |
bccolsum 35204 | A column-sum rule for bino... |
iprodefisumlem 35205 | Lemma for ~ iprodefisum . ... |
iprodefisum 35206 | Applying the exponential f... |
iprodgam 35207 | An infinite product versio... |
faclimlem1 35208 | Lemma for ~ faclim . Clos... |
faclimlem2 35209 | Lemma for ~ faclim . Show... |
faclimlem3 35210 | Lemma for ~ faclim . Alge... |
faclim 35211 | An infinite product expres... |
iprodfac 35212 | An infinite product expres... |
faclim2 35213 | Another factorial limit du... |
gcd32 35214 | Swap the second and third ... |
gcdabsorb 35215 | Absorption law for gcd. (... |
dftr6 35216 | A potential definition of ... |
coep 35217 | Composition with the membe... |
coepr 35218 | Composition with the conve... |
dffr5 35219 | A quantifier-free definiti... |
dfso2 35220 | Quantifier-free definition... |
br8 35221 | Substitution for an eight-... |
br6 35222 | Substitution for a six-pla... |
br4 35223 | Substitution for a four-pl... |
cnvco1 35224 | Another distributive law o... |
cnvco2 35225 | Another distributive law o... |
eldm3 35226 | Quantifier-free definition... |
elrn3 35227 | Quantifier-free definition... |
pocnv 35228 | The converse of a partial ... |
socnv 35229 | The converse of a strict o... |
sotrd 35230 | Transitivity law for stric... |
elintfv 35231 | Membership in an intersect... |
funpsstri 35232 | A condition for subset tri... |
fundmpss 35233 | If a class ` F ` is a prop... |
funsseq 35234 | Given two functions with e... |
fununiq 35235 | The uniqueness condition o... |
funbreq 35236 | An equality condition for ... |
br1steq 35237 | Uniqueness condition for t... |
br2ndeq 35238 | Uniqueness condition for t... |
dfdm5 35239 | Definition of domain in te... |
dfrn5 35240 | Definition of range in ter... |
opelco3 35241 | Alternate way of saying th... |
elima4 35242 | Quantifier-free expression... |
fv1stcnv 35243 | The value of the converse ... |
fv2ndcnv 35244 | The value of the converse ... |
setinds 35245 | Principle of set induction... |
setinds2f 35246 | ` _E ` induction schema, u... |
setinds2 35247 | ` _E ` induction schema, u... |
elpotr 35248 | A class of transitive sets... |
dford5reg 35249 | Given ~ ax-reg , an ordina... |
dfon2lem1 35250 | Lemma for ~ dfon2 . (Cont... |
dfon2lem2 35251 | Lemma for ~ dfon2 . (Cont... |
dfon2lem3 35252 | Lemma for ~ dfon2 . All s... |
dfon2lem4 35253 | Lemma for ~ dfon2 . If tw... |
dfon2lem5 35254 | Lemma for ~ dfon2 . Two s... |
dfon2lem6 35255 | Lemma for ~ dfon2 . A tra... |
dfon2lem7 35256 | Lemma for ~ dfon2 . All e... |
dfon2lem8 35257 | Lemma for ~ dfon2 . The i... |
dfon2lem9 35258 | Lemma for ~ dfon2 . A cla... |
dfon2 35259 | ` On ` consists of all set... |
rdgprc0 35260 | The value of the recursive... |
rdgprc 35261 | The value of the recursive... |
dfrdg2 35262 | Alternate definition of th... |
dfrdg3 35263 | Generalization of ~ dfrdg2... |
axextdfeq 35264 | A version of ~ ax-ext for ... |
ax8dfeq 35265 | A version of ~ ax-8 for us... |
axextdist 35266 | ~ ax-ext with distinctors ... |
axextbdist 35267 | ~ axextb with distinctors ... |
19.12b 35268 | Version of ~ 19.12vv with ... |
exnel 35269 | There is always a set not ... |
distel 35270 | Distinctors in terms of me... |
axextndbi 35271 | ~ axextnd as a bicondition... |
hbntg 35272 | A more general form of ~ h... |
hbimtg 35273 | A more general and closed ... |
hbaltg 35274 | A more general and closed ... |
hbng 35275 | A more general form of ~ h... |
hbimg 35276 | A more general form of ~ h... |
wsuceq123 35281 | Equality theorem for well-... |
wsuceq1 35282 | Equality theorem for well-... |
wsuceq2 35283 | Equality theorem for well-... |
wsuceq3 35284 | Equality theorem for well-... |
nfwsuc 35285 | Bound-variable hypothesis ... |
wlimeq12 35286 | Equality theorem for the l... |
wlimeq1 35287 | Equality theorem for the l... |
wlimeq2 35288 | Equality theorem for the l... |
nfwlim 35289 | Bound-variable hypothesis ... |
elwlim 35290 | Membership in the limit cl... |
wzel 35291 | The zero of a well-founded... |
wsuclem 35292 | Lemma for the supremum pro... |
wsucex 35293 | Existence theorem for well... |
wsuccl 35294 | If ` X ` is a set with an ... |
wsuclb 35295 | A well-founded successor i... |
wlimss 35296 | The class of limit points ... |
txpss3v 35345 | A tail Cartesian product i... |
txprel 35346 | A tail Cartesian product i... |
brtxp 35347 | Characterize a ternary rel... |
brtxp2 35348 | The binary relation over a... |
dfpprod2 35349 | Expanded definition of par... |
pprodcnveq 35350 | A converse law for paralle... |
pprodss4v 35351 | The parallel product is a ... |
brpprod 35352 | Characterize a quaternary ... |
brpprod3a 35353 | Condition for parallel pro... |
brpprod3b 35354 | Condition for parallel pro... |
relsset 35355 | The subset class is a bina... |
brsset 35356 | For sets, the ` SSet ` bin... |
idsset 35357 | ` _I ` is equal to the int... |
eltrans 35358 | Membership in the class of... |
dfon3 35359 | A quantifier-free definiti... |
dfon4 35360 | Another quantifier-free de... |
brtxpsd 35361 | Expansion of a common form... |
brtxpsd2 35362 | Another common abbreviatio... |
brtxpsd3 35363 | A third common abbreviatio... |
relbigcup 35364 | The ` Bigcup ` relationshi... |
brbigcup 35365 | Binary relation over ` Big... |
dfbigcup2 35366 | ` Bigcup ` using maps-to n... |
fobigcup 35367 | ` Bigcup ` maps the univer... |
fnbigcup 35368 | ` Bigcup ` is a function o... |
fvbigcup 35369 | For sets, ` Bigcup ` yield... |
elfix 35370 | Membership in the fixpoint... |
elfix2 35371 | Alternative membership in ... |
dffix2 35372 | The fixpoints of a class i... |
fixssdm 35373 | The fixpoints of a class a... |
fixssrn 35374 | The fixpoints of a class a... |
fixcnv 35375 | The fixpoints of a class a... |
fixun 35376 | The fixpoint operator dist... |
ellimits 35377 | Membership in the class of... |
limitssson 35378 | The class of all limit ord... |
dfom5b 35379 | A quantifier-free definiti... |
sscoid 35380 | A condition for subset and... |
dffun10 35381 | Another potential definiti... |
elfuns 35382 | Membership in the class of... |
elfunsg 35383 | Closed form of ~ elfuns . ... |
brsingle 35384 | The binary relation form o... |
elsingles 35385 | Membership in the class of... |
fnsingle 35386 | The singleton relationship... |
fvsingle 35387 | The value of the singleton... |
dfsingles2 35388 | Alternate definition of th... |
snelsingles 35389 | A singleton is a member of... |
dfiota3 35390 | A definition of iota using... |
dffv5 35391 | Another quantifier-free de... |
unisnif 35392 | Express union of singleton... |
brimage 35393 | Binary relation form of th... |
brimageg 35394 | Closed form of ~ brimage .... |
funimage 35395 | ` Image A ` is a function.... |
fnimage 35396 | ` Image R ` is a function ... |
imageval 35397 | The image functor in maps-... |
fvimage 35398 | Value of the image functor... |
brcart 35399 | Binary relation form of th... |
brdomain 35400 | Binary relation form of th... |
brrange 35401 | Binary relation form of th... |
brdomaing 35402 | Closed form of ~ brdomain ... |
brrangeg 35403 | Closed form of ~ brrange .... |
brimg 35404 | Binary relation form of th... |
brapply 35405 | Binary relation form of th... |
brcup 35406 | Binary relation form of th... |
brcap 35407 | Binary relation form of th... |
brsuccf 35408 | Binary relation form of th... |
funpartlem 35409 | Lemma for ~ funpartfun . ... |
funpartfun 35410 | The functional part of ` F... |
funpartss 35411 | The functional part of ` F... |
funpartfv 35412 | The function value of the ... |
fullfunfnv 35413 | The full functional part o... |
fullfunfv 35414 | The function value of the ... |
brfullfun 35415 | A binary relation form con... |
brrestrict 35416 | Binary relation form of th... |
dfrecs2 35417 | A quantifier-free definiti... |
dfrdg4 35418 | A quantifier-free definiti... |
dfint3 35419 | Quantifier-free definition... |
imagesset 35420 | The Image functor applied ... |
brub 35421 | Binary relation form of th... |
brlb 35422 | Binary relation form of th... |
altopex 35427 | Alternative ordered pairs ... |
altopthsn 35428 | Two alternate ordered pair... |
altopeq12 35429 | Equality for alternate ord... |
altopeq1 35430 | Equality for alternate ord... |
altopeq2 35431 | Equality for alternate ord... |
altopth1 35432 | Equality of the first memb... |
altopth2 35433 | Equality of the second mem... |
altopthg 35434 | Alternate ordered pair the... |
altopthbg 35435 | Alternate ordered pair the... |
altopth 35436 | The alternate ordered pair... |
altopthb 35437 | Alternate ordered pair the... |
altopthc 35438 | Alternate ordered pair the... |
altopthd 35439 | Alternate ordered pair the... |
altxpeq1 35440 | Equality for alternate Car... |
altxpeq2 35441 | Equality for alternate Car... |
elaltxp 35442 | Membership in alternate Ca... |
altopelaltxp 35443 | Alternate ordered pair mem... |
altxpsspw 35444 | An inclusion rule for alte... |
altxpexg 35445 | The alternate Cartesian pr... |
rankaltopb 35446 | Compute the rank of an alt... |
nfaltop 35447 | Bound-variable hypothesis ... |
sbcaltop 35448 | Distribution of class subs... |
cgrrflx2d 35451 | Deduction form of ~ axcgrr... |
cgrtr4d 35452 | Deduction form of ~ axcgrt... |
cgrtr4and 35453 | Deduction form of ~ axcgrt... |
cgrrflx 35454 | Reflexivity law for congru... |
cgrrflxd 35455 | Deduction form of ~ cgrrfl... |
cgrcomim 35456 | Congruence commutes on the... |
cgrcom 35457 | Congruence commutes betwee... |
cgrcomand 35458 | Deduction form of ~ cgrcom... |
cgrtr 35459 | Transitivity law for congr... |
cgrtrand 35460 | Deduction form of ~ cgrtr ... |
cgrtr3 35461 | Transitivity law for congr... |
cgrtr3and 35462 | Deduction form of ~ cgrtr3... |
cgrcoml 35463 | Congruence commutes on the... |
cgrcomr 35464 | Congruence commutes on the... |
cgrcomlr 35465 | Congruence commutes on bot... |
cgrcomland 35466 | Deduction form of ~ cgrcom... |
cgrcomrand 35467 | Deduction form of ~ cgrcom... |
cgrcomlrand 35468 | Deduction form of ~ cgrcom... |
cgrtriv 35469 | Degenerate segments are co... |
cgrid2 35470 | Identity law for congruenc... |
cgrdegen 35471 | Two congruent segments are... |
brofs 35472 | Binary relation form of th... |
5segofs 35473 | Rephrase ~ ax5seg using th... |
ofscom 35474 | The outer five segment pre... |
cgrextend 35475 | Link congruence over a pai... |
cgrextendand 35476 | Deduction form of ~ cgrext... |
segconeq 35477 | Two points that satisfy th... |
segconeu 35478 | Existential uniqueness ver... |
btwntriv2 35479 | Betweenness always holds f... |
btwncomim 35480 | Betweenness commutes. Imp... |
btwncom 35481 | Betweenness commutes. (Co... |
btwncomand 35482 | Deduction form of ~ btwnco... |
btwntriv1 35483 | Betweenness always holds f... |
btwnswapid 35484 | If you can swap the first ... |
btwnswapid2 35485 | If you can swap arguments ... |
btwnintr 35486 | Inner transitivity law for... |
btwnexch3 35487 | Exchange the first endpoin... |
btwnexch3and 35488 | Deduction form of ~ btwnex... |
btwnouttr2 35489 | Outer transitivity law for... |
btwnexch2 35490 | Exchange the outer point o... |
btwnouttr 35491 | Outer transitivity law for... |
btwnexch 35492 | Outer transitivity law for... |
btwnexchand 35493 | Deduction form of ~ btwnex... |
btwndiff 35494 | There is always a ` c ` di... |
trisegint 35495 | A line segment between two... |
funtransport 35498 | The ` TransportTo ` relati... |
fvtransport 35499 | Calculate the value of the... |
transportcl 35500 | Closure law for segment tr... |
transportprops 35501 | Calculate the defining pro... |
brifs 35510 | Binary relation form of th... |
ifscgr 35511 | Inner five segment congrue... |
cgrsub 35512 | Removing identical parts f... |
brcgr3 35513 | Binary relation form of th... |
cgr3permute3 35514 | Permutation law for three-... |
cgr3permute1 35515 | Permutation law for three-... |
cgr3permute2 35516 | Permutation law for three-... |
cgr3permute4 35517 | Permutation law for three-... |
cgr3permute5 35518 | Permutation law for three-... |
cgr3tr4 35519 | Transitivity law for three... |
cgr3com 35520 | Commutativity law for thre... |
cgr3rflx 35521 | Identity law for three-pla... |
cgrxfr 35522 | A line segment can be divi... |
btwnxfr 35523 | A condition for extending ... |
colinrel 35524 | Colinearity is a relations... |
brcolinear2 35525 | Alternate colinearity bina... |
brcolinear 35526 | The binary relation form o... |
colinearex 35527 | The colinear predicate exi... |
colineardim1 35528 | If ` A ` is colinear with ... |
colinearperm1 35529 | Permutation law for coline... |
colinearperm3 35530 | Permutation law for coline... |
colinearperm2 35531 | Permutation law for coline... |
colinearperm4 35532 | Permutation law for coline... |
colinearperm5 35533 | Permutation law for coline... |
colineartriv1 35534 | Trivial case of colinearit... |
colineartriv2 35535 | Trivial case of colinearit... |
btwncolinear1 35536 | Betweenness implies coline... |
btwncolinear2 35537 | Betweenness implies coline... |
btwncolinear3 35538 | Betweenness implies coline... |
btwncolinear4 35539 | Betweenness implies coline... |
btwncolinear5 35540 | Betweenness implies coline... |
btwncolinear6 35541 | Betweenness implies coline... |
colinearxfr 35542 | Transfer law for colineari... |
lineext 35543 | Extend a line with a missi... |
brofs2 35544 | Change some conditions for... |
brifs2 35545 | Change some conditions for... |
brfs 35546 | Binary relation form of th... |
fscgr 35547 | Congruence law for the gen... |
linecgr 35548 | Congruence rule for lines.... |
linecgrand 35549 | Deduction form of ~ linecg... |
lineid 35550 | Identity law for points on... |
idinside 35551 | Law for finding a point in... |
endofsegid 35552 | If ` A ` , ` B ` , and ` C... |
endofsegidand 35553 | Deduction form of ~ endofs... |
btwnconn1lem1 35554 | Lemma for ~ btwnconn1 . T... |
btwnconn1lem2 35555 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem3 35556 | Lemma for ~ btwnconn1 . E... |
btwnconn1lem4 35557 | Lemma for ~ btwnconn1 . A... |
btwnconn1lem5 35558 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem6 35559 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem7 35560 | Lemma for ~ btwnconn1 . U... |
btwnconn1lem8 35561 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem9 35562 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem10 35563 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem11 35564 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem12 35565 | Lemma for ~ btwnconn1 . U... |
btwnconn1lem13 35566 | Lemma for ~ btwnconn1 . B... |
btwnconn1lem14 35567 | Lemma for ~ btwnconn1 . F... |
btwnconn1 35568 | Connectitivy law for betwe... |
btwnconn2 35569 | Another connectivity law f... |
btwnconn3 35570 | Inner connectivity law for... |
midofsegid 35571 | If two points fall in the ... |
segcon2 35572 | Generalization of ~ axsegc... |
brsegle 35575 | Binary relation form of th... |
brsegle2 35576 | Alternate characterization... |
seglecgr12im 35577 | Substitution law for segme... |
seglecgr12 35578 | Substitution law for segme... |
seglerflx 35579 | Segment comparison is refl... |
seglemin 35580 | Any segment is at least as... |
segletr 35581 | Segment less than is trans... |
segleantisym 35582 | Antisymmetry law for segme... |
seglelin 35583 | Linearity law for segment ... |
btwnsegle 35584 | If ` B ` falls between ` A... |
colinbtwnle 35585 | Given three colinear point... |
broutsideof 35588 | Binary relation form of ` ... |
broutsideof2 35589 | Alternate form of ` Outsid... |
outsidene1 35590 | Outsideness implies inequa... |
outsidene2 35591 | Outsideness implies inequa... |
btwnoutside 35592 | A principle linking outsid... |
broutsideof3 35593 | Characterization of outsid... |
outsideofrflx 35594 | Reflexivity of outsideness... |
outsideofcom 35595 | Commutativity law for outs... |
outsideoftr 35596 | Transitivity law for outsi... |
outsideofeq 35597 | Uniqueness law for ` Outsi... |
outsideofeu 35598 | Given a nondegenerate ray,... |
outsidele 35599 | Relate ` OutsideOf ` to ` ... |
outsideofcol 35600 | Outside of implies colinea... |
funray 35607 | Show that the ` Ray ` rela... |
fvray 35608 | Calculate the value of the... |
funline 35609 | Show that the ` Line ` rel... |
linedegen 35610 | When ` Line ` is applied w... |
fvline 35611 | Calculate the value of the... |
liness 35612 | A line is a subset of the ... |
fvline2 35613 | Alternate definition of a ... |
lineunray 35614 | A line is composed of a po... |
lineelsb2 35615 | If ` S ` lies on ` P Q ` ,... |
linerflx1 35616 | Reflexivity law for line m... |
linecom 35617 | Commutativity law for line... |
linerflx2 35618 | Reflexivity law for line m... |
ellines 35619 | Membership in the set of a... |
linethru 35620 | If ` A ` is a line contain... |
hilbert1.1 35621 | There is a line through an... |
hilbert1.2 35622 | There is at most one line ... |
linethrueu 35623 | There is a unique line goi... |
lineintmo 35624 | Two distinct lines interse... |
fwddifval 35629 | Calculate the value of the... |
fwddifnval 35630 | The value of the forward d... |
fwddifn0 35631 | The value of the n-iterate... |
fwddifnp1 35632 | The value of the n-iterate... |
rankung 35633 | The rank of the union of t... |
ranksng 35634 | The rank of a singleton. ... |
rankelg 35635 | The membership relation is... |
rankpwg 35636 | The rank of a power set. ... |
rank0 35637 | The rank of the empty set ... |
rankeq1o 35638 | The only set with rank ` 1... |
elhf 35641 | Membership in the heredita... |
elhf2 35642 | Alternate form of membersh... |
elhf2g 35643 | Hereditarily finiteness vi... |
0hf 35644 | The empty set is a heredit... |
hfun 35645 | The union of two HF sets i... |
hfsn 35646 | The singleton of an HF set... |
hfadj 35647 | Adjoining one HF element t... |
hfelhf 35648 | Any member of an HF set is... |
hftr 35649 | The class of all hereditar... |
hfext 35650 | Extensionality for HF sets... |
hfuni 35651 | The union of an HF set is ... |
hfpw 35652 | The power class of an HF s... |
hfninf 35653 | ` _om ` is not hereditaril... |
mpomulnzcnf 35654 | Multiplication maps nonzer... |
mpomulex 35655 | The multiplication operati... |
gg-cnfldex 35656 | The field of complex numbe... |
gg-taylthlem2 35657 | Lemma for ~ taylth . (Con... |
mpoaddf 35658 | Addition is an operation o... |
mpoaddex 35659 | The addition operation is ... |
gg-dfcnfld 35660 | Alternative definition of ... |
gg-cnfldstr 35661 | The field of complex numbe... |
gg-cnfldbas 35662 | The base set of the field ... |
mpocnfldadd 35663 | The addition operation of ... |
mpocnfldmul 35664 | The multiplication operati... |
gg-cnfldcj 35665 | The conjugation operation ... |
gg-cnfldtset 35666 | The topology component of ... |
gg-cnfldle 35667 | The ordering of the field ... |
gg-cnfldds 35668 | The metric of the field of... |
gg-cnfldunif 35669 | The uniform structure comp... |
gg-cnfldfun 35670 | The field of complex numbe... |
gg-cnfldfunALT 35671 | The field of complex numbe... |
gg-cffldtocusgr 35672 | The field of complex numbe... |
gg-cncrng 35673 | The complex numbers form a... |
gg-cnfld1 35674 | One is the unity element o... |
a1i14 35675 | Add two antecedents to a w... |
a1i24 35676 | Add two antecedents to a w... |
exp5d 35677 | An exportation inference. ... |
exp5g 35678 | An exportation inference. ... |
exp5k 35679 | An exportation inference. ... |
exp56 35680 | An exportation inference. ... |
exp58 35681 | An exportation inference. ... |
exp510 35682 | An exportation inference. ... |
exp511 35683 | An exportation inference. ... |
exp512 35684 | An exportation inference. ... |
3com12d 35685 | Commutation in consequent.... |
imp5p 35686 | A triple importation infer... |
imp5q 35687 | A triple importation infer... |
ecase13d 35688 | Deduction for elimination ... |
subtr 35689 | Transitivity of implicit s... |
subtr2 35690 | Transitivity of implicit s... |
trer 35691 | A relation intersected wit... |
elicc3 35692 | An equivalent membership c... |
finminlem 35693 | A useful lemma about finit... |
gtinf 35694 | Any number greater than an... |
opnrebl 35695 | A set is open in the stand... |
opnrebl2 35696 | A set is open in the stand... |
nn0prpwlem 35697 | Lemma for ~ nn0prpw . Use... |
nn0prpw 35698 | Two nonnegative integers a... |
topbnd 35699 | Two equivalent expressions... |
opnbnd 35700 | A set is open iff it is di... |
cldbnd 35701 | A set is closed iff it con... |
ntruni 35702 | A union of interiors is a ... |
clsun 35703 | A pairwise union of closur... |
clsint2 35704 | The closure of an intersec... |
opnregcld 35705 | A set is regularly closed ... |
cldregopn 35706 | A set if regularly open if... |
neiin 35707 | Two neighborhoods intersec... |
hmeoclda 35708 | Homeomorphisms preserve cl... |
hmeocldb 35709 | Homeomorphisms preserve cl... |
ivthALT 35710 | An alternate proof of the ... |
fnerel 35713 | Fineness is a relation. (... |
isfne 35714 | The predicate " ` B ` is f... |
isfne4 35715 | The predicate " ` B ` is f... |
isfne4b 35716 | A condition for a topology... |
isfne2 35717 | The predicate " ` B ` is f... |
isfne3 35718 | The predicate " ` B ` is f... |
fnebas 35719 | A finer cover covers the s... |
fnetg 35720 | A finer cover generates a ... |
fnessex 35721 | If ` B ` is finer than ` A... |
fneuni 35722 | If ` B ` is finer than ` A... |
fneint 35723 | If a cover is finer than a... |
fness 35724 | A cover is finer than its ... |
fneref 35725 | Reflexivity of the finenes... |
fnetr 35726 | Transitivity of the finene... |
fneval 35727 | Two covers are finer than ... |
fneer 35728 | Fineness intersected with ... |
topfne 35729 | Fineness for covers corres... |
topfneec 35730 | A cover is equivalent to a... |
topfneec2 35731 | A topology is precisely id... |
fnessref 35732 | A cover is finer iff it ha... |
refssfne 35733 | A cover is a refinement if... |
neibastop1 35734 | A collection of neighborho... |
neibastop2lem 35735 | Lemma for ~ neibastop2 . ... |
neibastop2 35736 | In the topology generated ... |
neibastop3 35737 | The topology generated by ... |
topmtcl 35738 | The meet of a collection o... |
topmeet 35739 | Two equivalent formulation... |
topjoin 35740 | Two equivalent formulation... |
fnemeet1 35741 | The meet of a collection o... |
fnemeet2 35742 | The meet of equivalence cl... |
fnejoin1 35743 | Join of equivalence classe... |
fnejoin2 35744 | Join of equivalence classe... |
fgmin 35745 | Minimality property of a g... |
neifg 35746 | The neighborhood filter of... |
tailfval 35747 | The tail function for a di... |
tailval 35748 | The tail of an element in ... |
eltail 35749 | An element of a tail. (Co... |
tailf 35750 | The tail function of a dir... |
tailini 35751 | A tail contains its initia... |
tailfb 35752 | The collection of tails of... |
filnetlem1 35753 | Lemma for ~ filnet . Chan... |
filnetlem2 35754 | Lemma for ~ filnet . The ... |
filnetlem3 35755 | Lemma for ~ filnet . (Con... |
filnetlem4 35756 | Lemma for ~ filnet . (Con... |
filnet 35757 | A filter has the same conv... |
tb-ax1 35758 | The first of three axioms ... |
tb-ax2 35759 | The second of three axioms... |
tb-ax3 35760 | The third of three axioms ... |
tbsyl 35761 | The weak syllogism from Ta... |
re1ax2lem 35762 | Lemma for ~ re1ax2 . (Con... |
re1ax2 35763 | ~ ax-2 rederived from the ... |
naim1 35764 | Constructor theorem for ` ... |
naim2 35765 | Constructor theorem for ` ... |
naim1i 35766 | Constructor rule for ` -/\... |
naim2i 35767 | Constructor rule for ` -/\... |
naim12i 35768 | Constructor rule for ` -/\... |
nabi1i 35769 | Constructor rule for ` -/\... |
nabi2i 35770 | Constructor rule for ` -/\... |
nabi12i 35771 | Constructor rule for ` -/\... |
df3nandALT1 35774 | The double nand expressed ... |
df3nandALT2 35775 | The double nand expressed ... |
andnand1 35776 | Double and in terms of dou... |
imnand2 35777 | An ` -> ` nand relation. ... |
nalfal 35778 | Not all sets hold ` F. ` a... |
nexntru 35779 | There does not exist a set... |
nexfal 35780 | There does not exist a set... |
neufal 35781 | There does not exist exact... |
neutru 35782 | There does not exist exact... |
nmotru 35783 | There does not exist at mo... |
mofal 35784 | There exist at most one se... |
nrmo 35785 | "At most one" restricted e... |
meran1 35786 | A single axiom for proposi... |
meran2 35787 | A single axiom for proposi... |
meran3 35788 | A single axiom for proposi... |
waj-ax 35789 | A single axiom for proposi... |
lukshef-ax2 35790 | A single axiom for proposi... |
arg-ax 35791 | A single axiom for proposi... |
negsym1 35792 | In the paper "On Variable ... |
imsym1 35793 | A symmetry with ` -> ` . ... |
bisym1 35794 | A symmetry with ` <-> ` . ... |
consym1 35795 | A symmetry with ` /\ ` . ... |
dissym1 35796 | A symmetry with ` \/ ` . ... |
nandsym1 35797 | A symmetry with ` -/\ ` . ... |
unisym1 35798 | A symmetry with ` A. ` . ... |
exisym1 35799 | A symmetry with ` E. ` . ... |
unqsym1 35800 | A symmetry with ` E! ` . ... |
amosym1 35801 | A symmetry with ` E* ` . ... |
subsym1 35802 | A symmetry with ` [ x / y ... |
ontopbas 35803 | An ordinal number is a top... |
onsstopbas 35804 | The class of ordinal numbe... |
onpsstopbas 35805 | The class of ordinal numbe... |
ontgval 35806 | The topology generated fro... |
ontgsucval 35807 | The topology generated fro... |
onsuctop 35808 | A successor ordinal number... |
onsuctopon 35809 | One of the topologies on a... |
ordtoplem 35810 | Membership of the class of... |
ordtop 35811 | An ordinal is a topology i... |
onsucconni 35812 | A successor ordinal number... |
onsucconn 35813 | A successor ordinal number... |
ordtopconn 35814 | An ordinal topology is con... |
onintopssconn 35815 | An ordinal topology is con... |
onsuct0 35816 | A successor ordinal number... |
ordtopt0 35817 | An ordinal topology is T_0... |
onsucsuccmpi 35818 | The successor of a success... |
onsucsuccmp 35819 | The successor of a success... |
limsucncmpi 35820 | The successor of a limit o... |
limsucncmp 35821 | The successor of a limit o... |
ordcmp 35822 | An ordinal topology is com... |
ssoninhaus 35823 | The ordinal topologies ` 1... |
onint1 35824 | The ordinal T_1 spaces are... |
oninhaus 35825 | The ordinal Hausdorff spac... |
fveleq 35826 | Please add description her... |
findfvcl 35827 | Please add description her... |
findreccl 35828 | Please add description her... |
findabrcl 35829 | Please add description her... |
nnssi2 35830 | Convert a theorem for real... |
nnssi3 35831 | Convert a theorem for real... |
nndivsub 35832 | Please add description her... |
nndivlub 35833 | A factor of a positive int... |
ee7.2aOLD 35836 | Lemma for Euclid's Element... |
dnival 35837 | Value of the "distance to ... |
dnicld1 35838 | Closure theorem for the "d... |
dnicld2 35839 | Closure theorem for the "d... |
dnif 35840 | The "distance to nearest i... |
dnizeq0 35841 | The distance to nearest in... |
dnizphlfeqhlf 35842 | The distance to nearest in... |
rddif2 35843 | Variant of ~ rddif . (Con... |
dnibndlem1 35844 | Lemma for ~ dnibnd . (Con... |
dnibndlem2 35845 | Lemma for ~ dnibnd . (Con... |
dnibndlem3 35846 | Lemma for ~ dnibnd . (Con... |
dnibndlem4 35847 | Lemma for ~ dnibnd . (Con... |
dnibndlem5 35848 | Lemma for ~ dnibnd . (Con... |
dnibndlem6 35849 | Lemma for ~ dnibnd . (Con... |
dnibndlem7 35850 | Lemma for ~ dnibnd . (Con... |
dnibndlem8 35851 | Lemma for ~ dnibnd . (Con... |
dnibndlem9 35852 | Lemma for ~ dnibnd . (Con... |
dnibndlem10 35853 | Lemma for ~ dnibnd . (Con... |
dnibndlem11 35854 | Lemma for ~ dnibnd . (Con... |
dnibndlem12 35855 | Lemma for ~ dnibnd . (Con... |
dnibndlem13 35856 | Lemma for ~ dnibnd . (Con... |
dnibnd 35857 | The "distance to nearest i... |
dnicn 35858 | The "distance to nearest i... |
knoppcnlem1 35859 | Lemma for ~ knoppcn . (Co... |
knoppcnlem2 35860 | Lemma for ~ knoppcn . (Co... |
knoppcnlem3 35861 | Lemma for ~ knoppcn . (Co... |
knoppcnlem4 35862 | Lemma for ~ knoppcn . (Co... |
knoppcnlem5 35863 | Lemma for ~ knoppcn . (Co... |
knoppcnlem6 35864 | Lemma for ~ knoppcn . (Co... |
knoppcnlem7 35865 | Lemma for ~ knoppcn . (Co... |
knoppcnlem8 35866 | Lemma for ~ knoppcn . (Co... |
knoppcnlem9 35867 | Lemma for ~ knoppcn . (Co... |
knoppcnlem10 35868 | Lemma for ~ knoppcn . (Co... |
knoppcnlem11 35869 | Lemma for ~ knoppcn . (Co... |
knoppcn 35870 | The continuous nowhere dif... |
knoppcld 35871 | Closure theorem for Knopp'... |
unblimceq0lem 35872 | Lemma for ~ unblimceq0 . ... |
unblimceq0 35873 | If ` F ` is unbounded near... |
unbdqndv1 35874 | If the difference quotient... |
unbdqndv2lem1 35875 | Lemma for ~ unbdqndv2 . (... |
unbdqndv2lem2 35876 | Lemma for ~ unbdqndv2 . (... |
unbdqndv2 35877 | Variant of ~ unbdqndv1 wit... |
knoppndvlem1 35878 | Lemma for ~ knoppndv . (C... |
knoppndvlem2 35879 | Lemma for ~ knoppndv . (C... |
knoppndvlem3 35880 | Lemma for ~ knoppndv . (C... |
knoppndvlem4 35881 | Lemma for ~ knoppndv . (C... |
knoppndvlem5 35882 | Lemma for ~ knoppndv . (C... |
knoppndvlem6 35883 | Lemma for ~ knoppndv . (C... |
knoppndvlem7 35884 | Lemma for ~ knoppndv . (C... |
knoppndvlem8 35885 | Lemma for ~ knoppndv . (C... |
knoppndvlem9 35886 | Lemma for ~ knoppndv . (C... |
knoppndvlem10 35887 | Lemma for ~ knoppndv . (C... |
knoppndvlem11 35888 | Lemma for ~ knoppndv . (C... |
knoppndvlem12 35889 | Lemma for ~ knoppndv . (C... |
knoppndvlem13 35890 | Lemma for ~ knoppndv . (C... |
knoppndvlem14 35891 | Lemma for ~ knoppndv . (C... |
knoppndvlem15 35892 | Lemma for ~ knoppndv . (C... |
knoppndvlem16 35893 | Lemma for ~ knoppndv . (C... |
knoppndvlem17 35894 | Lemma for ~ knoppndv . (C... |
knoppndvlem18 35895 | Lemma for ~ knoppndv . (C... |
knoppndvlem19 35896 | Lemma for ~ knoppndv . (C... |
knoppndvlem20 35897 | Lemma for ~ knoppndv . (C... |
knoppndvlem21 35898 | Lemma for ~ knoppndv . (C... |
knoppndvlem22 35899 | Lemma for ~ knoppndv . (C... |
knoppndv 35900 | The continuous nowhere dif... |
knoppf 35901 | Knopp's function is a func... |
knoppcn2 35902 | Variant of ~ knoppcn with ... |
cnndvlem1 35903 | Lemma for ~ cnndv . (Cont... |
cnndvlem2 35904 | Lemma for ~ cnndv . (Cont... |
cnndv 35905 | There exists a continuous ... |
bj-mp2c 35906 | A double modus ponens infe... |
bj-mp2d 35907 | A double modus ponens infe... |
bj-0 35908 | A syntactic theorem. See ... |
bj-1 35909 | In this proof, the use of ... |
bj-a1k 35910 | Weakening of ~ ax-1 . As ... |
bj-poni 35911 | Inference associated with ... |
bj-nnclav 35912 | When ` F. ` is substituted... |
bj-nnclavi 35913 | Inference associated with ... |
bj-nnclavc 35914 | Commuted form of ~ bj-nncl... |
bj-nnclavci 35915 | Inference associated with ... |
bj-jarrii 35916 | Inference associated with ... |
bj-imim21 35917 | The propositional function... |
bj-imim21i 35918 | Inference associated with ... |
bj-peircestab 35919 | Over minimal implicational... |
bj-stabpeirce 35920 | This minimal implicational... |
bj-syl66ib 35921 | A mixed syllogism inferenc... |
bj-orim2 35922 | Proof of ~ orim2 from the ... |
bj-currypeirce 35923 | Curry's axiom ~ curryax (a... |
bj-peircecurry 35924 | Peirce's axiom ~ peirce im... |
bj-animbi 35925 | Conjunction in terms of im... |
bj-currypara 35926 | Curry's paradox. Note tha... |
bj-con2com 35927 | A commuted form of the con... |
bj-con2comi 35928 | Inference associated with ... |
bj-pm2.01i 35929 | Inference associated with ... |
bj-nimn 35930 | If a formula is true, then... |
bj-nimni 35931 | Inference associated with ... |
bj-peircei 35932 | Inference associated with ... |
bj-looinvi 35933 | Inference associated with ... |
bj-looinvii 35934 | Inference associated with ... |
bj-mt2bi 35935 | Version of ~ mt2 where the... |
bj-ntrufal 35936 | The negation of a theorem ... |
bj-fal 35937 | Shortening of ~ fal using ... |
bj-jaoi1 35938 | Shortens ~ orfa2 (58>53), ... |
bj-jaoi2 35939 | Shortens ~ consensus (110>... |
bj-dfbi4 35940 | Alternate definition of th... |
bj-dfbi5 35941 | Alternate definition of th... |
bj-dfbi6 35942 | Alternate definition of th... |
bj-bijust0ALT 35943 | Alternate proof of ~ bijus... |
bj-bijust00 35944 | A self-implication does no... |
bj-consensus 35945 | Version of ~ consensus exp... |
bj-consensusALT 35946 | Alternate proof of ~ bj-co... |
bj-df-ifc 35947 | Candidate definition for t... |
bj-dfif 35948 | Alternate definition of th... |
bj-ififc 35949 | A biconditional connecting... |
bj-imbi12 35950 | Uncurried (imported) form ... |
bj-biorfi 35951 | This should be labeled "bi... |
bj-falor 35952 | Dual of ~ truan (which has... |
bj-falor2 35953 | Dual of ~ truan . (Contri... |
bj-bibibi 35954 | A property of the bicondit... |
bj-imn3ani 35955 | Duplication of ~ bnj1224 .... |
bj-andnotim 35956 | Two ways of expressing a c... |
bj-bi3ant 35957 | This used to be in the mai... |
bj-bisym 35958 | This used to be in the mai... |
bj-bixor 35959 | Equivalence of two ternary... |
bj-axdd2 35960 | This implication, proved u... |
bj-axd2d 35961 | This implication, proved u... |
bj-axtd 35962 | This implication, proved f... |
bj-gl4 35963 | In a normal modal logic, t... |
bj-axc4 35964 | Over minimal calculus, the... |
prvlem1 35969 | An elementary property of ... |
prvlem2 35970 | An elementary property of ... |
bj-babygodel 35971 | See the section header com... |
bj-babylob 35972 | See the section header com... |
bj-godellob 35973 | Proof of Gödel's theo... |
bj-genr 35974 | Generalization rule on the... |
bj-genl 35975 | Generalization rule on the... |
bj-genan 35976 | Generalization rule on a c... |
bj-mpgs 35977 | From a closed form theorem... |
bj-2alim 35978 | Closed form of ~ 2alimi . ... |
bj-2exim 35979 | Closed form of ~ 2eximi . ... |
bj-alanim 35980 | Closed form of ~ alanimi .... |
bj-2albi 35981 | Closed form of ~ 2albii . ... |
bj-notalbii 35982 | Equivalence of universal q... |
bj-2exbi 35983 | Closed form of ~ 2exbii . ... |
bj-3exbi 35984 | Closed form of ~ 3exbii . ... |
bj-sylgt2 35985 | Uncurried (imported) form ... |
bj-alrimg 35986 | The general form of the *a... |
bj-alrimd 35987 | A slightly more general ~ ... |
bj-sylget 35988 | Dual statement of ~ sylgt ... |
bj-sylget2 35989 | Uncurried (imported) form ... |
bj-exlimg 35990 | The general form of the *e... |
bj-sylge 35991 | Dual statement of ~ sylg (... |
bj-exlimd 35992 | A slightly more general ~ ... |
bj-nfimexal 35993 | A weak from of nonfreeness... |
bj-alexim 35994 | Closed form of ~ aleximi .... |
bj-nexdh 35995 | Closed form of ~ nexdh (ac... |
bj-nexdh2 35996 | Uncurried (imported) form ... |
bj-hbxfrbi 35997 | Closed form of ~ hbxfrbi .... |
bj-hbyfrbi 35998 | Version of ~ bj-hbxfrbi wi... |
bj-exalim 35999 | Distribute quantifiers ove... |
bj-exalimi 36000 | An inference for distribut... |
bj-exalims 36001 | Distributing quantifiers o... |
bj-exalimsi 36002 | An inference for distribut... |
bj-ax12ig 36003 | A lemma used to prove a we... |
bj-ax12i 36004 | A weakening of ~ bj-ax12ig... |
bj-nfimt 36005 | Closed form of ~ nfim and ... |
bj-cbvalimt 36006 | A lemma in closed form use... |
bj-cbveximt 36007 | A lemma in closed form use... |
bj-eximALT 36008 | Alternate proof of ~ exim ... |
bj-aleximiALT 36009 | Alternate proof of ~ alexi... |
bj-eximcom 36010 | A commuted form of ~ exim ... |
bj-ax12wlem 36011 | A lemma used to prove a we... |
bj-cbvalim 36012 | A lemma used to prove ~ bj... |
bj-cbvexim 36013 | A lemma used to prove ~ bj... |
bj-cbvalimi 36014 | An equality-free general i... |
bj-cbveximi 36015 | An equality-free general i... |
bj-cbval 36016 | Changing a bound variable ... |
bj-cbvex 36017 | Changing a bound variable ... |
bj-ssbeq 36020 | Substitution in an equalit... |
bj-ssblem1 36021 | A lemma for the definiens ... |
bj-ssblem2 36022 | An instance of ~ ax-11 pro... |
bj-ax12v 36023 | A weaker form of ~ ax-12 a... |
bj-ax12 36024 | Remove a DV condition from... |
bj-ax12ssb 36025 | Axiom ~ bj-ax12 expressed ... |
bj-19.41al 36026 | Special case of ~ 19.41 pr... |
bj-equsexval 36027 | Special case of ~ equsexv ... |
bj-subst 36028 | Proof of ~ sbalex from cor... |
bj-ssbid2 36029 | A special case of ~ sbequ2... |
bj-ssbid2ALT 36030 | Alternate proof of ~ bj-ss... |
bj-ssbid1 36031 | A special case of ~ sbequ1... |
bj-ssbid1ALT 36032 | Alternate proof of ~ bj-ss... |
bj-ax6elem1 36033 | Lemma for ~ bj-ax6e . (Co... |
bj-ax6elem2 36034 | Lemma for ~ bj-ax6e . (Co... |
bj-ax6e 36035 | Proof of ~ ax6e (hence ~ a... |
bj-spimvwt 36036 | Closed form of ~ spimvw . ... |
bj-spnfw 36037 | Theorem close to a closed ... |
bj-cbvexiw 36038 | Change bound variable. Th... |
bj-cbvexivw 36039 | Change bound variable. Th... |
bj-modald 36040 | A short form of the axiom ... |
bj-denot 36041 | A weakening of ~ ax-6 and ... |
bj-eqs 36042 | A lemma for substitutions,... |
bj-cbvexw 36043 | Change bound variable. Th... |
bj-ax12w 36044 | The general statement that... |
bj-ax89 36045 | A theorem which could be u... |
bj-elequ12 36046 | An identity law for the no... |
bj-cleljusti 36047 | One direction of ~ cleljus... |
bj-alcomexcom 36048 | Commutation of two existen... |
bj-hbalt 36049 | Closed form of ~ hbal . W... |
axc11n11 36050 | Proof of ~ axc11n from { ~... |
axc11n11r 36051 | Proof of ~ axc11n from { ~... |
bj-axc16g16 36052 | Proof of ~ axc16g from { ~... |
bj-ax12v3 36053 | A weak version of ~ ax-12 ... |
bj-ax12v3ALT 36054 | Alternate proof of ~ bj-ax... |
bj-sb 36055 | A weak variant of ~ sbid2 ... |
bj-modalbe 36056 | The predicate-calculus ver... |
bj-spst 36057 | Closed form of ~ sps . On... |
bj-19.21bit 36058 | Closed form of ~ 19.21bi .... |
bj-19.23bit 36059 | Closed form of ~ 19.23bi .... |
bj-nexrt 36060 | Closed form of ~ nexr . C... |
bj-alrim 36061 | Closed form of ~ alrimi . ... |
bj-alrim2 36062 | Uncurried (imported) form ... |
bj-nfdt0 36063 | A theorem close to a close... |
bj-nfdt 36064 | Closed form of ~ nf5d and ... |
bj-nexdt 36065 | Closed form of ~ nexd . (... |
bj-nexdvt 36066 | Closed form of ~ nexdv . ... |
bj-alexbiex 36067 | Adding a second quantifier... |
bj-exexbiex 36068 | Adding a second quantifier... |
bj-alalbial 36069 | Adding a second quantifier... |
bj-exalbial 36070 | Adding a second quantifier... |
bj-19.9htbi 36071 | Strengthening ~ 19.9ht by ... |
bj-hbntbi 36072 | Strengthening ~ hbnt by re... |
bj-biexal1 36073 | A general FOL biconditiona... |
bj-biexal2 36074 | When ` ph ` is substituted... |
bj-biexal3 36075 | When ` ph ` is substituted... |
bj-bialal 36076 | When ` ph ` is substituted... |
bj-biexex 36077 | When ` ph ` is substituted... |
bj-hbext 36078 | Closed form of ~ hbex . (... |
bj-nfalt 36079 | Closed form of ~ nfal . (... |
bj-nfext 36080 | Closed form of ~ nfex . (... |
bj-eeanvw 36081 | Version of ~ exdistrv with... |
bj-modal4 36082 | First-order logic form of ... |
bj-modal4e 36083 | First-order logic form of ... |
bj-modalb 36084 | A short form of the axiom ... |
bj-wnf1 36085 | When ` ph ` is substituted... |
bj-wnf2 36086 | When ` ph ` is substituted... |
bj-wnfanf 36087 | When ` ph ` is substituted... |
bj-wnfenf 36088 | When ` ph ` is substituted... |
bj-substax12 36089 | Equivalent form of the axi... |
bj-substw 36090 | Weak form of the LHS of ~ ... |
bj-nnfbi 36093 | If two formulas are equiva... |
bj-nnfbd 36094 | If two formulas are equiva... |
bj-nnfbii 36095 | If two formulas are equiva... |
bj-nnfa 36096 | Nonfreeness implies the eq... |
bj-nnfad 36097 | Nonfreeness implies the eq... |
bj-nnfai 36098 | Nonfreeness implies the eq... |
bj-nnfe 36099 | Nonfreeness implies the eq... |
bj-nnfed 36100 | Nonfreeness implies the eq... |
bj-nnfei 36101 | Nonfreeness implies the eq... |
bj-nnfea 36102 | Nonfreeness implies the eq... |
bj-nnfead 36103 | Nonfreeness implies the eq... |
bj-nnfeai 36104 | Nonfreeness implies the eq... |
bj-dfnnf2 36105 | Alternate definition of ~ ... |
bj-nnfnfTEMP 36106 | New nonfreeness implies ol... |
bj-wnfnf 36107 | When ` ph ` is substituted... |
bj-nnfnt 36108 | A variable is nonfree in a... |
bj-nnftht 36109 | A variable is nonfree in a... |
bj-nnfth 36110 | A variable is nonfree in a... |
bj-nnfnth 36111 | A variable is nonfree in t... |
bj-nnfim1 36112 | A consequence of nonfreene... |
bj-nnfim2 36113 | A consequence of nonfreene... |
bj-nnfim 36114 | Nonfreeness in the anteced... |
bj-nnfimd 36115 | Nonfreeness in the anteced... |
bj-nnfan 36116 | Nonfreeness in both conjun... |
bj-nnfand 36117 | Nonfreeness in both conjun... |
bj-nnfor 36118 | Nonfreeness in both disjun... |
bj-nnford 36119 | Nonfreeness in both disjun... |
bj-nnfbit 36120 | Nonfreeness in both sides ... |
bj-nnfbid 36121 | Nonfreeness in both sides ... |
bj-nnfv 36122 | A non-occurring variable i... |
bj-nnf-alrim 36123 | Proof of the closed form o... |
bj-nnf-exlim 36124 | Proof of the closed form o... |
bj-dfnnf3 36125 | Alternate definition of no... |
bj-nfnnfTEMP 36126 | New nonfreeness is equival... |
bj-nnfa1 36127 | See ~ nfa1 . (Contributed... |
bj-nnfe1 36128 | See ~ nfe1 . (Contributed... |
bj-19.12 36129 | See ~ 19.12 . Could be la... |
bj-nnflemaa 36130 | One of four lemmas for non... |
bj-nnflemee 36131 | One of four lemmas for non... |
bj-nnflemae 36132 | One of four lemmas for non... |
bj-nnflemea 36133 | One of four lemmas for non... |
bj-nnfalt 36134 | See ~ nfal and ~ bj-nfalt ... |
bj-nnfext 36135 | See ~ nfex and ~ bj-nfext ... |
bj-stdpc5t 36136 | Alias of ~ bj-nnf-alrim fo... |
bj-19.21t 36137 | Statement ~ 19.21t proved ... |
bj-19.23t 36138 | Statement ~ 19.23t proved ... |
bj-19.36im 36139 | One direction of ~ 19.36 f... |
bj-19.37im 36140 | One direction of ~ 19.37 f... |
bj-19.42t 36141 | Closed form of ~ 19.42 fro... |
bj-19.41t 36142 | Closed form of ~ 19.41 fro... |
bj-sbft 36143 | Version of ~ sbft using ` ... |
bj-pm11.53vw 36144 | Version of ~ pm11.53v with... |
bj-pm11.53v 36145 | Version of ~ pm11.53v with... |
bj-pm11.53a 36146 | A variant of ~ pm11.53v . ... |
bj-equsvt 36147 | A variant of ~ equsv . (C... |
bj-equsalvwd 36148 | Variant of ~ equsalvw . (... |
bj-equsexvwd 36149 | Variant of ~ equsexvw . (... |
bj-sbievwd 36150 | Variant of ~ sbievw . (Co... |
bj-axc10 36151 | Alternate proof of ~ axc10... |
bj-alequex 36152 | A fol lemma. See ~ aleque... |
bj-spimt2 36153 | A step in the proof of ~ s... |
bj-cbv3ta 36154 | Closed form of ~ cbv3 . (... |
bj-cbv3tb 36155 | Closed form of ~ cbv3 . (... |
bj-hbsb3t 36156 | A theorem close to a close... |
bj-hbsb3 36157 | Shorter proof of ~ hbsb3 .... |
bj-nfs1t 36158 | A theorem close to a close... |
bj-nfs1t2 36159 | A theorem close to a close... |
bj-nfs1 36160 | Shorter proof of ~ nfs1 (t... |
bj-axc10v 36161 | Version of ~ axc10 with a ... |
bj-spimtv 36162 | Version of ~ spimt with a ... |
bj-cbv3hv2 36163 | Version of ~ cbv3h with tw... |
bj-cbv1hv 36164 | Version of ~ cbv1h with a ... |
bj-cbv2hv 36165 | Version of ~ cbv2h with a ... |
bj-cbv2v 36166 | Version of ~ cbv2 with a d... |
bj-cbvaldv 36167 | Version of ~ cbvald with a... |
bj-cbvexdv 36168 | Version of ~ cbvexd with a... |
bj-cbval2vv 36169 | Version of ~ cbval2vv with... |
bj-cbvex2vv 36170 | Version of ~ cbvex2vv with... |
bj-cbvaldvav 36171 | Version of ~ cbvaldva with... |
bj-cbvexdvav 36172 | Version of ~ cbvexdva with... |
bj-cbvex4vv 36173 | Version of ~ cbvex4v with ... |
bj-equsalhv 36174 | Version of ~ equsalh with ... |
bj-axc11nv 36175 | Version of ~ axc11n with a... |
bj-aecomsv 36176 | Version of ~ aecoms with a... |
bj-axc11v 36177 | Version of ~ axc11 with a ... |
bj-drnf2v 36178 | Version of ~ drnf2 with a ... |
bj-equs45fv 36179 | Version of ~ equs45f with ... |
bj-hbs1 36180 | Version of ~ hbsb2 with a ... |
bj-nfs1v 36181 | Version of ~ nfsb2 with a ... |
bj-hbsb2av 36182 | Version of ~ hbsb2a with a... |
bj-hbsb3v 36183 | Version of ~ hbsb3 with a ... |
bj-nfsab1 36184 | Remove dependency on ~ ax-... |
bj-dtrucor2v 36185 | Version of ~ dtrucor2 with... |
bj-hbaeb2 36186 | Biconditional version of a... |
bj-hbaeb 36187 | Biconditional version of ~... |
bj-hbnaeb 36188 | Biconditional version of ~... |
bj-dvv 36189 | A special instance of ~ bj... |
bj-equsal1t 36190 | Duplication of ~ wl-equsal... |
bj-equsal1ti 36191 | Inference associated with ... |
bj-equsal1 36192 | One direction of ~ equsal ... |
bj-equsal2 36193 | One direction of ~ equsal ... |
bj-equsal 36194 | Shorter proof of ~ equsal ... |
stdpc5t 36195 | Closed form of ~ stdpc5 . ... |
bj-stdpc5 36196 | More direct proof of ~ std... |
2stdpc5 36197 | A double ~ stdpc5 (one dir... |
bj-19.21t0 36198 | Proof of ~ 19.21t from ~ s... |
exlimii 36199 | Inference associated with ... |
ax11-pm 36200 | Proof of ~ ax-11 similar t... |
ax6er 36201 | Commuted form of ~ ax6e . ... |
exlimiieq1 36202 | Inferring a theorem when i... |
exlimiieq2 36203 | Inferring a theorem when i... |
ax11-pm2 36204 | Proof of ~ ax-11 from the ... |
bj-sbsb 36205 | Biconditional showing two ... |
bj-dfsb2 36206 | Alternate (dual) definitio... |
bj-sbf3 36207 | Substitution has no effect... |
bj-sbf4 36208 | Substitution has no effect... |
bj-sbnf 36209 | Move nonfree predicate in ... |
bj-eu3f 36210 | Version of ~ eu3v where th... |
bj-sblem1 36211 | Lemma for substitution. (... |
bj-sblem2 36212 | Lemma for substitution. (... |
bj-sblem 36213 | Lemma for substitution. (... |
bj-sbievw1 36214 | Lemma for substitution. (... |
bj-sbievw2 36215 | Lemma for substitution. (... |
bj-sbievw 36216 | Lemma for substitution. C... |
bj-sbievv 36217 | Version of ~ sbie with a s... |
bj-moeub 36218 | Uniqueness is equivalent t... |
bj-sbidmOLD 36219 | Obsolete proof of ~ sbidm ... |
bj-dvelimdv 36220 | Deduction form of ~ dvelim... |
bj-dvelimdv1 36221 | Curried (exported) form of... |
bj-dvelimv 36222 | A version of ~ dvelim usin... |
bj-nfeel2 36223 | Nonfreeness in a membershi... |
bj-axc14nf 36224 | Proof of a version of ~ ax... |
bj-axc14 36225 | Alternate proof of ~ axc14... |
mobidvALT 36226 | Alternate proof of ~ mobid... |
sbn1ALT 36227 | Alternate proof of ~ sbn1 ... |
eliminable1 36228 | A theorem used to prove th... |
eliminable2a 36229 | A theorem used to prove th... |
eliminable2b 36230 | A theorem used to prove th... |
eliminable2c 36231 | A theorem used to prove th... |
eliminable3a 36232 | A theorem used to prove th... |
eliminable3b 36233 | A theorem used to prove th... |
eliminable-velab 36234 | A theorem used to prove th... |
eliminable-veqab 36235 | A theorem used to prove th... |
eliminable-abeqv 36236 | A theorem used to prove th... |
eliminable-abeqab 36237 | A theorem used to prove th... |
eliminable-abelv 36238 | A theorem used to prove th... |
eliminable-abelab 36239 | A theorem used to prove th... |
bj-denoteslem 36240 | Lemma for ~ bj-denotes . ... |
bj-denotes 36241 | This would be the justific... |
bj-issettru 36242 | Weak version of ~ isset wi... |
bj-elabtru 36243 | This is as close as we can... |
bj-issetwt 36244 | Closed form of ~ bj-issetw... |
bj-issetw 36245 | The closest one can get to... |
bj-elissetALT 36246 | Alternate proof of ~ eliss... |
bj-issetiv 36247 | Version of ~ bj-isseti wit... |
bj-isseti 36248 | Version of ~ isseti with a... |
bj-ralvw 36249 | A weak version of ~ ralv n... |
bj-rexvw 36250 | A weak version of ~ rexv n... |
bj-rababw 36251 | A weak version of ~ rabab ... |
bj-rexcom4bv 36252 | Version of ~ rexcom4b and ... |
bj-rexcom4b 36253 | Remove from ~ rexcom4b dep... |
bj-ceqsalt0 36254 | The FOL content of ~ ceqsa... |
bj-ceqsalt1 36255 | The FOL content of ~ ceqsa... |
bj-ceqsalt 36256 | Remove from ~ ceqsalt depe... |
bj-ceqsaltv 36257 | Version of ~ bj-ceqsalt wi... |
bj-ceqsalg0 36258 | The FOL content of ~ ceqsa... |
bj-ceqsalg 36259 | Remove from ~ ceqsalg depe... |
bj-ceqsalgALT 36260 | Alternate proof of ~ bj-ce... |
bj-ceqsalgv 36261 | Version of ~ bj-ceqsalg wi... |
bj-ceqsalgvALT 36262 | Alternate proof of ~ bj-ce... |
bj-ceqsal 36263 | Remove from ~ ceqsal depen... |
bj-ceqsalv 36264 | Remove from ~ ceqsalv depe... |
bj-spcimdv 36265 | Remove from ~ spcimdv depe... |
bj-spcimdvv 36266 | Remove from ~ spcimdv depe... |
elelb 36267 | Equivalence between two co... |
bj-pwvrelb 36268 | Characterization of the el... |
bj-nfcsym 36269 | The nonfreeness quantifier... |
bj-sbeqALT 36270 | Substitution in an equalit... |
bj-sbeq 36271 | Distribute proper substitu... |
bj-sbceqgALT 36272 | Distribute proper substitu... |
bj-csbsnlem 36273 | Lemma for ~ bj-csbsn (in t... |
bj-csbsn 36274 | Substitution in a singleto... |
bj-sbel1 36275 | Version of ~ sbcel1g when ... |
bj-abv 36276 | The class of sets verifyin... |
bj-abvALT 36277 | Alternate version of ~ bj-... |
bj-ab0 36278 | The class of sets verifyin... |
bj-abf 36279 | Shorter proof of ~ abf (wh... |
bj-csbprc 36280 | More direct proof of ~ csb... |
bj-exlimvmpi 36281 | A Fol lemma ( ~ exlimiv fo... |
bj-exlimmpi 36282 | Lemma for ~ bj-vtoclg1f1 (... |
bj-exlimmpbi 36283 | Lemma for theorems of the ... |
bj-exlimmpbir 36284 | Lemma for theorems of the ... |
bj-vtoclf 36285 | Remove dependency on ~ ax-... |
bj-vtocl 36286 | Remove dependency on ~ ax-... |
bj-vtoclg1f1 36287 | The FOL content of ~ vtocl... |
bj-vtoclg1f 36288 | Reprove ~ vtoclg1f from ~ ... |
bj-vtoclg1fv 36289 | Version of ~ bj-vtoclg1f w... |
bj-vtoclg 36290 | A version of ~ vtoclg with... |
bj-rabeqbid 36291 | Version of ~ rabeqbidv wit... |
bj-seex 36292 | Version of ~ seex with a d... |
bj-nfcf 36293 | Version of ~ df-nfc with a... |
bj-zfauscl 36294 | General version of ~ zfaus... |
bj-elabd2ALT 36295 | Alternate proof of ~ elabd... |
bj-unrab 36296 | Generalization of ~ unrab ... |
bj-inrab 36297 | Generalization of ~ inrab ... |
bj-inrab2 36298 | Shorter proof of ~ inrab .... |
bj-inrab3 36299 | Generalization of ~ dfrab3... |
bj-rabtr 36300 | Restricted class abstracti... |
bj-rabtrALT 36301 | Alternate proof of ~ bj-ra... |
bj-rabtrAUTO 36302 | Proof of ~ bj-rabtr found ... |
bj-gabss 36305 | Inclusion of generalized c... |
bj-gabssd 36306 | Inclusion of generalized c... |
bj-gabeqd 36307 | Equality of generalized cl... |
bj-gabeqis 36308 | Equality of generalized cl... |
bj-elgab 36309 | Elements of a generalized ... |
bj-gabima 36310 | Generalized class abstract... |
bj-ru0 36313 | The FOL part of Russell's ... |
bj-ru1 36314 | A version of Russell's par... |
bj-ru 36315 | Remove dependency on ~ ax-... |
currysetlem 36316 | Lemma for ~ currysetlem , ... |
curryset 36317 | Curry's paradox in set the... |
currysetlem1 36318 | Lemma for ~ currysetALT . ... |
currysetlem2 36319 | Lemma for ~ currysetALT . ... |
currysetlem3 36320 | Lemma for ~ currysetALT . ... |
currysetALT 36321 | Alternate proof of ~ curry... |
bj-n0i 36322 | Inference associated with ... |
bj-disjsn01 36323 | Disjointness of the single... |
bj-0nel1 36324 | The empty set does not bel... |
bj-1nel0 36325 | ` 1o ` does not belong to ... |
bj-xpimasn 36326 | The image of a singleton, ... |
bj-xpima1sn 36327 | The image of a singleton b... |
bj-xpima1snALT 36328 | Alternate proof of ~ bj-xp... |
bj-xpima2sn 36329 | The image of a singleton b... |
bj-xpnzex 36330 | If the first factor of a p... |
bj-xpexg2 36331 | Curried (exported) form of... |
bj-xpnzexb 36332 | If the first factor of a p... |
bj-cleq 36333 | Substitution property for ... |
bj-snsetex 36334 | The class of sets "whose s... |
bj-clexab 36335 | Sethood of certain classes... |
bj-sngleq 36338 | Substitution property for ... |
bj-elsngl 36339 | Characterization of the el... |
bj-snglc 36340 | Characterization of the el... |
bj-snglss 36341 | The singletonization of a ... |
bj-0nelsngl 36342 | The empty set is not a mem... |
bj-snglinv 36343 | Inverse of singletonizatio... |
bj-snglex 36344 | A class is a set if and on... |
bj-tageq 36347 | Substitution property for ... |
bj-eltag 36348 | Characterization of the el... |
bj-0eltag 36349 | The empty set belongs to t... |
bj-tagn0 36350 | The tagging of a class is ... |
bj-tagss 36351 | The tagging of a class is ... |
bj-snglsstag 36352 | The singletonization is in... |
bj-sngltagi 36353 | The singletonization is in... |
bj-sngltag 36354 | The singletonization and t... |
bj-tagci 36355 | Characterization of the el... |
bj-tagcg 36356 | Characterization of the el... |
bj-taginv 36357 | Inverse of tagging. (Cont... |
bj-tagex 36358 | A class is a set if and on... |
bj-xtageq 36359 | The products of a given cl... |
bj-xtagex 36360 | The product of a set and t... |
bj-projeq 36363 | Substitution property for ... |
bj-projeq2 36364 | Substitution property for ... |
bj-projun 36365 | The class projection on a ... |
bj-projex 36366 | Sethood of the class proje... |
bj-projval 36367 | Value of the class project... |
bj-1upleq 36370 | Substitution property for ... |
bj-pr1eq 36373 | Substitution property for ... |
bj-pr1un 36374 | The first projection prese... |
bj-pr1val 36375 | Value of the first project... |
bj-pr11val 36376 | Value of the first project... |
bj-pr1ex 36377 | Sethood of the first proje... |
bj-1uplth 36378 | The characteristic propert... |
bj-1uplex 36379 | A monuple is a set if and ... |
bj-1upln0 36380 | A monuple is nonempty. (C... |
bj-2upleq 36383 | Substitution property for ... |
bj-pr21val 36384 | Value of the first project... |
bj-pr2eq 36387 | Substitution property for ... |
bj-pr2un 36388 | The second projection pres... |
bj-pr2val 36389 | Value of the second projec... |
bj-pr22val 36390 | Value of the second projec... |
bj-pr2ex 36391 | Sethood of the second proj... |
bj-2uplth 36392 | The characteristic propert... |
bj-2uplex 36393 | A couple is a set if and o... |
bj-2upln0 36394 | A couple is nonempty. (Co... |
bj-2upln1upl 36395 | A couple is never equal to... |
bj-rcleqf 36396 | Relative version of ~ cleq... |
bj-rcleq 36397 | Relative version of ~ dfcl... |
bj-reabeq 36398 | Relative form of ~ eqabb .... |
bj-disj2r 36399 | Relative version of ~ ssdi... |
bj-sscon 36400 | Contraposition law for rel... |
bj-abex 36401 | Two ways of stating that t... |
bj-clex 36402 | Two ways of stating that a... |
bj-axsn 36403 | Two ways of stating the ax... |
bj-snexg 36405 | A singleton built on a set... |
bj-snex 36406 | A singleton is a set. See... |
bj-axbun 36407 | Two ways of stating the ax... |
bj-unexg 36409 | Existence of binary unions... |
bj-prexg 36410 | Existence of unordered pai... |
bj-prex 36411 | Existence of unordered pai... |
bj-axadj 36412 | Two ways of stating the ax... |
bj-adjg1 36414 | Existence of the result of... |
bj-snfromadj 36415 | Singleton from adjunction ... |
bj-prfromadj 36416 | Unordered pair from adjunc... |
bj-adjfrombun 36417 | Adjunction from singleton ... |
eleq2w2ALT 36418 | Alternate proof of ~ eleq2... |
bj-clel3gALT 36419 | Alternate proof of ~ clel3... |
bj-pw0ALT 36420 | Alternate proof of ~ pw0 .... |
bj-sselpwuni 36421 | Quantitative version of ~ ... |
bj-unirel 36422 | Quantitative version of ~ ... |
bj-elpwg 36423 | If the intersection of two... |
bj-velpwALT 36424 | This theorem ~ bj-velpwALT... |
bj-elpwgALT 36425 | Alternate proof of ~ elpwg... |
bj-vjust 36426 | Justification theorem for ... |
bj-nul 36427 | Two formulations of the ax... |
bj-nuliota 36428 | Definition of the empty se... |
bj-nuliotaALT 36429 | Alternate proof of ~ bj-nu... |
bj-vtoclgfALT 36430 | Alternate proof of ~ vtocl... |
bj-elsn12g 36431 | Join of ~ elsng and ~ elsn... |
bj-elsnb 36432 | Biconditional version of ~... |
bj-pwcfsdom 36433 | Remove hypothesis from ~ p... |
bj-grur1 36434 | Remove hypothesis from ~ g... |
bj-bm1.3ii 36435 | The extension of a predica... |
bj-dfid2ALT 36436 | Alternate version of ~ dfi... |
bj-0nelopab 36437 | The empty set is never an ... |
bj-brrelex12ALT 36438 | Two classes related by a b... |
bj-epelg 36439 | The membership relation an... |
bj-epelb 36440 | Two classes are related by... |
bj-nsnid 36441 | A set does not contain the... |
bj-rdg0gALT 36442 | Alternate proof of ~ rdg0g... |
bj-evaleq 36443 | Equality theorem for the `... |
bj-evalfun 36444 | The evaluation at a class ... |
bj-evalfn 36445 | The evaluation at a class ... |
bj-evalval 36446 | Value of the evaluation at... |
bj-evalid 36447 | The evaluation at a set of... |
bj-ndxarg 36448 | Proof of ~ ndxarg from ~ b... |
bj-evalidval 36449 | Closed general form of ~ s... |
bj-rest00 36452 | An elementwise intersectio... |
bj-restsn 36453 | An elementwise intersectio... |
bj-restsnss 36454 | Special case of ~ bj-rests... |
bj-restsnss2 36455 | Special case of ~ bj-rests... |
bj-restsn0 36456 | An elementwise intersectio... |
bj-restsn10 36457 | Special case of ~ bj-rests... |
bj-restsnid 36458 | The elementwise intersecti... |
bj-rest10 36459 | An elementwise intersectio... |
bj-rest10b 36460 | Alternate version of ~ bj-... |
bj-restn0 36461 | An elementwise intersectio... |
bj-restn0b 36462 | Alternate version of ~ bj-... |
bj-restpw 36463 | The elementwise intersecti... |
bj-rest0 36464 | An elementwise intersectio... |
bj-restb 36465 | An elementwise intersectio... |
bj-restv 36466 | An elementwise intersectio... |
bj-resta 36467 | An elementwise intersectio... |
bj-restuni 36468 | The union of an elementwis... |
bj-restuni2 36469 | The union of an elementwis... |
bj-restreg 36470 | A reformulation of the axi... |
bj-raldifsn 36471 | All elements in a set sati... |
bj-0int 36472 | If ` A ` is a collection o... |
bj-mooreset 36473 | A Moore collection is a se... |
bj-ismoore 36476 | Characterization of Moore ... |
bj-ismoored0 36477 | Necessary condition to be ... |
bj-ismoored 36478 | Necessary condition to be ... |
bj-ismoored2 36479 | Necessary condition to be ... |
bj-ismooredr 36480 | Sufficient condition to be... |
bj-ismooredr2 36481 | Sufficient condition to be... |
bj-discrmoore 36482 | The powerclass ` ~P A ` is... |
bj-0nmoore 36483 | The empty set is not a Moo... |
bj-snmoore 36484 | A singleton is a Moore col... |
bj-snmooreb 36485 | A singleton is a Moore col... |
bj-prmoore 36486 | A pair formed of two neste... |
bj-0nelmpt 36487 | The empty set is not an el... |
bj-mptval 36488 | Value of a function given ... |
bj-dfmpoa 36489 | An equivalent definition o... |
bj-mpomptALT 36490 | Alternate proof of ~ mpomp... |
setsstrset 36507 | Relation between ~ df-sets... |
bj-nfald 36508 | Variant of ~ nfald . (Con... |
bj-nfexd 36509 | Variant of ~ nfexd . (Con... |
copsex2d 36510 | Implicit substitution dedu... |
copsex2b 36511 | Biconditional form of ~ co... |
opelopabd 36512 | Membership of an ordere pa... |
opelopabb 36513 | Membership of an ordered p... |
opelopabbv 36514 | Membership of an ordered p... |
bj-opelrelex 36515 | The coordinates of an orde... |
bj-opelresdm 36516 | If an ordered pair is in a... |
bj-brresdm 36517 | If two classes are related... |
brabd0 36518 | Expressing that two sets a... |
brabd 36519 | Expressing that two sets a... |
bj-brab2a1 36520 | "Unbounded" version of ~ b... |
bj-opabssvv 36521 | A variant of ~ relopabiv (... |
bj-funidres 36522 | The restricted identity re... |
bj-opelidb 36523 | Characterization of the or... |
bj-opelidb1 36524 | Characterization of the or... |
bj-inexeqex 36525 | Lemma for ~ bj-opelid (but... |
bj-elsn0 36526 | If the intersection of two... |
bj-opelid 36527 | Characterization of the or... |
bj-ideqg 36528 | Characterization of the cl... |
bj-ideqgALT 36529 | Alternate proof of ~ bj-id... |
bj-ideqb 36530 | Characterization of classe... |
bj-idres 36531 | Alternate expression for t... |
bj-opelidres 36532 | Characterization of the or... |
bj-idreseq 36533 | Sufficient condition for t... |
bj-idreseqb 36534 | Characterization for two c... |
bj-ideqg1 36535 | For sets, the identity rel... |
bj-ideqg1ALT 36536 | Alternate proof of bj-ideq... |
bj-opelidb1ALT 36537 | Characterization of the co... |
bj-elid3 36538 | Characterization of the co... |
bj-elid4 36539 | Characterization of the el... |
bj-elid5 36540 | Characterization of the el... |
bj-elid6 36541 | Characterization of the el... |
bj-elid7 36542 | Characterization of the el... |
bj-diagval 36545 | Value of the functionalize... |
bj-diagval2 36546 | Value of the functionalize... |
bj-eldiag 36547 | Characterization of the el... |
bj-eldiag2 36548 | Characterization of the el... |
bj-imdirvallem 36551 | Lemma for ~ bj-imdirval an... |
bj-imdirval 36552 | Value of the functionalize... |
bj-imdirval2lem 36553 | Lemma for ~ bj-imdirval2 a... |
bj-imdirval2 36554 | Value of the functionalize... |
bj-imdirval3 36555 | Value of the functionalize... |
bj-imdiridlem 36556 | Lemma for ~ bj-imdirid and... |
bj-imdirid 36557 | Functorial property of the... |
bj-opelopabid 36558 | Membership in an ordered-p... |
bj-opabco 36559 | Composition of ordered-pai... |
bj-xpcossxp 36560 | The composition of two Car... |
bj-imdirco 36561 | Functorial property of the... |
bj-iminvval 36564 | Value of the functionalize... |
bj-iminvval2 36565 | Value of the functionalize... |
bj-iminvid 36566 | Functorial property of the... |
bj-inftyexpitaufo 36573 | The function ` inftyexpita... |
bj-inftyexpitaudisj 36576 | An element of the circle a... |
bj-inftyexpiinv 36579 | Utility theorem for the in... |
bj-inftyexpiinj 36580 | Injectivity of the paramet... |
bj-inftyexpidisj 36581 | An element of the circle a... |
bj-ccinftydisj 36584 | The circle at infinity is ... |
bj-elccinfty 36585 | A lemma for infinite exten... |
bj-ccssccbar 36588 | Complex numbers are extend... |
bj-ccinftyssccbar 36589 | Infinite extended complex ... |
bj-pinftyccb 36592 | The class ` pinfty ` is an... |
bj-pinftynrr 36593 | The extended complex numbe... |
bj-minftyccb 36596 | The class ` minfty ` is an... |
bj-minftynrr 36597 | The extended complex numbe... |
bj-pinftynminfty 36598 | The extended complex numbe... |
bj-rrhatsscchat 36607 | The real projective line i... |
bj-imafv 36622 | If the direct image of a s... |
bj-funun 36623 | Value of a function expres... |
bj-fununsn1 36624 | Value of a function expres... |
bj-fununsn2 36625 | Value of a function expres... |
bj-fvsnun1 36626 | The value of a function wi... |
bj-fvsnun2 36627 | The value of a function wi... |
bj-fvmptunsn1 36628 | Value of a function expres... |
bj-fvmptunsn2 36629 | Value of a function expres... |
bj-iomnnom 36630 | The canonical bijection fr... |
bj-smgrpssmgm 36639 | Semigroups are magmas. (C... |
bj-smgrpssmgmel 36640 | Semigroups are magmas (ele... |
bj-mndsssmgrp 36641 | Monoids are semigroups. (... |
bj-mndsssmgrpel 36642 | Monoids are semigroups (el... |
bj-cmnssmnd 36643 | Commutative monoids are mo... |
bj-cmnssmndel 36644 | Commutative monoids are mo... |
bj-grpssmnd 36645 | Groups are monoids. (Cont... |
bj-grpssmndel 36646 | Groups are monoids (elemen... |
bj-ablssgrp 36647 | Abelian groups are groups.... |
bj-ablssgrpel 36648 | Abelian groups are groups ... |
bj-ablsscmn 36649 | Abelian groups are commuta... |
bj-ablsscmnel 36650 | Abelian groups are commuta... |
bj-modssabl 36651 | (The additive groups of) m... |
bj-vecssmod 36652 | Vector spaces are modules.... |
bj-vecssmodel 36653 | Vector spaces are modules ... |
bj-finsumval0 36656 | Value of a finite sum. (C... |
bj-fvimacnv0 36657 | Variant of ~ fvimacnv wher... |
bj-isvec 36658 | The predicate "is a vector... |
bj-fldssdrng 36659 | Fields are division rings.... |
bj-flddrng 36660 | Fields are division rings ... |
bj-rrdrg 36661 | The field of real numbers ... |
bj-isclm 36662 | The predicate "is a subcom... |
bj-isrvec 36665 | The predicate "is a real v... |
bj-rvecmod 36666 | Real vector spaces are mod... |
bj-rvecssmod 36667 | Real vector spaces are mod... |
bj-rvecrr 36668 | The field of scalars of a ... |
bj-isrvecd 36669 | The predicate "is a real v... |
bj-rvecvec 36670 | Real vector spaces are vec... |
bj-isrvec2 36671 | The predicate "is a real v... |
bj-rvecssvec 36672 | Real vector spaces are vec... |
bj-rveccmod 36673 | Real vector spaces are sub... |
bj-rvecsscmod 36674 | Real vector spaces are sub... |
bj-rvecsscvec 36675 | Real vector spaces are sub... |
bj-rveccvec 36676 | Real vector spaces are sub... |
bj-rvecssabl 36677 | (The additive groups of) r... |
bj-rvecabl 36678 | (The additive groups of) r... |
bj-subcom 36679 | A consequence of commutati... |
bj-lineqi 36680 | Solution of a (scalar) lin... |
bj-bary1lem 36681 | Lemma for ~ bj-bary1 : exp... |
bj-bary1lem1 36682 | Lemma for bj-bary1: comput... |
bj-bary1 36683 | Barycentric coordinates in... |
bj-endval 36686 | Value of the monoid of end... |
bj-endbase 36687 | Base set of the monoid of ... |
bj-endcomp 36688 | Composition law of the mon... |
bj-endmnd 36689 | The monoid of endomorphism... |
taupilem3 36690 | Lemma for tau-related theo... |
taupilemrplb 36691 | A set of positive reals ha... |
taupilem1 36692 | Lemma for ~ taupi . A pos... |
taupilem2 36693 | Lemma for ~ taupi . The s... |
taupi 36694 | Relationship between ` _ta... |
dfgcd3 36695 | Alternate definition of th... |
irrdifflemf 36696 | Lemma for ~ irrdiff . The... |
irrdiff 36697 | The irrationals are exactl... |
iccioo01 36698 | The closed unit interval i... |
csbrecsg 36699 | Move class substitution in... |
csbrdgg 36700 | Move class substitution in... |
csboprabg 36701 | Move class substitution in... |
csbmpo123 36702 | Move class substitution in... |
con1bii2 36703 | A contraposition inference... |
con2bii2 36704 | A contraposition inference... |
vtoclefex 36705 | Implicit substitution of a... |
rnmptsn 36706 | The range of a function ma... |
f1omptsnlem 36707 | This is the core of the pr... |
f1omptsn 36708 | A function mapping to sing... |
mptsnunlem 36709 | This is the core of the pr... |
mptsnun 36710 | A class ` B ` is equal to ... |
dissneqlem 36711 | This is the core of the pr... |
dissneq 36712 | Any topology that contains... |
exlimim 36713 | Closed form of ~ exlimimd ... |
exlimimd 36714 | Existential elimination ru... |
exellim 36715 | Closed form of ~ exellimdd... |
exellimddv 36716 | Eliminate an antecedent wh... |
topdifinfindis 36717 | Part of Exercise 3 of [Mun... |
topdifinffinlem 36718 | This is the core of the pr... |
topdifinffin 36719 | Part of Exercise 3 of [Mun... |
topdifinf 36720 | Part of Exercise 3 of [Mun... |
topdifinfeq 36721 | Two different ways of defi... |
icorempo 36722 | Closed-below, open-above i... |
icoreresf 36723 | Closed-below, open-above i... |
icoreval 36724 | Value of the closed-below,... |
icoreelrnab 36725 | Elementhood in the set of ... |
isbasisrelowllem1 36726 | Lemma for ~ isbasisrelowl ... |
isbasisrelowllem2 36727 | Lemma for ~ isbasisrelowl ... |
icoreclin 36728 | The set of closed-below, o... |
isbasisrelowl 36729 | The set of all closed-belo... |
icoreunrn 36730 | The union of all closed-be... |
istoprelowl 36731 | The set of all closed-belo... |
icoreelrn 36732 | A class abstraction which ... |
iooelexlt 36733 | An element of an open inte... |
relowlssretop 36734 | The lower limit topology o... |
relowlpssretop 36735 | The lower limit topology o... |
sucneqond 36736 | Inequality of an ordinal s... |
sucneqoni 36737 | Inequality of an ordinal s... |
onsucuni3 36738 | If an ordinal number has a... |
1oequni2o 36739 | The ordinal number ` 1o ` ... |
rdgsucuni 36740 | If an ordinal number has a... |
rdgeqoa 36741 | If a recursive function wi... |
elxp8 36742 | Membership in a Cartesian ... |
cbveud 36743 | Deduction used to change b... |
cbvreud 36744 | Deduction used to change b... |
difunieq 36745 | The difference of unions i... |
inunissunidif 36746 | Theorem about subsets of t... |
rdgellim 36747 | Elementhood in a recursive... |
rdglimss 36748 | A recursive definition at ... |
rdgssun 36749 | In a recursive definition ... |
exrecfnlem 36750 | Lemma for ~ exrecfn . (Co... |
exrecfn 36751 | Theorem about the existenc... |
exrecfnpw 36752 | For any base set, a set wh... |
finorwe 36753 | If the Axiom of Infinity i... |
dffinxpf 36756 | This theorem is the same a... |
finxpeq1 36757 | Equality theorem for Carte... |
finxpeq2 36758 | Equality theorem for Carte... |
csbfinxpg 36759 | Distribute proper substitu... |
finxpreclem1 36760 | Lemma for ` ^^ ` recursion... |
finxpreclem2 36761 | Lemma for ` ^^ ` recursion... |
finxp0 36762 | The value of Cartesian exp... |
finxp1o 36763 | The value of Cartesian exp... |
finxpreclem3 36764 | Lemma for ` ^^ ` recursion... |
finxpreclem4 36765 | Lemma for ` ^^ ` recursion... |
finxpreclem5 36766 | Lemma for ` ^^ ` recursion... |
finxpreclem6 36767 | Lemma for ` ^^ ` recursion... |
finxpsuclem 36768 | Lemma for ~ finxpsuc . (C... |
finxpsuc 36769 | The value of Cartesian exp... |
finxp2o 36770 | The value of Cartesian exp... |
finxp3o 36771 | The value of Cartesian exp... |
finxpnom 36772 | Cartesian exponentiation w... |
finxp00 36773 | Cartesian exponentiation o... |
iunctb2 36774 | Using the axiom of countab... |
domalom 36775 | A class which dominates ev... |
isinf2 36776 | The converse of ~ isinf . ... |
ctbssinf 36777 | Using the axiom of choice,... |
ralssiun 36778 | The index set of an indexe... |
nlpineqsn 36779 | For every point ` p ` of a... |
nlpfvineqsn 36780 | Given a subset ` A ` of ` ... |
fvineqsnf1 36781 | A theorem about functions ... |
fvineqsneu 36782 | A theorem about functions ... |
fvineqsneq 36783 | A theorem about functions ... |
pibp16 36784 | Property P000016 of pi-bas... |
pibp19 36785 | Property P000019 of pi-bas... |
pibp21 36786 | Property P000021 of pi-bas... |
pibt1 36787 | Theorem T000001 of pi-base... |
pibt2 36788 | Theorem T000002 of pi-base... |
wl-section-prop 36789 | Intuitionistic logic is no... |
wl-section-boot 36793 | In this section, I provide... |
wl-luk-imim1i 36794 | Inference adding common co... |
wl-luk-syl 36795 | An inference version of th... |
wl-luk-imtrid 36796 | A syllogism rule of infere... |
wl-luk-pm2.18d 36797 | Deduction based on reducti... |
wl-luk-con4i 36798 | Inference rule. Copy of ~... |
wl-luk-pm2.24i 36799 | Inference rule. Copy of ~... |
wl-luk-a1i 36800 | Inference rule. Copy of ~... |
wl-luk-mpi 36801 | A nested modus ponens infe... |
wl-luk-imim2i 36802 | Inference adding common an... |
wl-luk-imtrdi 36803 | A syllogism rule of infere... |
wl-luk-ax3 36804 | ~ ax-3 proved from Lukasie... |
wl-luk-ax1 36805 | ~ ax-1 proved from Lukasie... |
wl-luk-pm2.27 36806 | This theorem, called "Asse... |
wl-luk-com12 36807 | Inference that swaps (comm... |
wl-luk-pm2.21 36808 | From a wff and its negatio... |
wl-luk-con1i 36809 | A contraposition inference... |
wl-luk-ja 36810 | Inference joining the ante... |
wl-luk-imim2 36811 | A closed form of syllogism... |
wl-luk-a1d 36812 | Deduction introducing an e... |
wl-luk-ax2 36813 | ~ ax-2 proved from Lukasie... |
wl-luk-id 36814 | Principle of identity. Th... |
wl-luk-notnotr 36815 | Converse of double negatio... |
wl-luk-pm2.04 36816 | Swap antecedents. Theorem... |
wl-section-impchain 36817 | An implication like ` ( ps... |
wl-impchain-mp-x 36818 | This series of theorems pr... |
wl-impchain-mp-0 36819 | This theorem is the start ... |
wl-impchain-mp-1 36820 | This theorem is in fact a ... |
wl-impchain-mp-2 36821 | This theorem is in fact a ... |
wl-impchain-com-1.x 36822 | It is often convenient to ... |
wl-impchain-com-1.1 36823 | A degenerate form of antec... |
wl-impchain-com-1.2 36824 | This theorem is in fact a ... |
wl-impchain-com-1.3 36825 | This theorem is in fact a ... |
wl-impchain-com-1.4 36826 | This theorem is in fact a ... |
wl-impchain-com-n.m 36827 | This series of theorems al... |
wl-impchain-com-2.3 36828 | This theorem is in fact a ... |
wl-impchain-com-2.4 36829 | This theorem is in fact a ... |
wl-impchain-com-3.2.1 36830 | This theorem is in fact a ... |
wl-impchain-a1-x 36831 | If an implication chain is... |
wl-impchain-a1-1 36832 | Inference rule, a copy of ... |
wl-impchain-a1-2 36833 | Inference rule, a copy of ... |
wl-impchain-a1-3 36834 | Inference rule, a copy of ... |
wl-ifp-ncond1 36835 | If one case of an ` if- ` ... |
wl-ifp-ncond2 36836 | If one case of an ` if- ` ... |
wl-ifpimpr 36837 | If one case of an ` if- ` ... |
wl-ifp4impr 36838 | If one case of an ` if- ` ... |
wl-df-3xor 36839 | Alternative definition of ... |
wl-df3xor2 36840 | Alternative definition of ... |
wl-df3xor3 36841 | Alternative form of ~ wl-d... |
wl-3xortru 36842 | If the first input is true... |
wl-3xorfal 36843 | If the first input is fals... |
wl-3xorbi 36844 | Triple xor can be replaced... |
wl-3xorbi2 36845 | Alternative form of ~ wl-3... |
wl-3xorbi123d 36846 | Equivalence theorem for tr... |
wl-3xorbi123i 36847 | Equivalence theorem for tr... |
wl-3xorrot 36848 | Rotation law for triple xo... |
wl-3xorcoma 36849 | Commutative law for triple... |
wl-3xorcomb 36850 | Commutative law for triple... |
wl-3xornot1 36851 | Flipping the first input f... |
wl-3xornot 36852 | Triple xor distributes ove... |
wl-1xor 36853 | In the recursive scheme ... |
wl-2xor 36854 | In the recursive scheme ... |
wl-df-3mintru2 36855 | Alternative definition of ... |
wl-df2-3mintru2 36856 | The adder carry in disjunc... |
wl-df3-3mintru2 36857 | The adder carry in conjunc... |
wl-df4-3mintru2 36858 | An alternative definition ... |
wl-1mintru1 36859 | Using the recursion formul... |
wl-1mintru2 36860 | Using the recursion formul... |
wl-2mintru1 36861 | Using the recursion formul... |
wl-2mintru2 36862 | Using the recursion formul... |
wl-df3maxtru1 36863 | Assuming "(n+1)-maxtru1" `... |
wl-ax13lem1 36865 | A version of ~ ax-wl-13v w... |
wl-mps 36866 | Replacing a nested consequ... |
wl-syls1 36867 | Replacing a nested consequ... |
wl-syls2 36868 | Replacing a nested anteced... |
wl-embant 36869 | A true wff can always be a... |
wl-orel12 36870 | In a conjunctive normal fo... |
wl-cases2-dnf 36871 | A particular instance of ~... |
wl-cbvmotv 36872 | Change bound variable. Us... |
wl-moteq 36873 | Change bound variable. Us... |
wl-motae 36874 | Change bound variable. Us... |
wl-moae 36875 | Two ways to express "at mo... |
wl-euae 36876 | Two ways to express "exact... |
wl-nax6im 36877 | The following series of th... |
wl-hbae1 36878 | This specialization of ~ h... |
wl-naevhba1v 36879 | An instance of ~ hbn1w app... |
wl-spae 36880 | Prove an instance of ~ sp ... |
wl-speqv 36881 | Under the assumption ` -. ... |
wl-19.8eqv 36882 | Under the assumption ` -. ... |
wl-19.2reqv 36883 | Under the assumption ` -. ... |
wl-nfalv 36884 | If ` x ` is not present in... |
wl-nfimf1 36885 | An antecedent is irrelevan... |
wl-nfae1 36886 | Unlike ~ nfae , this speci... |
wl-nfnae1 36887 | Unlike ~ nfnae , this spec... |
wl-aetr 36888 | A transitive law for varia... |
wl-axc11r 36889 | Same as ~ axc11r , but usi... |
wl-dral1d 36890 | A version of ~ dral1 with ... |
wl-cbvalnaed 36891 | ~ wl-cbvalnae with a conte... |
wl-cbvalnae 36892 | A more general version of ... |
wl-exeq 36893 | The semantics of ` E. x y ... |
wl-aleq 36894 | The semantics of ` A. x y ... |
wl-nfeqfb 36895 | Extend ~ nfeqf to an equiv... |
wl-nfs1t 36896 | If ` y ` is not free in ` ... |
wl-equsalvw 36897 | Version of ~ equsalv with ... |
wl-equsald 36898 | Deduction version of ~ equ... |
wl-equsal 36899 | A useful equivalence relat... |
wl-equsal1t 36900 | The expression ` x = y ` i... |
wl-equsalcom 36901 | This simple equivalence ea... |
wl-equsal1i 36902 | The antecedent ` x = y ` i... |
wl-sb6rft 36903 | A specialization of ~ wl-e... |
wl-cbvalsbi 36904 | Change bounded variables i... |
wl-sbrimt 36905 | Substitution with a variab... |
wl-sblimt 36906 | Substitution with a variab... |
wl-sb8t 36907 | Substitution of variable i... |
wl-sb8et 36908 | Substitution of variable i... |
wl-sbhbt 36909 | Closed form of ~ sbhb . C... |
wl-sbnf1 36910 | Two ways expressing that `... |
wl-equsb3 36911 | ~ equsb3 with a distinctor... |
wl-equsb4 36912 | Substitution applied to an... |
wl-2sb6d 36913 | Version of ~ 2sb6 with a c... |
wl-sbcom2d-lem1 36914 | Lemma used to prove ~ wl-s... |
wl-sbcom2d-lem2 36915 | Lemma used to prove ~ wl-s... |
wl-sbcom2d 36916 | Version of ~ sbcom2 with a... |
wl-sbalnae 36917 | A theorem used in eliminat... |
wl-sbal1 36918 | A theorem used in eliminat... |
wl-sbal2 36919 | Move quantifier in and out... |
wl-2spsbbi 36920 | ~ spsbbi applied twice. (... |
wl-lem-exsb 36921 | This theorem provides a ba... |
wl-lem-nexmo 36922 | This theorem provides a ba... |
wl-lem-moexsb 36923 | The antecedent ` A. x ( ph... |
wl-alanbii 36924 | This theorem extends ~ ala... |
wl-mo2df 36925 | Version of ~ mof with a co... |
wl-mo2tf 36926 | Closed form of ~ mof with ... |
wl-eudf 36927 | Version of ~ eu6 with a co... |
wl-eutf 36928 | Closed form of ~ eu6 with ... |
wl-euequf 36929 | ~ euequ proved with a dist... |
wl-mo2t 36930 | Closed form of ~ mof . (C... |
wl-mo3t 36931 | Closed form of ~ mo3 . (C... |
wl-sb8eut 36932 | Substitution of variable i... |
wl-sb8mot 36933 | Substitution of variable i... |
wl-issetft 36934 | A closed form of ~ issetf ... |
wl-axc11rc11 36935 | Proving ~ axc11r from ~ ax... |
wl-ax11-lem1 36937 | A transitive law for varia... |
wl-ax11-lem2 36938 | Lemma. (Contributed by Wo... |
wl-ax11-lem3 36939 | Lemma. (Contributed by Wo... |
wl-ax11-lem4 36940 | Lemma. (Contributed by Wo... |
wl-ax11-lem5 36941 | Lemma. (Contributed by Wo... |
wl-ax11-lem6 36942 | Lemma. (Contributed by Wo... |
wl-ax11-lem7 36943 | Lemma. (Contributed by Wo... |
wl-ax11-lem8 36944 | Lemma. (Contributed by Wo... |
wl-ax11-lem9 36945 | The easy part when ` x ` c... |
wl-ax11-lem10 36946 | We now have prepared every... |
wl-clabv 36947 | Variant of ~ df-clab , whe... |
wl-dfclab 36948 | Rederive ~ df-clab from ~ ... |
wl-clabtv 36949 | Using class abstraction in... |
wl-clabt 36950 | Using class abstraction in... |
rabiun 36951 | Abstraction restricted to ... |
iundif1 36952 | Indexed union of class dif... |
imadifss 36953 | The difference of images i... |
cureq 36954 | Equality theorem for curry... |
unceq 36955 | Equality theorem for uncur... |
curf 36956 | Functional property of cur... |
uncf 36957 | Functional property of unc... |
curfv 36958 | Value of currying. (Contr... |
uncov 36959 | Value of uncurrying. (Con... |
curunc 36960 | Currying of uncurrying. (... |
unccur 36961 | Uncurrying of currying. (... |
phpreu 36962 | Theorem related to pigeonh... |
finixpnum 36963 | A finite Cartesian product... |
fin2solem 36964 | Lemma for ~ fin2so . (Con... |
fin2so 36965 | Any totally ordered Tarski... |
ltflcei 36966 | Theorem to move the floor ... |
leceifl 36967 | Theorem to move the floor ... |
sin2h 36968 | Half-angle rule for sine. ... |
cos2h 36969 | Half-angle rule for cosine... |
tan2h 36970 | Half-angle rule for tangen... |
lindsadd 36971 | In a vector space, the uni... |
lindsdom 36972 | A linearly independent set... |
lindsenlbs 36973 | A maximal linearly indepen... |
matunitlindflem1 36974 | One direction of ~ matunit... |
matunitlindflem2 36975 | One direction of ~ matunit... |
matunitlindf 36976 | A matrix over a field is i... |
ptrest 36977 | Expressing a restriction o... |
ptrecube 36978 | Any point in an open set o... |
poimirlem1 36979 | Lemma for ~ poimir - the v... |
poimirlem2 36980 | Lemma for ~ poimir - conse... |
poimirlem3 36981 | Lemma for ~ poimir to add ... |
poimirlem4 36982 | Lemma for ~ poimir connect... |
poimirlem5 36983 | Lemma for ~ poimir to esta... |
poimirlem6 36984 | Lemma for ~ poimir establi... |
poimirlem7 36985 | Lemma for ~ poimir , simil... |
poimirlem8 36986 | Lemma for ~ poimir , estab... |
poimirlem9 36987 | Lemma for ~ poimir , estab... |
poimirlem10 36988 | Lemma for ~ poimir establi... |
poimirlem11 36989 | Lemma for ~ poimir connect... |
poimirlem12 36990 | Lemma for ~ poimir connect... |
poimirlem13 36991 | Lemma for ~ poimir - for a... |
poimirlem14 36992 | Lemma for ~ poimir - for a... |
poimirlem15 36993 | Lemma for ~ poimir , that ... |
poimirlem16 36994 | Lemma for ~ poimir establi... |
poimirlem17 36995 | Lemma for ~ poimir establi... |
poimirlem18 36996 | Lemma for ~ poimir stating... |
poimirlem19 36997 | Lemma for ~ poimir establi... |
poimirlem20 36998 | Lemma for ~ poimir establi... |
poimirlem21 36999 | Lemma for ~ poimir stating... |
poimirlem22 37000 | Lemma for ~ poimir , that ... |
poimirlem23 37001 | Lemma for ~ poimir , two w... |
poimirlem24 37002 | Lemma for ~ poimir , two w... |
poimirlem25 37003 | Lemma for ~ poimir stating... |
poimirlem26 37004 | Lemma for ~ poimir showing... |
poimirlem27 37005 | Lemma for ~ poimir showing... |
poimirlem28 37006 | Lemma for ~ poimir , a var... |
poimirlem29 37007 | Lemma for ~ poimir connect... |
poimirlem30 37008 | Lemma for ~ poimir combini... |
poimirlem31 37009 | Lemma for ~ poimir , assig... |
poimirlem32 37010 | Lemma for ~ poimir , combi... |
poimir 37011 | Poincare-Miranda theorem. ... |
broucube 37012 | Brouwer - or as Kulpa call... |
heicant 37013 | Heine-Cantor theorem: a co... |
opnmbllem0 37014 | Lemma for ~ ismblfin ; cou... |
mblfinlem1 37015 | Lemma for ~ ismblfin , ord... |
mblfinlem2 37016 | Lemma for ~ ismblfin , eff... |
mblfinlem3 37017 | The difference between two... |
mblfinlem4 37018 | Backward direction of ~ is... |
ismblfin 37019 | Measurability in terms of ... |
ovoliunnfl 37020 | ~ ovoliun is incompatible ... |
ex-ovoliunnfl 37021 | Demonstration of ~ ovoliun... |
voliunnfl 37022 | ~ voliun is incompatible w... |
volsupnfl 37023 | ~ volsup is incompatible w... |
mbfresfi 37024 | Measurability of a piecewi... |
mbfposadd 37025 | If the sum of two measurab... |
cnambfre 37026 | A real-valued, a.e. contin... |
dvtanlem 37027 | Lemma for ~ dvtan - the do... |
dvtan 37028 | Derivative of tangent. (C... |
itg2addnclem 37029 | An alternate expression fo... |
itg2addnclem2 37030 | Lemma for ~ itg2addnc . T... |
itg2addnclem3 37031 | Lemma incomprehensible in ... |
itg2addnc 37032 | Alternate proof of ~ itg2a... |
itg2gt0cn 37033 | ~ itg2gt0 holds on functio... |
ibladdnclem 37034 | Lemma for ~ ibladdnc ; cf ... |
ibladdnc 37035 | Choice-free analogue of ~ ... |
itgaddnclem1 37036 | Lemma for ~ itgaddnc ; cf.... |
itgaddnclem2 37037 | Lemma for ~ itgaddnc ; cf.... |
itgaddnc 37038 | Choice-free analogue of ~ ... |
iblsubnc 37039 | Choice-free analogue of ~ ... |
itgsubnc 37040 | Choice-free analogue of ~ ... |
iblabsnclem 37041 | Lemma for ~ iblabsnc ; cf.... |
iblabsnc 37042 | Choice-free analogue of ~ ... |
iblmulc2nc 37043 | Choice-free analogue of ~ ... |
itgmulc2nclem1 37044 | Lemma for ~ itgmulc2nc ; c... |
itgmulc2nclem2 37045 | Lemma for ~ itgmulc2nc ; c... |
itgmulc2nc 37046 | Choice-free analogue of ~ ... |
itgabsnc 37047 | Choice-free analogue of ~ ... |
itggt0cn 37048 | ~ itggt0 holds for continu... |
ftc1cnnclem 37049 | Lemma for ~ ftc1cnnc ; cf.... |
ftc1cnnc 37050 | Choice-free proof of ~ ftc... |
ftc1anclem1 37051 | Lemma for ~ ftc1anc - the ... |
ftc1anclem2 37052 | Lemma for ~ ftc1anc - rest... |
ftc1anclem3 37053 | Lemma for ~ ftc1anc - the ... |
ftc1anclem4 37054 | Lemma for ~ ftc1anc . (Co... |
ftc1anclem5 37055 | Lemma for ~ ftc1anc , the ... |
ftc1anclem6 37056 | Lemma for ~ ftc1anc - cons... |
ftc1anclem7 37057 | Lemma for ~ ftc1anc . (Co... |
ftc1anclem8 37058 | Lemma for ~ ftc1anc . (Co... |
ftc1anc 37059 | ~ ftc1a holds for function... |
ftc2nc 37060 | Choice-free proof of ~ ftc... |
asindmre 37061 | Real part of domain of dif... |
dvasin 37062 | Derivative of arcsine. (C... |
dvacos 37063 | Derivative of arccosine. ... |
dvreasin 37064 | Real derivative of arcsine... |
dvreacos 37065 | Real derivative of arccosi... |
areacirclem1 37066 | Antiderivative of cross-se... |
areacirclem2 37067 | Endpoint-inclusive continu... |
areacirclem3 37068 | Integrability of cross-sec... |
areacirclem4 37069 | Endpoint-inclusive continu... |
areacirclem5 37070 | Finding the cross-section ... |
areacirc 37071 | The area of a circle of ra... |
unirep 37072 | Define a quantity whose de... |
cover2 37073 | Two ways of expressing the... |
cover2g 37074 | Two ways of expressing the... |
brabg2 37075 | Relation by a binary relat... |
opelopab3 37076 | Ordered pair membership in... |
cocanfo 37077 | Cancellation of a surjecti... |
brresi2 37078 | Restriction of a binary re... |
fnopabeqd 37079 | Equality deduction for fun... |
fvopabf4g 37080 | Function value of an opera... |
fnopabco 37081 | Composition of a function ... |
opropabco 37082 | Composition of an operator... |
cocnv 37083 | Composition with a functio... |
f1ocan1fv 37084 | Cancel a composition by a ... |
f1ocan2fv 37085 | Cancel a composition by th... |
inixp 37086 | Intersection of Cartesian ... |
upixp 37087 | Universal property of the ... |
abrexdom 37088 | An indexed set is dominate... |
abrexdom2 37089 | An indexed set is dominate... |
ac6gf 37090 | Axiom of Choice. (Contrib... |
indexa 37091 | If for every element of an... |
indexdom 37092 | If for every element of an... |
frinfm 37093 | A subset of a well-founded... |
welb 37094 | A nonempty subset of a wel... |
supex2g 37095 | Existence of supremum. (C... |
supclt 37096 | Closure of supremum. (Con... |
supubt 37097 | Upper bound property of su... |
filbcmb 37098 | Combine a finite set of lo... |
fzmul 37099 | Membership of a product in... |
sdclem2 37100 | Lemma for ~ sdc . (Contri... |
sdclem1 37101 | Lemma for ~ sdc . (Contri... |
sdc 37102 | Strong dependent choice. ... |
fdc 37103 | Finite version of dependen... |
fdc1 37104 | Variant of ~ fdc with no s... |
seqpo 37105 | Two ways to say that a seq... |
incsequz 37106 | An increasing sequence of ... |
incsequz2 37107 | An increasing sequence of ... |
nnubfi 37108 | A bounded above set of pos... |
nninfnub 37109 | An infinite set of positiv... |
subspopn 37110 | An open set is open in the... |
neificl 37111 | Neighborhoods are closed u... |
lpss2 37112 | Limit points of a subset a... |
metf1o 37113 | Use a bijection with a met... |
blssp 37114 | A ball in the subspace met... |
mettrifi 37115 | Generalized triangle inequ... |
lmclim2 37116 | A sequence in a metric spa... |
geomcau 37117 | If the distance between co... |
caures 37118 | The restriction of a Cauch... |
caushft 37119 | A shifted Cauchy sequence ... |
constcncf 37120 | A constant function is a c... |
cnres2 37121 | The restriction of a conti... |
cnresima 37122 | A continuous function is c... |
cncfres 37123 | A continuous function on c... |
istotbnd 37127 | The predicate "is a totall... |
istotbnd2 37128 | The predicate "is a totall... |
istotbnd3 37129 | A metric space is totally ... |
totbndmet 37130 | The predicate "totally bou... |
0totbnd 37131 | The metric (there is only ... |
sstotbnd2 37132 | Condition for a subset of ... |
sstotbnd 37133 | Condition for a subset of ... |
sstotbnd3 37134 | Use a net that is not nece... |
totbndss 37135 | A subset of a totally boun... |
equivtotbnd 37136 | If the metric ` M ` is "st... |
isbnd 37138 | The predicate "is a bounde... |
bndmet 37139 | A bounded metric space is ... |
isbndx 37140 | A "bounded extended metric... |
isbnd2 37141 | The predicate "is a bounde... |
isbnd3 37142 | A metric space is bounded ... |
isbnd3b 37143 | A metric space is bounded ... |
bndss 37144 | A subset of a bounded metr... |
blbnd 37145 | A ball is bounded. (Contr... |
ssbnd 37146 | A subset of a metric space... |
totbndbnd 37147 | A totally bounded metric s... |
equivbnd 37148 | If the metric ` M ` is "st... |
bnd2lem 37149 | Lemma for ~ equivbnd2 and ... |
equivbnd2 37150 | If balls are totally bound... |
prdsbnd 37151 | The product metric over fi... |
prdstotbnd 37152 | The product metric over fi... |
prdsbnd2 37153 | If balls are totally bound... |
cntotbnd 37154 | A subset of the complex nu... |
cnpwstotbnd 37155 | A subset of ` A ^ I ` , wh... |
ismtyval 37158 | The set of isometries betw... |
isismty 37159 | The condition "is an isome... |
ismtycnv 37160 | The inverse of an isometry... |
ismtyima 37161 | The image of a ball under ... |
ismtyhmeolem 37162 | Lemma for ~ ismtyhmeo . (... |
ismtyhmeo 37163 | An isometry is a homeomorp... |
ismtybndlem 37164 | Lemma for ~ ismtybnd . (C... |
ismtybnd 37165 | Isometries preserve bounde... |
ismtyres 37166 | A restriction of an isomet... |
heibor1lem 37167 | Lemma for ~ heibor1 . A c... |
heibor1 37168 | One half of ~ heibor , tha... |
heiborlem1 37169 | Lemma for ~ heibor . We w... |
heiborlem2 37170 | Lemma for ~ heibor . Subs... |
heiborlem3 37171 | Lemma for ~ heibor . Usin... |
heiborlem4 37172 | Lemma for ~ heibor . Usin... |
heiborlem5 37173 | Lemma for ~ heibor . The ... |
heiborlem6 37174 | Lemma for ~ heibor . Sinc... |
heiborlem7 37175 | Lemma for ~ heibor . Sinc... |
heiborlem8 37176 | Lemma for ~ heibor . The ... |
heiborlem9 37177 | Lemma for ~ heibor . Disc... |
heiborlem10 37178 | Lemma for ~ heibor . The ... |
heibor 37179 | Generalized Heine-Borel Th... |
bfplem1 37180 | Lemma for ~ bfp . The seq... |
bfplem2 37181 | Lemma for ~ bfp . Using t... |
bfp 37182 | Banach fixed point theorem... |
rrnval 37185 | The n-dimensional Euclidea... |
rrnmval 37186 | The value of the Euclidean... |
rrnmet 37187 | Euclidean space is a metri... |
rrndstprj1 37188 | The distance between two p... |
rrndstprj2 37189 | Bound on the distance betw... |
rrncmslem 37190 | Lemma for ~ rrncms . (Con... |
rrncms 37191 | Euclidean space is complet... |
repwsmet 37192 | The supremum metric on ` R... |
rrnequiv 37193 | The supremum metric on ` R... |
rrntotbnd 37194 | A set in Euclidean space i... |
rrnheibor 37195 | Heine-Borel theorem for Eu... |
ismrer1 37196 | An isometry between ` RR `... |
reheibor 37197 | Heine-Borel theorem for re... |
iccbnd 37198 | A closed interval in ` RR ... |
icccmpALT 37199 | A closed interval in ` RR ... |
isass 37204 | The predicate "is an assoc... |
isexid 37205 | The predicate ` G ` has a ... |
ismgmOLD 37208 | Obsolete version of ~ ismg... |
clmgmOLD 37209 | Obsolete version of ~ mgmc... |
opidonOLD 37210 | Obsolete version of ~ mndp... |
rngopidOLD 37211 | Obsolete version of ~ mndp... |
opidon2OLD 37212 | Obsolete version of ~ mndp... |
isexid2 37213 | If ` G e. ( Magma i^i ExId... |
exidu1 37214 | Uniqueness of the left and... |
idrval 37215 | The value of the identity ... |
iorlid 37216 | A magma right and left ide... |
cmpidelt 37217 | A magma right and left ide... |
smgrpismgmOLD 37220 | Obsolete version of ~ sgrp... |
issmgrpOLD 37221 | Obsolete version of ~ issg... |
smgrpmgm 37222 | A semigroup is a magma. (... |
smgrpassOLD 37223 | Obsolete version of ~ sgrp... |
mndoissmgrpOLD 37226 | Obsolete version of ~ mnds... |
mndoisexid 37227 | A monoid has an identity e... |
mndoismgmOLD 37228 | Obsolete version of ~ mndm... |
mndomgmid 37229 | A monoid is a magma with a... |
ismndo 37230 | The predicate "is a monoid... |
ismndo1 37231 | The predicate "is a monoid... |
ismndo2 37232 | The predicate "is a monoid... |
grpomndo 37233 | A group is a monoid. (Con... |
exidcl 37234 | Closure of the binary oper... |
exidreslem 37235 | Lemma for ~ exidres and ~ ... |
exidres 37236 | The restriction of a binar... |
exidresid 37237 | The restriction of a binar... |
ablo4pnp 37238 | A commutative/associative ... |
grpoeqdivid 37239 | Two group elements are equ... |
grposnOLD 37240 | The group operation for th... |
elghomlem1OLD 37243 | Obsolete as of 15-Mar-2020... |
elghomlem2OLD 37244 | Obsolete as of 15-Mar-2020... |
elghomOLD 37245 | Obsolete version of ~ isgh... |
ghomlinOLD 37246 | Obsolete version of ~ ghml... |
ghomidOLD 37247 | Obsolete version of ~ ghmi... |
ghomf 37248 | Mapping property of a grou... |
ghomco 37249 | The composition of two gro... |
ghomdiv 37250 | Group homomorphisms preser... |
grpokerinj 37251 | A group homomorphism is in... |
relrngo 37254 | The class of all unital ri... |
isrngo 37255 | The predicate "is a (unita... |
isrngod 37256 | Conditions that determine ... |
rngoi 37257 | The properties of a unital... |
rngosm 37258 | Functionality of the multi... |
rngocl 37259 | Closure of the multiplicat... |
rngoid 37260 | The multiplication operati... |
rngoideu 37261 | The unity element of a rin... |
rngodi 37262 | Distributive law for the m... |
rngodir 37263 | Distributive law for the m... |
rngoass 37264 | Associative law for the mu... |
rngo2 37265 | A ring element plus itself... |
rngoablo 37266 | A ring's addition operatio... |
rngoablo2 37267 | In a unital ring the addit... |
rngogrpo 37268 | A ring's addition operatio... |
rngone0 37269 | The base set of a ring is ... |
rngogcl 37270 | Closure law for the additi... |
rngocom 37271 | The addition operation of ... |
rngoaass 37272 | The addition operation of ... |
rngoa32 37273 | The addition operation of ... |
rngoa4 37274 | Rearrangement of 4 terms i... |
rngorcan 37275 | Right cancellation law for... |
rngolcan 37276 | Left cancellation law for ... |
rngo0cl 37277 | A ring has an additive ide... |
rngo0rid 37278 | The additive identity of a... |
rngo0lid 37279 | The additive identity of a... |
rngolz 37280 | The zero of a unital ring ... |
rngorz 37281 | The zero of a unital ring ... |
rngosn3 37282 | Obsolete as of 25-Jan-2020... |
rngosn4 37283 | Obsolete as of 25-Jan-2020... |
rngosn6 37284 | Obsolete as of 25-Jan-2020... |
rngonegcl 37285 | A ring is closed under neg... |
rngoaddneg1 37286 | Adding the negative in a r... |
rngoaddneg2 37287 | Adding the negative in a r... |
rngosub 37288 | Subtraction in a ring, in ... |
rngmgmbs4 37289 | The range of an internal o... |
rngodm1dm2 37290 | In a unital ring the domai... |
rngorn1 37291 | In a unital ring the range... |
rngorn1eq 37292 | In a unital ring the range... |
rngomndo 37293 | In a unital ring the multi... |
rngoidmlem 37294 | The unity element of a rin... |
rngolidm 37295 | The unity element of a rin... |
rngoridm 37296 | The unity element of a rin... |
rngo1cl 37297 | The unity element of a rin... |
rngoueqz 37298 | Obsolete as of 23-Jan-2020... |
rngonegmn1l 37299 | Negation in a ring is the ... |
rngonegmn1r 37300 | Negation in a ring is the ... |
rngoneglmul 37301 | Negation of a product in a... |
rngonegrmul 37302 | Negation of a product in a... |
rngosubdi 37303 | Ring multiplication distri... |
rngosubdir 37304 | Ring multiplication distri... |
zerdivemp1x 37305 | In a unital ring a left in... |
isdivrngo 37308 | The predicate "is a divisi... |
drngoi 37309 | The properties of a divisi... |
gidsn 37310 | Obsolete as of 23-Jan-2020... |
zrdivrng 37311 | The zero ring is not a div... |
dvrunz 37312 | In a division ring the rin... |
isgrpda 37313 | Properties that determine ... |
isdrngo1 37314 | The predicate "is a divisi... |
divrngcl 37315 | The product of two nonzero... |
isdrngo2 37316 | A division ring is a ring ... |
isdrngo3 37317 | A division ring is a ring ... |
rngohomval 37322 | The set of ring homomorphi... |
isrngohom 37323 | The predicate "is a ring h... |
rngohomf 37324 | A ring homomorphism is a f... |
rngohomcl 37325 | Closure law for a ring hom... |
rngohom1 37326 | A ring homomorphism preser... |
rngohomadd 37327 | Ring homomorphisms preserv... |
rngohommul 37328 | Ring homomorphisms preserv... |
rngogrphom 37329 | A ring homomorphism is a g... |
rngohom0 37330 | A ring homomorphism preser... |
rngohomsub 37331 | Ring homomorphisms preserv... |
rngohomco 37332 | The composition of two rin... |
rngokerinj 37333 | A ring homomorphism is inj... |
rngoisoval 37335 | The set of ring isomorphis... |
isrngoiso 37336 | The predicate "is a ring i... |
rngoiso1o 37337 | A ring isomorphism is a bi... |
rngoisohom 37338 | A ring isomorphism is a ri... |
rngoisocnv 37339 | The inverse of a ring isom... |
rngoisoco 37340 | The composition of two rin... |
isriscg 37342 | The ring isomorphism relat... |
isrisc 37343 | The ring isomorphism relat... |
risc 37344 | The ring isomorphism relat... |
risci 37345 | Determine that two rings a... |
riscer 37346 | Ring isomorphism is an equ... |
iscom2 37353 | A device to add commutativ... |
iscrngo 37354 | The predicate "is a commut... |
iscrngo2 37355 | The predicate "is a commut... |
iscringd 37356 | Conditions that determine ... |
flddivrng 37357 | A field is a division ring... |
crngorngo 37358 | A commutative ring is a ri... |
crngocom 37359 | The multiplication operati... |
crngm23 37360 | Commutative/associative la... |
crngm4 37361 | Commutative/associative la... |
fldcrngo 37362 | A field is a commutative r... |
isfld2 37363 | The predicate "is a field"... |
crngohomfo 37364 | The image of a homomorphis... |
idlval 37371 | The class of ideals of a r... |
isidl 37372 | The predicate "is an ideal... |
isidlc 37373 | The predicate "is an ideal... |
idlss 37374 | An ideal of ` R ` is a sub... |
idlcl 37375 | An element of an ideal is ... |
idl0cl 37376 | An ideal contains ` 0 ` . ... |
idladdcl 37377 | An ideal is closed under a... |
idllmulcl 37378 | An ideal is closed under m... |
idlrmulcl 37379 | An ideal is closed under m... |
idlnegcl 37380 | An ideal is closed under n... |
idlsubcl 37381 | An ideal is closed under s... |
rngoidl 37382 | A ring ` R ` is an ` R ` i... |
0idl 37383 | The set containing only ` ... |
1idl 37384 | Two ways of expressing the... |
0rngo 37385 | In a ring, ` 0 = 1 ` iff t... |
divrngidl 37386 | The only ideals in a divis... |
intidl 37387 | The intersection of a none... |
inidl 37388 | The intersection of two id... |
unichnidl 37389 | The union of a nonempty ch... |
keridl 37390 | The kernel of a ring homom... |
pridlval 37391 | The class of prime ideals ... |
ispridl 37392 | The predicate "is a prime ... |
pridlidl 37393 | A prime ideal is an ideal.... |
pridlnr 37394 | A prime ideal is a proper ... |
pridl 37395 | The main property of a pri... |
ispridl2 37396 | A condition that shows an ... |
maxidlval 37397 | The set of maximal ideals ... |
ismaxidl 37398 | The predicate "is a maxima... |
maxidlidl 37399 | A maximal ideal is an idea... |
maxidlnr 37400 | A maximal ideal is proper.... |
maxidlmax 37401 | A maximal ideal is a maxim... |
maxidln1 37402 | One is not contained in an... |
maxidln0 37403 | A ring with a maximal idea... |
isprrngo 37408 | The predicate "is a prime ... |
prrngorngo 37409 | A prime ring is a ring. (... |
smprngopr 37410 | A simple ring (one whose o... |
divrngpr 37411 | A division ring is a prime... |
isdmn 37412 | The predicate "is a domain... |
isdmn2 37413 | The predicate "is a domain... |
dmncrng 37414 | A domain is a commutative ... |
dmnrngo 37415 | A domain is a ring. (Cont... |
flddmn 37416 | A field is a domain. (Con... |
igenval 37419 | The ideal generated by a s... |
igenss 37420 | A set is a subset of the i... |
igenidl 37421 | The ideal generated by a s... |
igenmin 37422 | The ideal generated by a s... |
igenidl2 37423 | The ideal generated by an ... |
igenval2 37424 | The ideal generated by a s... |
prnc 37425 | A principal ideal (an idea... |
isfldidl 37426 | Determine if a ring is a f... |
isfldidl2 37427 | Determine if a ring is a f... |
ispridlc 37428 | The predicate "is a prime ... |
pridlc 37429 | Property of a prime ideal ... |
pridlc2 37430 | Property of a prime ideal ... |
pridlc3 37431 | Property of a prime ideal ... |
isdmn3 37432 | The predicate "is a domain... |
dmnnzd 37433 | A domain has no zero-divis... |
dmncan1 37434 | Cancellation law for domai... |
dmncan2 37435 | Cancellation law for domai... |
efald2 37436 | A proof by contradiction. ... |
notbinot1 37437 | Simplification rule of neg... |
bicontr 37438 | Biconditional of its own n... |
impor 37439 | An equivalent formula for ... |
orfa 37440 | The falsum ` F. ` can be r... |
notbinot2 37441 | Commutation rule between n... |
biimpor 37442 | A rewriting rule for bicon... |
orfa1 37443 | Add a contradicting disjun... |
orfa2 37444 | Remove a contradicting dis... |
bifald 37445 | Infer the equivalence to a... |
orsild 37446 | A lemma for not-or-not eli... |
orsird 37447 | A lemma for not-or-not eli... |
cnf1dd 37448 | A lemma for Conjunctive No... |
cnf2dd 37449 | A lemma for Conjunctive No... |
cnfn1dd 37450 | A lemma for Conjunctive No... |
cnfn2dd 37451 | A lemma for Conjunctive No... |
or32dd 37452 | A rearrangement of disjunc... |
notornotel1 37453 | A lemma for not-or-not eli... |
notornotel2 37454 | A lemma for not-or-not eli... |
contrd 37455 | A proof by contradiction, ... |
an12i 37456 | An inference from commutin... |
exmid2 37457 | An excluded middle law. (... |
selconj 37458 | An inference for selecting... |
truconj 37459 | Add true as a conjunct. (... |
orel 37460 | An inference for disjuncti... |
negel 37461 | An inference for negation ... |
botel 37462 | An inference for bottom el... |
tradd 37463 | Add top ad a conjunct. (C... |
gm-sbtru 37464 | Substitution does not chan... |
sbfal 37465 | Substitution does not chan... |
sbcani 37466 | Distribution of class subs... |
sbcori 37467 | Distribution of class subs... |
sbcimi 37468 | Distribution of class subs... |
sbcni 37469 | Move class substitution in... |
sbali 37470 | Discard class substitution... |
sbexi 37471 | Discard class substitution... |
sbcalf 37472 | Move universal quantifier ... |
sbcexf 37473 | Move existential quantifie... |
sbcalfi 37474 | Move universal quantifier ... |
sbcexfi 37475 | Move existential quantifie... |
spsbcdi 37476 | A lemma for eliminating a ... |
alrimii 37477 | A lemma for introducing a ... |
spesbcdi 37478 | A lemma for introducing an... |
exlimddvf 37479 | A lemma for eliminating an... |
exlimddvfi 37480 | A lemma for eliminating an... |
sbceq1ddi 37481 | A lemma for eliminating in... |
sbccom2lem 37482 | Lemma for ~ sbccom2 . (Co... |
sbccom2 37483 | Commutative law for double... |
sbccom2f 37484 | Commutative law for double... |
sbccom2fi 37485 | Commutative law for double... |
csbcom2fi 37486 | Commutative law for double... |
fald 37487 | Refutation of falsity, in ... |
tsim1 37488 | A Tseitin axiom for logica... |
tsim2 37489 | A Tseitin axiom for logica... |
tsim3 37490 | A Tseitin axiom for logica... |
tsbi1 37491 | A Tseitin axiom for logica... |
tsbi2 37492 | A Tseitin axiom for logica... |
tsbi3 37493 | A Tseitin axiom for logica... |
tsbi4 37494 | A Tseitin axiom for logica... |
tsxo1 37495 | A Tseitin axiom for logica... |
tsxo2 37496 | A Tseitin axiom for logica... |
tsxo3 37497 | A Tseitin axiom for logica... |
tsxo4 37498 | A Tseitin axiom for logica... |
tsan1 37499 | A Tseitin axiom for logica... |
tsan2 37500 | A Tseitin axiom for logica... |
tsan3 37501 | A Tseitin axiom for logica... |
tsna1 37502 | A Tseitin axiom for logica... |
tsna2 37503 | A Tseitin axiom for logica... |
tsna3 37504 | A Tseitin axiom for logica... |
tsor1 37505 | A Tseitin axiom for logica... |
tsor2 37506 | A Tseitin axiom for logica... |
tsor3 37507 | A Tseitin axiom for logica... |
ts3an1 37508 | A Tseitin axiom for triple... |
ts3an2 37509 | A Tseitin axiom for triple... |
ts3an3 37510 | A Tseitin axiom for triple... |
ts3or1 37511 | A Tseitin axiom for triple... |
ts3or2 37512 | A Tseitin axiom for triple... |
ts3or3 37513 | A Tseitin axiom for triple... |
iuneq2f 37514 | Equality deduction for ind... |
rabeq12f 37515 | Equality deduction for res... |
csbeq12 37516 | Equality deduction for sub... |
sbeqi 37517 | Equality deduction for sub... |
ralbi12f 37518 | Equality deduction for res... |
oprabbi 37519 | Equality deduction for cla... |
mpobi123f 37520 | Equality deduction for map... |
iuneq12f 37521 | Equality deduction for ind... |
iineq12f 37522 | Equality deduction for ind... |
opabbi 37523 | Equality deduction for cla... |
mptbi12f 37524 | Equality deduction for map... |
orcomdd 37525 | Commutativity of logic dis... |
scottexf 37526 | A version of ~ scottex wit... |
scott0f 37527 | A version of ~ scott0 with... |
scottn0f 37528 | A version of ~ scott0f wit... |
ac6s3f 37529 | Generalization of the Axio... |
ac6s6 37530 | Generalization of the Axio... |
ac6s6f 37531 | Generalization of the Axio... |
el2v1 37575 | New way ( ~ elv , and the ... |
el3v 37576 | New way ( ~ elv , and the ... |
el3v1 37577 | New way ( ~ elv , and the ... |
el3v2 37578 | New way ( ~ elv , and the ... |
el3v3 37579 | New way ( ~ elv , and the ... |
el3v12 37580 | New way ( ~ elv , and the ... |
el3v13 37581 | New way ( ~ elv , and the ... |
el3v23 37582 | New way ( ~ elv , and the ... |
anan 37583 | Multiple commutations in c... |
triantru3 37584 | A wff is equivalent to its... |
bianbi 37585 | Exchanging conjunction in ... |
bianim 37586 | Exchanging conjunction in ... |
biorfd 37587 | A wff is equivalent to its... |
eqbrtr 37588 | Substitution of equal clas... |
eqbrb 37589 | Substitution of equal clas... |
eqeltr 37590 | Substitution of equal clas... |
eqelb 37591 | Substitution of equal clas... |
eqeqan2d 37592 | Implication of introducing... |
suceqsneq 37593 | One-to-one relationship be... |
sucdifsn2 37594 | Absorption of union with a... |
sucdifsn 37595 | The difference between the... |
disjresin 37596 | The restriction to a disjo... |
disjresdisj 37597 | The intersection of restri... |
disjresdif 37598 | The difference between res... |
disjresundif 37599 | Lemma for ~ ressucdifsn2 .... |
ressucdifsn2 37600 | The difference between res... |
ressucdifsn 37601 | The difference between res... |
inres2 37602 | Two ways of expressing the... |
coideq 37603 | Equality theorem for compo... |
nexmo1 37604 | If there is no case where ... |
ralin 37605 | Restricted universal quant... |
r2alan 37606 | Double restricted universa... |
ssrabi 37607 | Inference of restricted ab... |
rabbieq 37608 | Equivalent wff's correspon... |
rabimbieq 37609 | Restricted equivalent wff'... |
abeqin 37610 | Intersection with class ab... |
abeqinbi 37611 | Intersection with class ab... |
rabeqel 37612 | Class element of a restric... |
eqrelf 37613 | The equality connective be... |
br1cnvinxp 37614 | Binary relation on the con... |
releleccnv 37615 | Elementhood in a converse ... |
releccnveq 37616 | Equality of converse ` R `... |
opelvvdif 37617 | Negated elementhood of ord... |
vvdifopab 37618 | Ordered-pair class abstrac... |
brvdif 37619 | Binary relation with unive... |
brvdif2 37620 | Binary relation with unive... |
brvvdif 37621 | Binary relation with the c... |
brvbrvvdif 37622 | Binary relation with the c... |
brcnvep 37623 | The converse of the binary... |
elecALTV 37624 | Elementhood in the ` R ` -... |
brcnvepres 37625 | Restricted converse epsilo... |
brres2 37626 | Binary relation on a restr... |
br1cnvres 37627 | Binary relation on the con... |
eldmres 37628 | Elementhood in the domain ... |
elrnres 37629 | Element of the range of a ... |
eldmressnALTV 37630 | Element of the domain of a... |
elrnressn 37631 | Element of the range of a ... |
eldm4 37632 | Elementhood in a domain. ... |
eldmres2 37633 | Elementhood in the domain ... |
eceq1i 37634 | Equality theorem for ` C `... |
elecres 37635 | Elementhood in the restric... |
ecres 37636 | Restricted coset of ` B ` ... |
ecres2 37637 | The restricted coset of ` ... |
eccnvepres 37638 | Restricted converse epsilo... |
eleccnvep 37639 | Elementhood in the convers... |
eccnvep 37640 | The converse epsilon coset... |
extep 37641 | Property of epsilon relati... |
disjeccnvep 37642 | Property of the epsilon re... |
eccnvepres2 37643 | The restricted converse ep... |
eccnvepres3 37644 | Condition for a restricted... |
eldmqsres 37645 | Elementhood in a restricte... |
eldmqsres2 37646 | Elementhood in a restricte... |
qsss1 37647 | Subclass theorem for quoti... |
qseq1i 37648 | Equality theorem for quoti... |
qseq1d 37649 | Equality theorem for quoti... |
brinxprnres 37650 | Binary relation on a restr... |
inxprnres 37651 | Restriction of a class as ... |
dfres4 37652 | Alternate definition of th... |
exan3 37653 | Equivalent expressions wit... |
exanres 37654 | Equivalent expressions wit... |
exanres3 37655 | Equivalent expressions wit... |
exanres2 37656 | Equivalent expressions wit... |
cnvepres 37657 | Restricted converse epsilo... |
eqrel2 37658 | Equality of relations. (C... |
rncnv 37659 | Range of converse is the d... |
dfdm6 37660 | Alternate definition of do... |
dfrn6 37661 | Alternate definition of ra... |
rncnvepres 37662 | The range of the restricte... |
dmecd 37663 | Equality of the coset of `... |
dmec2d 37664 | Equality of the coset of `... |
brid 37665 | Property of the identity b... |
ideq2 37666 | For sets, the identity bin... |
idresssidinxp 37667 | Condition for the identity... |
idreseqidinxp 37668 | Condition for the identity... |
extid 37669 | Property of identity relat... |
inxpss 37670 | Two ways to say that an in... |
idinxpss 37671 | Two ways to say that an in... |
ref5 37672 | Two ways to say that an in... |
inxpss3 37673 | Two ways to say that an in... |
inxpss2 37674 | Two ways to say that inter... |
inxpssidinxp 37675 | Two ways to say that inter... |
idinxpssinxp 37676 | Two ways to say that inter... |
idinxpssinxp2 37677 | Identity intersection with... |
idinxpssinxp3 37678 | Identity intersection with... |
idinxpssinxp4 37679 | Identity intersection with... |
relcnveq3 37680 | Two ways of saying a relat... |
relcnveq 37681 | Two ways of saying a relat... |
relcnveq2 37682 | Two ways of saying a relat... |
relcnveq4 37683 | Two ways of saying a relat... |
qsresid 37684 | Simplification of a specia... |
n0elqs 37685 | Two ways of expressing tha... |
n0elqs2 37686 | Two ways of expressing tha... |
ecex2 37687 | Condition for a coset to b... |
uniqsALTV 37688 | The union of a quotient se... |
imaexALTV 37689 | Existence of an image of a... |
ecexALTV 37690 | Existence of a coset, like... |
rnresequniqs 37691 | The range of a restriction... |
n0el2 37692 | Two ways of expressing tha... |
cnvepresex 37693 | Sethood condition for the ... |
eccnvepex 37694 | The converse epsilon coset... |
cnvepimaex 37695 | The image of converse epsi... |
cnvepima 37696 | The image of converse epsi... |
inex3 37697 | Sufficient condition for t... |
inxpex 37698 | Sufficient condition for a... |
eqres 37699 | Converting a class constan... |
brrabga 37700 | The law of concretion for ... |
brcnvrabga 37701 | The law of concretion for ... |
opideq 37702 | Equality conditions for or... |
iss2 37703 | A subclass of the identity... |
eldmcnv 37704 | Elementhood in a domain of... |
dfrel5 37705 | Alternate definition of th... |
dfrel6 37706 | Alternate definition of th... |
cnvresrn 37707 | Converse restricted to ran... |
relssinxpdmrn 37708 | Subset of restriction, spe... |
cnvref4 37709 | Two ways to say that a rel... |
cnvref5 37710 | Two ways to say that a rel... |
ecin0 37711 | Two ways of saying that th... |
ecinn0 37712 | Two ways of saying that th... |
ineleq 37713 | Equivalence of restricted ... |
inecmo 37714 | Equivalence of a double re... |
inecmo2 37715 | Equivalence of a double re... |
ineccnvmo 37716 | Equivalence of a double re... |
alrmomorn 37717 | Equivalence of an "at most... |
alrmomodm 37718 | Equivalence of an "at most... |
ineccnvmo2 37719 | Equivalence of a double un... |
inecmo3 37720 | Equivalence of a double un... |
moeu2 37721 | Uniqueness is equivalent t... |
mopickr 37722 | "At most one" picks a vari... |
moantr 37723 | Sufficient condition for t... |
brabidgaw 37724 | The law of concretion for ... |
brabidga 37725 | The law of concretion for ... |
inxp2 37726 | Intersection with a Cartes... |
opabf 37727 | A class abstraction of a c... |
ec0 37728 | The empty-coset of a class... |
0qs 37729 | Quotient set with the empt... |
brcnvin 37730 | Intersection with a conver... |
xrnss3v 37732 | A range Cartesian product ... |
xrnrel 37733 | A range Cartesian product ... |
brxrn 37734 | Characterize a ternary rel... |
brxrn2 37735 | A characterization of the ... |
dfxrn2 37736 | Alternate definition of th... |
xrneq1 37737 | Equality theorem for the r... |
xrneq1i 37738 | Equality theorem for the r... |
xrneq1d 37739 | Equality theorem for the r... |
xrneq2 37740 | Equality theorem for the r... |
xrneq2i 37741 | Equality theorem for the r... |
xrneq2d 37742 | Equality theorem for the r... |
xrneq12 37743 | Equality theorem for the r... |
xrneq12i 37744 | Equality theorem for the r... |
xrneq12d 37745 | Equality theorem for the r... |
elecxrn 37746 | Elementhood in the ` ( R |... |
ecxrn 37747 | The ` ( R |X. S ) ` -coset... |
disjressuc2 37748 | Double restricted quantifi... |
disjecxrn 37749 | Two ways of saying that ` ... |
disjecxrncnvep 37750 | Two ways of saying that co... |
disjsuc2 37751 | Double restricted quantifi... |
xrninxp 37752 | Intersection of a range Ca... |
xrninxp2 37753 | Intersection of a range Ca... |
xrninxpex 37754 | Sufficient condition for t... |
inxpxrn 37755 | Two ways to express the in... |
br1cnvxrn2 37756 | The converse of a binary r... |
elec1cnvxrn2 37757 | Elementhood in the convers... |
rnxrn 37758 | Range of the range Cartesi... |
rnxrnres 37759 | Range of a range Cartesian... |
rnxrncnvepres 37760 | Range of a range Cartesian... |
rnxrnidres 37761 | Range of a range Cartesian... |
xrnres 37762 | Two ways to express restri... |
xrnres2 37763 | Two ways to express restri... |
xrnres3 37764 | Two ways to express restri... |
xrnres4 37765 | Two ways to express restri... |
xrnresex 37766 | Sufficient condition for a... |
xrnidresex 37767 | Sufficient condition for a... |
xrncnvepresex 37768 | Sufficient condition for a... |
brin2 37769 | Binary relation on an inte... |
brin3 37770 | Binary relation on an inte... |
dfcoss2 37773 | Alternate definition of th... |
dfcoss3 37774 | Alternate definition of th... |
dfcoss4 37775 | Alternate definition of th... |
cosscnv 37776 | Class of cosets by the con... |
coss1cnvres 37777 | Class of cosets by the con... |
coss2cnvepres 37778 | Special case of ~ coss1cnv... |
cossex 37779 | If ` A ` is a set then the... |
cosscnvex 37780 | If ` A ` is a set then the... |
1cosscnvepresex 37781 | Sufficient condition for a... |
1cossxrncnvepresex 37782 | Sufficient condition for a... |
relcoss 37783 | Cosets by ` R ` is a relat... |
relcoels 37784 | Coelements on ` A ` is a r... |
cossss 37785 | Subclass theorem for the c... |
cosseq 37786 | Equality theorem for the c... |
cosseqi 37787 | Equality theorem for the c... |
cosseqd 37788 | Equality theorem for the c... |
1cossres 37789 | The class of cosets by a r... |
dfcoels 37790 | Alternate definition of th... |
brcoss 37791 | ` A ` and ` B ` are cosets... |
brcoss2 37792 | Alternate form of the ` A ... |
brcoss3 37793 | Alternate form of the ` A ... |
brcosscnvcoss 37794 | For sets, the ` A ` and ` ... |
brcoels 37795 | ` B ` and ` C ` are coelem... |
cocossss 37796 | Two ways of saying that co... |
cnvcosseq 37797 | The converse of cosets by ... |
br2coss 37798 | Cosets by ` ,~ R ` binary ... |
br1cossres 37799 | ` B ` and ` C ` are cosets... |
br1cossres2 37800 | ` B ` and ` C ` are cosets... |
brressn 37801 | Binary relation on a restr... |
ressn2 37802 | A class ' R ' restricted t... |
refressn 37803 | Any class ' R ' restricted... |
antisymressn 37804 | Every class ' R ' restrict... |
trressn 37805 | Any class ' R ' restricted... |
relbrcoss 37806 | ` A ` and ` B ` are cosets... |
br1cossinres 37807 | ` B ` and ` C ` are cosets... |
br1cossxrnres 37808 | ` <. B , C >. ` and ` <. D... |
br1cossinidres 37809 | ` B ` and ` C ` are cosets... |
br1cossincnvepres 37810 | ` B ` and ` C ` are cosets... |
br1cossxrnidres 37811 | ` <. B , C >. ` and ` <. D... |
br1cossxrncnvepres 37812 | ` <. B , C >. ` and ` <. D... |
dmcoss3 37813 | The domain of cosets is th... |
dmcoss2 37814 | The domain of cosets is th... |
rncossdmcoss 37815 | The range of cosets is the... |
dm1cosscnvepres 37816 | The domain of cosets of th... |
dmcoels 37817 | The domain of coelements i... |
eldmcoss 37818 | Elementhood in the domain ... |
eldmcoss2 37819 | Elementhood in the domain ... |
eldm1cossres 37820 | Elementhood in the domain ... |
eldm1cossres2 37821 | Elementhood in the domain ... |
refrelcosslem 37822 | Lemma for the left side of... |
refrelcoss3 37823 | The class of cosets by ` R... |
refrelcoss2 37824 | The class of cosets by ` R... |
symrelcoss3 37825 | The class of cosets by ` R... |
symrelcoss2 37826 | The class of cosets by ` R... |
cossssid 37827 | Equivalent expressions for... |
cossssid2 37828 | Equivalent expressions for... |
cossssid3 37829 | Equivalent expressions for... |
cossssid4 37830 | Equivalent expressions for... |
cossssid5 37831 | Equivalent expressions for... |
brcosscnv 37832 | ` A ` and ` B ` are cosets... |
brcosscnv2 37833 | ` A ` and ` B ` are cosets... |
br1cosscnvxrn 37834 | ` A ` and ` B ` are cosets... |
1cosscnvxrn 37835 | Cosets by the converse ran... |
cosscnvssid3 37836 | Equivalent expressions for... |
cosscnvssid4 37837 | Equivalent expressions for... |
cosscnvssid5 37838 | Equivalent expressions for... |
coss0 37839 | Cosets by the empty set ar... |
cossid 37840 | Cosets by the identity rel... |
cosscnvid 37841 | Cosets by the converse ide... |
trcoss 37842 | Sufficient condition for t... |
eleccossin 37843 | Two ways of saying that th... |
trcoss2 37844 | Equivalent expressions for... |
elrels2 37846 | The element of the relatio... |
elrelsrel 37847 | The element of the relatio... |
elrelsrelim 37848 | The element of the relatio... |
elrels5 37849 | Equivalent expressions for... |
elrels6 37850 | Equivalent expressions for... |
elrelscnveq3 37851 | Two ways of saying a relat... |
elrelscnveq 37852 | Two ways of saying a relat... |
elrelscnveq2 37853 | Two ways of saying a relat... |
elrelscnveq4 37854 | Two ways of saying a relat... |
cnvelrels 37855 | The converse of a set is a... |
cosselrels 37856 | Cosets of sets are element... |
cosscnvelrels 37857 | Cosets of converse sets ar... |
dfssr2 37859 | Alternate definition of th... |
relssr 37860 | The subset relation is a r... |
brssr 37861 | The subset relation and su... |
brssrid 37862 | Any set is a subset of its... |
issetssr 37863 | Two ways of expressing set... |
brssrres 37864 | Restricted subset binary r... |
br1cnvssrres 37865 | Restricted converse subset... |
brcnvssr 37866 | The converse of a subset r... |
brcnvssrid 37867 | Any set is a converse subs... |
br1cossxrncnvssrres 37868 | ` <. B , C >. ` and ` <. D... |
extssr 37869 | Property of subset relatio... |
dfrefrels2 37873 | Alternate definition of th... |
dfrefrels3 37874 | Alternate definition of th... |
dfrefrel2 37875 | Alternate definition of th... |
dfrefrel3 37876 | Alternate definition of th... |
dfrefrel5 37877 | Alternate definition of th... |
elrefrels2 37878 | Element of the class of re... |
elrefrels3 37879 | Element of the class of re... |
elrefrelsrel 37880 | For sets, being an element... |
refreleq 37881 | Equality theorem for refle... |
refrelid 37882 | Identity relation is refle... |
refrelcoss 37883 | The class of cosets by ` R... |
refrelressn 37884 | Any class ' R ' restricted... |
dfcnvrefrels2 37888 | Alternate definition of th... |
dfcnvrefrels3 37889 | Alternate definition of th... |
dfcnvrefrel2 37890 | Alternate definition of th... |
dfcnvrefrel3 37891 | Alternate definition of th... |
dfcnvrefrel4 37892 | Alternate definition of th... |
dfcnvrefrel5 37893 | Alternate definition of th... |
elcnvrefrels2 37894 | Element of the class of co... |
elcnvrefrels3 37895 | Element of the class of co... |
elcnvrefrelsrel 37896 | For sets, being an element... |
cnvrefrelcoss2 37897 | Necessary and sufficient c... |
cosselcnvrefrels2 37898 | Necessary and sufficient c... |
cosselcnvrefrels3 37899 | Necessary and sufficient c... |
cosselcnvrefrels4 37900 | Necessary and sufficient c... |
cosselcnvrefrels5 37901 | Necessary and sufficient c... |
dfsymrels2 37905 | Alternate definition of th... |
dfsymrels3 37906 | Alternate definition of th... |
dfsymrels4 37907 | Alternate definition of th... |
dfsymrels5 37908 | Alternate definition of th... |
dfsymrel2 37909 | Alternate definition of th... |
dfsymrel3 37910 | Alternate definition of th... |
dfsymrel4 37911 | Alternate definition of th... |
dfsymrel5 37912 | Alternate definition of th... |
elsymrels2 37913 | Element of the class of sy... |
elsymrels3 37914 | Element of the class of sy... |
elsymrels4 37915 | Element of the class of sy... |
elsymrels5 37916 | Element of the class of sy... |
elsymrelsrel 37917 | For sets, being an element... |
symreleq 37918 | Equality theorem for symme... |
symrelim 37919 | Symmetric relation implies... |
symrelcoss 37920 | The class of cosets by ` R... |
idsymrel 37921 | The identity relation is s... |
epnsymrel 37922 | The membership (epsilon) r... |
symrefref2 37923 | Symmetry is a sufficient c... |
symrefref3 37924 | Symmetry is a sufficient c... |
refsymrels2 37925 | Elements of the class of r... |
refsymrels3 37926 | Elements of the class of r... |
refsymrel2 37927 | A relation which is reflex... |
refsymrel3 37928 | A relation which is reflex... |
elrefsymrels2 37929 | Elements of the class of r... |
elrefsymrels3 37930 | Elements of the class of r... |
elrefsymrelsrel 37931 | For sets, being an element... |
dftrrels2 37935 | Alternate definition of th... |
dftrrels3 37936 | Alternate definition of th... |
dftrrel2 37937 | Alternate definition of th... |
dftrrel3 37938 | Alternate definition of th... |
eltrrels2 37939 | Element of the class of tr... |
eltrrels3 37940 | Element of the class of tr... |
eltrrelsrel 37941 | For sets, being an element... |
trreleq 37942 | Equality theorem for the t... |
trrelressn 37943 | Any class ' R ' restricted... |
dfeqvrels2 37948 | Alternate definition of th... |
dfeqvrels3 37949 | Alternate definition of th... |
dfeqvrel2 37950 | Alternate definition of th... |
dfeqvrel3 37951 | Alternate definition of th... |
eleqvrels2 37952 | Element of the class of eq... |
eleqvrels3 37953 | Element of the class of eq... |
eleqvrelsrel 37954 | For sets, being an element... |
elcoeleqvrels 37955 | Elementhood in the coeleme... |
elcoeleqvrelsrel 37956 | For sets, being an element... |
eqvrelrel 37957 | An equivalence relation is... |
eqvrelrefrel 37958 | An equivalence relation is... |
eqvrelsymrel 37959 | An equivalence relation is... |
eqvreltrrel 37960 | An equivalence relation is... |
eqvrelim 37961 | Equivalence relation impli... |
eqvreleq 37962 | Equality theorem for equiv... |
eqvreleqi 37963 | Equality theorem for equiv... |
eqvreleqd 37964 | Equality theorem for equiv... |
eqvrelsym 37965 | An equivalence relation is... |
eqvrelsymb 37966 | An equivalence relation is... |
eqvreltr 37967 | An equivalence relation is... |
eqvreltrd 37968 | A transitivity relation fo... |
eqvreltr4d 37969 | A transitivity relation fo... |
eqvrelref 37970 | An equivalence relation is... |
eqvrelth 37971 | Basic property of equivale... |
eqvrelcl 37972 | Elementhood in the field o... |
eqvrelthi 37973 | Basic property of equivale... |
eqvreldisj 37974 | Equivalence classes do not... |
qsdisjALTV 37975 | Elements of a quotient set... |
eqvrelqsel 37976 | If an element of a quotien... |
eqvrelcoss 37977 | Two ways to express equiva... |
eqvrelcoss3 37978 | Two ways to express equiva... |
eqvrelcoss2 37979 | Two ways to express equiva... |
eqvrelcoss4 37980 | Two ways to express equiva... |
dfcoeleqvrels 37981 | Alternate definition of th... |
dfcoeleqvrel 37982 | Alternate definition of th... |
brredunds 37986 | Binary relation on the cla... |
brredundsredund 37987 | For sets, binary relation ... |
redundss3 37988 | Implication of redundancy ... |
redundeq1 37989 | Equivalence of redundancy ... |
redundpim3 37990 | Implication of redundancy ... |
redundpbi1 37991 | Equivalence of redundancy ... |
refrelsredund4 37992 | The naive version of the c... |
refrelsredund2 37993 | The naive version of the c... |
refrelsredund3 37994 | The naive version of the c... |
refrelredund4 37995 | The naive version of the d... |
refrelredund2 37996 | The naive version of the d... |
refrelredund3 37997 | The naive version of the d... |
dmqseq 38000 | Equality theorem for domai... |
dmqseqi 38001 | Equality theorem for domai... |
dmqseqd 38002 | Equality theorem for domai... |
dmqseqeq1 38003 | Equality theorem for domai... |
dmqseqeq1i 38004 | Equality theorem for domai... |
dmqseqeq1d 38005 | Equality theorem for domai... |
brdmqss 38006 | The domain quotient binary... |
brdmqssqs 38007 | If ` A ` and ` R ` are set... |
n0eldmqs 38008 | The empty set is not an el... |
n0eldmqseq 38009 | The empty set is not an el... |
n0elim 38010 | Implication of that the em... |
n0el3 38011 | Two ways of expressing tha... |
cnvepresdmqss 38012 | The domain quotient binary... |
cnvepresdmqs 38013 | The domain quotient predic... |
unidmqs 38014 | The range of a relation is... |
unidmqseq 38015 | The union of the domain qu... |
dmqseqim 38016 | If the domain quotient of ... |
dmqseqim2 38017 | Lemma for ~ erimeq2 . (Co... |
releldmqs 38018 | Elementhood in the domain ... |
eldmqs1cossres 38019 | Elementhood in the domain ... |
releldmqscoss 38020 | Elementhood in the domain ... |
dmqscoelseq 38021 | Two ways to express the eq... |
dmqs1cosscnvepreseq 38022 | Two ways to express the eq... |
brers 38027 | Binary equivalence relatio... |
dferALTV2 38028 | Equivalence relation with ... |
erALTVeq1 38029 | Equality theorem for equiv... |
erALTVeq1i 38030 | Equality theorem for equiv... |
erALTVeq1d 38031 | Equality theorem for equiv... |
dfcomember 38032 | Alternate definition of th... |
dfcomember2 38033 | Alternate definition of th... |
dfcomember3 38034 | Alternate definition of th... |
eqvreldmqs 38035 | Two ways to express comemb... |
eqvreldmqs2 38036 | Two ways to express comemb... |
brerser 38037 | Binary equivalence relatio... |
erimeq2 38038 | Equivalence relation on it... |
erimeq 38039 | Equivalence relation on it... |
dffunsALTV 38043 | Alternate definition of th... |
dffunsALTV2 38044 | Alternate definition of th... |
dffunsALTV3 38045 | Alternate definition of th... |
dffunsALTV4 38046 | Alternate definition of th... |
dffunsALTV5 38047 | Alternate definition of th... |
dffunALTV2 38048 | Alternate definition of th... |
dffunALTV3 38049 | Alternate definition of th... |
dffunALTV4 38050 | Alternate definition of th... |
dffunALTV5 38051 | Alternate definition of th... |
elfunsALTV 38052 | Elementhood in the class o... |
elfunsALTV2 38053 | Elementhood in the class o... |
elfunsALTV3 38054 | Elementhood in the class o... |
elfunsALTV4 38055 | Elementhood in the class o... |
elfunsALTV5 38056 | Elementhood in the class o... |
elfunsALTVfunALTV 38057 | The element of the class o... |
funALTVfun 38058 | Our definition of the func... |
funALTVss 38059 | Subclass theorem for funct... |
funALTVeq 38060 | Equality theorem for funct... |
funALTVeqi 38061 | Equality inference for the... |
funALTVeqd 38062 | Equality deduction for the... |
dfdisjs 38068 | Alternate definition of th... |
dfdisjs2 38069 | Alternate definition of th... |
dfdisjs3 38070 | Alternate definition of th... |
dfdisjs4 38071 | Alternate definition of th... |
dfdisjs5 38072 | Alternate definition of th... |
dfdisjALTV 38073 | Alternate definition of th... |
dfdisjALTV2 38074 | Alternate definition of th... |
dfdisjALTV3 38075 | Alternate definition of th... |
dfdisjALTV4 38076 | Alternate definition of th... |
dfdisjALTV5 38077 | Alternate definition of th... |
dfeldisj2 38078 | Alternate definition of th... |
dfeldisj3 38079 | Alternate definition of th... |
dfeldisj4 38080 | Alternate definition of th... |
dfeldisj5 38081 | Alternate definition of th... |
eldisjs 38082 | Elementhood in the class o... |
eldisjs2 38083 | Elementhood in the class o... |
eldisjs3 38084 | Elementhood in the class o... |
eldisjs4 38085 | Elementhood in the class o... |
eldisjs5 38086 | Elementhood in the class o... |
eldisjsdisj 38087 | The element of the class o... |
eleldisjs 38088 | Elementhood in the disjoin... |
eleldisjseldisj 38089 | The element of the disjoin... |
disjrel 38090 | Disjoint relation is a rel... |
disjss 38091 | Subclass theorem for disjo... |
disjssi 38092 | Subclass theorem for disjo... |
disjssd 38093 | Subclass theorem for disjo... |
disjeq 38094 | Equality theorem for disjo... |
disjeqi 38095 | Equality theorem for disjo... |
disjeqd 38096 | Equality theorem for disjo... |
disjdmqseqeq1 38097 | Lemma for the equality the... |
eldisjss 38098 | Subclass theorem for disjo... |
eldisjssi 38099 | Subclass theorem for disjo... |
eldisjssd 38100 | Subclass theorem for disjo... |
eldisjeq 38101 | Equality theorem for disjo... |
eldisjeqi 38102 | Equality theorem for disjo... |
eldisjeqd 38103 | Equality theorem for disjo... |
disjres 38104 | Disjoint restriction. (Co... |
eldisjn0elb 38105 | Two forms of disjoint elem... |
disjxrn 38106 | Two ways of saying that a ... |
disjxrnres5 38107 | Disjoint range Cartesian p... |
disjorimxrn 38108 | Disjointness condition for... |
disjimxrn 38109 | Disjointness condition for... |
disjimres 38110 | Disjointness condition for... |
disjimin 38111 | Disjointness condition for... |
disjiminres 38112 | Disjointness condition for... |
disjimxrnres 38113 | Disjointness condition for... |
disjALTV0 38114 | The null class is disjoint... |
disjALTVid 38115 | The class of identity rela... |
disjALTVidres 38116 | The class of identity rela... |
disjALTVinidres 38117 | The intersection with rest... |
disjALTVxrnidres 38118 | The class of range Cartesi... |
disjsuc 38119 | Disjoint range Cartesian p... |
dfantisymrel4 38121 | Alternate definition of th... |
dfantisymrel5 38122 | Alternate definition of th... |
antisymrelres 38123 | (Contributed by Peter Mazs... |
antisymrelressn 38124 | (Contributed by Peter Mazs... |
dfpart2 38129 | Alternate definition of th... |
dfmembpart2 38130 | Alternate definition of th... |
brparts 38131 | Binary partitions relation... |
brparts2 38132 | Binary partitions relation... |
brpartspart 38133 | Binary partition and the p... |
parteq1 38134 | Equality theorem for parti... |
parteq2 38135 | Equality theorem for parti... |
parteq12 38136 | Equality theorem for parti... |
parteq1i 38137 | Equality theorem for parti... |
parteq1d 38138 | Equality theorem for parti... |
partsuc2 38139 | Property of the partition.... |
partsuc 38140 | Property of the partition.... |
disjim 38141 | The "Divide et Aequivalere... |
disjimi 38142 | Every disjoint relation ge... |
detlem 38143 | If a relation is disjoint,... |
eldisjim 38144 | If the elements of ` A ` a... |
eldisjim2 38145 | Alternate form of ~ eldisj... |
eqvrel0 38146 | The null class is an equiv... |
det0 38147 | The cosets by the null cla... |
eqvrelcoss0 38148 | The cosets by the null cla... |
eqvrelid 38149 | The identity relation is a... |
eqvrel1cossidres 38150 | The cosets by a restricted... |
eqvrel1cossinidres 38151 | The cosets by an intersect... |
eqvrel1cossxrnidres 38152 | The cosets by a range Cart... |
detid 38153 | The cosets by the identity... |
eqvrelcossid 38154 | The cosets by the identity... |
detidres 38155 | The cosets by the restrict... |
detinidres 38156 | The cosets by the intersec... |
detxrnidres 38157 | The cosets by the range Ca... |
disjlem14 38158 | Lemma for ~ disjdmqseq , ~... |
disjlem17 38159 | Lemma for ~ disjdmqseq , ~... |
disjlem18 38160 | Lemma for ~ disjdmqseq , ~... |
disjlem19 38161 | Lemma for ~ disjdmqseq , ~... |
disjdmqsss 38162 | Lemma for ~ disjdmqseq via... |
disjdmqscossss 38163 | Lemma for ~ disjdmqseq via... |
disjdmqs 38164 | If a relation is disjoint,... |
disjdmqseq 38165 | If a relation is disjoint,... |
eldisjn0el 38166 | Special case of ~ disjdmqs... |
partim2 38167 | Disjoint relation on its n... |
partim 38168 | Partition implies equivale... |
partimeq 38169 | Partition implies that the... |
eldisjlem19 38170 | Special case of ~ disjlem1... |
membpartlem19 38171 | Together with ~ disjlem19 ... |
petlem 38172 | If you can prove that the ... |
petlemi 38173 | If you can prove disjointn... |
pet02 38174 | Class ` A ` is a partition... |
pet0 38175 | Class ` A ` is a partition... |
petid2 38176 | Class ` A ` is a partition... |
petid 38177 | A class is a partition by ... |
petidres2 38178 | Class ` A ` is a partition... |
petidres 38179 | A class is a partition by ... |
petinidres2 38180 | Class ` A ` is a partition... |
petinidres 38181 | A class is a partition by ... |
petxrnidres2 38182 | Class ` A ` is a partition... |
petxrnidres 38183 | A class is a partition by ... |
eqvreldisj1 38184 | The elements of the quotie... |
eqvreldisj2 38185 | The elements of the quotie... |
eqvreldisj3 38186 | The elements of the quotie... |
eqvreldisj4 38187 | Intersection with the conv... |
eqvreldisj5 38188 | Range Cartesian product wi... |
eqvrelqseqdisj2 38189 | Implication of ~ eqvreldis... |
fences3 38190 | Implication of ~ eqvrelqse... |
eqvrelqseqdisj3 38191 | Implication of ~ eqvreldis... |
eqvrelqseqdisj4 38192 | Lemma for ~ petincnvepres2... |
eqvrelqseqdisj5 38193 | Lemma for the Partition-Eq... |
mainer 38194 | The Main Theorem of Equiva... |
partimcomember 38195 | Partition with general ` R... |
mpet3 38196 | Member Partition-Equivalen... |
cpet2 38197 | The conventional form of t... |
cpet 38198 | The conventional form of M... |
mpet 38199 | Member Partition-Equivalen... |
mpet2 38200 | Member Partition-Equivalen... |
mpets2 38201 | Member Partition-Equivalen... |
mpets 38202 | Member Partition-Equivalen... |
mainpart 38203 | Partition with general ` R... |
fences 38204 | The Theorem of Fences by E... |
fences2 38205 | The Theorem of Fences by E... |
mainer2 38206 | The Main Theorem of Equiva... |
mainerim 38207 | Every equivalence relation... |
petincnvepres2 38208 | A partition-equivalence th... |
petincnvepres 38209 | The shortest form of a par... |
pet2 38210 | Partition-Equivalence Theo... |
pet 38211 | Partition-Equivalence Theo... |
pets 38212 | Partition-Equivalence Theo... |
prtlem60 38213 | Lemma for ~ prter3 . (Con... |
bicomdd 38214 | Commute two sides of a bic... |
jca2r 38215 | Inference conjoining the c... |
jca3 38216 | Inference conjoining the c... |
prtlem70 38217 | Lemma for ~ prter3 : a rea... |
ibdr 38218 | Reverse of ~ ibd . (Contr... |
prtlem100 38219 | Lemma for ~ prter3 . (Con... |
prtlem5 38220 | Lemma for ~ prter1 , ~ prt... |
prtlem80 38221 | Lemma for ~ prter2 . (Con... |
brabsb2 38222 | A closed form of ~ brabsb ... |
eqbrrdv2 38223 | Other version of ~ eqbrrdi... |
prtlem9 38224 | Lemma for ~ prter3 . (Con... |
prtlem10 38225 | Lemma for ~ prter3 . (Con... |
prtlem11 38226 | Lemma for ~ prter2 . (Con... |
prtlem12 38227 | Lemma for ~ prtex and ~ pr... |
prtlem13 38228 | Lemma for ~ prter1 , ~ prt... |
prtlem16 38229 | Lemma for ~ prtex , ~ prte... |
prtlem400 38230 | Lemma for ~ prter2 and als... |
erprt 38233 | The quotient set of an equ... |
prtlem14 38234 | Lemma for ~ prter1 , ~ prt... |
prtlem15 38235 | Lemma for ~ prter1 and ~ p... |
prtlem17 38236 | Lemma for ~ prter2 . (Con... |
prtlem18 38237 | Lemma for ~ prter2 . (Con... |
prtlem19 38238 | Lemma for ~ prter2 . (Con... |
prter1 38239 | Every partition generates ... |
prtex 38240 | The equivalence relation g... |
prter2 38241 | The quotient set of the eq... |
prter3 38242 | For every partition there ... |
axc5 38253 | This theorem repeats ~ sp ... |
ax4fromc4 38254 | Rederivation of Axiom ~ ax... |
ax10fromc7 38255 | Rederivation of Axiom ~ ax... |
ax6fromc10 38256 | Rederivation of Axiom ~ ax... |
hba1-o 38257 | The setvar ` x ` is not fr... |
axc4i-o 38258 | Inference version of ~ ax-... |
equid1 38259 | Proof of ~ equid from our ... |
equcomi1 38260 | Proof of ~ equcomi from ~ ... |
aecom-o 38261 | Commutation law for identi... |
aecoms-o 38262 | A commutation rule for ide... |
hbae-o 38263 | All variables are effectiv... |
dral1-o 38264 | Formula-building lemma for... |
ax12fromc15 38265 | Rederivation of Axiom ~ ax... |
ax13fromc9 38266 | Derive ~ ax-13 from ~ ax-c... |
ax5ALT 38267 | Axiom to quantify a variab... |
sps-o 38268 | Generalization of antecede... |
hbequid 38269 | Bound-variable hypothesis ... |
nfequid-o 38270 | Bound-variable hypothesis ... |
axc5c7 38271 | Proof of a single axiom th... |
axc5c7toc5 38272 | Rederivation of ~ ax-c5 fr... |
axc5c7toc7 38273 | Rederivation of ~ ax-c7 fr... |
axc711 38274 | Proof of a single axiom th... |
nfa1-o 38275 | ` x ` is not free in ` A. ... |
axc711toc7 38276 | Rederivation of ~ ax-c7 fr... |
axc711to11 38277 | Rederivation of ~ ax-11 fr... |
axc5c711 38278 | Proof of a single axiom th... |
axc5c711toc5 38279 | Rederivation of ~ ax-c5 fr... |
axc5c711toc7 38280 | Rederivation of ~ ax-c7 fr... |
axc5c711to11 38281 | Rederivation of ~ ax-11 fr... |
equidqe 38282 | ~ equid with existential q... |
axc5sp1 38283 | A special case of ~ ax-c5 ... |
equidq 38284 | ~ equid with universal qua... |
equid1ALT 38285 | Alternate proof of ~ equid... |
axc11nfromc11 38286 | Rederivation of ~ ax-c11n ... |
naecoms-o 38287 | A commutation rule for dis... |
hbnae-o 38288 | All variables are effectiv... |
dvelimf-o 38289 | Proof of ~ dvelimh that us... |
dral2-o 38290 | Formula-building lemma for... |
aev-o 38291 | A "distinctor elimination"... |
ax5eq 38292 | Theorem to add distinct qu... |
dveeq2-o 38293 | Quantifier introduction wh... |
axc16g-o 38294 | A generalization of Axiom ... |
dveeq1-o 38295 | Quantifier introduction wh... |
dveeq1-o16 38296 | Version of ~ dveeq1 using ... |
ax5el 38297 | Theorem to add distinct qu... |
axc11n-16 38298 | This theorem shows that, g... |
dveel2ALT 38299 | Alternate proof of ~ dveel... |
ax12f 38300 | Basis step for constructin... |
ax12eq 38301 | Basis step for constructin... |
ax12el 38302 | Basis step for constructin... |
ax12indn 38303 | Induction step for constru... |
ax12indi 38304 | Induction step for constru... |
ax12indalem 38305 | Lemma for ~ ax12inda2 and ... |
ax12inda2ALT 38306 | Alternate proof of ~ ax12i... |
ax12inda2 38307 | Induction step for constru... |
ax12inda 38308 | Induction step for constru... |
ax12v2-o 38309 | Rederivation of ~ ax-c15 f... |
ax12a2-o 38310 | Derive ~ ax-c15 from a hyp... |
axc11-o 38311 | Show that ~ ax-c11 can be ... |
fsumshftd 38312 | Index shift of a finite su... |
riotaclbgBAD 38314 | Closure of restricted iota... |
riotaclbBAD 38315 | Closure of restricted iota... |
riotasvd 38316 | Deduction version of ~ rio... |
riotasv2d 38317 | Value of description binde... |
riotasv2s 38318 | The value of description b... |
riotasv 38319 | Value of description binde... |
riotasv3d 38320 | A property ` ch ` holding ... |
elimhyps 38321 | A version of ~ elimhyp usi... |
dedths 38322 | A version of weak deductio... |
renegclALT 38323 | Closure law for negative o... |
elimhyps2 38324 | Generalization of ~ elimhy... |
dedths2 38325 | Generalization of ~ dedths... |
nfcxfrdf 38326 | A utility lemma to transfe... |
nfded 38327 | A deduction theorem that c... |
nfded2 38328 | A deduction theorem that c... |
nfunidALT2 38329 | Deduction version of ~ nfu... |
nfunidALT 38330 | Deduction version of ~ nfu... |
nfopdALT 38331 | Deduction version of bound... |
cnaddcom 38332 | Recover the commutative la... |
toycom 38333 | Show the commutative law f... |
lshpset 38338 | The set of all hyperplanes... |
islshp 38339 | The predicate "is a hyperp... |
islshpsm 38340 | Hyperplane properties expr... |
lshplss 38341 | A hyperplane is a subspace... |
lshpne 38342 | A hyperplane is not equal ... |
lshpnel 38343 | A hyperplane's generating ... |
lshpnelb 38344 | The subspace sum of a hype... |
lshpnel2N 38345 | Condition that determines ... |
lshpne0 38346 | The member of the span in ... |
lshpdisj 38347 | A hyperplane and the span ... |
lshpcmp 38348 | If two hyperplanes are com... |
lshpinN 38349 | The intersection of two di... |
lsatset 38350 | The set of all 1-dim subsp... |
islsat 38351 | The predicate "is a 1-dim ... |
lsatlspsn2 38352 | The span of a nonzero sing... |
lsatlspsn 38353 | The span of a nonzero sing... |
islsati 38354 | A 1-dim subspace (atom) (o... |
lsateln0 38355 | A 1-dim subspace (atom) (o... |
lsatlss 38356 | The set of 1-dim subspaces... |
lsatlssel 38357 | An atom is a subspace. (C... |
lsatssv 38358 | An atom is a set of vector... |
lsatn0 38359 | A 1-dim subspace (atom) of... |
lsatspn0 38360 | The span of a vector is an... |
lsator0sp 38361 | The span of a vector is ei... |
lsatssn0 38362 | A subspace (or any class) ... |
lsatcmp 38363 | If two atoms are comparabl... |
lsatcmp2 38364 | If an atom is included in ... |
lsatel 38365 | A nonzero vector in an ato... |
lsatelbN 38366 | A nonzero vector in an ato... |
lsat2el 38367 | Two atoms sharing a nonzer... |
lsmsat 38368 | Convert comparison of atom... |
lsatfixedN 38369 | Show equality with the spa... |
lsmsatcv 38370 | Subspace sum has the cover... |
lssatomic 38371 | The lattice of subspaces i... |
lssats 38372 | The lattice of subspaces i... |
lpssat 38373 | Two subspaces in a proper ... |
lrelat 38374 | Subspaces are relatively a... |
lssatle 38375 | The ordering of two subspa... |
lssat 38376 | Two subspaces in a proper ... |
islshpat 38377 | Hyperplane properties expr... |
lcvfbr 38380 | The covers relation for a ... |
lcvbr 38381 | The covers relation for a ... |
lcvbr2 38382 | The covers relation for a ... |
lcvbr3 38383 | The covers relation for a ... |
lcvpss 38384 | The covers relation implie... |
lcvnbtwn 38385 | The covers relation implie... |
lcvntr 38386 | The covers relation is not... |
lcvnbtwn2 38387 | The covers relation implie... |
lcvnbtwn3 38388 | The covers relation implie... |
lsmcv2 38389 | Subspace sum has the cover... |
lcvat 38390 | If a subspace covers anoth... |
lsatcv0 38391 | An atom covers the zero su... |
lsatcveq0 38392 | A subspace covered by an a... |
lsat0cv 38393 | A subspace is an atom iff ... |
lcvexchlem1 38394 | Lemma for ~ lcvexch . (Co... |
lcvexchlem2 38395 | Lemma for ~ lcvexch . (Co... |
lcvexchlem3 38396 | Lemma for ~ lcvexch . (Co... |
lcvexchlem4 38397 | Lemma for ~ lcvexch . (Co... |
lcvexchlem5 38398 | Lemma for ~ lcvexch . (Co... |
lcvexch 38399 | Subspaces satisfy the exch... |
lcvp 38400 | Covering property of Defin... |
lcv1 38401 | Covering property of a sub... |
lcv2 38402 | Covering property of a sub... |
lsatexch 38403 | The atom exchange property... |
lsatnle 38404 | The meet of a subspace and... |
lsatnem0 38405 | The meet of distinct atoms... |
lsatexch1 38406 | The atom exch1ange propert... |
lsatcv0eq 38407 | If the sum of two atoms co... |
lsatcv1 38408 | Two atoms covering the zer... |
lsatcvatlem 38409 | Lemma for ~ lsatcvat . (C... |
lsatcvat 38410 | A nonzero subspace less th... |
lsatcvat2 38411 | A subspace covered by the ... |
lsatcvat3 38412 | A condition implying that ... |
islshpcv 38413 | Hyperplane properties expr... |
l1cvpat 38414 | A subspace covered by the ... |
l1cvat 38415 | Create an atom under an el... |
lshpat 38416 | Create an atom under a hyp... |
lflset 38419 | The set of linear function... |
islfl 38420 | The predicate "is a linear... |
lfli 38421 | Property of a linear funct... |
islfld 38422 | Properties that determine ... |
lflf 38423 | A linear functional is a f... |
lflcl 38424 | A linear functional value ... |
lfl0 38425 | A linear functional is zer... |
lfladd 38426 | Property of a linear funct... |
lflsub 38427 | Property of a linear funct... |
lflmul 38428 | Property of a linear funct... |
lfl0f 38429 | The zero function is a fun... |
lfl1 38430 | A nonzero functional has a... |
lfladdcl 38431 | Closure of addition of two... |
lfladdcom 38432 | Commutativity of functiona... |
lfladdass 38433 | Associativity of functiona... |
lfladd0l 38434 | Functional addition with t... |
lflnegcl 38435 | Closure of the negative of... |
lflnegl 38436 | A functional plus its nega... |
lflvscl 38437 | Closure of a scalar produc... |
lflvsdi1 38438 | Distributive law for (righ... |
lflvsdi2 38439 | Reverse distributive law f... |
lflvsdi2a 38440 | Reverse distributive law f... |
lflvsass 38441 | Associative law for (right... |
lfl0sc 38442 | The (right vector space) s... |
lflsc0N 38443 | The scalar product with th... |
lfl1sc 38444 | The (right vector space) s... |
lkrfval 38447 | The kernel of a functional... |
lkrval 38448 | Value of the kernel of a f... |
ellkr 38449 | Membership in the kernel o... |
lkrval2 38450 | Value of the kernel of a f... |
ellkr2 38451 | Membership in the kernel o... |
lkrcl 38452 | A member of the kernel of ... |
lkrf0 38453 | The value of a functional ... |
lkr0f 38454 | The kernel of the zero fun... |
lkrlss 38455 | The kernel of a linear fun... |
lkrssv 38456 | The kernel of a linear fun... |
lkrsc 38457 | The kernel of a nonzero sc... |
lkrscss 38458 | The kernel of a scalar pro... |
eqlkr 38459 | Two functionals with the s... |
eqlkr2 38460 | Two functionals with the s... |
eqlkr3 38461 | Two functionals with the s... |
lkrlsp 38462 | The subspace sum of a kern... |
lkrlsp2 38463 | The subspace sum of a kern... |
lkrlsp3 38464 | The subspace sum of a kern... |
lkrshp 38465 | The kernel of a nonzero fu... |
lkrshp3 38466 | The kernels of nonzero fun... |
lkrshpor 38467 | The kernel of a functional... |
lkrshp4 38468 | A kernel is a hyperplane i... |
lshpsmreu 38469 | Lemma for ~ lshpkrex . Sh... |
lshpkrlem1 38470 | Lemma for ~ lshpkrex . Th... |
lshpkrlem2 38471 | Lemma for ~ lshpkrex . Th... |
lshpkrlem3 38472 | Lemma for ~ lshpkrex . De... |
lshpkrlem4 38473 | Lemma for ~ lshpkrex . Pa... |
lshpkrlem5 38474 | Lemma for ~ lshpkrex . Pa... |
lshpkrlem6 38475 | Lemma for ~ lshpkrex . Sh... |
lshpkrcl 38476 | The set ` G ` defined by h... |
lshpkr 38477 | The kernel of functional `... |
lshpkrex 38478 | There exists a functional ... |
lshpset2N 38479 | The set of all hyperplanes... |
islshpkrN 38480 | The predicate "is a hyperp... |
lfl1dim 38481 | Equivalent expressions for... |
lfl1dim2N 38482 | Equivalent expressions for... |
ldualset 38485 | Define the (left) dual of ... |
ldualvbase 38486 | The vectors of a dual spac... |
ldualelvbase 38487 | Utility theorem for conver... |
ldualfvadd 38488 | Vector addition in the dua... |
ldualvadd 38489 | Vector addition in the dua... |
ldualvaddcl 38490 | The value of vector additi... |
ldualvaddval 38491 | The value of the value of ... |
ldualsca 38492 | The ring of scalars of the... |
ldualsbase 38493 | Base set of scalar ring fo... |
ldualsaddN 38494 | Scalar addition for the du... |
ldualsmul 38495 | Scalar multiplication for ... |
ldualfvs 38496 | Scalar product operation f... |
ldualvs 38497 | Scalar product operation v... |
ldualvsval 38498 | Value of scalar product op... |
ldualvscl 38499 | The scalar product operati... |
ldualvaddcom 38500 | Commutative law for vector... |
ldualvsass 38501 | Associative law for scalar... |
ldualvsass2 38502 | Associative law for scalar... |
ldualvsdi1 38503 | Distributive law for scala... |
ldualvsdi2 38504 | Reverse distributive law f... |
ldualgrplem 38505 | Lemma for ~ ldualgrp . (C... |
ldualgrp 38506 | The dual of a vector space... |
ldual0 38507 | The zero scalar of the dua... |
ldual1 38508 | The unit scalar of the dua... |
ldualneg 38509 | The negative of a scalar o... |
ldual0v 38510 | The zero vector of the dua... |
ldual0vcl 38511 | The dual zero vector is a ... |
lduallmodlem 38512 | Lemma for ~ lduallmod . (... |
lduallmod 38513 | The dual of a left module ... |
lduallvec 38514 | The dual of a left vector ... |
ldualvsub 38515 | The value of vector subtra... |
ldualvsubcl 38516 | Closure of vector subtract... |
ldualvsubval 38517 | The value of the value of ... |
ldualssvscl 38518 | Closure of scalar product ... |
ldualssvsubcl 38519 | Closure of vector subtract... |
ldual0vs 38520 | Scalar zero times a functi... |
lkr0f2 38521 | The kernel of the zero fun... |
lduallkr3 38522 | The kernels of nonzero fun... |
lkrpssN 38523 | Proper subset relation bet... |
lkrin 38524 | Intersection of the kernel... |
eqlkr4 38525 | Two functionals with the s... |
ldual1dim 38526 | Equivalent expressions for... |
ldualkrsc 38527 | The kernel of a nonzero sc... |
lkrss 38528 | The kernel of a scalar pro... |
lkrss2N 38529 | Two functionals with kerne... |
lkreqN 38530 | Proportional functionals h... |
lkrlspeqN 38531 | Condition for colinear fun... |
isopos 38540 | The predicate "is an ortho... |
opposet 38541 | Every orthoposet is a pose... |
oposlem 38542 | Lemma for orthoposet prope... |
op01dm 38543 | Conditions necessary for z... |
op0cl 38544 | An orthoposet has a zero e... |
op1cl 38545 | An orthoposet has a unity ... |
op0le 38546 | Orthoposet zero is less th... |
ople0 38547 | An element less than or eq... |
opnlen0 38548 | An element not less than a... |
lub0N 38549 | The least upper bound of t... |
opltn0 38550 | A lattice element greater ... |
ople1 38551 | Any element is less than t... |
op1le 38552 | If the orthoposet unity is... |
glb0N 38553 | The greatest lower bound o... |
opoccl 38554 | Closure of orthocomplement... |
opococ 38555 | Double negative law for or... |
opcon3b 38556 | Contraposition law for ort... |
opcon2b 38557 | Orthocomplement contraposi... |
opcon1b 38558 | Orthocomplement contraposi... |
oplecon3 38559 | Contraposition law for ort... |
oplecon3b 38560 | Contraposition law for ort... |
oplecon1b 38561 | Contraposition law for str... |
opoc1 38562 | Orthocomplement of orthopo... |
opoc0 38563 | Orthocomplement of orthopo... |
opltcon3b 38564 | Contraposition law for str... |
opltcon1b 38565 | Contraposition law for str... |
opltcon2b 38566 | Contraposition law for str... |
opexmid 38567 | Law of excluded middle for... |
opnoncon 38568 | Law of contradiction for o... |
riotaocN 38569 | The orthocomplement of the... |
cmtfvalN 38570 | Value of commutes relation... |
cmtvalN 38571 | Equivalence for commutes r... |
isolat 38572 | The predicate "is an ortho... |
ollat 38573 | An ortholattice is a latti... |
olop 38574 | An ortholattice is an orth... |
olposN 38575 | An ortholattice is a poset... |
isolatiN 38576 | Properties that determine ... |
oldmm1 38577 | De Morgan's law for meet i... |
oldmm2 38578 | De Morgan's law for meet i... |
oldmm3N 38579 | De Morgan's law for meet i... |
oldmm4 38580 | De Morgan's law for meet i... |
oldmj1 38581 | De Morgan's law for join i... |
oldmj2 38582 | De Morgan's law for join i... |
oldmj3 38583 | De Morgan's law for join i... |
oldmj4 38584 | De Morgan's law for join i... |
olj01 38585 | An ortholattice element jo... |
olj02 38586 | An ortholattice element jo... |
olm11 38587 | The meet of an ortholattic... |
olm12 38588 | The meet of an ortholattic... |
latmassOLD 38589 | Ortholattice meet is assoc... |
latm12 38590 | A rearrangement of lattice... |
latm32 38591 | A rearrangement of lattice... |
latmrot 38592 | Rotate lattice meet of 3 c... |
latm4 38593 | Rearrangement of lattice m... |
latmmdiN 38594 | Lattice meet distributes o... |
latmmdir 38595 | Lattice meet distributes o... |
olm01 38596 | Meet with lattice zero is ... |
olm02 38597 | Meet with lattice zero is ... |
isoml 38598 | The predicate "is an ortho... |
isomliN 38599 | Properties that determine ... |
omlol 38600 | An orthomodular lattice is... |
omlop 38601 | An orthomodular lattice is... |
omllat 38602 | An orthomodular lattice is... |
omllaw 38603 | The orthomodular law. (Co... |
omllaw2N 38604 | Variation of orthomodular ... |
omllaw3 38605 | Orthomodular law equivalen... |
omllaw4 38606 | Orthomodular law equivalen... |
omllaw5N 38607 | The orthomodular law. Rem... |
cmtcomlemN 38608 | Lemma for ~ cmtcomN . ( ~... |
cmtcomN 38609 | Commutation is symmetric. ... |
cmt2N 38610 | Commutation with orthocomp... |
cmt3N 38611 | Commutation with orthocomp... |
cmt4N 38612 | Commutation with orthocomp... |
cmtbr2N 38613 | Alternate definition of th... |
cmtbr3N 38614 | Alternate definition for t... |
cmtbr4N 38615 | Alternate definition for t... |
lecmtN 38616 | Ordered elements commute. ... |
cmtidN 38617 | Any element commutes with ... |
omlfh1N 38618 | Foulis-Holland Theorem, pa... |
omlfh3N 38619 | Foulis-Holland Theorem, pa... |
omlmod1i2N 38620 | Analogue of modular law ~ ... |
omlspjN 38621 | Contraction of a Sasaki pr... |
cvrfval 38628 | Value of covers relation "... |
cvrval 38629 | Binary relation expressing... |
cvrlt 38630 | The covers relation implie... |
cvrnbtwn 38631 | There is no element betwee... |
ncvr1 38632 | No element covers the latt... |
cvrletrN 38633 | Property of an element abo... |
cvrval2 38634 | Binary relation expressing... |
cvrnbtwn2 38635 | The covers relation implie... |
cvrnbtwn3 38636 | The covers relation implie... |
cvrcon3b 38637 | Contraposition law for the... |
cvrle 38638 | The covers relation implie... |
cvrnbtwn4 38639 | The covers relation implie... |
cvrnle 38640 | The covers relation implie... |
cvrne 38641 | The covers relation implie... |
cvrnrefN 38642 | The covers relation is not... |
cvrcmp 38643 | If two lattice elements th... |
cvrcmp2 38644 | If two lattice elements co... |
pats 38645 | The set of atoms in a pose... |
isat 38646 | The predicate "is an atom"... |
isat2 38647 | The predicate "is an atom"... |
atcvr0 38648 | An atom covers zero. ( ~ ... |
atbase 38649 | An atom is a member of the... |
atssbase 38650 | The set of atoms is a subs... |
0ltat 38651 | An atom is greater than ze... |
leatb 38652 | A poset element less than ... |
leat 38653 | A poset element less than ... |
leat2 38654 | A nonzero poset element le... |
leat3 38655 | A poset element less than ... |
meetat 38656 | The meet of any element wi... |
meetat2 38657 | The meet of any element wi... |
isatl 38659 | The predicate "is an atomi... |
atllat 38660 | An atomic lattice is a lat... |
atlpos 38661 | An atomic lattice is a pos... |
atl0dm 38662 | Condition necessary for ze... |
atl0cl 38663 | An atomic lattice has a ze... |
atl0le 38664 | Orthoposet zero is less th... |
atlle0 38665 | An element less than or eq... |
atlltn0 38666 | A lattice element greater ... |
isat3 38667 | The predicate "is an atom"... |
atn0 38668 | An atom is not zero. ( ~ ... |
atnle0 38669 | An atom is not less than o... |
atlen0 38670 | A lattice element is nonze... |
atcmp 38671 | If two atoms are comparabl... |
atncmp 38672 | Frequently-used variation ... |
atnlt 38673 | Two atoms cannot satisfy t... |
atcvreq0 38674 | An element covered by an a... |
atncvrN 38675 | Two atoms cannot satisfy t... |
atlex 38676 | Every nonzero element of a... |
atnle 38677 | Two ways of expressing "an... |
atnem0 38678 | The meet of distinct atoms... |
atlatmstc 38679 | An atomic, complete, ortho... |
atlatle 38680 | The ordering of two Hilber... |
atlrelat1 38681 | An atomistic lattice with ... |
iscvlat 38683 | The predicate "is an atomi... |
iscvlat2N 38684 | The predicate "is an atomi... |
cvlatl 38685 | An atomic lattice with the... |
cvllat 38686 | An atomic lattice with the... |
cvlposN 38687 | An atomic lattice with the... |
cvlexch1 38688 | An atomic covering lattice... |
cvlexch2 38689 | An atomic covering lattice... |
cvlexchb1 38690 | An atomic covering lattice... |
cvlexchb2 38691 | An atomic covering lattice... |
cvlexch3 38692 | An atomic covering lattice... |
cvlexch4N 38693 | An atomic covering lattice... |
cvlatexchb1 38694 | A version of ~ cvlexchb1 f... |
cvlatexchb2 38695 | A version of ~ cvlexchb2 f... |
cvlatexch1 38696 | Atom exchange property. (... |
cvlatexch2 38697 | Atom exchange property. (... |
cvlatexch3 38698 | Atom exchange property. (... |
cvlcvr1 38699 | The covering property. Pr... |
cvlcvrp 38700 | A Hilbert lattice satisfie... |
cvlatcvr1 38701 | An atom is covered by its ... |
cvlatcvr2 38702 | An atom is covered by its ... |
cvlsupr2 38703 | Two equivalent ways of exp... |
cvlsupr3 38704 | Two equivalent ways of exp... |
cvlsupr4 38705 | Consequence of superpositi... |
cvlsupr5 38706 | Consequence of superpositi... |
cvlsupr6 38707 | Consequence of superpositi... |
cvlsupr7 38708 | Consequence of superpositi... |
cvlsupr8 38709 | Consequence of superpositi... |
ishlat1 38712 | The predicate "is a Hilber... |
ishlat2 38713 | The predicate "is a Hilber... |
ishlat3N 38714 | The predicate "is a Hilber... |
ishlatiN 38715 | Properties that determine ... |
hlomcmcv 38716 | A Hilbert lattice is ortho... |
hloml 38717 | A Hilbert lattice is ortho... |
hlclat 38718 | A Hilbert lattice is compl... |
hlcvl 38719 | A Hilbert lattice is an at... |
hlatl 38720 | A Hilbert lattice is atomi... |
hlol 38721 | A Hilbert lattice is an or... |
hlop 38722 | A Hilbert lattice is an or... |
hllat 38723 | A Hilbert lattice is a lat... |
hllatd 38724 | Deduction form of ~ hllat ... |
hlomcmat 38725 | A Hilbert lattice is ortho... |
hlpos 38726 | A Hilbert lattice is a pos... |
hlatjcl 38727 | Closure of join operation.... |
hlatjcom 38728 | Commutatitivity of join op... |
hlatjidm 38729 | Idempotence of join operat... |
hlatjass 38730 | Lattice join is associativ... |
hlatj12 38731 | Swap 1st and 2nd members o... |
hlatj32 38732 | Swap 2nd and 3rd members o... |
hlatjrot 38733 | Rotate lattice join of 3 c... |
hlatj4 38734 | Rearrangement of lattice j... |
hlatlej1 38735 | A join's first argument is... |
hlatlej2 38736 | A join's second argument i... |
glbconN 38737 | De Morgan's law for GLB an... |
glbconNOLD 38738 | Obsolete version of ~ glbc... |
glbconxN 38739 | De Morgan's law for GLB an... |
atnlej1 38740 | If an atom is not less tha... |
atnlej2 38741 | If an atom is not less tha... |
hlsuprexch 38742 | A Hilbert lattice has the ... |
hlexch1 38743 | A Hilbert lattice has the ... |
hlexch2 38744 | A Hilbert lattice has the ... |
hlexchb1 38745 | A Hilbert lattice has the ... |
hlexchb2 38746 | A Hilbert lattice has the ... |
hlsupr 38747 | A Hilbert lattice has the ... |
hlsupr2 38748 | A Hilbert lattice has the ... |
hlhgt4 38749 | A Hilbert lattice has a he... |
hlhgt2 38750 | A Hilbert lattice has a he... |
hl0lt1N 38751 | Lattice 0 is less than lat... |
hlexch3 38752 | A Hilbert lattice has the ... |
hlexch4N 38753 | A Hilbert lattice has the ... |
hlatexchb1 38754 | A version of ~ hlexchb1 fo... |
hlatexchb2 38755 | A version of ~ hlexchb2 fo... |
hlatexch1 38756 | Atom exchange property. (... |
hlatexch2 38757 | Atom exchange property. (... |
hlatmstcOLDN 38758 | An atomic, complete, ortho... |
hlatle 38759 | The ordering of two Hilber... |
hlateq 38760 | The equality of two Hilber... |
hlrelat1 38761 | An atomistic lattice with ... |
hlrelat5N 38762 | An atomistic lattice with ... |
hlrelat 38763 | A Hilbert lattice is relat... |
hlrelat2 38764 | A consequence of relative ... |
exatleN 38765 | A condition for an atom to... |
hl2at 38766 | A Hilbert lattice has at l... |
atex 38767 | At least one atom exists. ... |
intnatN 38768 | If the intersection with a... |
2llnne2N 38769 | Condition implying that tw... |
2llnneN 38770 | Condition implying that tw... |
cvr1 38771 | A Hilbert lattice has the ... |
cvr2N 38772 | Less-than and covers equiv... |
hlrelat3 38773 | The Hilbert lattice is rel... |
cvrval3 38774 | Binary relation expressing... |
cvrval4N 38775 | Binary relation expressing... |
cvrval5 38776 | Binary relation expressing... |
cvrp 38777 | A Hilbert lattice satisfie... |
atcvr1 38778 | An atom is covered by its ... |
atcvr2 38779 | An atom is covered by its ... |
cvrexchlem 38780 | Lemma for ~ cvrexch . ( ~... |
cvrexch 38781 | A Hilbert lattice satisfie... |
cvratlem 38782 | Lemma for ~ cvrat . ( ~ a... |
cvrat 38783 | A nonzero Hilbert lattice ... |
ltltncvr 38784 | A chained strong ordering ... |
ltcvrntr 38785 | Non-transitive condition f... |
cvrntr 38786 | The covers relation is not... |
atcvr0eq 38787 | The covers relation is not... |
lnnat 38788 | A line (the join of two di... |
atcvrj0 38789 | Two atoms covering the zer... |
cvrat2 38790 | A Hilbert lattice element ... |
atcvrneN 38791 | Inequality derived from at... |
atcvrj1 38792 | Condition for an atom to b... |
atcvrj2b 38793 | Condition for an atom to b... |
atcvrj2 38794 | Condition for an atom to b... |
atleneN 38795 | Inequality derived from at... |
atltcvr 38796 | An equivalence of less-tha... |
atle 38797 | Any nonzero element has an... |
atlt 38798 | Two atoms are unequal iff ... |
atlelt 38799 | Transfer less-than relatio... |
2atlt 38800 | Given an atom less than an... |
atexchcvrN 38801 | Atom exchange property. V... |
atexchltN 38802 | Atom exchange property. V... |
cvrat3 38803 | A condition implying that ... |
cvrat4 38804 | A condition implying exist... |
cvrat42 38805 | Commuted version of ~ cvra... |
2atjm 38806 | The meet of a line (expres... |
atbtwn 38807 | Property of a 3rd atom ` R... |
atbtwnexOLDN 38808 | There exists a 3rd atom ` ... |
atbtwnex 38809 | Given atoms ` P ` in ` X `... |
3noncolr2 38810 | Two ways to express 3 non-... |
3noncolr1N 38811 | Two ways to express 3 non-... |
hlatcon3 38812 | Atom exchange combined wit... |
hlatcon2 38813 | Atom exchange combined wit... |
4noncolr3 38814 | A way to express 4 non-col... |
4noncolr2 38815 | A way to express 4 non-col... |
4noncolr1 38816 | A way to express 4 non-col... |
athgt 38817 | A Hilbert lattice, whose h... |
3dim0 38818 | There exists a 3-dimension... |
3dimlem1 38819 | Lemma for ~ 3dim1 . (Cont... |
3dimlem2 38820 | Lemma for ~ 3dim1 . (Cont... |
3dimlem3a 38821 | Lemma for ~ 3dim3 . (Cont... |
3dimlem3 38822 | Lemma for ~ 3dim1 . (Cont... |
3dimlem3OLDN 38823 | Lemma for ~ 3dim1 . (Cont... |
3dimlem4a 38824 | Lemma for ~ 3dim3 . (Cont... |
3dimlem4 38825 | Lemma for ~ 3dim1 . (Cont... |
3dimlem4OLDN 38826 | Lemma for ~ 3dim1 . (Cont... |
3dim1lem5 38827 | Lemma for ~ 3dim1 . (Cont... |
3dim1 38828 | Construct a 3-dimensional ... |
3dim2 38829 | Construct 2 new layers on ... |
3dim3 38830 | Construct a new layer on t... |
2dim 38831 | Generate a height-3 elemen... |
1dimN 38832 | An atom is covered by a he... |
1cvrco 38833 | The orthocomplement of an ... |
1cvratex 38834 | There exists an atom less ... |
1cvratlt 38835 | An atom less than or equal... |
1cvrjat 38836 | An element covered by the ... |
1cvrat 38837 | Create an atom under an el... |
ps-1 38838 | The join of two atoms ` R ... |
ps-2 38839 | Lattice analogue for the p... |
2atjlej 38840 | Two atoms are different if... |
hlatexch3N 38841 | Rearrange join of atoms in... |
hlatexch4 38842 | Exchange 2 atoms. (Contri... |
ps-2b 38843 | Variation of projective ge... |
3atlem1 38844 | Lemma for ~ 3at . (Contri... |
3atlem2 38845 | Lemma for ~ 3at . (Contri... |
3atlem3 38846 | Lemma for ~ 3at . (Contri... |
3atlem4 38847 | Lemma for ~ 3at . (Contri... |
3atlem5 38848 | Lemma for ~ 3at . (Contri... |
3atlem6 38849 | Lemma for ~ 3at . (Contri... |
3atlem7 38850 | Lemma for ~ 3at . (Contri... |
3at 38851 | Any three non-colinear ato... |
llnset 38866 | The set of lattice lines i... |
islln 38867 | The predicate "is a lattic... |
islln4 38868 | The predicate "is a lattic... |
llni 38869 | Condition implying a latti... |
llnbase 38870 | A lattice line is a lattic... |
islln3 38871 | The predicate "is a lattic... |
islln2 38872 | The predicate "is a lattic... |
llni2 38873 | The join of two different ... |
llnnleat 38874 | An atom cannot majorize a ... |
llnneat 38875 | A lattice line is not an a... |
2atneat 38876 | The join of two distinct a... |
llnn0 38877 | A lattice line is nonzero.... |
islln2a 38878 | The predicate "is a lattic... |
llnle 38879 | Any element greater than 0... |
atcvrlln2 38880 | An atom under a line is co... |
atcvrlln 38881 | An element covering an ato... |
llnexatN 38882 | Given an atom on a line, t... |
llncmp 38883 | If two lattice lines are c... |
llnnlt 38884 | Two lattice lines cannot s... |
2llnmat 38885 | Two intersecting lines int... |
2at0mat0 38886 | Special case of ~ 2atmat0 ... |
2atmat0 38887 | The meet of two unequal li... |
2atm 38888 | An atom majorized by two d... |
ps-2c 38889 | Variation of projective ge... |
lplnset 38890 | The set of lattice planes ... |
islpln 38891 | The predicate "is a lattic... |
islpln4 38892 | The predicate "is a lattic... |
lplni 38893 | Condition implying a latti... |
islpln3 38894 | The predicate "is a lattic... |
lplnbase 38895 | A lattice plane is a latti... |
islpln5 38896 | The predicate "is a lattic... |
islpln2 38897 | The predicate "is a lattic... |
lplni2 38898 | The join of 3 different at... |
lvolex3N 38899 | There is an atom outside o... |
llnmlplnN 38900 | The intersection of a line... |
lplnle 38901 | Any element greater than 0... |
lplnnle2at 38902 | A lattice line (or atom) c... |
lplnnleat 38903 | A lattice plane cannot maj... |
lplnnlelln 38904 | A lattice plane is not les... |
2atnelpln 38905 | The join of two atoms is n... |
lplnneat 38906 | No lattice plane is an ato... |
lplnnelln 38907 | No lattice plane is a latt... |
lplnn0N 38908 | A lattice plane is nonzero... |
islpln2a 38909 | The predicate "is a lattic... |
islpln2ah 38910 | The predicate "is a lattic... |
lplnriaN 38911 | Property of a lattice plan... |
lplnribN 38912 | Property of a lattice plan... |
lplnric 38913 | Property of a lattice plan... |
lplnri1 38914 | Property of a lattice plan... |
lplnri2N 38915 | Property of a lattice plan... |
lplnri3N 38916 | Property of a lattice plan... |
lplnllnneN 38917 | Two lattice lines defined ... |
llncvrlpln2 38918 | A lattice line under a lat... |
llncvrlpln 38919 | An element covering a latt... |
2lplnmN 38920 | If the join of two lattice... |
2llnmj 38921 | The meet of two lattice li... |
2atmat 38922 | The meet of two intersecti... |
lplncmp 38923 | If two lattice planes are ... |
lplnexatN 38924 | Given a lattice line on a ... |
lplnexllnN 38925 | Given an atom on a lattice... |
lplnnlt 38926 | Two lattice planes cannot ... |
2llnjaN 38927 | The join of two different ... |
2llnjN 38928 | The join of two different ... |
2llnm2N 38929 | The meet of two different ... |
2llnm3N 38930 | Two lattice lines in a lat... |
2llnm4 38931 | Two lattice lines that maj... |
2llnmeqat 38932 | An atom equals the interse... |
lvolset 38933 | The set of 3-dim lattice v... |
islvol 38934 | The predicate "is a 3-dim ... |
islvol4 38935 | The predicate "is a 3-dim ... |
lvoli 38936 | Condition implying a 3-dim... |
islvol3 38937 | The predicate "is a 3-dim ... |
lvoli3 38938 | Condition implying a 3-dim... |
lvolbase 38939 | A 3-dim lattice volume is ... |
islvol5 38940 | The predicate "is a 3-dim ... |
islvol2 38941 | The predicate "is a 3-dim ... |
lvoli2 38942 | The join of 4 different at... |
lvolnle3at 38943 | A lattice plane (or lattic... |
lvolnleat 38944 | An atom cannot majorize a ... |
lvolnlelln 38945 | A lattice line cannot majo... |
lvolnlelpln 38946 | A lattice plane cannot maj... |
3atnelvolN 38947 | The join of 3 atoms is not... |
2atnelvolN 38948 | The join of two atoms is n... |
lvolneatN 38949 | No lattice volume is an at... |
lvolnelln 38950 | No lattice volume is a lat... |
lvolnelpln 38951 | No lattice volume is a lat... |
lvoln0N 38952 | A lattice volume is nonzer... |
islvol2aN 38953 | The predicate "is a lattic... |
4atlem0a 38954 | Lemma for ~ 4at . (Contri... |
4atlem0ae 38955 | Lemma for ~ 4at . (Contri... |
4atlem0be 38956 | Lemma for ~ 4at . (Contri... |
4atlem3 38957 | Lemma for ~ 4at . Break i... |
4atlem3a 38958 | Lemma for ~ 4at . Break i... |
4atlem3b 38959 | Lemma for ~ 4at . Break i... |
4atlem4a 38960 | Lemma for ~ 4at . Frequen... |
4atlem4b 38961 | Lemma for ~ 4at . Frequen... |
4atlem4c 38962 | Lemma for ~ 4at . Frequen... |
4atlem4d 38963 | Lemma for ~ 4at . Frequen... |
4atlem9 38964 | Lemma for ~ 4at . Substit... |
4atlem10a 38965 | Lemma for ~ 4at . Substit... |
4atlem10b 38966 | Lemma for ~ 4at . Substit... |
4atlem10 38967 | Lemma for ~ 4at . Combine... |
4atlem11a 38968 | Lemma for ~ 4at . Substit... |
4atlem11b 38969 | Lemma for ~ 4at . Substit... |
4atlem11 38970 | Lemma for ~ 4at . Combine... |
4atlem12a 38971 | Lemma for ~ 4at . Substit... |
4atlem12b 38972 | Lemma for ~ 4at . Substit... |
4atlem12 38973 | Lemma for ~ 4at . Combine... |
4at 38974 | Four atoms determine a lat... |
4at2 38975 | Four atoms determine a lat... |
lplncvrlvol2 38976 | A lattice line under a lat... |
lplncvrlvol 38977 | An element covering a latt... |
lvolcmp 38978 | If two lattice planes are ... |
lvolnltN 38979 | Two lattice volumes cannot... |
2lplnja 38980 | The join of two different ... |
2lplnj 38981 | The join of two different ... |
2lplnm2N 38982 | The meet of two different ... |
2lplnmj 38983 | The meet of two lattice pl... |
dalemkehl 38984 | Lemma for ~ dath . Freque... |
dalemkelat 38985 | Lemma for ~ dath . Freque... |
dalemkeop 38986 | Lemma for ~ dath . Freque... |
dalempea 38987 | Lemma for ~ dath . Freque... |
dalemqea 38988 | Lemma for ~ dath . Freque... |
dalemrea 38989 | Lemma for ~ dath . Freque... |
dalemsea 38990 | Lemma for ~ dath . Freque... |
dalemtea 38991 | Lemma for ~ dath . Freque... |
dalemuea 38992 | Lemma for ~ dath . Freque... |
dalemyeo 38993 | Lemma for ~ dath . Freque... |
dalemzeo 38994 | Lemma for ~ dath . Freque... |
dalemclpjs 38995 | Lemma for ~ dath . Freque... |
dalemclqjt 38996 | Lemma for ~ dath . Freque... |
dalemclrju 38997 | Lemma for ~ dath . Freque... |
dalem-clpjq 38998 | Lemma for ~ dath . Freque... |
dalemceb 38999 | Lemma for ~ dath . Freque... |
dalempeb 39000 | Lemma for ~ dath . Freque... |
dalemqeb 39001 | Lemma for ~ dath . Freque... |
dalemreb 39002 | Lemma for ~ dath . Freque... |
dalemseb 39003 | Lemma for ~ dath . Freque... |
dalemteb 39004 | Lemma for ~ dath . Freque... |
dalemueb 39005 | Lemma for ~ dath . Freque... |
dalempjqeb 39006 | Lemma for ~ dath . Freque... |
dalemsjteb 39007 | Lemma for ~ dath . Freque... |
dalemtjueb 39008 | Lemma for ~ dath . Freque... |
dalemqrprot 39009 | Lemma for ~ dath . Freque... |
dalemyeb 39010 | Lemma for ~ dath . Freque... |
dalemcnes 39011 | Lemma for ~ dath . Freque... |
dalempnes 39012 | Lemma for ~ dath . Freque... |
dalemqnet 39013 | Lemma for ~ dath . Freque... |
dalempjsen 39014 | Lemma for ~ dath . Freque... |
dalemply 39015 | Lemma for ~ dath . Freque... |
dalemsly 39016 | Lemma for ~ dath . Freque... |
dalemswapyz 39017 | Lemma for ~ dath . Swap t... |
dalemrot 39018 | Lemma for ~ dath . Rotate... |
dalemrotyz 39019 | Lemma for ~ dath . Rotate... |
dalem1 39020 | Lemma for ~ dath . Show t... |
dalemcea 39021 | Lemma for ~ dath . Freque... |
dalem2 39022 | Lemma for ~ dath . Show t... |
dalemdea 39023 | Lemma for ~ dath . Freque... |
dalemeea 39024 | Lemma for ~ dath . Freque... |
dalem3 39025 | Lemma for ~ dalemdnee . (... |
dalem4 39026 | Lemma for ~ dalemdnee . (... |
dalemdnee 39027 | Lemma for ~ dath . Axis o... |
dalem5 39028 | Lemma for ~ dath . Atom `... |
dalem6 39029 | Lemma for ~ dath . Analog... |
dalem7 39030 | Lemma for ~ dath . Analog... |
dalem8 39031 | Lemma for ~ dath . Plane ... |
dalem-cly 39032 | Lemma for ~ dalem9 . Cent... |
dalem9 39033 | Lemma for ~ dath . Since ... |
dalem10 39034 | Lemma for ~ dath . Atom `... |
dalem11 39035 | Lemma for ~ dath . Analog... |
dalem12 39036 | Lemma for ~ dath . Analog... |
dalem13 39037 | Lemma for ~ dalem14 . (Co... |
dalem14 39038 | Lemma for ~ dath . Planes... |
dalem15 39039 | Lemma for ~ dath . The ax... |
dalem16 39040 | Lemma for ~ dath . The at... |
dalem17 39041 | Lemma for ~ dath . When p... |
dalem18 39042 | Lemma for ~ dath . Show t... |
dalem19 39043 | Lemma for ~ dath . Show t... |
dalemccea 39044 | Lemma for ~ dath . Freque... |
dalemddea 39045 | Lemma for ~ dath . Freque... |
dalem-ccly 39046 | Lemma for ~ dath . Freque... |
dalem-ddly 39047 | Lemma for ~ dath . Freque... |
dalemccnedd 39048 | Lemma for ~ dath . Freque... |
dalemclccjdd 39049 | Lemma for ~ dath . Freque... |
dalemcceb 39050 | Lemma for ~ dath . Freque... |
dalemswapyzps 39051 | Lemma for ~ dath . Swap t... |
dalemrotps 39052 | Lemma for ~ dath . Rotate... |
dalemcjden 39053 | Lemma for ~ dath . Show t... |
dalem20 39054 | Lemma for ~ dath . Show t... |
dalem21 39055 | Lemma for ~ dath . Show t... |
dalem22 39056 | Lemma for ~ dath . Show t... |
dalem23 39057 | Lemma for ~ dath . Show t... |
dalem24 39058 | Lemma for ~ dath . Show t... |
dalem25 39059 | Lemma for ~ dath . Show t... |
dalem27 39060 | Lemma for ~ dath . Show t... |
dalem28 39061 | Lemma for ~ dath . Lemma ... |
dalem29 39062 | Lemma for ~ dath . Analog... |
dalem30 39063 | Lemma for ~ dath . Analog... |
dalem31N 39064 | Lemma for ~ dath . Analog... |
dalem32 39065 | Lemma for ~ dath . Analog... |
dalem33 39066 | Lemma for ~ dath . Analog... |
dalem34 39067 | Lemma for ~ dath . Analog... |
dalem35 39068 | Lemma for ~ dath . Analog... |
dalem36 39069 | Lemma for ~ dath . Analog... |
dalem37 39070 | Lemma for ~ dath . Analog... |
dalem38 39071 | Lemma for ~ dath . Plane ... |
dalem39 39072 | Lemma for ~ dath . Auxili... |
dalem40 39073 | Lemma for ~ dath . Analog... |
dalem41 39074 | Lemma for ~ dath . (Contr... |
dalem42 39075 | Lemma for ~ dath . Auxili... |
dalem43 39076 | Lemma for ~ dath . Planes... |
dalem44 39077 | Lemma for ~ dath . Dummy ... |
dalem45 39078 | Lemma for ~ dath . Dummy ... |
dalem46 39079 | Lemma for ~ dath . Analog... |
dalem47 39080 | Lemma for ~ dath . Analog... |
dalem48 39081 | Lemma for ~ dath . Analog... |
dalem49 39082 | Lemma for ~ dath . Analog... |
dalem50 39083 | Lemma for ~ dath . Analog... |
dalem51 39084 | Lemma for ~ dath . Constr... |
dalem52 39085 | Lemma for ~ dath . Lines ... |
dalem53 39086 | Lemma for ~ dath . The au... |
dalem54 39087 | Lemma for ~ dath . Line `... |
dalem55 39088 | Lemma for ~ dath . Lines ... |
dalem56 39089 | Lemma for ~ dath . Analog... |
dalem57 39090 | Lemma for ~ dath . Axis o... |
dalem58 39091 | Lemma for ~ dath . Analog... |
dalem59 39092 | Lemma for ~ dath . Analog... |
dalem60 39093 | Lemma for ~ dath . ` B ` i... |
dalem61 39094 | Lemma for ~ dath . Show t... |
dalem62 39095 | Lemma for ~ dath . Elimin... |
dalem63 39096 | Lemma for ~ dath . Combin... |
dath 39097 | Desargues's theorem of pro... |
dath2 39098 | Version of Desargues's the... |
lineset 39099 | The set of lines in a Hilb... |
isline 39100 | The predicate "is a line".... |
islinei 39101 | Condition implying "is a l... |
pointsetN 39102 | The set of points in a Hil... |
ispointN 39103 | The predicate "is a point"... |
atpointN 39104 | The singleton of an atom i... |
psubspset 39105 | The set of projective subs... |
ispsubsp 39106 | The predicate "is a projec... |
ispsubsp2 39107 | The predicate "is a projec... |
psubspi 39108 | Property of a projective s... |
psubspi2N 39109 | Property of a projective s... |
0psubN 39110 | The empty set is a project... |
snatpsubN 39111 | The singleton of an atom i... |
pointpsubN 39112 | A point (singleton of an a... |
linepsubN 39113 | A line is a projective sub... |
atpsubN 39114 | The set of all atoms is a ... |
psubssat 39115 | A projective subspace cons... |
psubatN 39116 | A member of a projective s... |
pmapfval 39117 | The projective map of a Hi... |
pmapval 39118 | Value of the projective ma... |
elpmap 39119 | Member of a projective map... |
pmapssat 39120 | The projective map of a Hi... |
pmapssbaN 39121 | A weakening of ~ pmapssat ... |
pmaple 39122 | The projective map of a Hi... |
pmap11 39123 | The projective map of a Hi... |
pmapat 39124 | The projective map of an a... |
elpmapat 39125 | Member of the projective m... |
pmap0 39126 | Value of the projective ma... |
pmapeq0 39127 | A projective map value is ... |
pmap1N 39128 | Value of the projective ma... |
pmapsub 39129 | The projective map of a Hi... |
pmapglbx 39130 | The projective map of the ... |
pmapglb 39131 | The projective map of the ... |
pmapglb2N 39132 | The projective map of the ... |
pmapglb2xN 39133 | The projective map of the ... |
pmapmeet 39134 | The projective map of a me... |
isline2 39135 | Definition of line in term... |
linepmap 39136 | A line described with a pr... |
isline3 39137 | Definition of line in term... |
isline4N 39138 | Definition of line in term... |
lneq2at 39139 | A line equals the join of ... |
lnatexN 39140 | There is an atom in a line... |
lnjatN 39141 | Given an atom in a line, t... |
lncvrelatN 39142 | A lattice element covered ... |
lncvrat 39143 | A line covers the atoms it... |
lncmp 39144 | If two lines are comparabl... |
2lnat 39145 | Two intersecting lines int... |
2atm2atN 39146 | Two joins with a common at... |
2llnma1b 39147 | Generalization of ~ 2llnma... |
2llnma1 39148 | Two different intersecting... |
2llnma3r 39149 | Two different intersecting... |
2llnma2 39150 | Two different intersecting... |
2llnma2rN 39151 | Two different intersecting... |
cdlema1N 39152 | A condition for required f... |
cdlema2N 39153 | A condition for required f... |
cdlemblem 39154 | Lemma for ~ cdlemb . (Con... |
cdlemb 39155 | Given two atoms not less t... |
paddfval 39158 | Projective subspace sum op... |
paddval 39159 | Projective subspace sum op... |
elpadd 39160 | Member of a projective sub... |
elpaddn0 39161 | Member of projective subsp... |
paddvaln0N 39162 | Projective subspace sum op... |
elpaddri 39163 | Condition implying members... |
elpaddatriN 39164 | Condition implying members... |
elpaddat 39165 | Membership in a projective... |
elpaddatiN 39166 | Consequence of membership ... |
elpadd2at 39167 | Membership in a projective... |
elpadd2at2 39168 | Membership in a projective... |
paddunssN 39169 | Projective subspace sum in... |
elpadd0 39170 | Member of projective subsp... |
paddval0 39171 | Projective subspace sum wi... |
padd01 39172 | Projective subspace sum wi... |
padd02 39173 | Projective subspace sum wi... |
paddcom 39174 | Projective subspace sum co... |
paddssat 39175 | A projective subspace sum ... |
sspadd1 39176 | A projective subspace sum ... |
sspadd2 39177 | A projective subspace sum ... |
paddss1 39178 | Subset law for projective ... |
paddss2 39179 | Subset law for projective ... |
paddss12 39180 | Subset law for projective ... |
paddasslem1 39181 | Lemma for ~ paddass . (Co... |
paddasslem2 39182 | Lemma for ~ paddass . (Co... |
paddasslem3 39183 | Lemma for ~ paddass . Res... |
paddasslem4 39184 | Lemma for ~ paddass . Com... |
paddasslem5 39185 | Lemma for ~ paddass . Sho... |
paddasslem6 39186 | Lemma for ~ paddass . (Co... |
paddasslem7 39187 | Lemma for ~ paddass . Com... |
paddasslem8 39188 | Lemma for ~ paddass . (Co... |
paddasslem9 39189 | Lemma for ~ paddass . Com... |
paddasslem10 39190 | Lemma for ~ paddass . Use... |
paddasslem11 39191 | Lemma for ~ paddass . The... |
paddasslem12 39192 | Lemma for ~ paddass . The... |
paddasslem13 39193 | Lemma for ~ paddass . The... |
paddasslem14 39194 | Lemma for ~ paddass . Rem... |
paddasslem15 39195 | Lemma for ~ paddass . Use... |
paddasslem16 39196 | Lemma for ~ paddass . Use... |
paddasslem17 39197 | Lemma for ~ paddass . The... |
paddasslem18 39198 | Lemma for ~ paddass . Com... |
paddass 39199 | Projective subspace sum is... |
padd12N 39200 | Commutative/associative la... |
padd4N 39201 | Rearrangement of 4 terms i... |
paddidm 39202 | Projective subspace sum is... |
paddclN 39203 | The projective sum of two ... |
paddssw1 39204 | Subset law for projective ... |
paddssw2 39205 | Subset law for projective ... |
paddss 39206 | Subset law for projective ... |
pmodlem1 39207 | Lemma for ~ pmod1i . (Con... |
pmodlem2 39208 | Lemma for ~ pmod1i . (Con... |
pmod1i 39209 | The modular law holds in a... |
pmod2iN 39210 | Dual of the modular law. ... |
pmodN 39211 | The modular law for projec... |
pmodl42N 39212 | Lemma derived from modular... |
pmapjoin 39213 | The projective map of the ... |
pmapjat1 39214 | The projective map of the ... |
pmapjat2 39215 | The projective map of the ... |
pmapjlln1 39216 | The projective map of the ... |
hlmod1i 39217 | A version of the modular l... |
atmod1i1 39218 | Version of modular law ~ p... |
atmod1i1m 39219 | Version of modular law ~ p... |
atmod1i2 39220 | Version of modular law ~ p... |
llnmod1i2 39221 | Version of modular law ~ p... |
atmod2i1 39222 | Version of modular law ~ p... |
atmod2i2 39223 | Version of modular law ~ p... |
llnmod2i2 39224 | Version of modular law ~ p... |
atmod3i1 39225 | Version of modular law tha... |
atmod3i2 39226 | Version of modular law tha... |
atmod4i1 39227 | Version of modular law tha... |
atmod4i2 39228 | Version of modular law tha... |
llnexchb2lem 39229 | Lemma for ~ llnexchb2 . (... |
llnexchb2 39230 | Line exchange property (co... |
llnexch2N 39231 | Line exchange property (co... |
dalawlem1 39232 | Lemma for ~ dalaw . Speci... |
dalawlem2 39233 | Lemma for ~ dalaw . Utili... |
dalawlem3 39234 | Lemma for ~ dalaw . First... |
dalawlem4 39235 | Lemma for ~ dalaw . Secon... |
dalawlem5 39236 | Lemma for ~ dalaw . Speci... |
dalawlem6 39237 | Lemma for ~ dalaw . First... |
dalawlem7 39238 | Lemma for ~ dalaw . Secon... |
dalawlem8 39239 | Lemma for ~ dalaw . Speci... |
dalawlem9 39240 | Lemma for ~ dalaw . Speci... |
dalawlem10 39241 | Lemma for ~ dalaw . Combi... |
dalawlem11 39242 | Lemma for ~ dalaw . First... |
dalawlem12 39243 | Lemma for ~ dalaw . Secon... |
dalawlem13 39244 | Lemma for ~ dalaw . Speci... |
dalawlem14 39245 | Lemma for ~ dalaw . Combi... |
dalawlem15 39246 | Lemma for ~ dalaw . Swap ... |
dalaw 39247 | Desargues's law, derived f... |
pclfvalN 39250 | The projective subspace cl... |
pclvalN 39251 | Value of the projective su... |
pclclN 39252 | Closure of the projective ... |
elpclN 39253 | Membership in the projecti... |
elpcliN 39254 | Implication of membership ... |
pclssN 39255 | Ordering is preserved by s... |
pclssidN 39256 | A set of atoms is included... |
pclidN 39257 | The projective subspace cl... |
pclbtwnN 39258 | A projective subspace sand... |
pclunN 39259 | The projective subspace cl... |
pclun2N 39260 | The projective subspace cl... |
pclfinN 39261 | The projective subspace cl... |
pclcmpatN 39262 | The set of projective subs... |
polfvalN 39265 | The projective subspace po... |
polvalN 39266 | Value of the projective su... |
polval2N 39267 | Alternate expression for v... |
polsubN 39268 | The polarity of a set of a... |
polssatN 39269 | The polarity of a set of a... |
pol0N 39270 | The polarity of the empty ... |
pol1N 39271 | The polarity of the whole ... |
2pol0N 39272 | The closed subspace closur... |
polpmapN 39273 | The polarity of a projecti... |
2polpmapN 39274 | Double polarity of a proje... |
2polvalN 39275 | Value of double polarity. ... |
2polssN 39276 | A set of atoms is a subset... |
3polN 39277 | Triple polarity cancels to... |
polcon3N 39278 | Contraposition law for pol... |
2polcon4bN 39279 | Contraposition law for pol... |
polcon2N 39280 | Contraposition law for pol... |
polcon2bN 39281 | Contraposition law for pol... |
pclss2polN 39282 | The projective subspace cl... |
pcl0N 39283 | The projective subspace cl... |
pcl0bN 39284 | The projective subspace cl... |
pmaplubN 39285 | The LUB of a projective ma... |
sspmaplubN 39286 | A set of atoms is a subset... |
2pmaplubN 39287 | Double projective map of a... |
paddunN 39288 | The closure of the project... |
poldmj1N 39289 | De Morgan's law for polari... |
pmapj2N 39290 | The projective map of the ... |
pmapocjN 39291 | The projective map of the ... |
polatN 39292 | The polarity of the single... |
2polatN 39293 | Double polarity of the sin... |
pnonsingN 39294 | The intersection of a set ... |
psubclsetN 39297 | The set of closed projecti... |
ispsubclN 39298 | The predicate "is a closed... |
psubcliN 39299 | Property of a closed proje... |
psubcli2N 39300 | Property of a closed proje... |
psubclsubN 39301 | A closed projective subspa... |
psubclssatN 39302 | A closed projective subspa... |
pmapidclN 39303 | Projective map of the LUB ... |
0psubclN 39304 | The empty set is a closed ... |
1psubclN 39305 | The set of all atoms is a ... |
atpsubclN 39306 | A point (singleton of an a... |
pmapsubclN 39307 | A projective map value is ... |
ispsubcl2N 39308 | Alternate predicate for "i... |
psubclinN 39309 | The intersection of two cl... |
paddatclN 39310 | The projective sum of a cl... |
pclfinclN 39311 | The projective subspace cl... |
linepsubclN 39312 | A line is a closed project... |
polsubclN 39313 | A polarity is a closed pro... |
poml4N 39314 | Orthomodular law for proje... |
poml5N 39315 | Orthomodular law for proje... |
poml6N 39316 | Orthomodular law for proje... |
osumcllem1N 39317 | Lemma for ~ osumclN . (Co... |
osumcllem2N 39318 | Lemma for ~ osumclN . (Co... |
osumcllem3N 39319 | Lemma for ~ osumclN . (Co... |
osumcllem4N 39320 | Lemma for ~ osumclN . (Co... |
osumcllem5N 39321 | Lemma for ~ osumclN . (Co... |
osumcllem6N 39322 | Lemma for ~ osumclN . Use... |
osumcllem7N 39323 | Lemma for ~ osumclN . (Co... |
osumcllem8N 39324 | Lemma for ~ osumclN . (Co... |
osumcllem9N 39325 | Lemma for ~ osumclN . (Co... |
osumcllem10N 39326 | Lemma for ~ osumclN . Con... |
osumcllem11N 39327 | Lemma for ~ osumclN . (Co... |
osumclN 39328 | Closure of orthogonal sum.... |
pmapojoinN 39329 | For orthogonal elements, p... |
pexmidN 39330 | Excluded middle law for cl... |
pexmidlem1N 39331 | Lemma for ~ pexmidN . Hol... |
pexmidlem2N 39332 | Lemma for ~ pexmidN . (Co... |
pexmidlem3N 39333 | Lemma for ~ pexmidN . Use... |
pexmidlem4N 39334 | Lemma for ~ pexmidN . (Co... |
pexmidlem5N 39335 | Lemma for ~ pexmidN . (Co... |
pexmidlem6N 39336 | Lemma for ~ pexmidN . (Co... |
pexmidlem7N 39337 | Lemma for ~ pexmidN . Con... |
pexmidlem8N 39338 | Lemma for ~ pexmidN . The... |
pexmidALTN 39339 | Excluded middle law for cl... |
pl42lem1N 39340 | Lemma for ~ pl42N . (Cont... |
pl42lem2N 39341 | Lemma for ~ pl42N . (Cont... |
pl42lem3N 39342 | Lemma for ~ pl42N . (Cont... |
pl42lem4N 39343 | Lemma for ~ pl42N . (Cont... |
pl42N 39344 | Law holding in a Hilbert l... |
watfvalN 39353 | The W atoms function. (Co... |
watvalN 39354 | Value of the W atoms funct... |
iswatN 39355 | The predicate "is a W atom... |
lhpset 39356 | The set of co-atoms (latti... |
islhp 39357 | The predicate "is a co-ato... |
islhp2 39358 | The predicate "is a co-ato... |
lhpbase 39359 | A co-atom is a member of t... |
lhp1cvr 39360 | The lattice unity covers a... |
lhplt 39361 | An atom under a co-atom is... |
lhp2lt 39362 | The join of two atoms unde... |
lhpexlt 39363 | There exists an atom less ... |
lhp0lt 39364 | A co-atom is greater than ... |
lhpn0 39365 | A co-atom is nonzero. TOD... |
lhpexle 39366 | There exists an atom under... |
lhpexnle 39367 | There exists an atom not u... |
lhpexle1lem 39368 | Lemma for ~ lhpexle1 and o... |
lhpexle1 39369 | There exists an atom under... |
lhpexle2lem 39370 | Lemma for ~ lhpexle2 . (C... |
lhpexle2 39371 | There exists atom under a ... |
lhpexle3lem 39372 | There exists atom under a ... |
lhpexle3 39373 | There exists atom under a ... |
lhpex2leN 39374 | There exist at least two d... |
lhpoc 39375 | The orthocomplement of a c... |
lhpoc2N 39376 | The orthocomplement of an ... |
lhpocnle 39377 | The orthocomplement of a c... |
lhpocat 39378 | The orthocomplement of a c... |
lhpocnel 39379 | The orthocomplement of a c... |
lhpocnel2 39380 | The orthocomplement of a c... |
lhpjat1 39381 | The join of a co-atom (hyp... |
lhpjat2 39382 | The join of a co-atom (hyp... |
lhpj1 39383 | The join of a co-atom (hyp... |
lhpmcvr 39384 | The meet of a lattice hype... |
lhpmcvr2 39385 | Alternate way to express t... |
lhpmcvr3 39386 | Specialization of ~ lhpmcv... |
lhpmcvr4N 39387 | Specialization of ~ lhpmcv... |
lhpmcvr5N 39388 | Specialization of ~ lhpmcv... |
lhpmcvr6N 39389 | Specialization of ~ lhpmcv... |
lhpm0atN 39390 | If the meet of a lattice h... |
lhpmat 39391 | An element covered by the ... |
lhpmatb 39392 | An element covered by the ... |
lhp2at0 39393 | Join and meet with differe... |
lhp2atnle 39394 | Inequality for 2 different... |
lhp2atne 39395 | Inequality for joins with ... |
lhp2at0nle 39396 | Inequality for 2 different... |
lhp2at0ne 39397 | Inequality for joins with ... |
lhpelim 39398 | Eliminate an atom not unde... |
lhpmod2i2 39399 | Modular law for hyperplane... |
lhpmod6i1 39400 | Modular law for hyperplane... |
lhprelat3N 39401 | The Hilbert lattice is rel... |
cdlemb2 39402 | Given two atoms not under ... |
lhple 39403 | Property of a lattice elem... |
lhpat 39404 | Create an atom under a co-... |
lhpat4N 39405 | Property of an atom under ... |
lhpat2 39406 | Create an atom under a co-... |
lhpat3 39407 | There is only one atom und... |
4atexlemk 39408 | Lemma for ~ 4atexlem7 . (... |
4atexlemw 39409 | Lemma for ~ 4atexlem7 . (... |
4atexlempw 39410 | Lemma for ~ 4atexlem7 . (... |
4atexlemp 39411 | Lemma for ~ 4atexlem7 . (... |
4atexlemq 39412 | Lemma for ~ 4atexlem7 . (... |
4atexlems 39413 | Lemma for ~ 4atexlem7 . (... |
4atexlemt 39414 | Lemma for ~ 4atexlem7 . (... |
4atexlemutvt 39415 | Lemma for ~ 4atexlem7 . (... |
4atexlempnq 39416 | Lemma for ~ 4atexlem7 . (... |
4atexlemnslpq 39417 | Lemma for ~ 4atexlem7 . (... |
4atexlemkl 39418 | Lemma for ~ 4atexlem7 . (... |
4atexlemkc 39419 | Lemma for ~ 4atexlem7 . (... |
4atexlemwb 39420 | Lemma for ~ 4atexlem7 . (... |
4atexlempsb 39421 | Lemma for ~ 4atexlem7 . (... |
4atexlemqtb 39422 | Lemma for ~ 4atexlem7 . (... |
4atexlempns 39423 | Lemma for ~ 4atexlem7 . (... |
4atexlemswapqr 39424 | Lemma for ~ 4atexlem7 . S... |
4atexlemu 39425 | Lemma for ~ 4atexlem7 . (... |
4atexlemv 39426 | Lemma for ~ 4atexlem7 . (... |
4atexlemunv 39427 | Lemma for ~ 4atexlem7 . (... |
4atexlemtlw 39428 | Lemma for ~ 4atexlem7 . (... |
4atexlemntlpq 39429 | Lemma for ~ 4atexlem7 . (... |
4atexlemc 39430 | Lemma for ~ 4atexlem7 . (... |
4atexlemnclw 39431 | Lemma for ~ 4atexlem7 . (... |
4atexlemex2 39432 | Lemma for ~ 4atexlem7 . S... |
4atexlemcnd 39433 | Lemma for ~ 4atexlem7 . (... |
4atexlemex4 39434 | Lemma for ~ 4atexlem7 . S... |
4atexlemex6 39435 | Lemma for ~ 4atexlem7 . (... |
4atexlem7 39436 | Whenever there are at leas... |
4atex 39437 | Whenever there are at leas... |
4atex2 39438 | More general version of ~ ... |
4atex2-0aOLDN 39439 | Same as ~ 4atex2 except th... |
4atex2-0bOLDN 39440 | Same as ~ 4atex2 except th... |
4atex2-0cOLDN 39441 | Same as ~ 4atex2 except th... |
4atex3 39442 | More general version of ~ ... |
lautset 39443 | The set of lattice automor... |
islaut 39444 | The predicate "is a lattic... |
lautle 39445 | Less-than or equal propert... |
laut1o 39446 | A lattice automorphism is ... |
laut11 39447 | One-to-one property of a l... |
lautcl 39448 | A lattice automorphism val... |
lautcnvclN 39449 | Reverse closure of a latti... |
lautcnvle 39450 | Less-than or equal propert... |
lautcnv 39451 | The converse of a lattice ... |
lautlt 39452 | Less-than property of a la... |
lautcvr 39453 | Covering property of a lat... |
lautj 39454 | Meet property of a lattice... |
lautm 39455 | Meet property of a lattice... |
lauteq 39456 | A lattice automorphism arg... |
idlaut 39457 | The identity function is a... |
lautco 39458 | The composition of two lat... |
pautsetN 39459 | The set of projective auto... |
ispautN 39460 | The predicate "is a projec... |
ldilfset 39469 | The mapping from fiducial ... |
ldilset 39470 | The set of lattice dilatio... |
isldil 39471 | The predicate "is a lattic... |
ldillaut 39472 | A lattice dilation is an a... |
ldil1o 39473 | A lattice dilation is a on... |
ldilval 39474 | Value of a lattice dilatio... |
idldil 39475 | The identity function is a... |
ldilcnv 39476 | The converse of a lattice ... |
ldilco 39477 | The composition of two lat... |
ltrnfset 39478 | The set of all lattice tra... |
ltrnset 39479 | The set of lattice transla... |
isltrn 39480 | The predicate "is a lattic... |
isltrn2N 39481 | The predicate "is a lattic... |
ltrnu 39482 | Uniqueness property of a l... |
ltrnldil 39483 | A lattice translation is a... |
ltrnlaut 39484 | A lattice translation is a... |
ltrn1o 39485 | A lattice translation is a... |
ltrncl 39486 | Closure of a lattice trans... |
ltrn11 39487 | One-to-one property of a l... |
ltrncnvnid 39488 | If a translation is differ... |
ltrncoidN 39489 | Two translations are equal... |
ltrnle 39490 | Less-than or equal propert... |
ltrncnvleN 39491 | Less-than or equal propert... |
ltrnm 39492 | Lattice translation of a m... |
ltrnj 39493 | Lattice translation of a m... |
ltrncvr 39494 | Covering property of a lat... |
ltrnval1 39495 | Value of a lattice transla... |
ltrnid 39496 | A lattice translation is t... |
ltrnnid 39497 | If a lattice translation i... |
ltrnatb 39498 | The lattice translation of... |
ltrncnvatb 39499 | The converse of the lattic... |
ltrnel 39500 | The lattice translation of... |
ltrnat 39501 | The lattice translation of... |
ltrncnvat 39502 | The converse of the lattic... |
ltrncnvel 39503 | The converse of the lattic... |
ltrncoelN 39504 | Composition of lattice tra... |
ltrncoat 39505 | Composition of lattice tra... |
ltrncoval 39506 | Two ways to express value ... |
ltrncnv 39507 | The converse of a lattice ... |
ltrn11at 39508 | Frequently used one-to-one... |
ltrneq2 39509 | The equality of two transl... |
ltrneq 39510 | The equality of two transl... |
idltrn 39511 | The identity function is a... |
ltrnmw 39512 | Property of lattice transl... |
dilfsetN 39513 | The mapping from fiducial ... |
dilsetN 39514 | The set of dilations for a... |
isdilN 39515 | The predicate "is a dilati... |
trnfsetN 39516 | The mapping from fiducial ... |
trnsetN 39517 | The set of translations fo... |
istrnN 39518 | The predicate "is a transl... |
trlfset 39521 | The set of all traces of l... |
trlset 39522 | The set of traces of latti... |
trlval 39523 | The value of the trace of ... |
trlval2 39524 | The value of the trace of ... |
trlcl 39525 | Closure of the trace of a ... |
trlcnv 39526 | The trace of the converse ... |
trljat1 39527 | The value of a translation... |
trljat2 39528 | The value of a translation... |
trljat3 39529 | The value of a translation... |
trlat 39530 | If an atom differs from it... |
trl0 39531 | If an atom not under the f... |
trlator0 39532 | The trace of a lattice tra... |
trlatn0 39533 | The trace of a lattice tra... |
trlnidat 39534 | The trace of a lattice tra... |
ltrnnidn 39535 | If a lattice translation i... |
ltrnideq 39536 | Property of the identity l... |
trlid0 39537 | The trace of the identity ... |
trlnidatb 39538 | A lattice translation is n... |
trlid0b 39539 | A lattice translation is t... |
trlnid 39540 | Different translations wit... |
ltrn2ateq 39541 | Property of the equality o... |
ltrnateq 39542 | If any atom (under ` W ` )... |
ltrnatneq 39543 | If any atom (under ` W ` )... |
ltrnatlw 39544 | If the value of an atom eq... |
trlle 39545 | The trace of a lattice tra... |
trlne 39546 | The trace of a lattice tra... |
trlnle 39547 | The atom not under the fid... |
trlval3 39548 | The value of the trace of ... |
trlval4 39549 | The value of the trace of ... |
trlval5 39550 | The value of the trace of ... |
arglem1N 39551 | Lemma for Desargues's law.... |
cdlemc1 39552 | Part of proof of Lemma C i... |
cdlemc2 39553 | Part of proof of Lemma C i... |
cdlemc3 39554 | Part of proof of Lemma C i... |
cdlemc4 39555 | Part of proof of Lemma C i... |
cdlemc5 39556 | Lemma for ~ cdlemc . (Con... |
cdlemc6 39557 | Lemma for ~ cdlemc . (Con... |
cdlemc 39558 | Lemma C in [Crawley] p. 11... |
cdlemd1 39559 | Part of proof of Lemma D i... |
cdlemd2 39560 | Part of proof of Lemma D i... |
cdlemd3 39561 | Part of proof of Lemma D i... |
cdlemd4 39562 | Part of proof of Lemma D i... |
cdlemd5 39563 | Part of proof of Lemma D i... |
cdlemd6 39564 | Part of proof of Lemma D i... |
cdlemd7 39565 | Part of proof of Lemma D i... |
cdlemd8 39566 | Part of proof of Lemma D i... |
cdlemd9 39567 | Part of proof of Lemma D i... |
cdlemd 39568 | If two translations agree ... |
ltrneq3 39569 | Two translations agree at ... |
cdleme00a 39570 | Part of proof of Lemma E i... |
cdleme0aa 39571 | Part of proof of Lemma E i... |
cdleme0a 39572 | Part of proof of Lemma E i... |
cdleme0b 39573 | Part of proof of Lemma E i... |
cdleme0c 39574 | Part of proof of Lemma E i... |
cdleme0cp 39575 | Part of proof of Lemma E i... |
cdleme0cq 39576 | Part of proof of Lemma E i... |
cdleme0dN 39577 | Part of proof of Lemma E i... |
cdleme0e 39578 | Part of proof of Lemma E i... |
cdleme0fN 39579 | Part of proof of Lemma E i... |
cdleme0gN 39580 | Part of proof of Lemma E i... |
cdlemeulpq 39581 | Part of proof of Lemma E i... |
cdleme01N 39582 | Part of proof of Lemma E i... |
cdleme02N 39583 | Part of proof of Lemma E i... |
cdleme0ex1N 39584 | Part of proof of Lemma E i... |
cdleme0ex2N 39585 | Part of proof of Lemma E i... |
cdleme0moN 39586 | Part of proof of Lemma E i... |
cdleme1b 39587 | Part of proof of Lemma E i... |
cdleme1 39588 | Part of proof of Lemma E i... |
cdleme2 39589 | Part of proof of Lemma E i... |
cdleme3b 39590 | Part of proof of Lemma E i... |
cdleme3c 39591 | Part of proof of Lemma E i... |
cdleme3d 39592 | Part of proof of Lemma E i... |
cdleme3e 39593 | Part of proof of Lemma E i... |
cdleme3fN 39594 | Part of proof of Lemma E i... |
cdleme3g 39595 | Part of proof of Lemma E i... |
cdleme3h 39596 | Part of proof of Lemma E i... |
cdleme3fa 39597 | Part of proof of Lemma E i... |
cdleme3 39598 | Part of proof of Lemma E i... |
cdleme4 39599 | Part of proof of Lemma E i... |
cdleme4a 39600 | Part of proof of Lemma E i... |
cdleme5 39601 | Part of proof of Lemma E i... |
cdleme6 39602 | Part of proof of Lemma E i... |
cdleme7aa 39603 | Part of proof of Lemma E i... |
cdleme7a 39604 | Part of proof of Lemma E i... |
cdleme7b 39605 | Part of proof of Lemma E i... |
cdleme7c 39606 | Part of proof of Lemma E i... |
cdleme7d 39607 | Part of proof of Lemma E i... |
cdleme7e 39608 | Part of proof of Lemma E i... |
cdleme7ga 39609 | Part of proof of Lemma E i... |
cdleme7 39610 | Part of proof of Lemma E i... |
cdleme8 39611 | Part of proof of Lemma E i... |
cdleme9a 39612 | Part of proof of Lemma E i... |
cdleme9b 39613 | Utility lemma for Lemma E ... |
cdleme9 39614 | Part of proof of Lemma E i... |
cdleme10 39615 | Part of proof of Lemma E i... |
cdleme8tN 39616 | Part of proof of Lemma E i... |
cdleme9taN 39617 | Part of proof of Lemma E i... |
cdleme9tN 39618 | Part of proof of Lemma E i... |
cdleme10tN 39619 | Part of proof of Lemma E i... |
cdleme16aN 39620 | Part of proof of Lemma E i... |
cdleme11a 39621 | Part of proof of Lemma E i... |
cdleme11c 39622 | Part of proof of Lemma E i... |
cdleme11dN 39623 | Part of proof of Lemma E i... |
cdleme11e 39624 | Part of proof of Lemma E i... |
cdleme11fN 39625 | Part of proof of Lemma E i... |
cdleme11g 39626 | Part of proof of Lemma E i... |
cdleme11h 39627 | Part of proof of Lemma E i... |
cdleme11j 39628 | Part of proof of Lemma E i... |
cdleme11k 39629 | Part of proof of Lemma E i... |
cdleme11l 39630 | Part of proof of Lemma E i... |
cdleme11 39631 | Part of proof of Lemma E i... |
cdleme12 39632 | Part of proof of Lemma E i... |
cdleme13 39633 | Part of proof of Lemma E i... |
cdleme14 39634 | Part of proof of Lemma E i... |
cdleme15a 39635 | Part of proof of Lemma E i... |
cdleme15b 39636 | Part of proof of Lemma E i... |
cdleme15c 39637 | Part of proof of Lemma E i... |
cdleme15d 39638 | Part of proof of Lemma E i... |
cdleme15 39639 | Part of proof of Lemma E i... |
cdleme16b 39640 | Part of proof of Lemma E i... |
cdleme16c 39641 | Part of proof of Lemma E i... |
cdleme16d 39642 | Part of proof of Lemma E i... |
cdleme16e 39643 | Part of proof of Lemma E i... |
cdleme16f 39644 | Part of proof of Lemma E i... |
cdleme16g 39645 | Part of proof of Lemma E i... |
cdleme16 39646 | Part of proof of Lemma E i... |
cdleme17a 39647 | Part of proof of Lemma E i... |
cdleme17b 39648 | Lemma leading to ~ cdleme1... |
cdleme17c 39649 | Part of proof of Lemma E i... |
cdleme17d1 39650 | Part of proof of Lemma E i... |
cdleme0nex 39651 | Part of proof of Lemma E i... |
cdleme18a 39652 | Part of proof of Lemma E i... |
cdleme18b 39653 | Part of proof of Lemma E i... |
cdleme18c 39654 | Part of proof of Lemma E i... |
cdleme22gb 39655 | Utility lemma for Lemma E ... |
cdleme18d 39656 | Part of proof of Lemma E i... |
cdlemesner 39657 | Part of proof of Lemma E i... |
cdlemedb 39658 | Part of proof of Lemma E i... |
cdlemeda 39659 | Part of proof of Lemma E i... |
cdlemednpq 39660 | Part of proof of Lemma E i... |
cdlemednuN 39661 | Part of proof of Lemma E i... |
cdleme20zN 39662 | Part of proof of Lemma E i... |
cdleme20y 39663 | Part of proof of Lemma E i... |
cdleme19a 39664 | Part of proof of Lemma E i... |
cdleme19b 39665 | Part of proof of Lemma E i... |
cdleme19c 39666 | Part of proof of Lemma E i... |
cdleme19d 39667 | Part of proof of Lemma E i... |
cdleme19e 39668 | Part of proof of Lemma E i... |
cdleme19f 39669 | Part of proof of Lemma E i... |
cdleme20aN 39670 | Part of proof of Lemma E i... |
cdleme20bN 39671 | Part of proof of Lemma E i... |
cdleme20c 39672 | Part of proof of Lemma E i... |
cdleme20d 39673 | Part of proof of Lemma E i... |
cdleme20e 39674 | Part of proof of Lemma E i... |
cdleme20f 39675 | Part of proof of Lemma E i... |
cdleme20g 39676 | Part of proof of Lemma E i... |
cdleme20h 39677 | Part of proof of Lemma E i... |
cdleme20i 39678 | Part of proof of Lemma E i... |
cdleme20j 39679 | Part of proof of Lemma E i... |
cdleme20k 39680 | Part of proof of Lemma E i... |
cdleme20l1 39681 | Part of proof of Lemma E i... |
cdleme20l2 39682 | Part of proof of Lemma E i... |
cdleme20l 39683 | Part of proof of Lemma E i... |
cdleme20m 39684 | Part of proof of Lemma E i... |
cdleme20 39685 | Combine ~ cdleme19f and ~ ... |
cdleme21a 39686 | Part of proof of Lemma E i... |
cdleme21b 39687 | Part of proof of Lemma E i... |
cdleme21c 39688 | Part of proof of Lemma E i... |
cdleme21at 39689 | Part of proof of Lemma E i... |
cdleme21ct 39690 | Part of proof of Lemma E i... |
cdleme21d 39691 | Part of proof of Lemma E i... |
cdleme21e 39692 | Part of proof of Lemma E i... |
cdleme21f 39693 | Part of proof of Lemma E i... |
cdleme21g 39694 | Part of proof of Lemma E i... |
cdleme21h 39695 | Part of proof of Lemma E i... |
cdleme21i 39696 | Part of proof of Lemma E i... |
cdleme21j 39697 | Combine ~ cdleme20 and ~ c... |
cdleme21 39698 | Part of proof of Lemma E i... |
cdleme21k 39699 | Eliminate ` S =/= T ` cond... |
cdleme22aa 39700 | Part of proof of Lemma E i... |
cdleme22a 39701 | Part of proof of Lemma E i... |
cdleme22b 39702 | Part of proof of Lemma E i... |
cdleme22cN 39703 | Part of proof of Lemma E i... |
cdleme22d 39704 | Part of proof of Lemma E i... |
cdleme22e 39705 | Part of proof of Lemma E i... |
cdleme22eALTN 39706 | Part of proof of Lemma E i... |
cdleme22f 39707 | Part of proof of Lemma E i... |
cdleme22f2 39708 | Part of proof of Lemma E i... |
cdleme22g 39709 | Part of proof of Lemma E i... |
cdleme23a 39710 | Part of proof of Lemma E i... |
cdleme23b 39711 | Part of proof of Lemma E i... |
cdleme23c 39712 | Part of proof of Lemma E i... |
cdleme24 39713 | Quantified version of ~ cd... |
cdleme25a 39714 | Lemma for ~ cdleme25b . (... |
cdleme25b 39715 | Transform ~ cdleme24 . TO... |
cdleme25c 39716 | Transform ~ cdleme25b . (... |
cdleme25dN 39717 | Transform ~ cdleme25c . (... |
cdleme25cl 39718 | Show closure of the unique... |
cdleme25cv 39719 | Change bound variables in ... |
cdleme26e 39720 | Part of proof of Lemma E i... |
cdleme26ee 39721 | Part of proof of Lemma E i... |
cdleme26eALTN 39722 | Part of proof of Lemma E i... |
cdleme26fALTN 39723 | Part of proof of Lemma E i... |
cdleme26f 39724 | Part of proof of Lemma E i... |
cdleme26f2ALTN 39725 | Part of proof of Lemma E i... |
cdleme26f2 39726 | Part of proof of Lemma E i... |
cdleme27cl 39727 | Part of proof of Lemma E i... |
cdleme27a 39728 | Part of proof of Lemma E i... |
cdleme27b 39729 | Lemma for ~ cdleme27N . (... |
cdleme27N 39730 | Part of proof of Lemma E i... |
cdleme28a 39731 | Lemma for ~ cdleme25b . T... |
cdleme28b 39732 | Lemma for ~ cdleme25b . T... |
cdleme28c 39733 | Part of proof of Lemma E i... |
cdleme28 39734 | Quantified version of ~ cd... |
cdleme29ex 39735 | Lemma for ~ cdleme29b . (... |
cdleme29b 39736 | Transform ~ cdleme28 . (C... |
cdleme29c 39737 | Transform ~ cdleme28b . (... |
cdleme29cl 39738 | Show closure of the unique... |
cdleme30a 39739 | Part of proof of Lemma E i... |
cdleme31so 39740 | Part of proof of Lemma E i... |
cdleme31sn 39741 | Part of proof of Lemma E i... |
cdleme31sn1 39742 | Part of proof of Lemma E i... |
cdleme31se 39743 | Part of proof of Lemma D i... |
cdleme31se2 39744 | Part of proof of Lemma D i... |
cdleme31sc 39745 | Part of proof of Lemma E i... |
cdleme31sde 39746 | Part of proof of Lemma D i... |
cdleme31snd 39747 | Part of proof of Lemma D i... |
cdleme31sdnN 39748 | Part of proof of Lemma E i... |
cdleme31sn1c 39749 | Part of proof of Lemma E i... |
cdleme31sn2 39750 | Part of proof of Lemma E i... |
cdleme31fv 39751 | Part of proof of Lemma E i... |
cdleme31fv1 39752 | Part of proof of Lemma E i... |
cdleme31fv1s 39753 | Part of proof of Lemma E i... |
cdleme31fv2 39754 | Part of proof of Lemma E i... |
cdleme31id 39755 | Part of proof of Lemma E i... |
cdlemefrs29pre00 39756 | ***START OF VALUE AT ATOM ... |
cdlemefrs29bpre0 39757 | TODO fix comment. (Contri... |
cdlemefrs29bpre1 39758 | TODO: FIX COMMENT. (Contr... |
cdlemefrs29cpre1 39759 | TODO: FIX COMMENT. (Contr... |
cdlemefrs29clN 39760 | TODO: NOT USED? Show clo... |
cdlemefrs32fva 39761 | Part of proof of Lemma E i... |
cdlemefrs32fva1 39762 | Part of proof of Lemma E i... |
cdlemefr29exN 39763 | Lemma for ~ cdlemefs29bpre... |
cdlemefr27cl 39764 | Part of proof of Lemma E i... |
cdlemefr32sn2aw 39765 | Show that ` [_ R / s ]_ N ... |
cdlemefr32snb 39766 | Show closure of ` [_ R / s... |
cdlemefr29bpre0N 39767 | TODO fix comment. (Contri... |
cdlemefr29clN 39768 | Show closure of the unique... |
cdleme43frv1snN 39769 | Value of ` [_ R / s ]_ N `... |
cdlemefr32fvaN 39770 | Part of proof of Lemma E i... |
cdlemefr32fva1 39771 | Part of proof of Lemma E i... |
cdlemefr31fv1 39772 | Value of ` ( F `` R ) ` wh... |
cdlemefs29pre00N 39773 | FIX COMMENT. TODO: see if ... |
cdlemefs27cl 39774 | Part of proof of Lemma E i... |
cdlemefs32sn1aw 39775 | Show that ` [_ R / s ]_ N ... |
cdlemefs32snb 39776 | Show closure of ` [_ R / s... |
cdlemefs29bpre0N 39777 | TODO: FIX COMMENT. (Contr... |
cdlemefs29bpre1N 39778 | TODO: FIX COMMENT. (Contr... |
cdlemefs29cpre1N 39779 | TODO: FIX COMMENT. (Contr... |
cdlemefs29clN 39780 | Show closure of the unique... |
cdleme43fsv1snlem 39781 | Value of ` [_ R / s ]_ N `... |
cdleme43fsv1sn 39782 | Value of ` [_ R / s ]_ N `... |
cdlemefs32fvaN 39783 | Part of proof of Lemma E i... |
cdlemefs32fva1 39784 | Part of proof of Lemma E i... |
cdlemefs31fv1 39785 | Value of ` ( F `` R ) ` wh... |
cdlemefr44 39786 | Value of f(r) when r is an... |
cdlemefs44 39787 | Value of f_s(r) when r is ... |
cdlemefr45 39788 | Value of f(r) when r is an... |
cdlemefr45e 39789 | Explicit expansion of ~ cd... |
cdlemefs45 39790 | Value of f_s(r) when r is ... |
cdlemefs45ee 39791 | Explicit expansion of ~ cd... |
cdlemefs45eN 39792 | Explicit expansion of ~ cd... |
cdleme32sn1awN 39793 | Show that ` [_ R / s ]_ N ... |
cdleme41sn3a 39794 | Show that ` [_ R / s ]_ N ... |
cdleme32sn2awN 39795 | Show that ` [_ R / s ]_ N ... |
cdleme32snaw 39796 | Show that ` [_ R / s ]_ N ... |
cdleme32snb 39797 | Show closure of ` [_ R / s... |
cdleme32fva 39798 | Part of proof of Lemma D i... |
cdleme32fva1 39799 | Part of proof of Lemma D i... |
cdleme32fvaw 39800 | Show that ` ( F `` R ) ` i... |
cdleme32fvcl 39801 | Part of proof of Lemma D i... |
cdleme32a 39802 | Part of proof of Lemma D i... |
cdleme32b 39803 | Part of proof of Lemma D i... |
cdleme32c 39804 | Part of proof of Lemma D i... |
cdleme32d 39805 | Part of proof of Lemma D i... |
cdleme32e 39806 | Part of proof of Lemma D i... |
cdleme32f 39807 | Part of proof of Lemma D i... |
cdleme32le 39808 | Part of proof of Lemma D i... |
cdleme35a 39809 | Part of proof of Lemma E i... |
cdleme35fnpq 39810 | Part of proof of Lemma E i... |
cdleme35b 39811 | Part of proof of Lemma E i... |
cdleme35c 39812 | Part of proof of Lemma E i... |
cdleme35d 39813 | Part of proof of Lemma E i... |
cdleme35e 39814 | Part of proof of Lemma E i... |
cdleme35f 39815 | Part of proof of Lemma E i... |
cdleme35g 39816 | Part of proof of Lemma E i... |
cdleme35h 39817 | Part of proof of Lemma E i... |
cdleme35h2 39818 | Part of proof of Lemma E i... |
cdleme35sn2aw 39819 | Part of proof of Lemma E i... |
cdleme35sn3a 39820 | Part of proof of Lemma E i... |
cdleme36a 39821 | Part of proof of Lemma E i... |
cdleme36m 39822 | Part of proof of Lemma E i... |
cdleme37m 39823 | Part of proof of Lemma E i... |
cdleme38m 39824 | Part of proof of Lemma E i... |
cdleme38n 39825 | Part of proof of Lemma E i... |
cdleme39a 39826 | Part of proof of Lemma E i... |
cdleme39n 39827 | Part of proof of Lemma E i... |
cdleme40m 39828 | Part of proof of Lemma E i... |
cdleme40n 39829 | Part of proof of Lemma E i... |
cdleme40v 39830 | Part of proof of Lemma E i... |
cdleme40w 39831 | Part of proof of Lemma E i... |
cdleme42a 39832 | Part of proof of Lemma E i... |
cdleme42c 39833 | Part of proof of Lemma E i... |
cdleme42d 39834 | Part of proof of Lemma E i... |
cdleme41sn3aw 39835 | Part of proof of Lemma E i... |
cdleme41sn4aw 39836 | Part of proof of Lemma E i... |
cdleme41snaw 39837 | Part of proof of Lemma E i... |
cdleme41fva11 39838 | Part of proof of Lemma E i... |
cdleme42b 39839 | Part of proof of Lemma E i... |
cdleme42e 39840 | Part of proof of Lemma E i... |
cdleme42f 39841 | Part of proof of Lemma E i... |
cdleme42g 39842 | Part of proof of Lemma E i... |
cdleme42h 39843 | Part of proof of Lemma E i... |
cdleme42i 39844 | Part of proof of Lemma E i... |
cdleme42k 39845 | Part of proof of Lemma E i... |
cdleme42ke 39846 | Part of proof of Lemma E i... |
cdleme42keg 39847 | Part of proof of Lemma E i... |
cdleme42mN 39848 | Part of proof of Lemma E i... |
cdleme42mgN 39849 | Part of proof of Lemma E i... |
cdleme43aN 39850 | Part of proof of Lemma E i... |
cdleme43bN 39851 | Lemma for Lemma E in [Craw... |
cdleme43cN 39852 | Part of proof of Lemma E i... |
cdleme43dN 39853 | Part of proof of Lemma E i... |
cdleme46f2g2 39854 | Conversion for ` G ` to re... |
cdleme46f2g1 39855 | Conversion for ` G ` to re... |
cdleme17d2 39856 | Part of proof of Lemma E i... |
cdleme17d3 39857 | TODO: FIX COMMENT. (Contr... |
cdleme17d4 39858 | TODO: FIX COMMENT. (Contr... |
cdleme17d 39859 | Part of proof of Lemma E i... |
cdleme48fv 39860 | Part of proof of Lemma D i... |
cdleme48fvg 39861 | Remove ` P =/= Q ` conditi... |
cdleme46fvaw 39862 | Show that ` ( F `` R ) ` i... |
cdleme48bw 39863 | TODO: fix comment. TODO: ... |
cdleme48b 39864 | TODO: fix comment. (Contr... |
cdleme46frvlpq 39865 | Show that ` ( F `` S ) ` i... |
cdleme46fsvlpq 39866 | Show that ` ( F `` R ) ` i... |
cdlemeg46fvcl 39867 | TODO: fix comment. (Contr... |
cdleme4gfv 39868 | Part of proof of Lemma D i... |
cdlemeg47b 39869 | TODO: FIX COMMENT. (Contr... |
cdlemeg47rv 39870 | Value of g_s(r) when r is ... |
cdlemeg47rv2 39871 | Value of g_s(r) when r is ... |
cdlemeg49le 39872 | Part of proof of Lemma D i... |
cdlemeg46bOLDN 39873 | TODO FIX COMMENT. (Contrib... |
cdlemeg46c 39874 | TODO FIX COMMENT. (Contrib... |
cdlemeg46rvOLDN 39875 | Value of g_s(r) when r is ... |
cdlemeg46rv2OLDN 39876 | Value of g_s(r) when r is ... |
cdlemeg46fvaw 39877 | Show that ` ( F `` R ) ` i... |
cdlemeg46nlpq 39878 | Show that ` ( G `` S ) ` i... |
cdlemeg46ngfr 39879 | TODO FIX COMMENT g(f(s))=s... |
cdlemeg46nfgr 39880 | TODO FIX COMMENT f(g(s))=s... |
cdlemeg46sfg 39881 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46fjgN 39882 | NOT NEEDED? TODO FIX COMM... |
cdlemeg46rjgN 39883 | NOT NEEDED? TODO FIX COMM... |
cdlemeg46fjv 39884 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46fsfv 39885 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46frv 39886 | TODO FIX COMMENT. (f(r) ` ... |
cdlemeg46v1v2 39887 | TODO FIX COMMENT v_1 = v_2... |
cdlemeg46vrg 39888 | TODO FIX COMMENT v_1 ` <_ ... |
cdlemeg46rgv 39889 | TODO FIX COMMENT r ` <_ ` ... |
cdlemeg46req 39890 | TODO FIX COMMENT r = (v_1 ... |
cdlemeg46gfv 39891 | TODO FIX COMMENT p. 115 pe... |
cdlemeg46gfr 39892 | TODO FIX COMMENT p. 116 pe... |
cdlemeg46gfre 39893 | TODO FIX COMMENT p. 116 pe... |
cdlemeg46gf 39894 | TODO FIX COMMENT Eliminate... |
cdlemeg46fgN 39895 | TODO FIX COMMENT p. 116 pe... |
cdleme48d 39896 | TODO: fix comment. (Contr... |
cdleme48gfv1 39897 | TODO: fix comment. (Contr... |
cdleme48gfv 39898 | TODO: fix comment. (Contr... |
cdleme48fgv 39899 | TODO: fix comment. (Contr... |
cdlemeg49lebilem 39900 | Part of proof of Lemma D i... |
cdleme50lebi 39901 | Part of proof of Lemma D i... |
cdleme50eq 39902 | Part of proof of Lemma D i... |
cdleme50f 39903 | Part of proof of Lemma D i... |
cdleme50f1 39904 | Part of proof of Lemma D i... |
cdleme50rnlem 39905 | Part of proof of Lemma D i... |
cdleme50rn 39906 | Part of proof of Lemma D i... |
cdleme50f1o 39907 | Part of proof of Lemma D i... |
cdleme50laut 39908 | Part of proof of Lemma D i... |
cdleme50ldil 39909 | Part of proof of Lemma D i... |
cdleme50trn1 39910 | Part of proof that ` F ` i... |
cdleme50trn2a 39911 | Part of proof that ` F ` i... |
cdleme50trn2 39912 | Part of proof that ` F ` i... |
cdleme50trn12 39913 | Part of proof that ` F ` i... |
cdleme50trn3 39914 | Part of proof that ` F ` i... |
cdleme50trn123 39915 | Part of proof that ` F ` i... |
cdleme51finvfvN 39916 | Part of proof of Lemma E i... |
cdleme51finvN 39917 | Part of proof of Lemma E i... |
cdleme50ltrn 39918 | Part of proof of Lemma E i... |
cdleme51finvtrN 39919 | Part of proof of Lemma E i... |
cdleme50ex 39920 | Part of Lemma E in [Crawle... |
cdleme 39921 | Lemma E in [Crawley] p. 11... |
cdlemf1 39922 | Part of Lemma F in [Crawle... |
cdlemf2 39923 | Part of Lemma F in [Crawle... |
cdlemf 39924 | Lemma F in [Crawley] p. 11... |
cdlemfnid 39925 | ~ cdlemf with additional c... |
cdlemftr3 39926 | Special case of ~ cdlemf s... |
cdlemftr2 39927 | Special case of ~ cdlemf s... |
cdlemftr1 39928 | Part of proof of Lemma G o... |
cdlemftr0 39929 | Special case of ~ cdlemf s... |
trlord 39930 | The ordering of two Hilber... |
cdlemg1a 39931 | Shorter expression for ` G... |
cdlemg1b2 39932 | This theorem can be used t... |
cdlemg1idlemN 39933 | Lemma for ~ cdlemg1idN . ... |
cdlemg1fvawlemN 39934 | Lemma for ~ ltrniotafvawN ... |
cdlemg1ltrnlem 39935 | Lemma for ~ ltrniotacl . ... |
cdlemg1finvtrlemN 39936 | Lemma for ~ ltrniotacnvN .... |
cdlemg1bOLDN 39937 | This theorem can be used t... |
cdlemg1idN 39938 | Version of ~ cdleme31id wi... |
ltrniotafvawN 39939 | Version of ~ cdleme46fvaw ... |
ltrniotacl 39940 | Version of ~ cdleme50ltrn ... |
ltrniotacnvN 39941 | Version of ~ cdleme51finvt... |
ltrniotaval 39942 | Value of the unique transl... |
ltrniotacnvval 39943 | Converse value of the uniq... |
ltrniotaidvalN 39944 | Value of the unique transl... |
ltrniotavalbN 39945 | Value of the unique transl... |
cdlemeiota 39946 | A translation is uniquely ... |
cdlemg1ci2 39947 | Any function of the form o... |
cdlemg1cN 39948 | Any translation belongs to... |
cdlemg1cex 39949 | Any translation is one of ... |
cdlemg2cN 39950 | Any translation belongs to... |
cdlemg2dN 39951 | This theorem can be used t... |
cdlemg2cex 39952 | Any translation is one of ... |
cdlemg2ce 39953 | Utility theorem to elimina... |
cdlemg2jlemOLDN 39954 | Part of proof of Lemma E i... |
cdlemg2fvlem 39955 | Lemma for ~ cdlemg2fv . (... |
cdlemg2klem 39956 | ~ cdleme42keg with simpler... |
cdlemg2idN 39957 | Version of ~ cdleme31id wi... |
cdlemg3a 39958 | Part of proof of Lemma G i... |
cdlemg2jOLDN 39959 | TODO: Replace this with ~... |
cdlemg2fv 39960 | Value of a translation in ... |
cdlemg2fv2 39961 | Value of a translation in ... |
cdlemg2k 39962 | ~ cdleme42keg with simpler... |
cdlemg2kq 39963 | ~ cdlemg2k with ` P ` and ... |
cdlemg2l 39964 | TODO: FIX COMMENT. (Contr... |
cdlemg2m 39965 | TODO: FIX COMMENT. (Contr... |
cdlemg5 39966 | TODO: Is there a simpler ... |
cdlemb3 39967 | Given two atoms not under ... |
cdlemg7fvbwN 39968 | Properties of a translatio... |
cdlemg4a 39969 | TODO: FIX COMMENT If fg(p... |
cdlemg4b1 39970 | TODO: FIX COMMENT. (Contr... |
cdlemg4b2 39971 | TODO: FIX COMMENT. (Contr... |
cdlemg4b12 39972 | TODO: FIX COMMENT. (Contr... |
cdlemg4c 39973 | TODO: FIX COMMENT. (Contr... |
cdlemg4d 39974 | TODO: FIX COMMENT. (Contr... |
cdlemg4e 39975 | TODO: FIX COMMENT. (Contr... |
cdlemg4f 39976 | TODO: FIX COMMENT. (Contr... |
cdlemg4g 39977 | TODO: FIX COMMENT. (Contr... |
cdlemg4 39978 | TODO: FIX COMMENT. (Contr... |
cdlemg6a 39979 | TODO: FIX COMMENT. TODO: ... |
cdlemg6b 39980 | TODO: FIX COMMENT. TODO: ... |
cdlemg6c 39981 | TODO: FIX COMMENT. (Contr... |
cdlemg6d 39982 | TODO: FIX COMMENT. (Contr... |
cdlemg6e 39983 | TODO: FIX COMMENT. (Contr... |
cdlemg6 39984 | TODO: FIX COMMENT. (Contr... |
cdlemg7fvN 39985 | Value of a translation com... |
cdlemg7aN 39986 | TODO: FIX COMMENT. (Contr... |
cdlemg7N 39987 | TODO: FIX COMMENT. (Contr... |
cdlemg8a 39988 | TODO: FIX COMMENT. (Contr... |
cdlemg8b 39989 | TODO: FIX COMMENT. (Contr... |
cdlemg8c 39990 | TODO: FIX COMMENT. (Contr... |
cdlemg8d 39991 | TODO: FIX COMMENT. (Contr... |
cdlemg8 39992 | TODO: FIX COMMENT. (Contr... |
cdlemg9a 39993 | TODO: FIX COMMENT. (Contr... |
cdlemg9b 39994 | The triples ` <. P , ( F `... |
cdlemg9 39995 | The triples ` <. P , ( F `... |
cdlemg10b 39996 | TODO: FIX COMMENT. TODO: ... |
cdlemg10bALTN 39997 | TODO: FIX COMMENT. TODO: ... |
cdlemg11a 39998 | TODO: FIX COMMENT. (Contr... |
cdlemg11aq 39999 | TODO: FIX COMMENT. TODO: ... |
cdlemg10c 40000 | TODO: FIX COMMENT. TODO: ... |
cdlemg10a 40001 | TODO: FIX COMMENT. (Contr... |
cdlemg10 40002 | TODO: FIX COMMENT. (Contr... |
cdlemg11b 40003 | TODO: FIX COMMENT. (Contr... |
cdlemg12a 40004 | TODO: FIX COMMENT. (Contr... |
cdlemg12b 40005 | The triples ` <. P , ( F `... |
cdlemg12c 40006 | The triples ` <. P , ( F `... |
cdlemg12d 40007 | TODO: FIX COMMENT. (Contr... |
cdlemg12e 40008 | TODO: FIX COMMENT. (Contr... |
cdlemg12f 40009 | TODO: FIX COMMENT. (Contr... |
cdlemg12g 40010 | TODO: FIX COMMENT. TODO: ... |
cdlemg12 40011 | TODO: FIX COMMENT. (Contr... |
cdlemg13a 40012 | TODO: FIX COMMENT. (Contr... |
cdlemg13 40013 | TODO: FIX COMMENT. (Contr... |
cdlemg14f 40014 | TODO: FIX COMMENT. (Contr... |
cdlemg14g 40015 | TODO: FIX COMMENT. (Contr... |
cdlemg15a 40016 | Eliminate the ` ( F `` P )... |
cdlemg15 40017 | Eliminate the ` ( (... |
cdlemg16 40018 | Part of proof of Lemma G o... |
cdlemg16ALTN 40019 | This version of ~ cdlemg16... |
cdlemg16z 40020 | Eliminate ` ( ( F `... |
cdlemg16zz 40021 | Eliminate ` P =/= Q ` from... |
cdlemg17a 40022 | TODO: FIX COMMENT. (Contr... |
cdlemg17b 40023 | Part of proof of Lemma G i... |
cdlemg17dN 40024 | TODO: fix comment. (Contr... |
cdlemg17dALTN 40025 | Same as ~ cdlemg17dN with ... |
cdlemg17e 40026 | TODO: fix comment. (Contr... |
cdlemg17f 40027 | TODO: fix comment. (Contr... |
cdlemg17g 40028 | TODO: fix comment. (Contr... |
cdlemg17h 40029 | TODO: fix comment. (Contr... |
cdlemg17i 40030 | TODO: fix comment. (Contr... |
cdlemg17ir 40031 | TODO: fix comment. (Contr... |
cdlemg17j 40032 | TODO: fix comment. (Contr... |
cdlemg17pq 40033 | Utility theorem for swappi... |
cdlemg17bq 40034 | ~ cdlemg17b with ` P ` and... |
cdlemg17iqN 40035 | ~ cdlemg17i with ` P ` and... |
cdlemg17irq 40036 | ~ cdlemg17ir with ` P ` an... |
cdlemg17jq 40037 | ~ cdlemg17j with ` P ` and... |
cdlemg17 40038 | Part of Lemma G of [Crawle... |
cdlemg18a 40039 | Show two lines are differe... |
cdlemg18b 40040 | Lemma for ~ cdlemg18c . T... |
cdlemg18c 40041 | Show two lines intersect a... |
cdlemg18d 40042 | Show two lines intersect a... |
cdlemg18 40043 | Show two lines intersect a... |
cdlemg19a 40044 | Show two lines intersect a... |
cdlemg19 40045 | Show two lines intersect a... |
cdlemg20 40046 | Show two lines intersect a... |
cdlemg21 40047 | Version of cdlemg19 with `... |
cdlemg22 40048 | ~ cdlemg21 with ` ( F `` P... |
cdlemg24 40049 | Combine ~ cdlemg16z and ~ ... |
cdlemg37 40050 | Use ~ cdlemg8 to eliminate... |
cdlemg25zz 40051 | ~ cdlemg16zz restated for ... |
cdlemg26zz 40052 | ~ cdlemg16zz restated for ... |
cdlemg27a 40053 | For use with case when ` (... |
cdlemg28a 40054 | Part of proof of Lemma G o... |
cdlemg31b0N 40055 | TODO: Fix comment. (Cont... |
cdlemg31b0a 40056 | TODO: Fix comment. (Cont... |
cdlemg27b 40057 | TODO: Fix comment. (Cont... |
cdlemg31a 40058 | TODO: fix comment. (Contr... |
cdlemg31b 40059 | TODO: fix comment. (Contr... |
cdlemg31c 40060 | Show that when ` N ` is an... |
cdlemg31d 40061 | Eliminate ` ( F `` P ) =/=... |
cdlemg33b0 40062 | TODO: Fix comment. (Cont... |
cdlemg33c0 40063 | TODO: Fix comment. (Cont... |
cdlemg28b 40064 | Part of proof of Lemma G o... |
cdlemg28 40065 | Part of proof of Lemma G o... |
cdlemg29 40066 | Eliminate ` ( F `` P ) =/=... |
cdlemg33a 40067 | TODO: Fix comment. (Cont... |
cdlemg33b 40068 | TODO: Fix comment. (Cont... |
cdlemg33c 40069 | TODO: Fix comment. (Cont... |
cdlemg33d 40070 | TODO: Fix comment. (Cont... |
cdlemg33e 40071 | TODO: Fix comment. (Cont... |
cdlemg33 40072 | Combine ~ cdlemg33b , ~ cd... |
cdlemg34 40073 | Use cdlemg33 to eliminate ... |
cdlemg35 40074 | TODO: Fix comment. TODO:... |
cdlemg36 40075 | Use cdlemg35 to eliminate ... |
cdlemg38 40076 | Use ~ cdlemg37 to eliminat... |
cdlemg39 40077 | Eliminate ` =/= ` conditio... |
cdlemg40 40078 | Eliminate ` P =/= Q ` cond... |
cdlemg41 40079 | Convert ~ cdlemg40 to func... |
ltrnco 40080 | The composition of two tra... |
trlcocnv 40081 | Swap the arguments of the ... |
trlcoabs 40082 | Absorption into a composit... |
trlcoabs2N 40083 | Absorption of the trace of... |
trlcoat 40084 | The trace of a composition... |
trlcocnvat 40085 | Commonly used special case... |
trlconid 40086 | The composition of two dif... |
trlcolem 40087 | Lemma for ~ trlco . (Cont... |
trlco 40088 | The trace of a composition... |
trlcone 40089 | If two translations have d... |
cdlemg42 40090 | Part of proof of Lemma G o... |
cdlemg43 40091 | Part of proof of Lemma G o... |
cdlemg44a 40092 | Part of proof of Lemma G o... |
cdlemg44b 40093 | Eliminate ` ( F `` P ) =/=... |
cdlemg44 40094 | Part of proof of Lemma G o... |
cdlemg47a 40095 | TODO: fix comment. TODO: ... |
cdlemg46 40096 | Part of proof of Lemma G o... |
cdlemg47 40097 | Part of proof of Lemma G o... |
cdlemg48 40098 | Eliminate ` h ` from ~ cdl... |
ltrncom 40099 | Composition is commutative... |
ltrnco4 40100 | Rearrange a composition of... |
trljco 40101 | Trace joined with trace of... |
trljco2 40102 | Trace joined with trace of... |
tgrpfset 40105 | The translation group maps... |
tgrpset 40106 | The translation group for ... |
tgrpbase 40107 | The base set of the transl... |
tgrpopr 40108 | The group operation of the... |
tgrpov 40109 | The group operation value ... |
tgrpgrplem 40110 | Lemma for ~ tgrpgrp . (Co... |
tgrpgrp 40111 | The translation group is a... |
tgrpabl 40112 | The translation group is a... |
tendofset 40119 | The set of all trace-prese... |
tendoset 40120 | The set of trace-preservin... |
istendo 40121 | The predicate "is a trace-... |
tendotp 40122 | Trace-preserving property ... |
istendod 40123 | Deduce the predicate "is a... |
tendof 40124 | Functionality of a trace-p... |
tendoeq1 40125 | Condition determining equa... |
tendovalco 40126 | Value of composition of tr... |
tendocoval 40127 | Value of composition of en... |
tendocl 40128 | Closure of a trace-preserv... |
tendoco2 40129 | Distribution of compositio... |
tendoidcl 40130 | The identity is a trace-pr... |
tendo1mul 40131 | Multiplicative identity mu... |
tendo1mulr 40132 | Multiplicative identity mu... |
tendococl 40133 | The composition of two tra... |
tendoid 40134 | The identity value of a tr... |
tendoeq2 40135 | Condition determining equa... |
tendoplcbv 40136 | Define sum operation for t... |
tendopl 40137 | Value of endomorphism sum ... |
tendopl2 40138 | Value of result of endomor... |
tendoplcl2 40139 | Value of result of endomor... |
tendoplco2 40140 | Value of result of endomor... |
tendopltp 40141 | Trace-preserving property ... |
tendoplcl 40142 | Endomorphism sum is a trac... |
tendoplcom 40143 | The endomorphism sum opera... |
tendoplass 40144 | The endomorphism sum opera... |
tendodi1 40145 | Endomorphism composition d... |
tendodi2 40146 | Endomorphism composition d... |
tendo0cbv 40147 | Define additive identity f... |
tendo02 40148 | Value of additive identity... |
tendo0co2 40149 | The additive identity trac... |
tendo0tp 40150 | Trace-preserving property ... |
tendo0cl 40151 | The additive identity is a... |
tendo0pl 40152 | Property of the additive i... |
tendo0plr 40153 | Property of the additive i... |
tendoicbv 40154 | Define inverse function fo... |
tendoi 40155 | Value of inverse endomorph... |
tendoi2 40156 | Value of additive inverse ... |
tendoicl 40157 | Closure of the additive in... |
tendoipl 40158 | Property of the additive i... |
tendoipl2 40159 | Property of the additive i... |
erngfset 40160 | The division rings on trac... |
erngset 40161 | The division ring on trace... |
erngbase 40162 | The base set of the divisi... |
erngfplus 40163 | Ring addition operation. ... |
erngplus 40164 | Ring addition operation. ... |
erngplus2 40165 | Ring addition operation. ... |
erngfmul 40166 | Ring multiplication operat... |
erngmul 40167 | Ring addition operation. ... |
erngfset-rN 40168 | The division rings on trac... |
erngset-rN 40169 | The division ring on trace... |
erngbase-rN 40170 | The base set of the divisi... |
erngfplus-rN 40171 | Ring addition operation. ... |
erngplus-rN 40172 | Ring addition operation. ... |
erngplus2-rN 40173 | Ring addition operation. ... |
erngfmul-rN 40174 | Ring multiplication operat... |
erngmul-rN 40175 | Ring addition operation. ... |
cdlemh1 40176 | Part of proof of Lemma H o... |
cdlemh2 40177 | Part of proof of Lemma H o... |
cdlemh 40178 | Lemma H of [Crawley] p. 11... |
cdlemi1 40179 | Part of proof of Lemma I o... |
cdlemi2 40180 | Part of proof of Lemma I o... |
cdlemi 40181 | Lemma I of [Crawley] p. 11... |
cdlemj1 40182 | Part of proof of Lemma J o... |
cdlemj2 40183 | Part of proof of Lemma J o... |
cdlemj3 40184 | Part of proof of Lemma J o... |
tendocan 40185 | Cancellation law: if the v... |
tendoid0 40186 | A trace-preserving endomor... |
tendo0mul 40187 | Additive identity multipli... |
tendo0mulr 40188 | Additive identity multipli... |
tendo1ne0 40189 | The identity (unity) is no... |
tendoconid 40190 | The composition (product) ... |
tendotr 40191 | The trace of the value of ... |
cdlemk1 40192 | Part of proof of Lemma K o... |
cdlemk2 40193 | Part of proof of Lemma K o... |
cdlemk3 40194 | Part of proof of Lemma K o... |
cdlemk4 40195 | Part of proof of Lemma K o... |
cdlemk5a 40196 | Part of proof of Lemma K o... |
cdlemk5 40197 | Part of proof of Lemma K o... |
cdlemk6 40198 | Part of proof of Lemma K o... |
cdlemk8 40199 | Part of proof of Lemma K o... |
cdlemk9 40200 | Part of proof of Lemma K o... |
cdlemk9bN 40201 | Part of proof of Lemma K o... |
cdlemki 40202 | Part of proof of Lemma K o... |
cdlemkvcl 40203 | Part of proof of Lemma K o... |
cdlemk10 40204 | Part of proof of Lemma K o... |
cdlemksv 40205 | Part of proof of Lemma K o... |
cdlemksel 40206 | Part of proof of Lemma K o... |
cdlemksat 40207 | Part of proof of Lemma K o... |
cdlemksv2 40208 | Part of proof of Lemma K o... |
cdlemk7 40209 | Part of proof of Lemma K o... |
cdlemk11 40210 | Part of proof of Lemma K o... |
cdlemk12 40211 | Part of proof of Lemma K o... |
cdlemkoatnle 40212 | Utility lemma. (Contribut... |
cdlemk13 40213 | Part of proof of Lemma K o... |
cdlemkole 40214 | Utility lemma. (Contribut... |
cdlemk14 40215 | Part of proof of Lemma K o... |
cdlemk15 40216 | Part of proof of Lemma K o... |
cdlemk16a 40217 | Part of proof of Lemma K o... |
cdlemk16 40218 | Part of proof of Lemma K o... |
cdlemk17 40219 | Part of proof of Lemma K o... |
cdlemk1u 40220 | Part of proof of Lemma K o... |
cdlemk5auN 40221 | Part of proof of Lemma K o... |
cdlemk5u 40222 | Part of proof of Lemma K o... |
cdlemk6u 40223 | Part of proof of Lemma K o... |
cdlemkj 40224 | Part of proof of Lemma K o... |
cdlemkuvN 40225 | Part of proof of Lemma K o... |
cdlemkuel 40226 | Part of proof of Lemma K o... |
cdlemkuat 40227 | Part of proof of Lemma K o... |
cdlemkuv2 40228 | Part of proof of Lemma K o... |
cdlemk18 40229 | Part of proof of Lemma K o... |
cdlemk19 40230 | Part of proof of Lemma K o... |
cdlemk7u 40231 | Part of proof of Lemma K o... |
cdlemk11u 40232 | Part of proof of Lemma K o... |
cdlemk12u 40233 | Part of proof of Lemma K o... |
cdlemk21N 40234 | Part of proof of Lemma K o... |
cdlemk20 40235 | Part of proof of Lemma K o... |
cdlemkoatnle-2N 40236 | Utility lemma. (Contribut... |
cdlemk13-2N 40237 | Part of proof of Lemma K o... |
cdlemkole-2N 40238 | Utility lemma. (Contribut... |
cdlemk14-2N 40239 | Part of proof of Lemma K o... |
cdlemk15-2N 40240 | Part of proof of Lemma K o... |
cdlemk16-2N 40241 | Part of proof of Lemma K o... |
cdlemk17-2N 40242 | Part of proof of Lemma K o... |
cdlemkj-2N 40243 | Part of proof of Lemma K o... |
cdlemkuv-2N 40244 | Part of proof of Lemma K o... |
cdlemkuel-2N 40245 | Part of proof of Lemma K o... |
cdlemkuv2-2 40246 | Part of proof of Lemma K o... |
cdlemk18-2N 40247 | Part of proof of Lemma K o... |
cdlemk19-2N 40248 | Part of proof of Lemma K o... |
cdlemk7u-2N 40249 | Part of proof of Lemma K o... |
cdlemk11u-2N 40250 | Part of proof of Lemma K o... |
cdlemk12u-2N 40251 | Part of proof of Lemma K o... |
cdlemk21-2N 40252 | Part of proof of Lemma K o... |
cdlemk20-2N 40253 | Part of proof of Lemma K o... |
cdlemk22 40254 | Part of proof of Lemma K o... |
cdlemk30 40255 | Part of proof of Lemma K o... |
cdlemkuu 40256 | Convert between function a... |
cdlemk31 40257 | Part of proof of Lemma K o... |
cdlemk32 40258 | Part of proof of Lemma K o... |
cdlemkuel-3 40259 | Part of proof of Lemma K o... |
cdlemkuv2-3N 40260 | Part of proof of Lemma K o... |
cdlemk18-3N 40261 | Part of proof of Lemma K o... |
cdlemk22-3 40262 | Part of proof of Lemma K o... |
cdlemk23-3 40263 | Part of proof of Lemma K o... |
cdlemk24-3 40264 | Part of proof of Lemma K o... |
cdlemk25-3 40265 | Part of proof of Lemma K o... |
cdlemk26b-3 40266 | Part of proof of Lemma K o... |
cdlemk26-3 40267 | Part of proof of Lemma K o... |
cdlemk27-3 40268 | Part of proof of Lemma K o... |
cdlemk28-3 40269 | Part of proof of Lemma K o... |
cdlemk33N 40270 | Part of proof of Lemma K o... |
cdlemk34 40271 | Part of proof of Lemma K o... |
cdlemk29-3 40272 | Part of proof of Lemma K o... |
cdlemk35 40273 | Part of proof of Lemma K o... |
cdlemk36 40274 | Part of proof of Lemma K o... |
cdlemk37 40275 | Part of proof of Lemma K o... |
cdlemk38 40276 | Part of proof of Lemma K o... |
cdlemk39 40277 | Part of proof of Lemma K o... |
cdlemk40 40278 | TODO: fix comment. (Contr... |
cdlemk40t 40279 | TODO: fix comment. (Contr... |
cdlemk40f 40280 | TODO: fix comment. (Contr... |
cdlemk41 40281 | Part of proof of Lemma K o... |
cdlemkfid1N 40282 | Lemma for ~ cdlemkfid3N . ... |
cdlemkid1 40283 | Lemma for ~ cdlemkid . (C... |
cdlemkfid2N 40284 | Lemma for ~ cdlemkfid3N . ... |
cdlemkid2 40285 | Lemma for ~ cdlemkid . (C... |
cdlemkfid3N 40286 | TODO: is this useful or sh... |
cdlemky 40287 | Part of proof of Lemma K o... |
cdlemkyu 40288 | Convert between function a... |
cdlemkyuu 40289 | ~ cdlemkyu with some hypot... |
cdlemk11ta 40290 | Part of proof of Lemma K o... |
cdlemk19ylem 40291 | Lemma for ~ cdlemk19y . (... |
cdlemk11tb 40292 | Part of proof of Lemma K o... |
cdlemk19y 40293 | ~ cdlemk19 with simpler hy... |
cdlemkid3N 40294 | Lemma for ~ cdlemkid . (C... |
cdlemkid4 40295 | Lemma for ~ cdlemkid . (C... |
cdlemkid5 40296 | Lemma for ~ cdlemkid . (C... |
cdlemkid 40297 | The value of the tau funct... |
cdlemk35s 40298 | Substitution version of ~ ... |
cdlemk35s-id 40299 | Substitution version of ~ ... |
cdlemk39s 40300 | Substitution version of ~ ... |
cdlemk39s-id 40301 | Substitution version of ~ ... |
cdlemk42 40302 | Part of proof of Lemma K o... |
cdlemk19xlem 40303 | Lemma for ~ cdlemk19x . (... |
cdlemk19x 40304 | ~ cdlemk19 with simpler hy... |
cdlemk42yN 40305 | Part of proof of Lemma K o... |
cdlemk11tc 40306 | Part of proof of Lemma K o... |
cdlemk11t 40307 | Part of proof of Lemma K o... |
cdlemk45 40308 | Part of proof of Lemma K o... |
cdlemk46 40309 | Part of proof of Lemma K o... |
cdlemk47 40310 | Part of proof of Lemma K o... |
cdlemk48 40311 | Part of proof of Lemma K o... |
cdlemk49 40312 | Part of proof of Lemma K o... |
cdlemk50 40313 | Part of proof of Lemma K o... |
cdlemk51 40314 | Part of proof of Lemma K o... |
cdlemk52 40315 | Part of proof of Lemma K o... |
cdlemk53a 40316 | Lemma for ~ cdlemk53 . (C... |
cdlemk53b 40317 | Lemma for ~ cdlemk53 . (C... |
cdlemk53 40318 | Part of proof of Lemma K o... |
cdlemk54 40319 | Part of proof of Lemma K o... |
cdlemk55a 40320 | Lemma for ~ cdlemk55 . (C... |
cdlemk55b 40321 | Lemma for ~ cdlemk55 . (C... |
cdlemk55 40322 | Part of proof of Lemma K o... |
cdlemkyyN 40323 | Part of proof of Lemma K o... |
cdlemk43N 40324 | Part of proof of Lemma K o... |
cdlemk35u 40325 | Substitution version of ~ ... |
cdlemk55u1 40326 | Lemma for ~ cdlemk55u . (... |
cdlemk55u 40327 | Part of proof of Lemma K o... |
cdlemk39u1 40328 | Lemma for ~ cdlemk39u . (... |
cdlemk39u 40329 | Part of proof of Lemma K o... |
cdlemk19u1 40330 | ~ cdlemk19 with simpler hy... |
cdlemk19u 40331 | Part of Lemma K of [Crawle... |
cdlemk56 40332 | Part of Lemma K of [Crawle... |
cdlemk19w 40333 | Use a fixed element to eli... |
cdlemk56w 40334 | Use a fixed element to eli... |
cdlemk 40335 | Lemma K of [Crawley] p. 11... |
tendoex 40336 | Generalization of Lemma K ... |
cdleml1N 40337 | Part of proof of Lemma L o... |
cdleml2N 40338 | Part of proof of Lemma L o... |
cdleml3N 40339 | Part of proof of Lemma L o... |
cdleml4N 40340 | Part of proof of Lemma L o... |
cdleml5N 40341 | Part of proof of Lemma L o... |
cdleml6 40342 | Part of proof of Lemma L o... |
cdleml7 40343 | Part of proof of Lemma L o... |
cdleml8 40344 | Part of proof of Lemma L o... |
cdleml9 40345 | Part of proof of Lemma L o... |
dva1dim 40346 | Two expressions for the 1-... |
dvhb1dimN 40347 | Two expressions for the 1-... |
erng1lem 40348 | Value of the endomorphism ... |
erngdvlem1 40349 | Lemma for ~ eringring . (... |
erngdvlem2N 40350 | Lemma for ~ eringring . (... |
erngdvlem3 40351 | Lemma for ~ eringring . (... |
erngdvlem4 40352 | Lemma for ~ erngdv . (Con... |
eringring 40353 | An endomorphism ring is a ... |
erngdv 40354 | An endomorphism ring is a ... |
erng0g 40355 | The division ring zero of ... |
erng1r 40356 | The division ring unity of... |
erngdvlem1-rN 40357 | Lemma for ~ eringring . (... |
erngdvlem2-rN 40358 | Lemma for ~ eringring . (... |
erngdvlem3-rN 40359 | Lemma for ~ eringring . (... |
erngdvlem4-rN 40360 | Lemma for ~ erngdv . (Con... |
erngring-rN 40361 | An endomorphism ring is a ... |
erngdv-rN 40362 | An endomorphism ring is a ... |
dvafset 40365 | The constructed partial ve... |
dvaset 40366 | The constructed partial ve... |
dvasca 40367 | The ring base set of the c... |
dvabase 40368 | The ring base set of the c... |
dvafplusg 40369 | Ring addition operation fo... |
dvaplusg 40370 | Ring addition operation fo... |
dvaplusgv 40371 | Ring addition operation fo... |
dvafmulr 40372 | Ring multiplication operat... |
dvamulr 40373 | Ring multiplication operat... |
dvavbase 40374 | The vectors (vector base s... |
dvafvadd 40375 | The vector sum operation f... |
dvavadd 40376 | Ring addition operation fo... |
dvafvsca 40377 | Ring addition operation fo... |
dvavsca 40378 | Ring addition operation fo... |
tendospcl 40379 | Closure of endomorphism sc... |
tendospass 40380 | Associative law for endomo... |
tendospdi1 40381 | Forward distributive law f... |
tendocnv 40382 | Converse of a trace-preser... |
tendospdi2 40383 | Reverse distributive law f... |
tendospcanN 40384 | Cancellation law for trace... |
dvaabl 40385 | The constructed partial ve... |
dvalveclem 40386 | Lemma for ~ dvalvec . (Co... |
dvalvec 40387 | The constructed partial ve... |
dva0g 40388 | The zero vector of partial... |
diaffval 40391 | The partial isomorphism A ... |
diafval 40392 | The partial isomorphism A ... |
diaval 40393 | The partial isomorphism A ... |
diaelval 40394 | Member of the partial isom... |
diafn 40395 | Functionality and domain o... |
diadm 40396 | Domain of the partial isom... |
diaeldm 40397 | Member of domain of the pa... |
diadmclN 40398 | A member of domain of the ... |
diadmleN 40399 | A member of domain of the ... |
dian0 40400 | The value of the partial i... |
dia0eldmN 40401 | The lattice zero belongs t... |
dia1eldmN 40402 | The fiducial hyperplane (t... |
diass 40403 | The value of the partial i... |
diael 40404 | A member of the value of t... |
diatrl 40405 | Trace of a member of the p... |
diaelrnN 40406 | Any value of the partial i... |
dialss 40407 | The value of partial isomo... |
diaord 40408 | The partial isomorphism A ... |
dia11N 40409 | The partial isomorphism A ... |
diaf11N 40410 | The partial isomorphism A ... |
diaclN 40411 | Closure of partial isomorp... |
diacnvclN 40412 | Closure of partial isomorp... |
dia0 40413 | The value of the partial i... |
dia1N 40414 | The value of the partial i... |
dia1elN 40415 | The largest subspace in th... |
diaglbN 40416 | Partial isomorphism A of a... |
diameetN 40417 | Partial isomorphism A of a... |
diainN 40418 | Inverse partial isomorphis... |
diaintclN 40419 | The intersection of partia... |
diasslssN 40420 | The partial isomorphism A ... |
diassdvaN 40421 | The partial isomorphism A ... |
dia1dim 40422 | Two expressions for the 1-... |
dia1dim2 40423 | Two expressions for a 1-di... |
dia1dimid 40424 | A vector (translation) bel... |
dia2dimlem1 40425 | Lemma for ~ dia2dim . Sho... |
dia2dimlem2 40426 | Lemma for ~ dia2dim . Def... |
dia2dimlem3 40427 | Lemma for ~ dia2dim . Def... |
dia2dimlem4 40428 | Lemma for ~ dia2dim . Sho... |
dia2dimlem5 40429 | Lemma for ~ dia2dim . The... |
dia2dimlem6 40430 | Lemma for ~ dia2dim . Eli... |
dia2dimlem7 40431 | Lemma for ~ dia2dim . Eli... |
dia2dimlem8 40432 | Lemma for ~ dia2dim . Eli... |
dia2dimlem9 40433 | Lemma for ~ dia2dim . Eli... |
dia2dimlem10 40434 | Lemma for ~ dia2dim . Con... |
dia2dimlem11 40435 | Lemma for ~ dia2dim . Con... |
dia2dimlem12 40436 | Lemma for ~ dia2dim . Obt... |
dia2dimlem13 40437 | Lemma for ~ dia2dim . Eli... |
dia2dim 40438 | A two-dimensional subspace... |
dvhfset 40441 | The constructed full vecto... |
dvhset 40442 | The constructed full vecto... |
dvhsca 40443 | The ring of scalars of the... |
dvhbase 40444 | The ring base set of the c... |
dvhfplusr 40445 | Ring addition operation fo... |
dvhfmulr 40446 | Ring multiplication operat... |
dvhmulr 40447 | Ring multiplication operat... |
dvhvbase 40448 | The vectors (vector base s... |
dvhelvbasei 40449 | Vector membership in the c... |
dvhvaddcbv 40450 | Change bound variables to ... |
dvhvaddval 40451 | The vector sum operation f... |
dvhfvadd 40452 | The vector sum operation f... |
dvhvadd 40453 | The vector sum operation f... |
dvhopvadd 40454 | The vector sum operation f... |
dvhopvadd2 40455 | The vector sum operation f... |
dvhvaddcl 40456 | Closure of the vector sum ... |
dvhvaddcomN 40457 | Commutativity of vector su... |
dvhvaddass 40458 | Associativity of vector su... |
dvhvscacbv 40459 | Change bound variables to ... |
dvhvscaval 40460 | The scalar product operati... |
dvhfvsca 40461 | Scalar product operation f... |
dvhvsca 40462 | Scalar product operation f... |
dvhopvsca 40463 | Scalar product operation f... |
dvhvscacl 40464 | Closure of the scalar prod... |
tendoinvcl 40465 | Closure of multiplicative ... |
tendolinv 40466 | Left multiplicative invers... |
tendorinv 40467 | Right multiplicative inver... |
dvhgrp 40468 | The full vector space ` U ... |
dvhlveclem 40469 | Lemma for ~ dvhlvec . TOD... |
dvhlvec 40470 | The full vector space ` U ... |
dvhlmod 40471 | The full vector space ` U ... |
dvh0g 40472 | The zero vector of vector ... |
dvheveccl 40473 | Properties of a unit vecto... |
dvhopclN 40474 | Closure of a ` DVecH ` vec... |
dvhopaddN 40475 | Sum of ` DVecH ` vectors e... |
dvhopspN 40476 | Scalar product of ` DVecH ... |
dvhopN 40477 | Decompose a ` DVecH ` vect... |
dvhopellsm 40478 | Ordered pair membership in... |
cdlemm10N 40479 | The image of the map ` G `... |
docaffvalN 40482 | Subspace orthocomplement f... |
docafvalN 40483 | Subspace orthocomplement f... |
docavalN 40484 | Subspace orthocomplement f... |
docaclN 40485 | Closure of subspace orthoc... |
diaocN 40486 | Value of partial isomorphi... |
doca2N 40487 | Double orthocomplement of ... |
doca3N 40488 | Double orthocomplement of ... |
dvadiaN 40489 | Any closed subspace is a m... |
diarnN 40490 | Partial isomorphism A maps... |
diaf1oN 40491 | The partial isomorphism A ... |
djaffvalN 40494 | Subspace join for ` DVecA ... |
djafvalN 40495 | Subspace join for ` DVecA ... |
djavalN 40496 | Subspace join for ` DVecA ... |
djaclN 40497 | Closure of subspace join f... |
djajN 40498 | Transfer lattice join to `... |
dibffval 40501 | The partial isomorphism B ... |
dibfval 40502 | The partial isomorphism B ... |
dibval 40503 | The partial isomorphism B ... |
dibopelvalN 40504 | Member of the partial isom... |
dibval2 40505 | Value of the partial isomo... |
dibopelval2 40506 | Member of the partial isom... |
dibval3N 40507 | Value of the partial isomo... |
dibelval3 40508 | Member of the partial isom... |
dibopelval3 40509 | Member of the partial isom... |
dibelval1st 40510 | Membership in value of the... |
dibelval1st1 40511 | Membership in value of the... |
dibelval1st2N 40512 | Membership in value of the... |
dibelval2nd 40513 | Membership in value of the... |
dibn0 40514 | The value of the partial i... |
dibfna 40515 | Functionality and domain o... |
dibdiadm 40516 | Domain of the partial isom... |
dibfnN 40517 | Functionality and domain o... |
dibdmN 40518 | Domain of the partial isom... |
dibeldmN 40519 | Member of domain of the pa... |
dibord 40520 | The isomorphism B for a la... |
dib11N 40521 | The isomorphism B for a la... |
dibf11N 40522 | The partial isomorphism A ... |
dibclN 40523 | Closure of partial isomorp... |
dibvalrel 40524 | The value of partial isomo... |
dib0 40525 | The value of partial isomo... |
dib1dim 40526 | Two expressions for the 1-... |
dibglbN 40527 | Partial isomorphism B of a... |
dibintclN 40528 | The intersection of partia... |
dib1dim2 40529 | Two expressions for a 1-di... |
dibss 40530 | The partial isomorphism B ... |
diblss 40531 | The value of partial isomo... |
diblsmopel 40532 | Membership in subspace sum... |
dicffval 40535 | The partial isomorphism C ... |
dicfval 40536 | The partial isomorphism C ... |
dicval 40537 | The partial isomorphism C ... |
dicopelval 40538 | Membership in value of the... |
dicelvalN 40539 | Membership in value of the... |
dicval2 40540 | The partial isomorphism C ... |
dicelval3 40541 | Member of the partial isom... |
dicopelval2 40542 | Membership in value of the... |
dicelval2N 40543 | Membership in value of the... |
dicfnN 40544 | Functionality and domain o... |
dicdmN 40545 | Domain of the partial isom... |
dicvalrelN 40546 | The value of partial isomo... |
dicssdvh 40547 | The partial isomorphism C ... |
dicelval1sta 40548 | Membership in value of the... |
dicelval1stN 40549 | Membership in value of the... |
dicelval2nd 40550 | Membership in value of the... |
dicvaddcl 40551 | Membership in value of the... |
dicvscacl 40552 | Membership in value of the... |
dicn0 40553 | The value of the partial i... |
diclss 40554 | The value of partial isomo... |
diclspsn 40555 | The value of isomorphism C... |
cdlemn2 40556 | Part of proof of Lemma N o... |
cdlemn2a 40557 | Part of proof of Lemma N o... |
cdlemn3 40558 | Part of proof of Lemma N o... |
cdlemn4 40559 | Part of proof of Lemma N o... |
cdlemn4a 40560 | Part of proof of Lemma N o... |
cdlemn5pre 40561 | Part of proof of Lemma N o... |
cdlemn5 40562 | Part of proof of Lemma N o... |
cdlemn6 40563 | Part of proof of Lemma N o... |
cdlemn7 40564 | Part of proof of Lemma N o... |
cdlemn8 40565 | Part of proof of Lemma N o... |
cdlemn9 40566 | Part of proof of Lemma N o... |
cdlemn10 40567 | Part of proof of Lemma N o... |
cdlemn11a 40568 | Part of proof of Lemma N o... |
cdlemn11b 40569 | Part of proof of Lemma N o... |
cdlemn11c 40570 | Part of proof of Lemma N o... |
cdlemn11pre 40571 | Part of proof of Lemma N o... |
cdlemn11 40572 | Part of proof of Lemma N o... |
cdlemn 40573 | Lemma N of [Crawley] p. 12... |
dihordlem6 40574 | Part of proof of Lemma N o... |
dihordlem7 40575 | Part of proof of Lemma N o... |
dihordlem7b 40576 | Part of proof of Lemma N o... |
dihjustlem 40577 | Part of proof after Lemma ... |
dihjust 40578 | Part of proof after Lemma ... |
dihord1 40579 | Part of proof after Lemma ... |
dihord2a 40580 | Part of proof after Lemma ... |
dihord2b 40581 | Part of proof after Lemma ... |
dihord2cN 40582 | Part of proof after Lemma ... |
dihord11b 40583 | Part of proof after Lemma ... |
dihord10 40584 | Part of proof after Lemma ... |
dihord11c 40585 | Part of proof after Lemma ... |
dihord2pre 40586 | Part of proof after Lemma ... |
dihord2pre2 40587 | Part of proof after Lemma ... |
dihord2 40588 | Part of proof after Lemma ... |
dihffval 40591 | The isomorphism H for a la... |
dihfval 40592 | Isomorphism H for a lattic... |
dihval 40593 | Value of isomorphism H for... |
dihvalc 40594 | Value of isomorphism H for... |
dihlsscpre 40595 | Closure of isomorphism H f... |
dihvalcqpre 40596 | Value of isomorphism H for... |
dihvalcq 40597 | Value of isomorphism H for... |
dihvalb 40598 | Value of isomorphism H for... |
dihopelvalbN 40599 | Ordered pair member of the... |
dihvalcqat 40600 | Value of isomorphism H for... |
dih1dimb 40601 | Two expressions for a 1-di... |
dih1dimb2 40602 | Isomorphism H at an atom u... |
dih1dimc 40603 | Isomorphism H at an atom n... |
dib2dim 40604 | Extend ~ dia2dim to partia... |
dih2dimb 40605 | Extend ~ dib2dim to isomor... |
dih2dimbALTN 40606 | Extend ~ dia2dim to isomor... |
dihopelvalcqat 40607 | Ordered pair member of the... |
dihvalcq2 40608 | Value of isomorphism H for... |
dihopelvalcpre 40609 | Member of value of isomorp... |
dihopelvalc 40610 | Member of value of isomorp... |
dihlss 40611 | The value of isomorphism H... |
dihss 40612 | The value of isomorphism H... |
dihssxp 40613 | An isomorphism H value is ... |
dihopcl 40614 | Closure of an ordered pair... |
xihopellsmN 40615 | Ordered pair membership in... |
dihopellsm 40616 | Ordered pair membership in... |
dihord6apre 40617 | Part of proof that isomorp... |
dihord3 40618 | The isomorphism H for a la... |
dihord4 40619 | The isomorphism H for a la... |
dihord5b 40620 | Part of proof that isomorp... |
dihord6b 40621 | Part of proof that isomorp... |
dihord6a 40622 | Part of proof that isomorp... |
dihord5apre 40623 | Part of proof that isomorp... |
dihord5a 40624 | Part of proof that isomorp... |
dihord 40625 | The isomorphism H is order... |
dih11 40626 | The isomorphism H is one-t... |
dihf11lem 40627 | Functionality of the isomo... |
dihf11 40628 | The isomorphism H for a la... |
dihfn 40629 | Functionality and domain o... |
dihdm 40630 | Domain of isomorphism H. (... |
dihcl 40631 | Closure of isomorphism H. ... |
dihcnvcl 40632 | Closure of isomorphism H c... |
dihcnvid1 40633 | The converse isomorphism o... |
dihcnvid2 40634 | The isomorphism of a conve... |
dihcnvord 40635 | Ordering property for conv... |
dihcnv11 40636 | The converse of isomorphis... |
dihsslss 40637 | The isomorphism H maps to ... |
dihrnlss 40638 | The isomorphism H maps to ... |
dihrnss 40639 | The isomorphism H maps to ... |
dihvalrel 40640 | The value of isomorphism H... |
dih0 40641 | The value of isomorphism H... |
dih0bN 40642 | A lattice element is zero ... |
dih0vbN 40643 | A vector is zero iff its s... |
dih0cnv 40644 | The isomorphism H converse... |
dih0rn 40645 | The zero subspace belongs ... |
dih0sb 40646 | A subspace is zero iff the... |
dih1 40647 | The value of isomorphism H... |
dih1rn 40648 | The full vector space belo... |
dih1cnv 40649 | The isomorphism H converse... |
dihwN 40650 | Value of isomorphism H at ... |
dihmeetlem1N 40651 | Isomorphism H of a conjunc... |
dihglblem5apreN 40652 | A conjunction property of ... |
dihglblem5aN 40653 | A conjunction property of ... |
dihglblem2aN 40654 | Lemma for isomorphism H of... |
dihglblem2N 40655 | The GLB of a set of lattic... |
dihglblem3N 40656 | Isomorphism H of a lattice... |
dihglblem3aN 40657 | Isomorphism H of a lattice... |
dihglblem4 40658 | Isomorphism H of a lattice... |
dihglblem5 40659 | Isomorphism H of a lattice... |
dihmeetlem2N 40660 | Isomorphism H of a conjunc... |
dihglbcpreN 40661 | Isomorphism H of a lattice... |
dihglbcN 40662 | Isomorphism H of a lattice... |
dihmeetcN 40663 | Isomorphism H of a lattice... |
dihmeetbN 40664 | Isomorphism H of a lattice... |
dihmeetbclemN 40665 | Lemma for isomorphism H of... |
dihmeetlem3N 40666 | Lemma for isomorphism H of... |
dihmeetlem4preN 40667 | Lemma for isomorphism H of... |
dihmeetlem4N 40668 | Lemma for isomorphism H of... |
dihmeetlem5 40669 | Part of proof that isomorp... |
dihmeetlem6 40670 | Lemma for isomorphism H of... |
dihmeetlem7N 40671 | Lemma for isomorphism H of... |
dihjatc1 40672 | Lemma for isomorphism H of... |
dihjatc2N 40673 | Isomorphism H of join with... |
dihjatc3 40674 | Isomorphism H of join with... |
dihmeetlem8N 40675 | Lemma for isomorphism H of... |
dihmeetlem9N 40676 | Lemma for isomorphism H of... |
dihmeetlem10N 40677 | Lemma for isomorphism H of... |
dihmeetlem11N 40678 | Lemma for isomorphism H of... |
dihmeetlem12N 40679 | Lemma for isomorphism H of... |
dihmeetlem13N 40680 | Lemma for isomorphism H of... |
dihmeetlem14N 40681 | Lemma for isomorphism H of... |
dihmeetlem15N 40682 | Lemma for isomorphism H of... |
dihmeetlem16N 40683 | Lemma for isomorphism H of... |
dihmeetlem17N 40684 | Lemma for isomorphism H of... |
dihmeetlem18N 40685 | Lemma for isomorphism H of... |
dihmeetlem19N 40686 | Lemma for isomorphism H of... |
dihmeetlem20N 40687 | Lemma for isomorphism H of... |
dihmeetALTN 40688 | Isomorphism H of a lattice... |
dih1dimatlem0 40689 | Lemma for ~ dih1dimat . (... |
dih1dimatlem 40690 | Lemma for ~ dih1dimat . (... |
dih1dimat 40691 | Any 1-dimensional subspace... |
dihlsprn 40692 | The span of a vector belon... |
dihlspsnssN 40693 | A subspace included in a 1... |
dihlspsnat 40694 | The inverse isomorphism H ... |
dihatlat 40695 | The isomorphism H of an at... |
dihat 40696 | There exists at least one ... |
dihpN 40697 | The value of isomorphism H... |
dihlatat 40698 | The reverse isomorphism H ... |
dihatexv 40699 | There is a nonzero vector ... |
dihatexv2 40700 | There is a nonzero vector ... |
dihglblem6 40701 | Isomorphism H of a lattice... |
dihglb 40702 | Isomorphism H of a lattice... |
dihglb2 40703 | Isomorphism H of a lattice... |
dihmeet 40704 | Isomorphism H of a lattice... |
dihintcl 40705 | The intersection of closed... |
dihmeetcl 40706 | Closure of closed subspace... |
dihmeet2 40707 | Reverse isomorphism H of a... |
dochffval 40710 | Subspace orthocomplement f... |
dochfval 40711 | Subspace orthocomplement f... |
dochval 40712 | Subspace orthocomplement f... |
dochval2 40713 | Subspace orthocomplement f... |
dochcl 40714 | Closure of subspace orthoc... |
dochlss 40715 | A subspace orthocomplement... |
dochssv 40716 | A subspace orthocomplement... |
dochfN 40717 | Domain and codomain of the... |
dochvalr 40718 | Orthocomplement of a close... |
doch0 40719 | Orthocomplement of the zer... |
doch1 40720 | Orthocomplement of the uni... |
dochoc0 40721 | The zero subspace is close... |
dochoc1 40722 | The unit subspace (all vec... |
dochvalr2 40723 | Orthocomplement of a close... |
dochvalr3 40724 | Orthocomplement of a close... |
doch2val2 40725 | Double orthocomplement for... |
dochss 40726 | Subset law for orthocomple... |
dochocss 40727 | Double negative law for or... |
dochoc 40728 | Double negative law for or... |
dochsscl 40729 | If a set of vectors is inc... |
dochoccl 40730 | A set of vectors is closed... |
dochord 40731 | Ordering law for orthocomp... |
dochord2N 40732 | Ordering law for orthocomp... |
dochord3 40733 | Ordering law for orthocomp... |
doch11 40734 | Orthocomplement is one-to-... |
dochsordN 40735 | Strict ordering law for or... |
dochn0nv 40736 | An orthocomplement is nonz... |
dihoml4c 40737 | Version of ~ dihoml4 with ... |
dihoml4 40738 | Orthomodular law for const... |
dochspss 40739 | The span of a set of vecto... |
dochocsp 40740 | The span of an orthocomple... |
dochspocN 40741 | The span of an orthocomple... |
dochocsn 40742 | The double orthocomplement... |
dochsncom 40743 | Swap vectors in an orthoco... |
dochsat 40744 | The double orthocomplement... |
dochshpncl 40745 | If a hyperplane is not clo... |
dochlkr 40746 | Equivalent conditions for ... |
dochkrshp 40747 | The closure of a kernel is... |
dochkrshp2 40748 | Properties of the closure ... |
dochkrshp3 40749 | Properties of the closure ... |
dochkrshp4 40750 | Properties of the closure ... |
dochdmj1 40751 | De Morgan-like law for sub... |
dochnoncon 40752 | Law of noncontradiction. ... |
dochnel2 40753 | A nonzero member of a subs... |
dochnel 40754 | A nonzero vector doesn't b... |
djhffval 40757 | Subspace join for ` DVecH ... |
djhfval 40758 | Subspace join for ` DVecH ... |
djhval 40759 | Subspace join for ` DVecH ... |
djhval2 40760 | Value of subspace join for... |
djhcl 40761 | Closure of subspace join f... |
djhlj 40762 | Transfer lattice join to `... |
djhljjN 40763 | Lattice join in terms of `... |
djhjlj 40764 | ` DVecH ` vector space clo... |
djhj 40765 | ` DVecH ` vector space clo... |
djhcom 40766 | Subspace join commutes. (... |
djhspss 40767 | Subspace span of union is ... |
djhsumss 40768 | Subspace sum is a subset o... |
dihsumssj 40769 | The subspace sum of two is... |
djhunssN 40770 | Subspace union is a subset... |
dochdmm1 40771 | De Morgan-like law for clo... |
djhexmid 40772 | Excluded middle property o... |
djh01 40773 | Closed subspace join with ... |
djh02 40774 | Closed subspace join with ... |
djhlsmcl 40775 | A closed subspace sum equa... |
djhcvat42 40776 | A covering property. ( ~ ... |
dihjatb 40777 | Isomorphism H of lattice j... |
dihjatc 40778 | Isomorphism H of lattice j... |
dihjatcclem1 40779 | Lemma for isomorphism H of... |
dihjatcclem2 40780 | Lemma for isomorphism H of... |
dihjatcclem3 40781 | Lemma for ~ dihjatcc . (C... |
dihjatcclem4 40782 | Lemma for isomorphism H of... |
dihjatcc 40783 | Isomorphism H of lattice j... |
dihjat 40784 | Isomorphism H of lattice j... |
dihprrnlem1N 40785 | Lemma for ~ dihprrn , show... |
dihprrnlem2 40786 | Lemma for ~ dihprrn . (Co... |
dihprrn 40787 | The span of a vector pair ... |
djhlsmat 40788 | The sum of two subspace at... |
dihjat1lem 40789 | Subspace sum of a closed s... |
dihjat1 40790 | Subspace sum of a closed s... |
dihsmsprn 40791 | Subspace sum of a closed s... |
dihjat2 40792 | The subspace sum of a clos... |
dihjat3 40793 | Isomorphism H of lattice j... |
dihjat4 40794 | Transfer the subspace sum ... |
dihjat6 40795 | Transfer the subspace sum ... |
dihsmsnrn 40796 | The subspace sum of two si... |
dihsmatrn 40797 | The subspace sum of a clos... |
dihjat5N 40798 | Transfer lattice join with... |
dvh4dimat 40799 | There is an atom that is o... |
dvh3dimatN 40800 | There is an atom that is o... |
dvh2dimatN 40801 | Given an atom, there exist... |
dvh1dimat 40802 | There exists an atom. (Co... |
dvh1dim 40803 | There exists a nonzero vec... |
dvh4dimlem 40804 | Lemma for ~ dvh4dimN . (C... |
dvhdimlem 40805 | Lemma for ~ dvh2dim and ~ ... |
dvh2dim 40806 | There is a vector that is ... |
dvh3dim 40807 | There is a vector that is ... |
dvh4dimN 40808 | There is a vector that is ... |
dvh3dim2 40809 | There is a vector that is ... |
dvh3dim3N 40810 | There is a vector that is ... |
dochsnnz 40811 | The orthocomplement of a s... |
dochsatshp 40812 | The orthocomplement of a s... |
dochsatshpb 40813 | The orthocomplement of a s... |
dochsnshp 40814 | The orthocomplement of a n... |
dochshpsat 40815 | A hyperplane is closed iff... |
dochkrsat 40816 | The orthocomplement of a k... |
dochkrsat2 40817 | The orthocomplement of a k... |
dochsat0 40818 | The orthocomplement of a k... |
dochkrsm 40819 | The subspace sum of a clos... |
dochexmidat 40820 | Special case of excluded m... |
dochexmidlem1 40821 | Lemma for ~ dochexmid . H... |
dochexmidlem2 40822 | Lemma for ~ dochexmid . (... |
dochexmidlem3 40823 | Lemma for ~ dochexmid . U... |
dochexmidlem4 40824 | Lemma for ~ dochexmid . (... |
dochexmidlem5 40825 | Lemma for ~ dochexmid . (... |
dochexmidlem6 40826 | Lemma for ~ dochexmid . (... |
dochexmidlem7 40827 | Lemma for ~ dochexmid . C... |
dochexmidlem8 40828 | Lemma for ~ dochexmid . T... |
dochexmid 40829 | Excluded middle law for cl... |
dochsnkrlem1 40830 | Lemma for ~ dochsnkr . (C... |
dochsnkrlem2 40831 | Lemma for ~ dochsnkr . (C... |
dochsnkrlem3 40832 | Lemma for ~ dochsnkr . (C... |
dochsnkr 40833 | A (closed) kernel expresse... |
dochsnkr2 40834 | Kernel of the explicit fun... |
dochsnkr2cl 40835 | The ` X ` determining func... |
dochflcl 40836 | Closure of the explicit fu... |
dochfl1 40837 | The value of the explicit ... |
dochfln0 40838 | The value of a functional ... |
dochkr1 40839 | A nonzero functional has a... |
dochkr1OLDN 40840 | A nonzero functional has a... |
lpolsetN 40843 | The set of polarities of a... |
islpolN 40844 | The predicate "is a polari... |
islpoldN 40845 | Properties that determine ... |
lpolfN 40846 | Functionality of a polarit... |
lpolvN 40847 | The polarity of the whole ... |
lpolconN 40848 | Contraposition property of... |
lpolsatN 40849 | The polarity of an atomic ... |
lpolpolsatN 40850 | Property of a polarity. (... |
dochpolN 40851 | The subspace orthocompleme... |
lcfl1lem 40852 | Property of a functional w... |
lcfl1 40853 | Property of a functional w... |
lcfl2 40854 | Property of a functional w... |
lcfl3 40855 | Property of a functional w... |
lcfl4N 40856 | Property of a functional w... |
lcfl5 40857 | Property of a functional w... |
lcfl5a 40858 | Property of a functional w... |
lcfl6lem 40859 | Lemma for ~ lcfl6 . A fun... |
lcfl7lem 40860 | Lemma for ~ lcfl7N . If t... |
lcfl6 40861 | Property of a functional w... |
lcfl7N 40862 | Property of a functional w... |
lcfl8 40863 | Property of a functional w... |
lcfl8a 40864 | Property of a functional w... |
lcfl8b 40865 | Property of a nonzero func... |
lcfl9a 40866 | Property implying that a f... |
lclkrlem1 40867 | The set of functionals hav... |
lclkrlem2a 40868 | Lemma for ~ lclkr . Use ~... |
lclkrlem2b 40869 | Lemma for ~ lclkr . (Cont... |
lclkrlem2c 40870 | Lemma for ~ lclkr . (Cont... |
lclkrlem2d 40871 | Lemma for ~ lclkr . (Cont... |
lclkrlem2e 40872 | Lemma for ~ lclkr . The k... |
lclkrlem2f 40873 | Lemma for ~ lclkr . Const... |
lclkrlem2g 40874 | Lemma for ~ lclkr . Compa... |
lclkrlem2h 40875 | Lemma for ~ lclkr . Elimi... |
lclkrlem2i 40876 | Lemma for ~ lclkr . Elimi... |
lclkrlem2j 40877 | Lemma for ~ lclkr . Kerne... |
lclkrlem2k 40878 | Lemma for ~ lclkr . Kerne... |
lclkrlem2l 40879 | Lemma for ~ lclkr . Elimi... |
lclkrlem2m 40880 | Lemma for ~ lclkr . Const... |
lclkrlem2n 40881 | Lemma for ~ lclkr . (Cont... |
lclkrlem2o 40882 | Lemma for ~ lclkr . When ... |
lclkrlem2p 40883 | Lemma for ~ lclkr . When ... |
lclkrlem2q 40884 | Lemma for ~ lclkr . The s... |
lclkrlem2r 40885 | Lemma for ~ lclkr . When ... |
lclkrlem2s 40886 | Lemma for ~ lclkr . Thus,... |
lclkrlem2t 40887 | Lemma for ~ lclkr . We el... |
lclkrlem2u 40888 | Lemma for ~ lclkr . ~ lclk... |
lclkrlem2v 40889 | Lemma for ~ lclkr . When ... |
lclkrlem2w 40890 | Lemma for ~ lclkr . This ... |
lclkrlem2x 40891 | Lemma for ~ lclkr . Elimi... |
lclkrlem2y 40892 | Lemma for ~ lclkr . Resta... |
lclkrlem2 40893 | The set of functionals hav... |
lclkr 40894 | The set of functionals wit... |
lcfls1lem 40895 | Property of a functional w... |
lcfls1N 40896 | Property of a functional w... |
lcfls1c 40897 | Property of a functional w... |
lclkrslem1 40898 | The set of functionals hav... |
lclkrslem2 40899 | The set of functionals hav... |
lclkrs 40900 | The set of functionals hav... |
lclkrs2 40901 | The set of functionals wit... |
lcfrvalsnN 40902 | Reconstruction from the du... |
lcfrlem1 40903 | Lemma for ~ lcfr . Note t... |
lcfrlem2 40904 | Lemma for ~ lcfr . (Contr... |
lcfrlem3 40905 | Lemma for ~ lcfr . (Contr... |
lcfrlem4 40906 | Lemma for ~ lcfr . (Contr... |
lcfrlem5 40907 | Lemma for ~ lcfr . The se... |
lcfrlem6 40908 | Lemma for ~ lcfr . Closur... |
lcfrlem7 40909 | Lemma for ~ lcfr . Closur... |
lcfrlem8 40910 | Lemma for ~ lcf1o and ~ lc... |
lcfrlem9 40911 | Lemma for ~ lcf1o . (This... |
lcf1o 40912 | Define a function ` J ` th... |
lcfrlem10 40913 | Lemma for ~ lcfr . (Contr... |
lcfrlem11 40914 | Lemma for ~ lcfr . (Contr... |
lcfrlem12N 40915 | Lemma for ~ lcfr . (Contr... |
lcfrlem13 40916 | Lemma for ~ lcfr . (Contr... |
lcfrlem14 40917 | Lemma for ~ lcfr . (Contr... |
lcfrlem15 40918 | Lemma for ~ lcfr . (Contr... |
lcfrlem16 40919 | Lemma for ~ lcfr . (Contr... |
lcfrlem17 40920 | Lemma for ~ lcfr . Condit... |
lcfrlem18 40921 | Lemma for ~ lcfr . (Contr... |
lcfrlem19 40922 | Lemma for ~ lcfr . (Contr... |
lcfrlem20 40923 | Lemma for ~ lcfr . (Contr... |
lcfrlem21 40924 | Lemma for ~ lcfr . (Contr... |
lcfrlem22 40925 | Lemma for ~ lcfr . (Contr... |
lcfrlem23 40926 | Lemma for ~ lcfr . TODO: ... |
lcfrlem24 40927 | Lemma for ~ lcfr . (Contr... |
lcfrlem25 40928 | Lemma for ~ lcfr . Specia... |
lcfrlem26 40929 | Lemma for ~ lcfr . Specia... |
lcfrlem27 40930 | Lemma for ~ lcfr . Specia... |
lcfrlem28 40931 | Lemma for ~ lcfr . TODO: ... |
lcfrlem29 40932 | Lemma for ~ lcfr . (Contr... |
lcfrlem30 40933 | Lemma for ~ lcfr . (Contr... |
lcfrlem31 40934 | Lemma for ~ lcfr . (Contr... |
lcfrlem32 40935 | Lemma for ~ lcfr . (Contr... |
lcfrlem33 40936 | Lemma for ~ lcfr . (Contr... |
lcfrlem34 40937 | Lemma for ~ lcfr . (Contr... |
lcfrlem35 40938 | Lemma for ~ lcfr . (Contr... |
lcfrlem36 40939 | Lemma for ~ lcfr . (Contr... |
lcfrlem37 40940 | Lemma for ~ lcfr . (Contr... |
lcfrlem38 40941 | Lemma for ~ lcfr . Combin... |
lcfrlem39 40942 | Lemma for ~ lcfr . Elimin... |
lcfrlem40 40943 | Lemma for ~ lcfr . Elimin... |
lcfrlem41 40944 | Lemma for ~ lcfr . Elimin... |
lcfrlem42 40945 | Lemma for ~ lcfr . Elimin... |
lcfr 40946 | Reconstruction of a subspa... |
lcdfval 40949 | Dual vector space of funct... |
lcdval 40950 | Dual vector space of funct... |
lcdval2 40951 | Dual vector space of funct... |
lcdlvec 40952 | The dual vector space of f... |
lcdlmod 40953 | The dual vector space of f... |
lcdvbase 40954 | Vector base set of a dual ... |
lcdvbasess 40955 | The vector base set of the... |
lcdvbaselfl 40956 | A vector in the base set o... |
lcdvbasecl 40957 | Closure of the value of a ... |
lcdvadd 40958 | Vector addition for the cl... |
lcdvaddval 40959 | The value of the value of ... |
lcdsca 40960 | The ring of scalars of the... |
lcdsbase 40961 | Base set of scalar ring fo... |
lcdsadd 40962 | Scalar addition for the cl... |
lcdsmul 40963 | Scalar multiplication for ... |
lcdvs 40964 | Scalar product for the clo... |
lcdvsval 40965 | Value of scalar product op... |
lcdvscl 40966 | The scalar product operati... |
lcdlssvscl 40967 | Closure of scalar product ... |
lcdvsass 40968 | Associative law for scalar... |
lcd0 40969 | The zero scalar of the clo... |
lcd1 40970 | The unit scalar of the clo... |
lcdneg 40971 | The unit scalar of the clo... |
lcd0v 40972 | The zero functional in the... |
lcd0v2 40973 | The zero functional in the... |
lcd0vvalN 40974 | Value of the zero function... |
lcd0vcl 40975 | Closure of the zero functi... |
lcd0vs 40976 | A scalar zero times a func... |
lcdvs0N 40977 | A scalar times the zero fu... |
lcdvsub 40978 | The value of vector subtra... |
lcdvsubval 40979 | The value of the value of ... |
lcdlss 40980 | Subspaces of a dual vector... |
lcdlss2N 40981 | Subspaces of a dual vector... |
lcdlsp 40982 | Span in the set of functio... |
lcdlkreqN 40983 | Colinear functionals have ... |
lcdlkreq2N 40984 | Colinear functionals have ... |
mapdffval 40987 | Projectivity from vector s... |
mapdfval 40988 | Projectivity from vector s... |
mapdval 40989 | Value of projectivity from... |
mapdvalc 40990 | Value of projectivity from... |
mapdval2N 40991 | Value of projectivity from... |
mapdval3N 40992 | Value of projectivity from... |
mapdval4N 40993 | Value of projectivity from... |
mapdval5N 40994 | Value of projectivity from... |
mapdordlem1a 40995 | Lemma for ~ mapdord . (Co... |
mapdordlem1bN 40996 | Lemma for ~ mapdord . (Co... |
mapdordlem1 40997 | Lemma for ~ mapdord . (Co... |
mapdordlem2 40998 | Lemma for ~ mapdord . Ord... |
mapdord 40999 | Ordering property of the m... |
mapd11 41000 | The map defined by ~ df-ma... |
mapddlssN 41001 | The mapping of a subspace ... |
mapdsn 41002 | Value of the map defined b... |
mapdsn2 41003 | Value of the map defined b... |
mapdsn3 41004 | Value of the map defined b... |
mapd1dim2lem1N 41005 | Value of the map defined b... |
mapdrvallem2 41006 | Lemma for ~ mapdrval . TO... |
mapdrvallem3 41007 | Lemma for ~ mapdrval . (C... |
mapdrval 41008 | Given a dual subspace ` R ... |
mapd1o 41009 | The map defined by ~ df-ma... |
mapdrn 41010 | Range of the map defined b... |
mapdunirnN 41011 | Union of the range of the ... |
mapdrn2 41012 | Range of the map defined b... |
mapdcnvcl 41013 | Closure of the converse of... |
mapdcl 41014 | Closure the value of the m... |
mapdcnvid1N 41015 | Converse of the value of t... |
mapdsord 41016 | Strong ordering property o... |
mapdcl2 41017 | The mapping of a subspace ... |
mapdcnvid2 41018 | Value of the converse of t... |
mapdcnvordN 41019 | Ordering property of the c... |
mapdcnv11N 41020 | The converse of the map de... |
mapdcv 41021 | Covering property of the c... |
mapdincl 41022 | Closure of dual subspace i... |
mapdin 41023 | Subspace intersection is p... |
mapdlsmcl 41024 | Closure of dual subspace s... |
mapdlsm 41025 | Subspace sum is preserved ... |
mapd0 41026 | Projectivity map of the ze... |
mapdcnvatN 41027 | Atoms are preserved by the... |
mapdat 41028 | Atoms are preserved by the... |
mapdspex 41029 | The map of a span equals t... |
mapdn0 41030 | Transfer nonzero property ... |
mapdncol 41031 | Transfer non-colinearity f... |
mapdindp 41032 | Transfer (part of) vector ... |
mapdpglem1 41033 | Lemma for ~ mapdpg . Baer... |
mapdpglem2 41034 | Lemma for ~ mapdpg . Baer... |
mapdpglem2a 41035 | Lemma for ~ mapdpg . (Con... |
mapdpglem3 41036 | Lemma for ~ mapdpg . Baer... |
mapdpglem4N 41037 | Lemma for ~ mapdpg . (Con... |
mapdpglem5N 41038 | Lemma for ~ mapdpg . (Con... |
mapdpglem6 41039 | Lemma for ~ mapdpg . Baer... |
mapdpglem8 41040 | Lemma for ~ mapdpg . Baer... |
mapdpglem9 41041 | Lemma for ~ mapdpg . Baer... |
mapdpglem10 41042 | Lemma for ~ mapdpg . Baer... |
mapdpglem11 41043 | Lemma for ~ mapdpg . (Con... |
mapdpglem12 41044 | Lemma for ~ mapdpg . TODO... |
mapdpglem13 41045 | Lemma for ~ mapdpg . (Con... |
mapdpglem14 41046 | Lemma for ~ mapdpg . (Con... |
mapdpglem15 41047 | Lemma for ~ mapdpg . (Con... |
mapdpglem16 41048 | Lemma for ~ mapdpg . Baer... |
mapdpglem17N 41049 | Lemma for ~ mapdpg . Baer... |
mapdpglem18 41050 | Lemma for ~ mapdpg . Baer... |
mapdpglem19 41051 | Lemma for ~ mapdpg . Baer... |
mapdpglem20 41052 | Lemma for ~ mapdpg . Baer... |
mapdpglem21 41053 | Lemma for ~ mapdpg . (Con... |
mapdpglem22 41054 | Lemma for ~ mapdpg . Baer... |
mapdpglem23 41055 | Lemma for ~ mapdpg . Baer... |
mapdpglem30a 41056 | Lemma for ~ mapdpg . (Con... |
mapdpglem30b 41057 | Lemma for ~ mapdpg . (Con... |
mapdpglem25 41058 | Lemma for ~ mapdpg . Baer... |
mapdpglem26 41059 | Lemma for ~ mapdpg . Baer... |
mapdpglem27 41060 | Lemma for ~ mapdpg . Baer... |
mapdpglem29 41061 | Lemma for ~ mapdpg . Baer... |
mapdpglem28 41062 | Lemma for ~ mapdpg . Baer... |
mapdpglem30 41063 | Lemma for ~ mapdpg . Baer... |
mapdpglem31 41064 | Lemma for ~ mapdpg . Baer... |
mapdpglem24 41065 | Lemma for ~ mapdpg . Exis... |
mapdpglem32 41066 | Lemma for ~ mapdpg . Uniq... |
mapdpg 41067 | Part 1 of proof of the fir... |
baerlem3lem1 41068 | Lemma for ~ baerlem3 . (C... |
baerlem5alem1 41069 | Lemma for ~ baerlem5a . (... |
baerlem5blem1 41070 | Lemma for ~ baerlem5b . (... |
baerlem3lem2 41071 | Lemma for ~ baerlem3 . (C... |
baerlem5alem2 41072 | Lemma for ~ baerlem5a . (... |
baerlem5blem2 41073 | Lemma for ~ baerlem5b . (... |
baerlem3 41074 | An equality that holds whe... |
baerlem5a 41075 | An equality that holds whe... |
baerlem5b 41076 | An equality that holds whe... |
baerlem5amN 41077 | An equality that holds whe... |
baerlem5bmN 41078 | An equality that holds whe... |
baerlem5abmN 41079 | An equality that holds whe... |
mapdindp0 41080 | Vector independence lemma.... |
mapdindp1 41081 | Vector independence lemma.... |
mapdindp2 41082 | Vector independence lemma.... |
mapdindp3 41083 | Vector independence lemma.... |
mapdindp4 41084 | Vector independence lemma.... |
mapdhval 41085 | Lemmma for ~~? mapdh . (C... |
mapdhval0 41086 | Lemmma for ~~? mapdh . (C... |
mapdhval2 41087 | Lemmma for ~~? mapdh . (C... |
mapdhcl 41088 | Lemmma for ~~? mapdh . (C... |
mapdheq 41089 | Lemmma for ~~? mapdh . Th... |
mapdheq2 41090 | Lemmma for ~~? mapdh . On... |
mapdheq2biN 41091 | Lemmma for ~~? mapdh . Pa... |
mapdheq4lem 41092 | Lemma for ~ mapdheq4 . Pa... |
mapdheq4 41093 | Lemma for ~~? mapdh . Par... |
mapdh6lem1N 41094 | Lemma for ~ mapdh6N . Par... |
mapdh6lem2N 41095 | Lemma for ~ mapdh6N . Par... |
mapdh6aN 41096 | Lemma for ~ mapdh6N . Par... |
mapdh6b0N 41097 | Lemmma for ~ mapdh6N . (C... |
mapdh6bN 41098 | Lemmma for ~ mapdh6N . (C... |
mapdh6cN 41099 | Lemmma for ~ mapdh6N . (C... |
mapdh6dN 41100 | Lemmma for ~ mapdh6N . (C... |
mapdh6eN 41101 | Lemmma for ~ mapdh6N . Pa... |
mapdh6fN 41102 | Lemmma for ~ mapdh6N . Pa... |
mapdh6gN 41103 | Lemmma for ~ mapdh6N . Pa... |
mapdh6hN 41104 | Lemmma for ~ mapdh6N . Pa... |
mapdh6iN 41105 | Lemmma for ~ mapdh6N . El... |
mapdh6jN 41106 | Lemmma for ~ mapdh6N . El... |
mapdh6kN 41107 | Lemmma for ~ mapdh6N . El... |
mapdh6N 41108 | Part (6) of [Baer] p. 47 l... |
mapdh7eN 41109 | Part (7) of [Baer] p. 48 l... |
mapdh7cN 41110 | Part (7) of [Baer] p. 48 l... |
mapdh7dN 41111 | Part (7) of [Baer] p. 48 l... |
mapdh7fN 41112 | Part (7) of [Baer] p. 48 l... |
mapdh75e 41113 | Part (7) of [Baer] p. 48 l... |
mapdh75cN 41114 | Part (7) of [Baer] p. 48 l... |
mapdh75d 41115 | Part (7) of [Baer] p. 48 l... |
mapdh75fN 41116 | Part (7) of [Baer] p. 48 l... |
hvmapffval 41119 | Map from nonzero vectors t... |
hvmapfval 41120 | Map from nonzero vectors t... |
hvmapval 41121 | Value of map from nonzero ... |
hvmapvalvalN 41122 | Value of value of map (i.e... |
hvmapidN 41123 | The value of the vector to... |
hvmap1o 41124 | The vector to functional m... |
hvmapclN 41125 | Closure of the vector to f... |
hvmap1o2 41126 | The vector to functional m... |
hvmapcl2 41127 | Closure of the vector to f... |
hvmaplfl 41128 | The vector to functional m... |
hvmaplkr 41129 | Kernel of the vector to fu... |
mapdhvmap 41130 | Relationship between ` map... |
lspindp5 41131 | Obtain an independent vect... |
hdmaplem1 41132 | Lemma to convert a frequen... |
hdmaplem2N 41133 | Lemma to convert a frequen... |
hdmaplem3 41134 | Lemma to convert a frequen... |
hdmaplem4 41135 | Lemma to convert a frequen... |
mapdh8a 41136 | Part of Part (8) in [Baer]... |
mapdh8aa 41137 | Part of Part (8) in [Baer]... |
mapdh8ab 41138 | Part of Part (8) in [Baer]... |
mapdh8ac 41139 | Part of Part (8) in [Baer]... |
mapdh8ad 41140 | Part of Part (8) in [Baer]... |
mapdh8b 41141 | Part of Part (8) in [Baer]... |
mapdh8c 41142 | Part of Part (8) in [Baer]... |
mapdh8d0N 41143 | Part of Part (8) in [Baer]... |
mapdh8d 41144 | Part of Part (8) in [Baer]... |
mapdh8e 41145 | Part of Part (8) in [Baer]... |
mapdh8g 41146 | Part of Part (8) in [Baer]... |
mapdh8i 41147 | Part of Part (8) in [Baer]... |
mapdh8j 41148 | Part of Part (8) in [Baer]... |
mapdh8 41149 | Part (8) in [Baer] p. 48. ... |
mapdh9a 41150 | Lemma for part (9) in [Bae... |
mapdh9aOLDN 41151 | Lemma for part (9) in [Bae... |
hdmap1ffval 41156 | Preliminary map from vecto... |
hdmap1fval 41157 | Preliminary map from vecto... |
hdmap1vallem 41158 | Value of preliminary map f... |
hdmap1val 41159 | Value of preliminary map f... |
hdmap1val0 41160 | Value of preliminary map f... |
hdmap1val2 41161 | Value of preliminary map f... |
hdmap1eq 41162 | The defining equation for ... |
hdmap1cbv 41163 | Frequently used lemma to c... |
hdmap1valc 41164 | Connect the value of the p... |
hdmap1cl 41165 | Convert closure theorem ~ ... |
hdmap1eq2 41166 | Convert ~ mapdheq2 to use ... |
hdmap1eq4N 41167 | Convert ~ mapdheq4 to use ... |
hdmap1l6lem1 41168 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6lem2 41169 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6a 41170 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6b0N 41171 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6b 41172 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6c 41173 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6d 41174 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6e 41175 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6f 41176 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6g 41177 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6h 41178 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6i 41179 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6j 41180 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6k 41181 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6 41182 | Part (6) of [Baer] p. 47 l... |
hdmap1eulem 41183 | Lemma for ~ hdmap1eu . TO... |
hdmap1eulemOLDN 41184 | Lemma for ~ hdmap1euOLDN .... |
hdmap1eu 41185 | Convert ~ mapdh9a to use t... |
hdmap1euOLDN 41186 | Convert ~ mapdh9aOLDN to u... |
hdmapffval 41187 | Map from vectors to functi... |
hdmapfval 41188 | Map from vectors to functi... |
hdmapval 41189 | Value of map from vectors ... |
hdmapfnN 41190 | Functionality of map from ... |
hdmapcl 41191 | Closure of map from vector... |
hdmapval2lem 41192 | Lemma for ~ hdmapval2 . (... |
hdmapval2 41193 | Value of map from vectors ... |
hdmapval0 41194 | Value of map from vectors ... |
hdmapeveclem 41195 | Lemma for ~ hdmapevec . T... |
hdmapevec 41196 | Value of map from vectors ... |
hdmapevec2 41197 | The inner product of the r... |
hdmapval3lemN 41198 | Value of map from vectors ... |
hdmapval3N 41199 | Value of map from vectors ... |
hdmap10lem 41200 | Lemma for ~ hdmap10 . (Co... |
hdmap10 41201 | Part 10 in [Baer] p. 48 li... |
hdmap11lem1 41202 | Lemma for ~ hdmapadd . (C... |
hdmap11lem2 41203 | Lemma for ~ hdmapadd . (C... |
hdmapadd 41204 | Part 11 in [Baer] p. 48 li... |
hdmapeq0 41205 | Part of proof of part 12 i... |
hdmapnzcl 41206 | Nonzero vector closure of ... |
hdmapneg 41207 | Part of proof of part 12 i... |
hdmapsub 41208 | Part of proof of part 12 i... |
hdmap11 41209 | Part of proof of part 12 i... |
hdmaprnlem1N 41210 | Part of proof of part 12 i... |
hdmaprnlem3N 41211 | Part of proof of part 12 i... |
hdmaprnlem3uN 41212 | Part of proof of part 12 i... |
hdmaprnlem4tN 41213 | Lemma for ~ hdmaprnN . TO... |
hdmaprnlem4N 41214 | Part of proof of part 12 i... |
hdmaprnlem6N 41215 | Part of proof of part 12 i... |
hdmaprnlem7N 41216 | Part of proof of part 12 i... |
hdmaprnlem8N 41217 | Part of proof of part 12 i... |
hdmaprnlem9N 41218 | Part of proof of part 12 i... |
hdmaprnlem3eN 41219 | Lemma for ~ hdmaprnN . (C... |
hdmaprnlem10N 41220 | Lemma for ~ hdmaprnN . Sh... |
hdmaprnlem11N 41221 | Lemma for ~ hdmaprnN . Sh... |
hdmaprnlem15N 41222 | Lemma for ~ hdmaprnN . El... |
hdmaprnlem16N 41223 | Lemma for ~ hdmaprnN . El... |
hdmaprnlem17N 41224 | Lemma for ~ hdmaprnN . In... |
hdmaprnN 41225 | Part of proof of part 12 i... |
hdmapf1oN 41226 | Part 12 in [Baer] p. 49. ... |
hdmap14lem1a 41227 | Prior to part 14 in [Baer]... |
hdmap14lem2a 41228 | Prior to part 14 in [Baer]... |
hdmap14lem1 41229 | Prior to part 14 in [Baer]... |
hdmap14lem2N 41230 | Prior to part 14 in [Baer]... |
hdmap14lem3 41231 | Prior to part 14 in [Baer]... |
hdmap14lem4a 41232 | Simplify ` ( A \ { Q } ) `... |
hdmap14lem4 41233 | Simplify ` ( A \ { Q } ) `... |
hdmap14lem6 41234 | Case where ` F ` is zero. ... |
hdmap14lem7 41235 | Combine cases of ` F ` . ... |
hdmap14lem8 41236 | Part of proof of part 14 i... |
hdmap14lem9 41237 | Part of proof of part 14 i... |
hdmap14lem10 41238 | Part of proof of part 14 i... |
hdmap14lem11 41239 | Part of proof of part 14 i... |
hdmap14lem12 41240 | Lemma for proof of part 14... |
hdmap14lem13 41241 | Lemma for proof of part 14... |
hdmap14lem14 41242 | Part of proof of part 14 i... |
hdmap14lem15 41243 | Part of proof of part 14 i... |
hgmapffval 41246 | Map from the scalar divisi... |
hgmapfval 41247 | Map from the scalar divisi... |
hgmapval 41248 | Value of map from the scal... |
hgmapfnN 41249 | Functionality of scalar si... |
hgmapcl 41250 | Closure of scalar sigma ma... |
hgmapdcl 41251 | Closure of the vector spac... |
hgmapvs 41252 | Part 15 of [Baer] p. 50 li... |
hgmapval0 41253 | Value of the scalar sigma ... |
hgmapval1 41254 | Value of the scalar sigma ... |
hgmapadd 41255 | Part 15 of [Baer] p. 50 li... |
hgmapmul 41256 | Part 15 of [Baer] p. 50 li... |
hgmaprnlem1N 41257 | Lemma for ~ hgmaprnN . (C... |
hgmaprnlem2N 41258 | Lemma for ~ hgmaprnN . Pa... |
hgmaprnlem3N 41259 | Lemma for ~ hgmaprnN . El... |
hgmaprnlem4N 41260 | Lemma for ~ hgmaprnN . El... |
hgmaprnlem5N 41261 | Lemma for ~ hgmaprnN . El... |
hgmaprnN 41262 | Part of proof of part 16 i... |
hgmap11 41263 | The scalar sigma map is on... |
hgmapf1oN 41264 | The scalar sigma map is a ... |
hgmapeq0 41265 | The scalar sigma map is ze... |
hdmapipcl 41266 | The inner product (Hermiti... |
hdmapln1 41267 | Linearity property that wi... |
hdmaplna1 41268 | Additive property of first... |
hdmaplns1 41269 | Subtraction property of fi... |
hdmaplnm1 41270 | Multiplicative property of... |
hdmaplna2 41271 | Additive property of secon... |
hdmapglnm2 41272 | g-linear property of secon... |
hdmapgln2 41273 | g-linear property that wil... |
hdmaplkr 41274 | Kernel of the vector to du... |
hdmapellkr 41275 | Membership in the kernel (... |
hdmapip0 41276 | Zero property that will be... |
hdmapip1 41277 | Construct a proportional v... |
hdmapip0com 41278 | Commutation property of Ba... |
hdmapinvlem1 41279 | Line 27 in [Baer] p. 110. ... |
hdmapinvlem2 41280 | Line 28 in [Baer] p. 110, ... |
hdmapinvlem3 41281 | Line 30 in [Baer] p. 110, ... |
hdmapinvlem4 41282 | Part 1.1 of Proposition 1 ... |
hdmapglem5 41283 | Part 1.2 in [Baer] p. 110 ... |
hgmapvvlem1 41284 | Involution property of sca... |
hgmapvvlem2 41285 | Lemma for ~ hgmapvv . Eli... |
hgmapvvlem3 41286 | Lemma for ~ hgmapvv . Eli... |
hgmapvv 41287 | Value of a double involuti... |
hdmapglem7a 41288 | Lemma for ~ hdmapg . (Con... |
hdmapglem7b 41289 | Lemma for ~ hdmapg . (Con... |
hdmapglem7 41290 | Lemma for ~ hdmapg . Line... |
hdmapg 41291 | Apply the scalar sigma fun... |
hdmapoc 41292 | Express our constructed or... |
hlhilset 41295 | The final Hilbert space co... |
hlhilsca 41296 | The scalar of the final co... |
hlhilbase 41297 | The base set of the final ... |
hlhilplus 41298 | The vector addition for th... |
hlhilslem 41299 | Lemma for ~ hlhilsbase etc... |
hlhilslemOLD 41300 | Obsolete version of ~ hlhi... |
hlhilsbase 41301 | The scalar base set of the... |
hlhilsbaseOLD 41302 | Obsolete version of ~ hlhi... |
hlhilsplus 41303 | Scalar addition for the fi... |
hlhilsplusOLD 41304 | Obsolete version of ~ hlhi... |
hlhilsmul 41305 | Scalar multiplication for ... |
hlhilsmulOLD 41306 | Obsolete version of ~ hlhi... |
hlhilsbase2 41307 | The scalar base set of the... |
hlhilsplus2 41308 | Scalar addition for the fi... |
hlhilsmul2 41309 | Scalar multiplication for ... |
hlhils0 41310 | The scalar ring zero for t... |
hlhils1N 41311 | The scalar ring unity for ... |
hlhilvsca 41312 | The scalar product for the... |
hlhilip 41313 | Inner product operation fo... |
hlhilipval 41314 | Value of inner product ope... |
hlhilnvl 41315 | The involution operation o... |
hlhillvec 41316 | The final constructed Hilb... |
hlhildrng 41317 | The star division ring for... |
hlhilsrnglem 41318 | Lemma for ~ hlhilsrng . (... |
hlhilsrng 41319 | The star division ring for... |
hlhil0 41320 | The zero vector for the fi... |
hlhillsm 41321 | The vector sum operation f... |
hlhilocv 41322 | The orthocomplement for th... |
hlhillcs 41323 | The closed subspaces of th... |
hlhilphllem 41324 | Lemma for ~ hlhil . (Cont... |
hlhilhillem 41325 | Lemma for ~ hlhil . (Cont... |
hlathil 41326 | Construction of a Hilbert ... |
iscsrg 41329 | A commutative semiring is ... |
leexp1ad 41330 | Weak base ordering relatio... |
relogbcld 41331 | Closure of the general log... |
relogbexpd 41332 | Identity law for general l... |
relogbzexpd 41333 | Power law for the general ... |
logblebd 41334 | The general logarithm is m... |
uzindd 41335 | Induction on the upper int... |
fzadd2d 41336 | Membership of a sum in a f... |
zltlem1d 41337 | Integer ordering relation,... |
zltp1led 41338 | Integer ordering relation,... |
fzne2d 41339 | Elementhood in a finite se... |
eqfnfv2d2 41340 | Equality of functions is d... |
fzsplitnd 41341 | Split a finite interval of... |
fzsplitnr 41342 | Split a finite interval of... |
addassnni 41343 | Associative law for additi... |
addcomnni 41344 | Commutative law for additi... |
mulassnni 41345 | Associative law for multip... |
mulcomnni 41346 | Commutative law for multip... |
gcdcomnni 41347 | Commutative law for gcd. ... |
gcdnegnni 41348 | Negation invariance for gc... |
neggcdnni 41349 | Negation invariance for gc... |
bccl2d 41350 | Closure of the binomial co... |
recbothd 41351 | Take reciprocal on both si... |
gcdmultiplei 41352 | The GCD of a multiple of a... |
gcdaddmzz2nni 41353 | Adding a multiple of one o... |
gcdaddmzz2nncomi 41354 | Adding a multiple of one o... |
gcdnncli 41355 | Closure of the gcd operato... |
muldvds1d 41356 | If a product divides an in... |
muldvds2d 41357 | If a product divides an in... |
nndivdvdsd 41358 | A positive integer divides... |
nnproddivdvdsd 41359 | A product of natural numbe... |
coprmdvds2d 41360 | If an integer is divisible... |
12gcd5e1 41361 | The gcd of 12 and 5 is 1. ... |
60gcd6e6 41362 | The gcd of 60 and 6 is 6. ... |
60gcd7e1 41363 | The gcd of 60 and 7 is 1. ... |
420gcd8e4 41364 | The gcd of 420 and 8 is 4.... |
lcmeprodgcdi 41365 | Calculate the least common... |
12lcm5e60 41366 | The lcm of 12 and 5 is 60.... |
60lcm6e60 41367 | The lcm of 60 and 6 is 60.... |
60lcm7e420 41368 | The lcm of 60 and 7 is 420... |
420lcm8e840 41369 | The lcm of 420 and 8 is 84... |
lcmfunnnd 41370 | Useful equation to calcula... |
lcm1un 41371 | Least common multiple of n... |
lcm2un 41372 | Least common multiple of n... |
lcm3un 41373 | Least common multiple of n... |
lcm4un 41374 | Least common multiple of n... |
lcm5un 41375 | Least common multiple of n... |
lcm6un 41376 | Least common multiple of n... |
lcm7un 41377 | Least common multiple of n... |
lcm8un 41378 | Least common multiple of n... |
3factsumint1 41379 | Move constants out of inte... |
3factsumint2 41380 | Move constants out of inte... |
3factsumint3 41381 | Move constants out of inte... |
3factsumint4 41382 | Move constants out of inte... |
3factsumint 41383 | Helpful equation for lcm i... |
resopunitintvd 41384 | Restrict continuous functi... |
resclunitintvd 41385 | Restrict continuous functi... |
resdvopclptsd 41386 | Restrict derivative on uni... |
lcmineqlem1 41387 | Part of lcm inequality lem... |
lcmineqlem2 41388 | Part of lcm inequality lem... |
lcmineqlem3 41389 | Part of lcm inequality lem... |
lcmineqlem4 41390 | Part of lcm inequality lem... |
lcmineqlem5 41391 | Technical lemma for recipr... |
lcmineqlem6 41392 | Part of lcm inequality lem... |
lcmineqlem7 41393 | Derivative of 1-x for chai... |
lcmineqlem8 41394 | Derivative of (1-x)^(N-M).... |
lcmineqlem9 41395 | (1-x)^(N-M) is continuous.... |
lcmineqlem10 41396 | Induction step of ~ lcmine... |
lcmineqlem11 41397 | Induction step, continuati... |
lcmineqlem12 41398 | Base case for induction. ... |
lcmineqlem13 41399 | Induction proof for lcm in... |
lcmineqlem14 41400 | Technical lemma for inequa... |
lcmineqlem15 41401 | F times the least common m... |
lcmineqlem16 41402 | Technical divisibility lem... |
lcmineqlem17 41403 | Inequality of 2^{2n}. (Co... |
lcmineqlem18 41404 | Technical lemma to shift f... |
lcmineqlem19 41405 | Dividing implies inequalit... |
lcmineqlem20 41406 | Inequality for lcm lemma. ... |
lcmineqlem21 41407 | The lcm inequality lemma w... |
lcmineqlem22 41408 | The lcm inequality lemma w... |
lcmineqlem23 41409 | Penultimate step to the lc... |
lcmineqlem 41410 | The least common multiple ... |
3exp7 41411 | 3 to the power of 7 equals... |
3lexlogpow5ineq1 41412 | First inequality in inequa... |
3lexlogpow5ineq2 41413 | Second inequality in inequ... |
3lexlogpow5ineq4 41414 | Sharper logarithm inequali... |
3lexlogpow5ineq3 41415 | Combined inequality chain ... |
3lexlogpow2ineq1 41416 | Result for bound in AKS in... |
3lexlogpow2ineq2 41417 | Result for bound in AKS in... |
3lexlogpow5ineq5 41418 | Result for bound in AKS in... |
intlewftc 41419 | Inequality inference by in... |
aks4d1lem1 41420 | Technical lemma to reduce ... |
aks4d1p1p1 41421 | Exponential law for finite... |
dvrelog2 41422 | The derivative of the loga... |
dvrelog3 41423 | The derivative of the loga... |
dvrelog2b 41424 | Derivative of the binary l... |
0nonelalab 41425 | Technical lemma for open i... |
dvrelogpow2b 41426 | Derivative of the power of... |
aks4d1p1p3 41427 | Bound of a ceiling of the ... |
aks4d1p1p2 41428 | Rewrite ` A ` in more suit... |
aks4d1p1p4 41429 | Technical step for inequal... |
dvle2 41430 | Collapsed ~ dvle . (Contr... |
aks4d1p1p6 41431 | Inequality lift to differe... |
aks4d1p1p7 41432 | Bound of intermediary of i... |
aks4d1p1p5 41433 | Show inequality for existe... |
aks4d1p1 41434 | Show inequality for existe... |
aks4d1p2 41435 | Technical lemma for existe... |
aks4d1p3 41436 | There exists a small enoug... |
aks4d1p4 41437 | There exists a small enoug... |
aks4d1p5 41438 | Show that ` N ` and ` R ` ... |
aks4d1p6 41439 | The maximal prime power ex... |
aks4d1p7d1 41440 | Technical step in AKS lemm... |
aks4d1p7 41441 | Technical step in AKS lemm... |
aks4d1p8d1 41442 | If a prime divides one num... |
aks4d1p8d2 41443 | Any prime power dividing a... |
aks4d1p8d3 41444 | The remainder of a divisio... |
aks4d1p8 41445 | Show that ` N ` and ` R ` ... |
aks4d1p9 41446 | Show that the order is bou... |
aks4d1 41447 | Lemma 4.1 from ~ https://w... |
fldhmf1 41448 | A field homomorphism is in... |
aks6d1c2p1 41449 | In the AKS-theorem the sub... |
aks6d1c2p2 41450 | Injective condition for co... |
5bc2eq10 41451 | The value of 5 choose 2. ... |
facp2 41452 | The factorial of a success... |
2np3bcnp1 41453 | Part of induction step for... |
2ap1caineq 41454 | Inequality for Theorem 6.6... |
sticksstones1 41455 | Different strictly monoton... |
sticksstones2 41456 | The range function on stri... |
sticksstones3 41457 | The range function on stri... |
sticksstones4 41458 | Equinumerosity lemma for s... |
sticksstones5 41459 | Count the number of strict... |
sticksstones6 41460 | Function induces an order ... |
sticksstones7 41461 | Closure property of sticks... |
sticksstones8 41462 | Establish mapping between ... |
sticksstones9 41463 | Establish mapping between ... |
sticksstones10 41464 | Establish mapping between ... |
sticksstones11 41465 | Establish bijective mappin... |
sticksstones12a 41466 | Establish bijective mappin... |
sticksstones12 41467 | Establish bijective mappin... |
sticksstones13 41468 | Establish bijective mappin... |
sticksstones14 41469 | Sticks and stones with def... |
sticksstones15 41470 | Sticks and stones with alm... |
sticksstones16 41471 | Sticks and stones with col... |
sticksstones17 41472 | Extend sticks and stones t... |
sticksstones18 41473 | Extend sticks and stones t... |
sticksstones19 41474 | Extend sticks and stones t... |
sticksstones20 41475 | Lift sticks and stones to ... |
sticksstones21 41476 | Lift sticks and stones to ... |
sticksstones22 41477 | Non-exhaustive sticks and ... |
metakunt1 41478 | A is an endomapping. (Con... |
metakunt2 41479 | A is an endomapping. (Con... |
metakunt3 41480 | Value of A. (Contributed b... |
metakunt4 41481 | Value of A. (Contributed b... |
metakunt5 41482 | C is the left inverse for ... |
metakunt6 41483 | C is the left inverse for ... |
metakunt7 41484 | C is the left inverse for ... |
metakunt8 41485 | C is the left inverse for ... |
metakunt9 41486 | C is the left inverse for ... |
metakunt10 41487 | C is the right inverse for... |
metakunt11 41488 | C is the right inverse for... |
metakunt12 41489 | C is the right inverse for... |
metakunt13 41490 | C is the right inverse for... |
metakunt14 41491 | A is a primitive permutati... |
metakunt15 41492 | Construction of another pe... |
metakunt16 41493 | Construction of another pe... |
metakunt17 41494 | The union of three disjoin... |
metakunt18 41495 | Disjoint domains and codom... |
metakunt19 41496 | Domains on restrictions of... |
metakunt20 41497 | Show that B coincides on t... |
metakunt21 41498 | Show that B coincides on t... |
metakunt22 41499 | Show that B coincides on t... |
metakunt23 41500 | B coincides on the union o... |
metakunt24 41501 | Technical condition such t... |
metakunt25 41502 | B is a permutation. (Cont... |
metakunt26 41503 | Construction of one soluti... |
metakunt27 41504 | Construction of one soluti... |
metakunt28 41505 | Construction of one soluti... |
metakunt29 41506 | Construction of one soluti... |
metakunt30 41507 | Construction of one soluti... |
metakunt31 41508 | Construction of one soluti... |
metakunt32 41509 | Construction of one soluti... |
metakunt33 41510 | Construction of one soluti... |
metakunt34 41511 | ` D ` is a permutation. (... |
andiff 41512 | Adding biconditional when ... |
fac2xp3 41513 | Factorial of 2x+3, sublemm... |
prodsplit 41514 | Product split into two fac... |
2xp3dxp2ge1d 41515 | 2x+3 is greater than or eq... |
factwoffsmonot 41516 | A factorial with offset is... |
ioin9i8 41517 | Miscellaneous inference cr... |
jaodd 41518 | Double deduction form of ~... |
syl3an12 41519 | A double syllogism inferen... |
sbtd 41520 | A true statement is true u... |
sbor2 41521 | One direction of ~ sbor , ... |
19.9dev 41522 | ~ 19.9d in the case of an ... |
3rspcedvdw 41523 | Triple application of ~ rs... |
3rspcedvd 41524 | Triple application of ~ rs... |
rabdif 41525 | Move difference in and out... |
sn-axrep5v 41526 | A condensed form of ~ axre... |
sn-axprlem3 41527 | ~ axprlem3 using only Tars... |
sn-exelALT 41528 | Alternate proof of ~ exel ... |
ss2ab1 41529 | Class abstractions in a su... |
ssabdv 41530 | Deduction of abstraction s... |
sn-iotalem 41531 | An unused lemma showing th... |
sn-iotalemcor 41532 | Corollary of ~ sn-iotalem ... |
abbi1sn 41533 | Originally part of ~ uniab... |
brif1 41534 | Move a relation inside and... |
brif2 41535 | Move a relation inside and... |
brif12 41536 | Move a relation inside and... |
pssexg 41537 | The proper subset of a set... |
pssn0 41538 | A proper superset is nonem... |
psspwb 41539 | Classes are proper subclas... |
xppss12 41540 | Proper subset theorem for ... |
coexd 41541 | The composition of two set... |
elpwbi 41542 | Membership in a power set,... |
imaopab 41543 | The image of a class of or... |
fnsnbt 41544 | A function's domain is a s... |
fnimasnd 41545 | The image of a function by... |
fvmptd4 41546 | Deduction version of ~ fvm... |
eqresfnbd 41547 | Property of being the rest... |
f1o2d2 41548 | Sufficient condition for a... |
fmpocos 41549 | Composition of two functio... |
ovmpogad 41550 | Value of an operation give... |
ofun 41551 | A function operation of un... |
dfqs2 41552 | Alternate definition of qu... |
dfqs3 41553 | Alternate definition of qu... |
qseq12d 41554 | Equality theorem for quoti... |
qsalrel 41555 | The quotient set is equal ... |
fsuppfund 41556 | A finitely supported funct... |
fsuppsssuppgd 41557 | If the support of a functi... |
fsuppss 41558 | A subset of a finitely sup... |
elmapssresd 41559 | A restricted mapping is a ... |
mapcod 41560 | Compose two mappings. (Co... |
fzosumm1 41561 | Separate out the last term... |
ccatcan2d 41562 | Cancellation law for conca... |
nelsubginvcld 41563 | The inverse of a non-subgr... |
nelsubgcld 41564 | A non-subgroup-member plus... |
nelsubgsubcld 41565 | A non-subgroup-member minu... |
rnasclg 41566 | The set of injected scalar... |
frlmfielbas 41567 | The vectors of a finite fr... |
frlmfzwrd 41568 | A vector of a module with ... |
frlmfzowrd 41569 | A vector of a module with ... |
frlmfzolen 41570 | The dimension of a vector ... |
frlmfzowrdb 41571 | The vectors of a module wi... |
frlmfzoccat 41572 | The concatenation of two v... |
frlmvscadiccat 41573 | Scalar multiplication dist... |
grpasscan2d 41574 | An associative cancellatio... |
grpcominv1 41575 | If two elements commute, t... |
grpcominv2 41576 | If two elements commute, t... |
finsubmsubg 41577 | A submonoid of a finite gr... |
crngcomd 41578 | Multiplication is commutat... |
crng12d 41579 | Commutative/associative la... |
imacrhmcl 41580 | The image of a commutative... |
rimrcl1 41581 | Reverse closure of a ring ... |
rimrcl2 41582 | Reverse closure of a ring ... |
rimcnv 41583 | The converse of a ring iso... |
rimco 41584 | The composition of ring is... |
ricsym 41585 | Ring isomorphism is symmet... |
rictr 41586 | Ring isomorphism is transi... |
riccrng1 41587 | Ring isomorphism preserves... |
riccrng 41588 | A ring is commutative if a... |
drnginvrn0d 41589 | A multiplicative inverse i... |
drngmulcanad 41590 | Cancellation of a nonzero ... |
drngmulcan2ad 41591 | Cancellation of a nonzero ... |
drnginvmuld 41592 | Inverse of a nonzero produ... |
ricdrng1 41593 | A ring isomorphism maps a ... |
ricdrng 41594 | A ring is a division ring ... |
ricfld 41595 | A ring is a field if and o... |
lvecgrp 41596 | A vector space is a group.... |
lvecring 41597 | The scalar component of a ... |
frlm0vald 41598 | All coordinates of the zer... |
frlmsnic 41599 | Given a free module with a... |
uvccl 41600 | A unit vector is a vector.... |
uvcn0 41601 | A unit vector is nonzero. ... |
pwselbasr 41602 | The reverse direction of ~... |
pwsgprod 41603 | Finite products in a power... |
psrbagres 41604 | Restrict a bag of variable... |
mpllmodd 41605 | The polynomial ring is a l... |
mplringd 41606 | The polynomial ring is a r... |
mplcrngd 41607 | The polynomial ring is a c... |
mplsubrgcl 41608 | An element of a polynomial... |
mhmcompl 41609 | The composition of a monoi... |
rhmmpllem1 41610 | Lemma for ~ rhmmpl . A su... |
rhmmpllem2 41611 | Lemma for ~ rhmmpl . A su... |
mhmcoaddmpl 41612 | Show that the ring homomor... |
rhmcomulmpl 41613 | Show that the ring homomor... |
rhmmpl 41614 | Provide a ring homomorphis... |
mplascl0 41615 | The zero scalar as a polyn... |
mplascl1 41616 | The one scalar as a polyno... |
mplmapghm 41617 | The function ` H ` mapping... |
evl0 41618 | The zero polynomial evalua... |
evlscl 41619 | A polynomial over the ring... |
evlsval3 41620 | Give a formula for the pol... |
evlsvval 41621 | Give a formula for the eva... |
evlsvvvallem 41622 | Lemma for ~ evlsvvval akin... |
evlsvvvallem2 41623 | Lemma for theorems using ~... |
evlsvvval 41624 | Give a formula for the eva... |
evlsscaval 41625 | Polynomial evaluation buil... |
evlsvarval 41626 | Polynomial evaluation buil... |
evlsbagval 41627 | Polynomial evaluation buil... |
evlsexpval 41628 | Polynomial evaluation buil... |
evlsaddval 41629 | Polynomial evaluation buil... |
evlsmulval 41630 | Polynomial evaluation buil... |
evlsmaprhm 41631 | The function ` F ` mapping... |
evlsevl 41632 | Evaluation in a subring is... |
evlcl 41633 | A polynomial over the ring... |
evlvvval 41634 | Give a formula for the eva... |
evlvvvallem 41635 | Lemma for theorems using ~... |
evladdval 41636 | Polynomial evaluation buil... |
evlmulval 41637 | Polynomial evaluation buil... |
selvcllem1 41638 | ` T ` is an associative al... |
selvcllem2 41639 | ` D ` is a ring homomorphi... |
selvcllem3 41640 | The third argument passed ... |
selvcllemh 41641 | Apply the third argument (... |
selvcllem4 41642 | The fourth argument passed... |
selvcllem5 41643 | The fifth argument passed ... |
selvcl 41644 | Closure of the "variable s... |
selvval2 41645 | Value of the "variable sel... |
selvvvval 41646 | Recover the original polyn... |
evlselvlem 41647 | Lemma for ~ evlselv . Use... |
evlselv 41648 | Evaluating a selection of ... |
selvadd 41649 | The "variable selection" f... |
selvmul 41650 | The "variable selection" f... |
fsuppind 41651 | Induction on functions ` F... |
fsuppssindlem1 41652 | Lemma for ~ fsuppssind . ... |
fsuppssindlem2 41653 | Lemma for ~ fsuppssind . ... |
fsuppssind 41654 | Induction on functions ` F... |
mhpind 41655 | The homogeneous polynomial... |
evlsmhpvvval 41656 | Give a formula for the eva... |
mhphflem 41657 | Lemma for ~ mhphf . Add s... |
mhphf 41658 | A homogeneous polynomial d... |
mhphf2 41659 | A homogeneous polynomial d... |
mhphf3 41660 | A homogeneous polynomial d... |
mhphf4 41661 | A homogeneous polynomial d... |
c0exALT 41662 | Alternate proof of ~ c0ex ... |
0cnALT3 41663 | Alternate proof of ~ 0cn u... |
elre0re 41664 | Specialized version of ~ 0... |
1t1e1ALT 41665 | Alternate proof of ~ 1t1e1... |
remulcan2d 41666 | ~ mulcan2d for real number... |
readdridaddlidd 41667 | Given some real number ` B... |
sn-1ne2 41668 | A proof of ~ 1ne2 without ... |
nnn1suc 41669 | A positive integer that is... |
nnadd1com 41670 | Addition with 1 is commuta... |
nnaddcom 41671 | Addition is commutative fo... |
nnaddcomli 41672 | Version of ~ addcomli for ... |
nnadddir 41673 | Right-distributivity for n... |
nnmul1com 41674 | Multiplication with 1 is c... |
nnmulcom 41675 | Multiplication is commutat... |
mvrrsubd 41676 | Move a subtraction in the ... |
laddrotrd 41677 | Rotate the variables right... |
raddcom12d 41678 | Swap the first two variabl... |
lsubrotld 41679 | Rotate the variables left ... |
lsubcom23d 41680 | Swap the second and third ... |
addsubeq4com 41681 | Relation between sums and ... |
sqsumi 41682 | A sum squared. (Contribut... |
negn0nposznnd 41683 | Lemma for ~ dffltz . (Con... |
sqmid3api 41684 | Value of the square of the... |
decaddcom 41685 | Commute ones place in addi... |
sqn5i 41686 | The square of a number end... |
sqn5ii 41687 | The square of a number end... |
decpmulnc 41688 | Partial products algorithm... |
decpmul 41689 | Partial products algorithm... |
sqdeccom12 41690 | The square of a number in ... |
sq3deccom12 41691 | Variant of ~ sqdeccom12 wi... |
4t5e20 41692 | 4 times 5 equals 20. (Con... |
sq9 41693 | The square of 9 is 81. (C... |
235t711 41694 | Calculate a product by lon... |
ex-decpmul 41695 | Example usage of ~ decpmul... |
fz1sumconst 41696 | The sum of ` N ` constant ... |
fz1sump1 41697 | Add one more term to a sum... |
oddnumth 41698 | The Odd Number Theorem. T... |
nicomachus 41699 | Nicomachus's Theorem. The... |
sumcubes 41700 | The sum of the first ` N `... |
oexpreposd 41701 | Lemma for ~ dffltz . TODO... |
ltexp1d 41702 | ~ ltmul1d for exponentiati... |
ltexp1dd 41703 | Raising both sides of 'les... |
exp11nnd 41704 | ~ sq11d for positive real ... |
exp11d 41705 | ~ exp11nnd for nonzero int... |
0dvds0 41706 | 0 divides 0. (Contributed... |
absdvdsabsb 41707 | Divisibility is invariant ... |
dvdsexpim 41708 | ~ dvdssqim generalized to ... |
gcdnn0id 41709 | The ` gcd ` of a nonnegati... |
gcdle1d 41710 | The greatest common diviso... |
gcdle2d 41711 | The greatest common diviso... |
dvdsexpad 41712 | Deduction associated with ... |
nn0rppwr 41713 | If ` A ` and ` B ` are rel... |
expgcd 41714 | Exponentiation distributes... |
nn0expgcd 41715 | Exponentiation distributes... |
zexpgcd 41716 | Exponentiation distributes... |
numdenexp 41717 | ~ numdensq extended to non... |
numexp 41718 | ~ numsq extended to nonneg... |
denexp 41719 | ~ densq extended to nonneg... |
dvdsexpnn 41720 | ~ dvdssqlem generalized to... |
dvdsexpnn0 41721 | ~ dvdsexpnn generalized to... |
dvdsexpb 41722 | ~ dvdssq generalized to po... |
posqsqznn 41723 | When a positive rational s... |
zrtelqelz 41724 | ~ zsqrtelqelz generalized ... |
zrtdvds 41725 | A positive integer root di... |
rtprmirr 41726 | The root of a prime number... |
resubval 41729 | Value of real subtraction,... |
renegeulemv 41730 | Lemma for ~ renegeu and si... |
renegeulem 41731 | Lemma for ~ renegeu and si... |
renegeu 41732 | Existential uniqueness of ... |
rernegcl 41733 | Closure law for negative r... |
renegadd 41734 | Relationship between real ... |
renegid 41735 | Addition of a real number ... |
reneg0addlid 41736 | Negative zero is a left ad... |
resubeulem1 41737 | Lemma for ~ resubeu . A v... |
resubeulem2 41738 | Lemma for ~ resubeu . A v... |
resubeu 41739 | Existential uniqueness of ... |
rersubcl 41740 | Closure for real subtracti... |
resubadd 41741 | Relation between real subt... |
resubaddd 41742 | Relationship between subtr... |
resubf 41743 | Real subtraction is an ope... |
repncan2 41744 | Addition and subtraction o... |
repncan3 41745 | Addition and subtraction o... |
readdsub 41746 | Law for addition and subtr... |
reladdrsub 41747 | Move LHS of a sum into RHS... |
reltsub1 41748 | Subtraction from both side... |
reltsubadd2 41749 | 'Less than' relationship b... |
resubcan2 41750 | Cancellation law for real ... |
resubsub4 41751 | Law for double subtraction... |
rennncan2 41752 | Cancellation law for real ... |
renpncan3 41753 | Cancellation law for real ... |
repnpcan 41754 | Cancellation law for addit... |
reppncan 41755 | Cancellation law for mixed... |
resubidaddlidlem 41756 | Lemma for ~ resubidaddlid ... |
resubidaddlid 41757 | Any real number subtracted... |
resubdi 41758 | Distribution of multiplica... |
re1m1e0m0 41759 | Equality of two left-addit... |
sn-00idlem1 41760 | Lemma for ~ sn-00id . (Co... |
sn-00idlem2 41761 | Lemma for ~ sn-00id . (Co... |
sn-00idlem3 41762 | Lemma for ~ sn-00id . (Co... |
sn-00id 41763 | ~ 00id proven without ~ ax... |
re0m0e0 41764 | Real number version of ~ 0... |
readdlid 41765 | Real number version of ~ a... |
sn-addlid 41766 | ~ addlid without ~ ax-mulc... |
remul02 41767 | Real number version of ~ m... |
sn-0ne2 41768 | ~ 0ne2 without ~ ax-mulcom... |
remul01 41769 | Real number version of ~ m... |
resubid 41770 | Subtraction of a real numb... |
readdrid 41771 | Real number version of ~ a... |
resubid1 41772 | Real number version of ~ s... |
renegneg 41773 | A real number is equal to ... |
readdcan2 41774 | Commuted version of ~ read... |
renegid2 41775 | Commuted version of ~ rene... |
remulneg2d 41776 | Product with negative is n... |
sn-it0e0 41777 | Proof of ~ it0e0 without ~... |
sn-negex12 41778 | A combination of ~ cnegex ... |
sn-negex 41779 | Proof of ~ cnegex without ... |
sn-negex2 41780 | Proof of ~ cnegex2 without... |
sn-addcand 41781 | ~ addcand without ~ ax-mul... |
sn-addrid 41782 | ~ addrid without ~ ax-mulc... |
sn-addcan2d 41783 | ~ addcan2d without ~ ax-mu... |
reixi 41784 | ~ ixi without ~ ax-mulcom ... |
rei4 41785 | ~ i4 without ~ ax-mulcom .... |
sn-addid0 41786 | A number that sums to itse... |
sn-mul01 41787 | ~ mul01 without ~ ax-mulco... |
sn-subeu 41788 | ~ negeu without ~ ax-mulco... |
sn-subcl 41789 | ~ subcl without ~ ax-mulco... |
sn-subf 41790 | ~ subf without ~ ax-mulcom... |
resubeqsub 41791 | Equivalence between real s... |
subresre 41792 | Subtraction restricted to ... |
addinvcom 41793 | A number commutes with its... |
remulinvcom 41794 | A left multiplicative inve... |
remullid 41795 | Commuted version of ~ ax-1... |
sn-1ticom 41796 | Lemma for ~ sn-mullid and ... |
sn-mullid 41797 | ~ mullid without ~ ax-mulc... |
it1ei 41798 | ` 1 ` is a multiplicative ... |
ipiiie0 41799 | The multiplicative inverse... |
remulcand 41800 | Commuted version of ~ remu... |
sn-0tie0 41801 | Lemma for ~ sn-mul02 . Co... |
sn-mul02 41802 | ~ mul02 without ~ ax-mulco... |
sn-ltaddpos 41803 | ~ ltaddpos without ~ ax-mu... |
sn-ltaddneg 41804 | ~ ltaddneg without ~ ax-mu... |
reposdif 41805 | Comparison of two numbers ... |
relt0neg1 41806 | Comparison of a real and i... |
relt0neg2 41807 | Comparison of a real and i... |
sn-addlt0d 41808 | The sum of negative number... |
sn-addgt0d 41809 | The sum of positive number... |
sn-nnne0 41810 | ~ nnne0 without ~ ax-mulco... |
reelznn0nn 41811 | ~ elznn0nn restated using ... |
nn0addcom 41812 | Addition is commutative fo... |
zaddcomlem 41813 | Lemma for ~ zaddcom . (Co... |
zaddcom 41814 | Addition is commutative fo... |
renegmulnnass 41815 | Move multiplication by a n... |
nn0mulcom 41816 | Multiplication is commutat... |
zmulcomlem 41817 | Lemma for ~ zmulcom . (Co... |
zmulcom 41818 | Multiplication is commutat... |
mulgt0con1dlem 41819 | Lemma for ~ mulgt0con1d . ... |
mulgt0con1d 41820 | Counterpart to ~ mulgt0con... |
mulgt0con2d 41821 | Lemma for ~ mulgt0b2d and ... |
mulgt0b2d 41822 | Biconditional, deductive f... |
sn-ltmul2d 41823 | ~ ltmul2d without ~ ax-mul... |
sn-0lt1 41824 | ~ 0lt1 without ~ ax-mulcom... |
sn-ltp1 41825 | ~ ltp1 without ~ ax-mulcom... |
reneg1lt0 41826 | Lemma for ~ sn-inelr . (C... |
sn-inelr 41827 | ~ inelr without ~ ax-mulco... |
itrere 41828 | ` _i ` times a real is rea... |
retire 41829 | Commuted version of ~ itre... |
cnreeu 41830 | The reals in the expressio... |
sn-sup2 41831 | ~ sup2 with exactly the sa... |
prjspval 41834 | Value of the projective sp... |
prjsprel 41835 | Utility theorem regarding ... |
prjspertr 41836 | The relation in ` PrjSp ` ... |
prjsperref 41837 | The relation in ` PrjSp ` ... |
prjspersym 41838 | The relation in ` PrjSp ` ... |
prjsper 41839 | The relation used to defin... |
prjspreln0 41840 | Two nonzero vectors are eq... |
prjspvs 41841 | A nonzero multiple of a ve... |
prjsprellsp 41842 | Two vectors are equivalent... |
prjspeclsp 41843 | The vectors equivalent to ... |
prjspval2 41844 | Alternate definition of pr... |
prjspnval 41847 | Value of the n-dimensional... |
prjspnerlem 41848 | A lemma showing that the e... |
prjspnval2 41849 | Value of the n-dimensional... |
prjspner 41850 | The relation used to defin... |
prjspnvs 41851 | A nonzero multiple of a ve... |
prjspnssbas 41852 | A projective point spans a... |
prjspnn0 41853 | A projective point is none... |
0prjspnlem 41854 | Lemma for ~ 0prjspn . The... |
prjspnfv01 41855 | Any vector is equivalent t... |
prjspner01 41856 | Any vector is equivalent t... |
prjspner1 41857 | Two vectors whose zeroth c... |
0prjspnrel 41858 | In the zero-dimensional pr... |
0prjspn 41859 | A zero-dimensional project... |
prjcrvfval 41862 | Value of the projective cu... |
prjcrvval 41863 | Value of the projective cu... |
prjcrv0 41864 | The "curve" (zero set) cor... |
dffltz 41865 | Fermat's Last Theorem (FLT... |
fltmul 41866 | A counterexample to FLT st... |
fltdiv 41867 | A counterexample to FLT st... |
flt0 41868 | A counterexample for FLT d... |
fltdvdsabdvdsc 41869 | Any factor of both ` A ` a... |
fltabcoprmex 41870 | A counterexample to FLT im... |
fltaccoprm 41871 | A counterexample to FLT wi... |
fltbccoprm 41872 | A counterexample to FLT wi... |
fltabcoprm 41873 | A counterexample to FLT wi... |
infdesc 41874 | Infinite descent. The hyp... |
fltne 41875 | If a counterexample to FLT... |
flt4lem 41876 | Raising a number to the fo... |
flt4lem1 41877 | Satisfy the antecedent use... |
flt4lem2 41878 | If ` A ` is even, ` B ` is... |
flt4lem3 41879 | Equivalent to ~ pythagtrip... |
flt4lem4 41880 | If the product of two copr... |
flt4lem5 41881 | In the context of the lemm... |
flt4lem5elem 41882 | Version of ~ fltaccoprm an... |
flt4lem5a 41883 | Part 1 of Equation 1 of ... |
flt4lem5b 41884 | Part 2 of Equation 1 of ... |
flt4lem5c 41885 | Part 2 of Equation 2 of ... |
flt4lem5d 41886 | Part 3 of Equation 2 of ... |
flt4lem5e 41887 | Satisfy the hypotheses of ... |
flt4lem5f 41888 | Final equation of ~... |
flt4lem6 41889 | Remove shared factors in a... |
flt4lem7 41890 | Convert ~ flt4lem5f into a... |
nna4b4nsq 41891 | Strengthening of Fermat's ... |
fltltc 41892 | ` ( C ^ N ) ` is the large... |
fltnltalem 41893 | Lemma for ~ fltnlta . A l... |
fltnlta 41894 | In a Fermat counterexample... |
iddii 41895 | Version of ~ a1ii with the... |
bicomdALT 41896 | Alternate proof of ~ bicom... |
elabgw 41897 | Membership in a class abst... |
elab2gw 41898 | Membership in a class abst... |
elrab2w 41899 | Membership in a restricted... |
ruvALT 41900 | Alternate proof of ~ ruv w... |
sn-wcdeq 41901 | Alternative to ~ wcdeq and... |
sq45 41902 | 45 squared is 2025. (Cont... |
sum9cubes 41903 | The sum of the first nine ... |
acos1half 41904 | The arccosine of ` 1 / 2 `... |
aprilfools2025 41905 | An abuse of notation. (Co... |
binom2d 41906 | Deduction form of binom2. ... |
cu3addd 41907 | Cube of sum of three numbe... |
sqnegd 41908 | The square of the negative... |
negexpidd 41909 | The sum of a real number t... |
rexlimdv3d 41910 | An extended version of ~ r... |
3cubeslem1 41911 | Lemma for ~ 3cubes . (Con... |
3cubeslem2 41912 | Lemma for ~ 3cubes . Used... |
3cubeslem3l 41913 | Lemma for ~ 3cubes . (Con... |
3cubeslem3r 41914 | Lemma for ~ 3cubes . (Con... |
3cubeslem3 41915 | Lemma for ~ 3cubes . (Con... |
3cubeslem4 41916 | Lemma for ~ 3cubes . This... |
3cubes 41917 | Every rational number is a... |
rntrclfvOAI 41918 | The range of the transitiv... |
moxfr 41919 | Transfer at-most-one betwe... |
imaiinfv 41920 | Indexed intersection of an... |
elrfi 41921 | Elementhood in a set of re... |
elrfirn 41922 | Elementhood in a set of re... |
elrfirn2 41923 | Elementhood in a set of re... |
cmpfiiin 41924 | In a compact topology, a s... |
ismrcd1 41925 | Any function from the subs... |
ismrcd2 41926 | Second half of ~ ismrcd1 .... |
istopclsd 41927 | A closure function which s... |
ismrc 41928 | A function is a Moore clos... |
isnacs 41931 | Expand definition of Noeth... |
nacsfg 41932 | In a Noetherian-type closu... |
isnacs2 41933 | Express Noetherian-type cl... |
mrefg2 41934 | Slight variation on finite... |
mrefg3 41935 | Slight variation on finite... |
nacsacs 41936 | A closure system of Noethe... |
isnacs3 41937 | A choice-free order equiva... |
incssnn0 41938 | Transitivity induction of ... |
nacsfix 41939 | An increasing sequence of ... |
constmap 41940 | A constant (represented wi... |
mapco2g 41941 | Renaming indices in a tupl... |
mapco2 41942 | Post-composition (renaming... |
mapfzcons 41943 | Extending a one-based mapp... |
mapfzcons1 41944 | Recover prefix mapping fro... |
mapfzcons1cl 41945 | A nonempty mapping has a p... |
mapfzcons2 41946 | Recover added element from... |
mptfcl 41947 | Interpret range of a maps-... |
mzpclval 41952 | Substitution lemma for ` m... |
elmzpcl 41953 | Double substitution lemma ... |
mzpclall 41954 | The set of all functions w... |
mzpcln0 41955 | Corollary of ~ mzpclall : ... |
mzpcl1 41956 | Defining property 1 of a p... |
mzpcl2 41957 | Defining property 2 of a p... |
mzpcl34 41958 | Defining properties 3 and ... |
mzpval 41959 | Value of the ` mzPoly ` fu... |
dmmzp 41960 | ` mzPoly ` is defined for ... |
mzpincl 41961 | Polynomial closedness is a... |
mzpconst 41962 | Constant functions are pol... |
mzpf 41963 | A polynomial function is a... |
mzpproj 41964 | A projection function is p... |
mzpadd 41965 | The pointwise sum of two p... |
mzpmul 41966 | The pointwise product of t... |
mzpconstmpt 41967 | A constant function expres... |
mzpaddmpt 41968 | Sum of polynomial function... |
mzpmulmpt 41969 | Product of polynomial func... |
mzpsubmpt 41970 | The difference of two poly... |
mzpnegmpt 41971 | Negation of a polynomial f... |
mzpexpmpt 41972 | Raise a polynomial functio... |
mzpindd 41973 | "Structural" induction to ... |
mzpmfp 41974 | Relationship between multi... |
mzpsubst 41975 | Substituting polynomials f... |
mzprename 41976 | Simplified version of ~ mz... |
mzpresrename 41977 | A polynomial is a polynomi... |
mzpcompact2lem 41978 | Lemma for ~ mzpcompact2 . ... |
mzpcompact2 41979 | Polynomials are finitary o... |
coeq0i 41980 | ~ coeq0 but without explic... |
fzsplit1nn0 41981 | Split a finite 1-based set... |
eldiophb 41984 | Initial expression of Diop... |
eldioph 41985 | Condition for a set to be ... |
diophrw 41986 | Renaming and adding unused... |
eldioph2lem1 41987 | Lemma for ~ eldioph2 . Co... |
eldioph2lem2 41988 | Lemma for ~ eldioph2 . Co... |
eldioph2 41989 | Construct a Diophantine se... |
eldioph2b 41990 | While Diophantine sets wer... |
eldiophelnn0 41991 | Remove antecedent on ` B `... |
eldioph3b 41992 | Define Diophantine sets in... |
eldioph3 41993 | Inference version of ~ eld... |
ellz1 41994 | Membership in a lower set ... |
lzunuz 41995 | The union of a lower set o... |
fz1eqin 41996 | Express a one-based finite... |
lzenom 41997 | Lower integers are countab... |
elmapresaunres2 41998 | ~ fresaunres2 transposed t... |
diophin 41999 | If two sets are Diophantin... |
diophun 42000 | If two sets are Diophantin... |
eldiophss 42001 | Diophantine sets are sets ... |
diophrex 42002 | Projecting a Diophantine s... |
eq0rabdioph 42003 | This is the first of a num... |
eqrabdioph 42004 | Diophantine set builder fo... |
0dioph 42005 | The null set is Diophantin... |
vdioph 42006 | The "universal" set (as la... |
anrabdioph 42007 | Diophantine set builder fo... |
orrabdioph 42008 | Diophantine set builder fo... |
3anrabdioph 42009 | Diophantine set builder fo... |
3orrabdioph 42010 | Diophantine set builder fo... |
2sbcrex 42011 | Exchange an existential qu... |
sbcrexgOLD 42012 | Interchange class substitu... |
2sbcrexOLD 42013 | Exchange an existential qu... |
sbc2rex 42014 | Exchange a substitution wi... |
sbc2rexgOLD 42015 | Exchange a substitution wi... |
sbc4rex 42016 | Exchange a substitution wi... |
sbc4rexgOLD 42017 | Exchange a substitution wi... |
sbcrot3 42018 | Rotate a sequence of three... |
sbcrot5 42019 | Rotate a sequence of five ... |
sbccomieg 42020 | Commute two explicit subst... |
rexrabdioph 42021 | Diophantine set builder fo... |
rexfrabdioph 42022 | Diophantine set builder fo... |
2rexfrabdioph 42023 | Diophantine set builder fo... |
3rexfrabdioph 42024 | Diophantine set builder fo... |
4rexfrabdioph 42025 | Diophantine set builder fo... |
6rexfrabdioph 42026 | Diophantine set builder fo... |
7rexfrabdioph 42027 | Diophantine set builder fo... |
rabdiophlem1 42028 | Lemma for arithmetic dioph... |
rabdiophlem2 42029 | Lemma for arithmetic dioph... |
elnn0rabdioph 42030 | Diophantine set builder fo... |
rexzrexnn0 42031 | Rewrite an existential qua... |
lerabdioph 42032 | Diophantine set builder fo... |
eluzrabdioph 42033 | Diophantine set builder fo... |
elnnrabdioph 42034 | Diophantine set builder fo... |
ltrabdioph 42035 | Diophantine set builder fo... |
nerabdioph 42036 | Diophantine set builder fo... |
dvdsrabdioph 42037 | Divisibility is a Diophant... |
eldioph4b 42038 | Membership in ` Dioph ` ex... |
eldioph4i 42039 | Forward-only version of ~ ... |
diophren 42040 | Change variables in a Diop... |
rabrenfdioph 42041 | Change variable numbers in... |
rabren3dioph 42042 | Change variable numbers in... |
fphpd 42043 | Pigeonhole principle expre... |
fphpdo 42044 | Pigeonhole principle for s... |
ctbnfien 42045 | An infinite subset of a co... |
fiphp3d 42046 | Infinite pigeonhole princi... |
rencldnfilem 42047 | Lemma for ~ rencldnfi . (... |
rencldnfi 42048 | A set of real numbers whic... |
irrapxlem1 42049 | Lemma for ~ irrapx1 . Div... |
irrapxlem2 42050 | Lemma for ~ irrapx1 . Two... |
irrapxlem3 42051 | Lemma for ~ irrapx1 . By ... |
irrapxlem4 42052 | Lemma for ~ irrapx1 . Eli... |
irrapxlem5 42053 | Lemma for ~ irrapx1 . Swi... |
irrapxlem6 42054 | Lemma for ~ irrapx1 . Exp... |
irrapx1 42055 | Dirichlet's approximation ... |
pellexlem1 42056 | Lemma for ~ pellex . Arit... |
pellexlem2 42057 | Lemma for ~ pellex . Arit... |
pellexlem3 42058 | Lemma for ~ pellex . To e... |
pellexlem4 42059 | Lemma for ~ pellex . Invo... |
pellexlem5 42060 | Lemma for ~ pellex . Invo... |
pellexlem6 42061 | Lemma for ~ pellex . Doin... |
pellex 42062 | Every Pell equation has a ... |
pell1qrval 42073 | Value of the set of first-... |
elpell1qr 42074 | Membership in a first-quad... |
pell14qrval 42075 | Value of the set of positi... |
elpell14qr 42076 | Membership in the set of p... |
pell1234qrval 42077 | Value of the set of genera... |
elpell1234qr 42078 | Membership in the set of g... |
pell1234qrre 42079 | General Pell solutions are... |
pell1234qrne0 42080 | No solution to a Pell equa... |
pell1234qrreccl 42081 | General solutions of the P... |
pell1234qrmulcl 42082 | General solutions of the P... |
pell14qrss1234 42083 | A positive Pell solution i... |
pell14qrre 42084 | A positive Pell solution i... |
pell14qrne0 42085 | A positive Pell solution i... |
pell14qrgt0 42086 | A positive Pell solution i... |
pell14qrrp 42087 | A positive Pell solution i... |
pell1234qrdich 42088 | A general Pell solution is... |
elpell14qr2 42089 | A number is a positive Pel... |
pell14qrmulcl 42090 | Positive Pell solutions ar... |
pell14qrreccl 42091 | Positive Pell solutions ar... |
pell14qrdivcl 42092 | Positive Pell solutions ar... |
pell14qrexpclnn0 42093 | Lemma for ~ pell14qrexpcl ... |
pell14qrexpcl 42094 | Positive Pell solutions ar... |
pell1qrss14 42095 | First-quadrant Pell soluti... |
pell14qrdich 42096 | A positive Pell solution i... |
pell1qrge1 42097 | A Pell solution in the fir... |
pell1qr1 42098 | 1 is a Pell solution and i... |
elpell1qr2 42099 | The first quadrant solutio... |
pell1qrgaplem 42100 | Lemma for ~ pell1qrgap . ... |
pell1qrgap 42101 | First-quadrant Pell soluti... |
pell14qrgap 42102 | Positive Pell solutions ar... |
pell14qrgapw 42103 | Positive Pell solutions ar... |
pellqrexplicit 42104 | Condition for a calculated... |
infmrgelbi 42105 | Any lower bound of a nonem... |
pellqrex 42106 | There is a nontrivial solu... |
pellfundval 42107 | Value of the fundamental s... |
pellfundre 42108 | The fundamental solution o... |
pellfundge 42109 | Lower bound on the fundame... |
pellfundgt1 42110 | Weak lower bound on the Pe... |
pellfundlb 42111 | A nontrivial first quadran... |
pellfundglb 42112 | If a real is larger than t... |
pellfundex 42113 | The fundamental solution a... |
pellfund14gap 42114 | There are no solutions bet... |
pellfundrp 42115 | The fundamental Pell solut... |
pellfundne1 42116 | The fundamental Pell solut... |
reglogcl 42117 | General logarithm is a rea... |
reglogltb 42118 | General logarithm preserve... |
reglogleb 42119 | General logarithm preserve... |
reglogmul 42120 | Multiplication law for gen... |
reglogexp 42121 | Power law for general log.... |
reglogbas 42122 | General log of the base is... |
reglog1 42123 | General log of 1 is 0. (C... |
reglogexpbas 42124 | General log of a power of ... |
pellfund14 42125 | Every positive Pell soluti... |
pellfund14b 42126 | The positive Pell solution... |
rmxfval 42131 | Value of the X sequence. ... |
rmyfval 42132 | Value of the Y sequence. ... |
rmspecsqrtnq 42133 | The discriminant used to d... |
rmspecnonsq 42134 | The discriminant used to d... |
qirropth 42135 | This lemma implements the ... |
rmspecfund 42136 | The base of exponent used ... |
rmxyelqirr 42137 | The solutions used to cons... |
rmxyelqirrOLD 42138 | Obsolete version of ~ rmxy... |
rmxypairf1o 42139 | The function used to extra... |
rmxyelxp 42140 | Lemma for ~ frmx and ~ frm... |
frmx 42141 | The X sequence is a nonneg... |
frmy 42142 | The Y sequence is an integ... |
rmxyval 42143 | Main definition of the X a... |
rmspecpos 42144 | The discriminant used to d... |
rmxycomplete 42145 | The X and Y sequences take... |
rmxynorm 42146 | The X and Y sequences defi... |
rmbaserp 42147 | The base of exponentiation... |
rmxyneg 42148 | Negation law for X and Y s... |
rmxyadd 42149 | Addition formula for X and... |
rmxy1 42150 | Value of the X and Y seque... |
rmxy0 42151 | Value of the X and Y seque... |
rmxneg 42152 | Negation law (even functio... |
rmx0 42153 | Value of X sequence at 0. ... |
rmx1 42154 | Value of X sequence at 1. ... |
rmxadd 42155 | Addition formula for X seq... |
rmyneg 42156 | Negation formula for Y seq... |
rmy0 42157 | Value of Y sequence at 0. ... |
rmy1 42158 | Value of Y sequence at 1. ... |
rmyadd 42159 | Addition formula for Y seq... |
rmxp1 42160 | Special addition-of-1 form... |
rmyp1 42161 | Special addition of 1 form... |
rmxm1 42162 | Subtraction of 1 formula f... |
rmym1 42163 | Subtraction of 1 formula f... |
rmxluc 42164 | The X sequence is a Lucas ... |
rmyluc 42165 | The Y sequence is a Lucas ... |
rmyluc2 42166 | Lucas sequence property of... |
rmxdbl 42167 | "Double-angle formula" for... |
rmydbl 42168 | "Double-angle formula" for... |
monotuz 42169 | A function defined on an u... |
monotoddzzfi 42170 | A function which is odd an... |
monotoddzz 42171 | A function (given implicit... |
oddcomabszz 42172 | An odd function which take... |
2nn0ind 42173 | Induction on nonnegative i... |
zindbi 42174 | Inductively transfer a pro... |
rmxypos 42175 | For all nonnegative indice... |
ltrmynn0 42176 | The Y-sequence is strictly... |
ltrmxnn0 42177 | The X-sequence is strictly... |
lermxnn0 42178 | The X-sequence is monotoni... |
rmxnn 42179 | The X-sequence is defined ... |
ltrmy 42180 | The Y-sequence is strictly... |
rmyeq0 42181 | Y is zero only at zero. (... |
rmyeq 42182 | Y is one-to-one. (Contrib... |
lermy 42183 | Y is monotonic (non-strict... |
rmynn 42184 | ` rmY ` is positive for po... |
rmynn0 42185 | ` rmY ` is nonnegative for... |
rmyabs 42186 | ` rmY ` commutes with ` ab... |
jm2.24nn 42187 | X(n) is strictly greater t... |
jm2.17a 42188 | First half of lemma 2.17 o... |
jm2.17b 42189 | Weak form of the second ha... |
jm2.17c 42190 | Second half of lemma 2.17 ... |
jm2.24 42191 | Lemma 2.24 of [JonesMatija... |
rmygeid 42192 | Y(n) increases faster than... |
congtr 42193 | A wff of the form ` A || (... |
congadd 42194 | If two pairs of numbers ar... |
congmul 42195 | If two pairs of numbers ar... |
congsym 42196 | Congruence mod ` A ` is a ... |
congneg 42197 | If two integers are congru... |
congsub 42198 | If two pairs of numbers ar... |
congid 42199 | Every integer is congruent... |
mzpcong 42200 | Polynomials commute with c... |
congrep 42201 | Every integer is congruent... |
congabseq 42202 | If two integers are congru... |
acongid 42203 | A wff like that in this th... |
acongsym 42204 | Symmetry of alternating co... |
acongneg2 42205 | Negate right side of alter... |
acongtr 42206 | Transitivity of alternatin... |
acongeq12d 42207 | Substitution deduction for... |
acongrep 42208 | Every integer is alternati... |
fzmaxdif 42209 | Bound on the difference be... |
fzneg 42210 | Reflection of a finite ran... |
acongeq 42211 | Two numbers in the fundame... |
dvdsacongtr 42212 | Alternating congruence pas... |
coprmdvdsb 42213 | Multiplication by a coprim... |
modabsdifz 42214 | Divisibility in terms of m... |
dvdsabsmod0 42215 | Divisibility in terms of m... |
jm2.18 42216 | Theorem 2.18 of [JonesMati... |
jm2.19lem1 42217 | Lemma for ~ jm2.19 . X an... |
jm2.19lem2 42218 | Lemma for ~ jm2.19 . (Con... |
jm2.19lem3 42219 | Lemma for ~ jm2.19 . (Con... |
jm2.19lem4 42220 | Lemma for ~ jm2.19 . Exte... |
jm2.19 42221 | Lemma 2.19 of [JonesMatija... |
jm2.21 42222 | Lemma for ~ jm2.20nn . Ex... |
jm2.22 42223 | Lemma for ~ jm2.20nn . Ap... |
jm2.23 42224 | Lemma for ~ jm2.20nn . Tr... |
jm2.20nn 42225 | Lemma 2.20 of [JonesMatija... |
jm2.25lem1 42226 | Lemma for ~ jm2.26 . (Con... |
jm2.25 42227 | Lemma for ~ jm2.26 . Rema... |
jm2.26a 42228 | Lemma for ~ jm2.26 . Reve... |
jm2.26lem3 42229 | Lemma for ~ jm2.26 . Use ... |
jm2.26 42230 | Lemma 2.26 of [JonesMatija... |
jm2.15nn0 42231 | Lemma 2.15 of [JonesMatija... |
jm2.16nn0 42232 | Lemma 2.16 of [JonesMatija... |
jm2.27a 42233 | Lemma for ~ jm2.27 . Reve... |
jm2.27b 42234 | Lemma for ~ jm2.27 . Expa... |
jm2.27c 42235 | Lemma for ~ jm2.27 . Forw... |
jm2.27 42236 | Lemma 2.27 of [JonesMatija... |
jm2.27dlem1 42237 | Lemma for ~ rmydioph . Su... |
jm2.27dlem2 42238 | Lemma for ~ rmydioph . Th... |
jm2.27dlem3 42239 | Lemma for ~ rmydioph . In... |
jm2.27dlem4 42240 | Lemma for ~ rmydioph . In... |
jm2.27dlem5 42241 | Lemma for ~ rmydioph . Us... |
rmydioph 42242 | ~ jm2.27 restated in terms... |
rmxdiophlem 42243 | X can be expressed in term... |
rmxdioph 42244 | X is a Diophantine functio... |
jm3.1lem1 42245 | Lemma for ~ jm3.1 . (Cont... |
jm3.1lem2 42246 | Lemma for ~ jm3.1 . (Cont... |
jm3.1lem3 42247 | Lemma for ~ jm3.1 . (Cont... |
jm3.1 42248 | Diophantine expression for... |
expdiophlem1 42249 | Lemma for ~ expdioph . Fu... |
expdiophlem2 42250 | Lemma for ~ expdioph . Ex... |
expdioph 42251 | The exponential function i... |
setindtr 42252 | Set induction for sets con... |
setindtrs 42253 | Set induction scheme witho... |
dford3lem1 42254 | Lemma for ~ dford3 . (Con... |
dford3lem2 42255 | Lemma for ~ dford3 . (Con... |
dford3 42256 | Ordinals are precisely the... |
dford4 42257 | ~ dford3 expressed in prim... |
wopprc 42258 | Unrelated: Wiener pairs t... |
rpnnen3lem 42259 | Lemma for ~ rpnnen3 . (Co... |
rpnnen3 42260 | Dedekind cut injection of ... |
axac10 42261 | Characterization of choice... |
harinf 42262 | The Hartogs number of an i... |
wdom2d2 42263 | Deduction for weak dominan... |
ttac 42264 | Tarski's theorem about cho... |
pw2f1ocnv 42265 | Define a bijection between... |
pw2f1o2 42266 | Define a bijection between... |
pw2f1o2val 42267 | Function value of the ~ pw... |
pw2f1o2val2 42268 | Membership in a mapped set... |
soeq12d 42269 | Equality deduction for tot... |
freq12d 42270 | Equality deduction for fou... |
weeq12d 42271 | Equality deduction for wel... |
limsuc2 42272 | Limit ordinals in the sens... |
wepwsolem 42273 | Transfer an ordering on ch... |
wepwso 42274 | A well-ordering induces a ... |
dnnumch1 42275 | Define an enumeration of a... |
dnnumch2 42276 | Define an enumeration (wea... |
dnnumch3lem 42277 | Value of the ordinal injec... |
dnnumch3 42278 | Define an injection from a... |
dnwech 42279 | Define a well-ordering fro... |
fnwe2val 42280 | Lemma for ~ fnwe2 . Subst... |
fnwe2lem1 42281 | Lemma for ~ fnwe2 . Subst... |
fnwe2lem2 42282 | Lemma for ~ fnwe2 . An el... |
fnwe2lem3 42283 | Lemma for ~ fnwe2 . Trich... |
fnwe2 42284 | A well-ordering can be con... |
aomclem1 42285 | Lemma for ~ dfac11 . This... |
aomclem2 42286 | Lemma for ~ dfac11 . Succ... |
aomclem3 42287 | Lemma for ~ dfac11 . Succ... |
aomclem4 42288 | Lemma for ~ dfac11 . Limi... |
aomclem5 42289 | Lemma for ~ dfac11 . Comb... |
aomclem6 42290 | Lemma for ~ dfac11 . Tran... |
aomclem7 42291 | Lemma for ~ dfac11 . ` ( R... |
aomclem8 42292 | Lemma for ~ dfac11 . Perf... |
dfac11 42293 | The right-hand side of thi... |
kelac1 42294 | Kelley's choice, basic for... |
kelac2lem 42295 | Lemma for ~ kelac2 and ~ d... |
kelac2 42296 | Kelley's choice, most comm... |
dfac21 42297 | Tychonoff's theorem is a c... |
islmodfg 42300 | Property of a finitely gen... |
islssfg 42301 | Property of a finitely gen... |
islssfg2 42302 | Property of a finitely gen... |
islssfgi 42303 | Finitely spanned subspaces... |
fglmod 42304 | Finitely generated left mo... |
lsmfgcl 42305 | The sum of two finitely ge... |
islnm 42308 | Property of being a Noethe... |
islnm2 42309 | Property of being a Noethe... |
lnmlmod 42310 | A Noetherian left module i... |
lnmlssfg 42311 | A submodule of Noetherian ... |
lnmlsslnm 42312 | All submodules of a Noethe... |
lnmfg 42313 | A Noetherian left module i... |
kercvrlsm 42314 | The domain of a linear fun... |
lmhmfgima 42315 | A homomorphism maps finite... |
lnmepi 42316 | Epimorphic images of Noeth... |
lmhmfgsplit 42317 | If the kernel and range of... |
lmhmlnmsplit 42318 | If the kernel and range of... |
lnmlmic 42319 | Noetherian is an invariant... |
pwssplit4 42320 | Splitting for structure po... |
filnm 42321 | Finite left modules are No... |
pwslnmlem0 42322 | Zeroeth powers are Noether... |
pwslnmlem1 42323 | First powers are Noetheria... |
pwslnmlem2 42324 | A sum of powers is Noether... |
pwslnm 42325 | Finite powers of Noetheria... |
unxpwdom3 42326 | Weaker version of ~ unxpwd... |
pwfi2f1o 42327 | The ~ pw2f1o bijection rel... |
pwfi2en 42328 | Finitely supported indicat... |
frlmpwfi 42329 | Formal linear combinations... |
gicabl 42330 | Being Abelian is a group i... |
imasgim 42331 | A relabeling of the elemen... |
isnumbasgrplem1 42332 | A set which is equipollent... |
harn0 42333 | The Hartogs number of a se... |
numinfctb 42334 | A numerable infinite set c... |
isnumbasgrplem2 42335 | If the (to be thought of a... |
isnumbasgrplem3 42336 | Every nonempty numerable s... |
isnumbasabl 42337 | A set is numerable iff it ... |
isnumbasgrp 42338 | A set is numerable iff it ... |
dfacbasgrp 42339 | A choice equivalent in abs... |
islnr 42342 | Property of a left-Noether... |
lnrring 42343 | Left-Noetherian rings are ... |
lnrlnm 42344 | Left-Noetherian rings have... |
islnr2 42345 | Property of being a left-N... |
islnr3 42346 | Relate left-Noetherian rin... |
lnr2i 42347 | Given an ideal in a left-N... |
lpirlnr 42348 | Left principal ideal rings... |
lnrfrlm 42349 | Finite-dimensional free mo... |
lnrfg 42350 | Finitely-generated modules... |
lnrfgtr 42351 | A submodule of a finitely ... |
hbtlem1 42354 | Value of the leading coeff... |
hbtlem2 42355 | Leading coefficient ideals... |
hbtlem7 42356 | Functionality of leading c... |
hbtlem4 42357 | The leading ideal function... |
hbtlem3 42358 | The leading ideal function... |
hbtlem5 42359 | The leading ideal function... |
hbtlem6 42360 | There is a finite set of p... |
hbt 42361 | The Hilbert Basis Theorem ... |
dgrsub2 42366 | Subtracting two polynomial... |
elmnc 42367 | Property of a monic polyno... |
mncply 42368 | A monic polynomial is a po... |
mnccoe 42369 | A monic polynomial has lea... |
mncn0 42370 | A monic polynomial is not ... |
dgraaval 42375 | Value of the degree functi... |
dgraalem 42376 | Properties of the degree o... |
dgraacl 42377 | Closure of the degree func... |
dgraaf 42378 | Degree function on algebra... |
dgraaub 42379 | Upper bound on degree of a... |
dgraa0p 42380 | A rational polynomial of d... |
mpaaeu 42381 | An algebraic number has ex... |
mpaaval 42382 | Value of the minimal polyn... |
mpaalem 42383 | Properties of the minimal ... |
mpaacl 42384 | Minimal polynomial is a po... |
mpaadgr 42385 | Minimal polynomial has deg... |
mpaaroot 42386 | The minimal polynomial of ... |
mpaamn 42387 | Minimal polynomial is moni... |
itgoval 42392 | Value of the integral-over... |
aaitgo 42393 | The standard algebraic num... |
itgoss 42394 | An integral element is int... |
itgocn 42395 | All integral elements are ... |
cnsrexpcl 42396 | Exponentiation is closed i... |
fsumcnsrcl 42397 | Finite sums are closed in ... |
cnsrplycl 42398 | Polynomials are closed in ... |
rgspnval 42399 | Value of the ring-span of ... |
rgspncl 42400 | The ring-span of a set is ... |
rgspnssid 42401 | The ring-span of a set con... |
rgspnmin 42402 | The ring-span is contained... |
rgspnid 42403 | The span of a subring is i... |
rngunsnply 42404 | Adjoining one element to a... |
flcidc 42405 | Finite linear combinations... |
algstr 42408 | Lemma to shorten proofs of... |
algbase 42409 | The base set of a construc... |
algaddg 42410 | The additive operation of ... |
algmulr 42411 | The multiplicative operati... |
algsca 42412 | The set of scalars of a co... |
algvsca 42413 | The scalar product operati... |
mendval 42414 | Value of the module endomo... |
mendbas 42415 | Base set of the module end... |
mendplusgfval 42416 | Addition in the module end... |
mendplusg 42417 | A specific addition in the... |
mendmulrfval 42418 | Multiplication in the modu... |
mendmulr 42419 | A specific multiplication ... |
mendsca 42420 | The module endomorphism al... |
mendvscafval 42421 | Scalar multiplication in t... |
mendvsca 42422 | A specific scalar multipli... |
mendring 42423 | The module endomorphism al... |
mendlmod 42424 | The module endomorphism al... |
mendassa 42425 | The module endomorphism al... |
idomrootle 42426 | No element of an integral ... |
idomodle 42427 | Limit on the number of ` N... |
fiuneneq 42428 | Two finite sets of equal s... |
idomsubgmo 42429 | The units of an integral d... |
proot1mul 42430 | Any primitive ` N ` -th ro... |
proot1hash 42431 | If an integral domain has ... |
proot1ex 42432 | The complex field has prim... |
isdomn3 42435 | Nonzero elements form a mu... |
mon1pid 42436 | Monicity and degree of the... |
mon1psubm 42437 | Monic polynomials are a mu... |
deg1mhm 42438 | Homomorphic property of th... |
cytpfn 42439 | Functionality of the cyclo... |
cytpval 42440 | Substitutions for the Nth ... |
fgraphopab 42441 | Express a function as a su... |
fgraphxp 42442 | Express a function as a su... |
hausgraph 42443 | The graph of a continuous ... |
r1sssucd 42448 | Deductive form of ~ r1sssu... |
iocunico 42449 | Split an open interval int... |
iocinico 42450 | The intersection of two se... |
iocmbl 42451 | An open-below, closed-abov... |
cnioobibld 42452 | A bounded, continuous func... |
arearect 42453 | The area of a rectangle wh... |
areaquad 42454 | The area of a quadrilatera... |
uniel 42455 | Two ways to say a union is... |
unielss 42456 | Two ways to say the union ... |
unielid 42457 | Two ways to say the union ... |
ssunib 42458 | Two ways to say a class is... |
rp-intrabeq 42459 | Equality theorem for supre... |
rp-unirabeq 42460 | Equality theorem for infim... |
onmaxnelsup 42461 | Two ways to say the maximu... |
onsupneqmaxlim0 42462 | If the supremum of a class... |
onsupcl2 42463 | The supremum of a set of o... |
onuniintrab 42464 | The union of a set of ordi... |
onintunirab 42465 | The intersection of a non-... |
onsupnmax 42466 | If the union of a class of... |
onsupuni 42467 | The supremum of a set of o... |
onsupuni2 42468 | The supremum of a set of o... |
onsupintrab 42469 | The supremum of a set of o... |
onsupintrab2 42470 | The supremum of a set of o... |
onsupcl3 42471 | The supremum of a set of o... |
onsupex3 42472 | The supremum of a set of o... |
onuniintrab2 42473 | The union of a set of ordi... |
oninfint 42474 | The infimum of a non-empty... |
oninfunirab 42475 | The infimum of a non-empty... |
oninfcl2 42476 | The infimum of a non-empty... |
onsupmaxb 42477 | The union of a class of or... |
onexgt 42478 | For any ordinal, there is ... |
onexomgt 42479 | For any ordinal, there is ... |
omlimcl2 42480 | The product of a limit ord... |
onexlimgt 42481 | For any ordinal, there is ... |
onexoegt 42482 | For any ordinal, there is ... |
oninfex2 42483 | The infimum of a non-empty... |
onsupeqmax 42484 | Condition when the supremu... |
onsupeqnmax 42485 | Condition when the supremu... |
onsuplub 42486 | The supremum of a set of o... |
onsupnub 42487 | An upper bound of a set of... |
onfisupcl 42488 | Sufficient condition when ... |
onelord 42489 | Every element of a ordinal... |
onepsuc 42490 | Every ordinal is less than... |
epsoon 42491 | The ordinals are strictly ... |
epirron 42492 | The strict order on the or... |
oneptr 42493 | The strict order on the or... |
oneltr 42494 | The elementhood relation o... |
oneptri 42495 | The strict, complete (line... |
oneltri 42496 | The elementhood relation o... |
ordeldif 42497 | Membership in the differen... |
ordeldifsucon 42498 | Membership in the differen... |
ordeldif1o 42499 | Membership in the differen... |
ordne0gt0 42500 | Ordinal zero is less than ... |
ondif1i 42501 | Ordinal zero is less than ... |
onsucelab 42502 | The successor of every ord... |
dflim6 42503 | A limit ordinal is a non-z... |
limnsuc 42504 | A limit ordinal is not an ... |
onsucss 42505 | If one ordinal is less tha... |
ordnexbtwnsuc 42506 | For any distinct pair of o... |
orddif0suc 42507 | For any distinct pair of o... |
onsucf1lem 42508 | For ordinals, the successo... |
onsucf1olem 42509 | The successor operation is... |
onsucrn 42510 | The successor operation is... |
onsucf1o 42511 | The successor operation is... |
dflim7 42512 | A limit ordinal is a non-z... |
onov0suclim 42513 | Compactly express rules fo... |
oa0suclim 42514 | Closed form expression of ... |
om0suclim 42515 | Closed form expression of ... |
oe0suclim 42516 | Closed form expression of ... |
oaomoecl 42517 | The operations of addition... |
onsupsucismax 42518 | If the union of a set of o... |
onsssupeqcond 42519 | If for every element of a ... |
limexissup 42520 | An ordinal which is a limi... |
limiun 42521 | A limit ordinal is the uni... |
limexissupab 42522 | An ordinal which is a limi... |
om1om1r 42523 | Ordinal one is both a left... |
oe0rif 42524 | Ordinal zero raised to any... |
oasubex 42525 | While subtraction can't be... |
nnamecl 42526 | Natural numbers are closed... |
onsucwordi 42527 | The successor operation pr... |
oalim2cl 42528 | The ordinal sum of any ord... |
oaltublim 42529 | Given ` C ` is a limit ord... |
oaordi3 42530 | Ordinal addition of the sa... |
oaord3 42531 | When the same ordinal is a... |
1oaomeqom 42532 | Ordinal one plus omega is ... |
oaabsb 42533 | The right addend absorbs t... |
oaordnrex 42534 | When omega is added on the... |
oaordnr 42535 | When the same ordinal is a... |
omge1 42536 | Any non-zero ordinal produ... |
omge2 42537 | Any non-zero ordinal produ... |
omlim2 42538 | The non-zero product with ... |
omord2lim 42539 | Given a limit ordinal, the... |
omord2i 42540 | Ordinal multiplication of ... |
omord2com 42541 | When the same non-zero ord... |
2omomeqom 42542 | Ordinal two times omega is... |
omnord1ex 42543 | When omega is multiplied o... |
omnord1 42544 | When the same non-zero ord... |
oege1 42545 | Any non-zero ordinal power... |
oege2 42546 | Any power of an ordinal at... |
rp-oelim2 42547 | The power of an ordinal at... |
oeord2lim 42548 | Given a limit ordinal, the... |
oeord2i 42549 | Ordinal exponentiation of ... |
oeord2com 42550 | When the same base at leas... |
nnoeomeqom 42551 | Any natural number at leas... |
df3o2 42552 | Ordinal 3 is the unordered... |
df3o3 42553 | Ordinal 3, fully expanded.... |
oenord1ex 42554 | When ordinals two and thre... |
oenord1 42555 | When two ordinals (both at... |
oaomoencom 42556 | Ordinal addition, multipli... |
oenassex 42557 | Ordinal two raised to two ... |
oenass 42558 | Ordinal exponentiation is ... |
cantnftermord 42559 | For terms of the form of a... |
cantnfub 42560 | Given a finite number of t... |
cantnfub2 42561 | Given a finite number of t... |
bropabg 42562 | Equivalence for two classe... |
cantnfresb 42563 | A Cantor normal form which... |
cantnf2 42564 | For every ordinal, ` A ` ,... |
oawordex2 42565 | If ` C ` is between ` A ` ... |
nnawordexg 42566 | If an ordinal, ` B ` , is ... |
succlg 42567 | Closure law for ordinal su... |
dflim5 42568 | A limit ordinal is either ... |
oacl2g 42569 | Closure law for ordinal ad... |
onmcl 42570 | If an ordinal is less than... |
omabs2 42571 | Ordinal multiplication by ... |
omcl2 42572 | Closure law for ordinal mu... |
omcl3g 42573 | Closure law for ordinal mu... |
ordsssucb 42574 | An ordinal number is less ... |
tfsconcatlem 42575 | Lemma for ~ tfsconcatun . ... |
tfsconcatun 42576 | The concatenation of two t... |
tfsconcatfn 42577 | The concatenation of two t... |
tfsconcatfv1 42578 | An early value of the conc... |
tfsconcatfv2 42579 | A latter value of the conc... |
tfsconcatfv 42580 | The value of the concatena... |
tfsconcatrn 42581 | The range of the concatena... |
tfsconcatfo 42582 | The concatenation of two t... |
tfsconcatb0 42583 | The concatentation with th... |
tfsconcat0i 42584 | The concatentation with th... |
tfsconcat0b 42585 | The concatentation with th... |
tfsconcat00 42586 | The concatentation of two ... |
tfsconcatrev 42587 | If the domain of a transfi... |
tfsconcatrnss12 42588 | The range of the concatena... |
tfsconcatrnss 42589 | The concatenation of trans... |
tfsconcatrnsson 42590 | The concatenation of trans... |
tfsnfin 42591 | A transfinite sequence is ... |
rp-tfslim 42592 | The limit of a sequence of... |
ofoafg 42593 | Addition operator for func... |
ofoaf 42594 | Addition operator for func... |
ofoafo 42595 | Addition operator for func... |
ofoacl 42596 | Closure law for component ... |
ofoaid1 42597 | Identity law for component... |
ofoaid2 42598 | Identity law for component... |
ofoaass 42599 | Component-wise addition of... |
ofoacom 42600 | Component-wise addition of... |
naddcnff 42601 | Addition operator for Cant... |
naddcnffn 42602 | Addition operator for Cant... |
naddcnffo 42603 | Addition of Cantor normal ... |
naddcnfcl 42604 | Closure law for component-... |
naddcnfcom 42605 | Component-wise ordinal add... |
naddcnfid1 42606 | Identity law for component... |
naddcnfid2 42607 | Identity law for component... |
naddcnfass 42608 | Component-wise addition of... |
onsucunifi 42609 | The successor to the union... |
sucunisn 42610 | The successor to the union... |
onsucunipr 42611 | The successor to the union... |
onsucunitp 42612 | The successor to the union... |
oaun3lem1 42613 | The class of all ordinal s... |
oaun3lem2 42614 | The class of all ordinal s... |
oaun3lem3 42615 | The class of all ordinal s... |
oaun3lem4 42616 | The class of all ordinal s... |
rp-abid 42617 | Two ways to express a clas... |
oadif1lem 42618 | Express the set difference... |
oadif1 42619 | Express the set difference... |
oaun2 42620 | Ordinal addition as a unio... |
oaun3 42621 | Ordinal addition as a unio... |
naddov4 42622 | Alternate expression for n... |
nadd2rabtr 42623 | The set of ordinals which ... |
nadd2rabord 42624 | The set of ordinals which ... |
nadd2rabex 42625 | The class of ordinals whic... |
nadd2rabon 42626 | The set of ordinals which ... |
nadd1rabtr 42627 | The set of ordinals which ... |
nadd1rabord 42628 | The set of ordinals which ... |
nadd1rabex 42629 | The class of ordinals whic... |
nadd1rabon 42630 | The set of ordinals which ... |
nadd1suc 42631 | Natural addition with 1 is... |
naddsuc2 42632 | Natural addition with succ... |
naddass1 42633 | Natural addition of ordina... |
naddgeoa 42634 | Natural addition results i... |
naddonnn 42635 | Natural addition with a na... |
naddwordnexlem0 42636 | When ` A ` is the sum of a... |
naddwordnexlem1 42637 | When ` A ` is the sum of a... |
naddwordnexlem2 42638 | When ` A ` is the sum of a... |
naddwordnexlem3 42639 | When ` A ` is the sum of a... |
oawordex3 42640 | When ` A ` is the sum of a... |
naddwordnexlem4 42641 | When ` A ` is the sum of a... |
ordsssucim 42642 | If an ordinal is less than... |
insucid 42643 | The intersection of a clas... |
om2 42644 | Two ways to double an ordi... |
oaltom 42645 | Multiplication eventually ... |
oe2 42646 | Two ways to square an ordi... |
omltoe 42647 | Exponentiation eventually ... |
abeqabi 42648 | Generalized condition for ... |
abpr 42649 | Condition for a class abst... |
abtp 42650 | Condition for a class abst... |
ralopabb 42651 | Restricted universal quant... |
fpwfvss 42652 | Functions into a powerset ... |
sdomne0 42653 | A class that strictly domi... |
sdomne0d 42654 | A class that strictly domi... |
safesnsupfiss 42655 | If ` B ` is a finite subse... |
safesnsupfiub 42656 | If ` B ` is a finite subse... |
safesnsupfidom1o 42657 | If ` B ` is a finite subse... |
safesnsupfilb 42658 | If ` B ` is a finite subse... |
isoeq145d 42659 | Equality deduction for iso... |
resisoeq45d 42660 | Equality deduction for equ... |
negslem1 42661 | An equivalence between ide... |
nvocnvb 42662 | Equivalence to saying the ... |
rp-brsslt 42663 | Binary relation form of a ... |
nla0002 42664 | Extending a linear order t... |
nla0003 42665 | Extending a linear order t... |
nla0001 42666 | Extending a linear order t... |
faosnf0.11b 42667 | ` B ` is called a non-limi... |
dfno2 42668 | A surreal number, in the f... |
onnog 42669 | Every ordinal maps to a su... |
onnobdayg 42670 | Every ordinal maps to a su... |
bdaybndex 42671 | Bounds formed from the bir... |
bdaybndbday 42672 | Bounds formed from the bir... |
onno 42673 | Every ordinal maps to a su... |
onnoi 42674 | Every ordinal maps to a su... |
0no 42675 | Ordinal zero maps to a sur... |
1no 42676 | Ordinal one maps to a surr... |
2no 42677 | Ordinal two maps to a surr... |
3no 42678 | Ordinal three maps to a su... |
4no 42679 | Ordinal four maps to a sur... |
fnimafnex 42680 | The functional image of a ... |
nlimsuc 42681 | A successor is not a limit... |
nlim1NEW 42682 | 1 is not a limit ordinal. ... |
nlim2NEW 42683 | 2 is not a limit ordinal. ... |
nlim3 42684 | 3 is not a limit ordinal. ... |
nlim4 42685 | 4 is not a limit ordinal. ... |
oa1un 42686 | Given ` A e. On ` , let ` ... |
oa1cl 42687 | ` A +o 1o ` is in ` On ` .... |
0finon 42688 | 0 is a finite ordinal. Se... |
1finon 42689 | 1 is a finite ordinal. Se... |
2finon 42690 | 2 is a finite ordinal. Se... |
3finon 42691 | 3 is a finite ordinal. Se... |
4finon 42692 | 4 is a finite ordinal. Se... |
finona1cl 42693 | The finite ordinals are cl... |
finonex 42694 | The finite ordinals are a ... |
fzunt 42695 | Union of two adjacent fini... |
fzuntd 42696 | Union of two adjacent fini... |
fzunt1d 42697 | Union of two overlapping f... |
fzuntgd 42698 | Union of two adjacent or o... |
ifpan123g 42699 | Conjunction of conditional... |
ifpan23 42700 | Conjunction of conditional... |
ifpdfor2 42701 | Define or in terms of cond... |
ifporcor 42702 | Corollary of commutation o... |
ifpdfan2 42703 | Define and with conditiona... |
ifpancor 42704 | Corollary of commutation o... |
ifpdfor 42705 | Define or in terms of cond... |
ifpdfan 42706 | Define and with conditiona... |
ifpbi2 42707 | Equivalence theorem for co... |
ifpbi3 42708 | Equivalence theorem for co... |
ifpim1 42709 | Restate implication as con... |
ifpnot 42710 | Restate negated wff as con... |
ifpid2 42711 | Restate wff as conditional... |
ifpim2 42712 | Restate implication as con... |
ifpbi23 42713 | Equivalence theorem for co... |
ifpbiidcor 42714 | Restatement of ~ biid . (... |
ifpbicor 42715 | Corollary of commutation o... |
ifpxorcor 42716 | Corollary of commutation o... |
ifpbi1 42717 | Equivalence theorem for co... |
ifpnot23 42718 | Negation of conditional lo... |
ifpnotnotb 42719 | Factor conditional logic o... |
ifpnorcor 42720 | Corollary of commutation o... |
ifpnancor 42721 | Corollary of commutation o... |
ifpnot23b 42722 | Negation of conditional lo... |
ifpbiidcor2 42723 | Restatement of ~ biid . (... |
ifpnot23c 42724 | Negation of conditional lo... |
ifpnot23d 42725 | Negation of conditional lo... |
ifpdfnan 42726 | Define nand as conditional... |
ifpdfxor 42727 | Define xor as conditional ... |
ifpbi12 42728 | Equivalence theorem for co... |
ifpbi13 42729 | Equivalence theorem for co... |
ifpbi123 42730 | Equivalence theorem for co... |
ifpidg 42731 | Restate wff as conditional... |
ifpid3g 42732 | Restate wff as conditional... |
ifpid2g 42733 | Restate wff as conditional... |
ifpid1g 42734 | Restate wff as conditional... |
ifpim23g 42735 | Restate implication as con... |
ifpim3 42736 | Restate implication as con... |
ifpnim1 42737 | Restate negated implicatio... |
ifpim4 42738 | Restate implication as con... |
ifpnim2 42739 | Restate negated implicatio... |
ifpim123g 42740 | Implication of conditional... |
ifpim1g 42741 | Implication of conditional... |
ifp1bi 42742 | Substitute the first eleme... |
ifpbi1b 42743 | When the first variable is... |
ifpimimb 42744 | Factor conditional logic o... |
ifpororb 42745 | Factor conditional logic o... |
ifpananb 42746 | Factor conditional logic o... |
ifpnannanb 42747 | Factor conditional logic o... |
ifpor123g 42748 | Disjunction of conditional... |
ifpimim 42749 | Consequnce of implication.... |
ifpbibib 42750 | Factor conditional logic o... |
ifpxorxorb 42751 | Factor conditional logic o... |
rp-fakeimass 42752 | A special case where impli... |
rp-fakeanorass 42753 | A special case where a mix... |
rp-fakeoranass 42754 | A special case where a mix... |
rp-fakeinunass 42755 | A special case where a mix... |
rp-fakeuninass 42756 | A special case where a mix... |
rp-isfinite5 42757 | A set is said to be finite... |
rp-isfinite6 42758 | A set is said to be finite... |
intabssd 42759 | When for each element ` y ... |
eu0 42760 | There is only one empty se... |
epelon2 42761 | Over the ordinal numbers, ... |
ontric3g 42762 | For all ` x , y e. On ` , ... |
dfsucon 42763 | ` A ` is called a successo... |
snen1g 42764 | A singleton is equinumerou... |
snen1el 42765 | A singleton is equinumerou... |
sn1dom 42766 | A singleton is dominated b... |
pr2dom 42767 | An unordered pair is domin... |
tr3dom 42768 | An unordered triple is dom... |
ensucne0 42769 | A class equinumerous to a ... |
ensucne0OLD 42770 | A class equinumerous to a ... |
dfom6 42771 | Let ` _om ` be defined to ... |
infordmin 42772 | ` _om ` is the smallest in... |
iscard4 42773 | Two ways to express the pr... |
minregex 42774 | Given any cardinal number ... |
minregex2 42775 | Given any cardinal number ... |
iscard5 42776 | Two ways to express the pr... |
elrncard 42777 | Let us define a cardinal n... |
harval3 42778 | ` ( har `` A ) ` is the le... |
harval3on 42779 | For any ordinal number ` A... |
omssrncard 42780 | All natural numbers are ca... |
0iscard 42781 | 0 is a cardinal number. (... |
1iscard 42782 | 1 is a cardinal number. (... |
omiscard 42783 | ` _om ` is a cardinal numb... |
sucomisnotcard 42784 | ` _om +o 1o ` is not a car... |
nna1iscard 42785 | For any natural number, th... |
har2o 42786 | The least cardinal greater... |
en2pr 42787 | A class is equinumerous to... |
pr2cv 42788 | If an unordered pair is eq... |
pr2el1 42789 | If an unordered pair is eq... |
pr2cv1 42790 | If an unordered pair is eq... |
pr2el2 42791 | If an unordered pair is eq... |
pr2cv2 42792 | If an unordered pair is eq... |
pren2 42793 | An unordered pair is equin... |
pr2eldif1 42794 | If an unordered pair is eq... |
pr2eldif2 42795 | If an unordered pair is eq... |
pren2d 42796 | A pair of two distinct set... |
aleph1min 42797 | ` ( aleph `` 1o ) ` is the... |
alephiso2 42798 | ` aleph ` is a strictly or... |
alephiso3 42799 | ` aleph ` is a strictly or... |
pwelg 42800 | The powerclass is an eleme... |
pwinfig 42801 | The powerclass of an infin... |
pwinfi2 42802 | The powerclass of an infin... |
pwinfi3 42803 | The powerclass of an infin... |
pwinfi 42804 | The powerclass of an infin... |
fipjust 42805 | A definition of the finite... |
cllem0 42806 | The class of all sets with... |
superficl 42807 | The class of all supersets... |
superuncl 42808 | The class of all supersets... |
ssficl 42809 | The class of all subsets o... |
ssuncl 42810 | The class of all subsets o... |
ssdifcl 42811 | The class of all subsets o... |
sssymdifcl 42812 | The class of all subsets o... |
fiinfi 42813 | If two classes have the fi... |
rababg 42814 | Condition when restricted ... |
elinintab 42815 | Two ways of saying a set i... |
elmapintrab 42816 | Two ways to say a set is a... |
elinintrab 42817 | Two ways of saying a set i... |
inintabss 42818 | Upper bound on intersectio... |
inintabd 42819 | Value of the intersection ... |
xpinintabd 42820 | Value of the intersection ... |
relintabex 42821 | If the intersection of a c... |
elcnvcnvintab 42822 | Two ways of saying a set i... |
relintab 42823 | Value of the intersection ... |
nonrel 42824 | A non-relation is equal to... |
elnonrel 42825 | Only an ordered pair where... |
cnvssb 42826 | Subclass theorem for conve... |
relnonrel 42827 | The non-relation part of a... |
cnvnonrel 42828 | The converse of the non-re... |
brnonrel 42829 | A non-relation cannot rela... |
dmnonrel 42830 | The domain of the non-rela... |
rnnonrel 42831 | The range of the non-relat... |
resnonrel 42832 | A restriction of the non-r... |
imanonrel 42833 | An image under the non-rel... |
cononrel1 42834 | Composition with the non-r... |
cononrel2 42835 | Composition with the non-r... |
elmapintab 42836 | Two ways to say a set is a... |
fvnonrel 42837 | The function value of any ... |
elinlem 42838 | Two ways to say a set is a... |
elcnvcnvlem 42839 | Two ways to say a set is a... |
cnvcnvintabd 42840 | Value of the relationship ... |
elcnvlem 42841 | Two ways to say a set is a... |
elcnvintab 42842 | Two ways of saying a set i... |
cnvintabd 42843 | Value of the converse of t... |
undmrnresiss 42844 | Two ways of saying the ide... |
reflexg 42845 | Two ways of saying a relat... |
cnvssco 42846 | A condition weaker than re... |
refimssco 42847 | Reflexive relations are su... |
cleq2lem 42848 | Equality implies bijection... |
cbvcllem 42849 | Change of bound variable i... |
clublem 42850 | If a superset ` Y ` of ` X... |
clss2lem 42851 | The closure of a property ... |
dfid7 42852 | Definition of identity rel... |
mptrcllem 42853 | Show two versions of a clo... |
cotrintab 42854 | The intersection of a clas... |
rclexi 42855 | The reflexive closure of a... |
rtrclexlem 42856 | Existence of relation impl... |
rtrclex 42857 | The reflexive-transitive c... |
trclubgNEW 42858 | If a relation exists then ... |
trclubNEW 42859 | If a relation exists then ... |
trclexi 42860 | The transitive closure of ... |
rtrclexi 42861 | The reflexive-transitive c... |
clrellem 42862 | When the property ` ps ` h... |
clcnvlem 42863 | When ` A ` , an upper boun... |
cnvtrucl0 42864 | The converse of the trivia... |
cnvrcl0 42865 | The converse of the reflex... |
cnvtrcl0 42866 | The converse of the transi... |
dmtrcl 42867 | The domain of the transiti... |
rntrcl 42868 | The range of the transitiv... |
dfrtrcl5 42869 | Definition of reflexive-tr... |
trcleq2lemRP 42870 | Equality implies bijection... |
sqrtcvallem1 42871 | Two ways of saying a compl... |
reabsifneg 42872 | Alternate expression for t... |
reabsifnpos 42873 | Alternate expression for t... |
reabsifpos 42874 | Alternate expression for t... |
reabsifnneg 42875 | Alternate expression for t... |
reabssgn 42876 | Alternate expression for t... |
sqrtcvallem2 42877 | Equivalent to saying that ... |
sqrtcvallem3 42878 | Equivalent to saying that ... |
sqrtcvallem4 42879 | Equivalent to saying that ... |
sqrtcvallem5 42880 | Equivalent to saying that ... |
sqrtcval 42881 | Explicit formula for the c... |
sqrtcval2 42882 | Explicit formula for the c... |
resqrtval 42883 | Real part of the complex s... |
imsqrtval 42884 | Imaginary part of the comp... |
resqrtvalex 42885 | Example for ~ resqrtval . ... |
imsqrtvalex 42886 | Example for ~ imsqrtval . ... |
al3im 42887 | Version of ~ ax-4 for a ne... |
intima0 42888 | Two ways of expressing the... |
elimaint 42889 | Element of image of inters... |
cnviun 42890 | Converse of indexed union.... |
imaiun1 42891 | The image of an indexed un... |
coiun1 42892 | Composition with an indexe... |
elintima 42893 | Element of intersection of... |
intimass 42894 | The image under the inters... |
intimass2 42895 | The image under the inters... |
intimag 42896 | Requirement for the image ... |
intimasn 42897 | Two ways to express the im... |
intimasn2 42898 | Two ways to express the im... |
ss2iundf 42899 | Subclass theorem for index... |
ss2iundv 42900 | Subclass theorem for index... |
cbviuneq12df 42901 | Rule used to change the bo... |
cbviuneq12dv 42902 | Rule used to change the bo... |
conrel1d 42903 | Deduction about compositio... |
conrel2d 42904 | Deduction about compositio... |
trrelind 42905 | The intersection of transi... |
xpintrreld 42906 | The intersection of a tran... |
restrreld 42907 | The restriction of a trans... |
trrelsuperreldg 42908 | Concrete construction of a... |
trficl 42909 | The class of all transitiv... |
cnvtrrel 42910 | The converse of a transiti... |
trrelsuperrel2dg 42911 | Concrete construction of a... |
dfrcl2 42914 | Reflexive closure of a rel... |
dfrcl3 42915 | Reflexive closure of a rel... |
dfrcl4 42916 | Reflexive closure of a rel... |
relexp2 42917 | A set operated on by the r... |
relexpnul 42918 | If the domain and range of... |
eliunov2 42919 | Membership in the indexed ... |
eltrclrec 42920 | Membership in the indexed ... |
elrtrclrec 42921 | Membership in the indexed ... |
briunov2 42922 | Two classes related by the... |
brmptiunrelexpd 42923 | If two elements are connec... |
fvmptiunrelexplb0d 42924 | If the indexed union range... |
fvmptiunrelexplb0da 42925 | If the indexed union range... |
fvmptiunrelexplb1d 42926 | If the indexed union range... |
brfvid 42927 | If two elements are connec... |
brfvidRP 42928 | If two elements are connec... |
fvilbd 42929 | A set is a subset of its i... |
fvilbdRP 42930 | A set is a subset of its i... |
brfvrcld 42931 | If two elements are connec... |
brfvrcld2 42932 | If two elements are connec... |
fvrcllb0d 42933 | A restriction of the ident... |
fvrcllb0da 42934 | A restriction of the ident... |
fvrcllb1d 42935 | A set is a subset of its i... |
brtrclrec 42936 | Two classes related by the... |
brrtrclrec 42937 | Two classes related by the... |
briunov2uz 42938 | Two classes related by the... |
eliunov2uz 42939 | Membership in the indexed ... |
ov2ssiunov2 42940 | Any particular operator va... |
relexp0eq 42941 | The zeroth power of relati... |
iunrelexp0 42942 | Simplification of zeroth p... |
relexpxpnnidm 42943 | Any positive power of a Ca... |
relexpiidm 42944 | Any power of any restricti... |
relexpss1d 42945 | The relational power of a ... |
comptiunov2i 42946 | The composition two indexe... |
corclrcl 42947 | The reflexive closure is i... |
iunrelexpmin1 42948 | The indexed union of relat... |
relexpmulnn 42949 | With exponents limited to ... |
relexpmulg 42950 | With ordered exponents, th... |
trclrelexplem 42951 | The union of relational po... |
iunrelexpmin2 42952 | The indexed union of relat... |
relexp01min 42953 | With exponents limited to ... |
relexp1idm 42954 | Repeated raising a relatio... |
relexp0idm 42955 | Repeated raising a relatio... |
relexp0a 42956 | Absorption law for zeroth ... |
relexpxpmin 42957 | The composition of powers ... |
relexpaddss 42958 | The composition of two pow... |
iunrelexpuztr 42959 | The indexed union of relat... |
dftrcl3 42960 | Transitive closure of a re... |
brfvtrcld 42961 | If two elements are connec... |
fvtrcllb1d 42962 | A set is a subset of its i... |
trclfvcom 42963 | The transitive closure of ... |
cnvtrclfv 42964 | The converse of the transi... |
cotrcltrcl 42965 | The transitive closure is ... |
trclimalb2 42966 | Lower bound for image unde... |
brtrclfv2 42967 | Two ways to indicate two e... |
trclfvdecomr 42968 | The transitive closure of ... |
trclfvdecoml 42969 | The transitive closure of ... |
dmtrclfvRP 42970 | The domain of the transiti... |
rntrclfvRP 42971 | The range of the transitiv... |
rntrclfv 42972 | The range of the transitiv... |
dfrtrcl3 42973 | Reflexive-transitive closu... |
brfvrtrcld 42974 | If two elements are connec... |
fvrtrcllb0d 42975 | A restriction of the ident... |
fvrtrcllb0da 42976 | A restriction of the ident... |
fvrtrcllb1d 42977 | A set is a subset of its i... |
dfrtrcl4 42978 | Reflexive-transitive closu... |
corcltrcl 42979 | The composition of the ref... |
cortrcltrcl 42980 | Composition with the refle... |
corclrtrcl 42981 | Composition with the refle... |
cotrclrcl 42982 | The composition of the ref... |
cortrclrcl 42983 | Composition with the refle... |
cotrclrtrcl 42984 | Composition with the refle... |
cortrclrtrcl 42985 | The reflexive-transitive c... |
frege77d 42986 | If the images of both ` { ... |
frege81d 42987 | If the image of ` U ` is a... |
frege83d 42988 | If the image of the union ... |
frege96d 42989 | If ` C ` follows ` A ` in ... |
frege87d 42990 | If the images of both ` { ... |
frege91d 42991 | If ` B ` follows ` A ` in ... |
frege97d 42992 | If ` A ` contains all elem... |
frege98d 42993 | If ` C ` follows ` A ` and... |
frege102d 42994 | If either ` A ` and ` C ` ... |
frege106d 42995 | If ` B ` follows ` A ` in ... |
frege108d 42996 | If either ` A ` and ` C ` ... |
frege109d 42997 | If ` A ` contains all elem... |
frege114d 42998 | If either ` R ` relates ` ... |
frege111d 42999 | If either ` A ` and ` C ` ... |
frege122d 43000 | If ` F ` is a function, ` ... |
frege124d 43001 | If ` F ` is a function, ` ... |
frege126d 43002 | If ` F ` is a function, ` ... |
frege129d 43003 | If ` F ` is a function and... |
frege131d 43004 | If ` F ` is a function and... |
frege133d 43005 | If ` F ` is a function and... |
dfxor4 43006 | Express exclusive-or in te... |
dfxor5 43007 | Express exclusive-or in te... |
df3or2 43008 | Express triple-or in terms... |
df3an2 43009 | Express triple-and in term... |
nev 43010 | Express that not every set... |
0pssin 43011 | Express that an intersecti... |
dfhe2 43014 | The property of relation `... |
dfhe3 43015 | The property of relation `... |
heeq12 43016 | Equality law for relations... |
heeq1 43017 | Equality law for relations... |
heeq2 43018 | Equality law for relations... |
sbcheg 43019 | Distribute proper substitu... |
hess 43020 | Subclass law for relations... |
xphe 43021 | Any Cartesian product is h... |
0he 43022 | The empty relation is here... |
0heALT 43023 | The empty relation is here... |
he0 43024 | Any relation is hereditary... |
unhe1 43025 | The union of two relations... |
snhesn 43026 | Any singleton is hereditar... |
idhe 43027 | The identity relation is h... |
psshepw 43028 | The relation between sets ... |
sshepw 43029 | The relation between sets ... |
rp-simp2-frege 43032 | Simplification of triple c... |
rp-simp2 43033 | Simplification of triple c... |
rp-frege3g 43034 | Add antecedent to ~ ax-fre... |
frege3 43035 | Add antecedent to ~ ax-fre... |
rp-misc1-frege 43036 | Double-use of ~ ax-frege2 ... |
rp-frege24 43037 | Introducing an embedded an... |
rp-frege4g 43038 | Deduction related to distr... |
frege4 43039 | Special case of closed for... |
frege5 43040 | A closed form of ~ syl . ... |
rp-7frege 43041 | Distribute antecedent and ... |
rp-4frege 43042 | Elimination of a nested an... |
rp-6frege 43043 | Elimination of a nested an... |
rp-8frege 43044 | Eliminate antecedent when ... |
rp-frege25 43045 | Closed form for ~ a1dd . ... |
frege6 43046 | A closed form of ~ imim2d ... |
axfrege8 43047 | Swap antecedents. Identic... |
frege7 43048 | A closed form of ~ syl6 . ... |
frege26 43050 | Identical to ~ idd . Prop... |
frege27 43051 | We cannot (at the same tim... |
frege9 43052 | Closed form of ~ syl with ... |
frege12 43053 | A closed form of ~ com23 .... |
frege11 43054 | Elimination of a nested an... |
frege24 43055 | Closed form for ~ a1d . D... |
frege16 43056 | A closed form of ~ com34 .... |
frege25 43057 | Closed form for ~ a1dd . ... |
frege18 43058 | Closed form of a syllogism... |
frege22 43059 | A closed form of ~ com45 .... |
frege10 43060 | Result commuting anteceden... |
frege17 43061 | A closed form of ~ com3l .... |
frege13 43062 | A closed form of ~ com3r .... |
frege14 43063 | Closed form of a deduction... |
frege19 43064 | A closed form of ~ syl6 . ... |
frege23 43065 | Syllogism followed by rota... |
frege15 43066 | A closed form of ~ com4r .... |
frege21 43067 | Replace antecedent in ante... |
frege20 43068 | A closed form of ~ syl8 . ... |
axfrege28 43069 | Contraposition. Identical... |
frege29 43071 | Closed form of ~ con3d . ... |
frege30 43072 | Commuted, closed form of ~... |
axfrege31 43073 | Identical to ~ notnotr . ... |
frege32 43075 | Deduce ~ con1 from ~ con3 ... |
frege33 43076 | If ` ph ` or ` ps ` takes ... |
frege34 43077 | If as a consequence of the... |
frege35 43078 | Commuted, closed form of ~... |
frege36 43079 | The case in which ` ps ` i... |
frege37 43080 | If ` ch ` is a necessary c... |
frege38 43081 | Identical to ~ pm2.21 . P... |
frege39 43082 | Syllogism between ~ pm2.18... |
frege40 43083 | Anything implies ~ pm2.18 ... |
axfrege41 43084 | Identical to ~ notnot . A... |
frege42 43086 | Not not ~ id . Propositio... |
frege43 43087 | If there is a choice only ... |
frege44 43088 | Similar to a commuted ~ pm... |
frege45 43089 | Deduce ~ pm2.6 from ~ con1... |
frege46 43090 | If ` ps ` holds when ` ph ... |
frege47 43091 | Deduce consequence follows... |
frege48 43092 | Closed form of syllogism w... |
frege49 43093 | Closed form of deduction w... |
frege50 43094 | Closed form of ~ jaoi . P... |
frege51 43095 | Compare with ~ jaod . Pro... |
axfrege52a 43096 | Justification for ~ ax-fre... |
frege52aid 43098 | The case when the content ... |
frege53aid 43099 | Specialization of ~ frege5... |
frege53a 43100 | Lemma for ~ frege55a . Pr... |
axfrege54a 43101 | Justification for ~ ax-fre... |
frege54cor0a 43103 | Synonym for logical equiva... |
frege54cor1a 43104 | Reflexive equality. (Cont... |
frege55aid 43105 | Lemma for ~ frege57aid . ... |
frege55lem1a 43106 | Necessary deduction regard... |
frege55lem2a 43107 | Core proof of Proposition ... |
frege55a 43108 | Proposition 55 of [Frege18... |
frege55cor1a 43109 | Proposition 55 of [Frege18... |
frege56aid 43110 | Lemma for ~ frege57aid . ... |
frege56a 43111 | Proposition 56 of [Frege18... |
frege57aid 43112 | This is the all imporant f... |
frege57a 43113 | Analogue of ~ frege57aid .... |
axfrege58a 43114 | Identical to ~ anifp . Ju... |
frege58acor 43116 | Lemma for ~ frege59a . (C... |
frege59a 43117 | A kind of Aristotelian inf... |
frege60a 43118 | Swap antecedents of ~ ax-f... |
frege61a 43119 | Lemma for ~ frege65a . Pr... |
frege62a 43120 | A kind of Aristotelian inf... |
frege63a 43121 | Proposition 63 of [Frege18... |
frege64a 43122 | Lemma for ~ frege65a . Pr... |
frege65a 43123 | A kind of Aristotelian inf... |
frege66a 43124 | Swap antecedents of ~ freg... |
frege67a 43125 | Lemma for ~ frege68a . Pr... |
frege68a 43126 | Combination of applying a ... |
axfrege52c 43127 | Justification for ~ ax-fre... |
frege52b 43129 | The case when the content ... |
frege53b 43130 | Lemma for frege102 (via ~ ... |
axfrege54c 43131 | Reflexive equality of clas... |
frege54b 43133 | Reflexive equality of sets... |
frege54cor1b 43134 | Reflexive equality. (Cont... |
frege55lem1b 43135 | Necessary deduction regard... |
frege55lem2b 43136 | Lemma for ~ frege55b . Co... |
frege55b 43137 | Lemma for ~ frege57b . Pr... |
frege56b 43138 | Lemma for ~ frege57b . Pr... |
frege57b 43139 | Analogue of ~ frege57aid .... |
axfrege58b 43140 | If ` A. x ph ` is affirmed... |
frege58bid 43142 | If ` A. x ph ` is affirmed... |
frege58bcor 43143 | Lemma for ~ frege59b . (C... |
frege59b 43144 | A kind of Aristotelian inf... |
frege60b 43145 | Swap antecedents of ~ ax-f... |
frege61b 43146 | Lemma for ~ frege65b . Pr... |
frege62b 43147 | A kind of Aristotelian inf... |
frege63b 43148 | Lemma for ~ frege91 . Pro... |
frege64b 43149 | Lemma for ~ frege65b . Pr... |
frege65b 43150 | A kind of Aristotelian inf... |
frege66b 43151 | Swap antecedents of ~ freg... |
frege67b 43152 | Lemma for ~ frege68b . Pr... |
frege68b 43153 | Combination of applying a ... |
frege53c 43154 | Proposition 53 of [Frege18... |
frege54cor1c 43155 | Reflexive equality. (Cont... |
frege55lem1c 43156 | Necessary deduction regard... |
frege55lem2c 43157 | Core proof of Proposition ... |
frege55c 43158 | Proposition 55 of [Frege18... |
frege56c 43159 | Lemma for ~ frege57c . Pr... |
frege57c 43160 | Swap order of implication ... |
frege58c 43161 | Principle related to ~ sp ... |
frege59c 43162 | A kind of Aristotelian inf... |
frege60c 43163 | Swap antecedents of ~ freg... |
frege61c 43164 | Lemma for ~ frege65c . Pr... |
frege62c 43165 | A kind of Aristotelian inf... |
frege63c 43166 | Analogue of ~ frege63b . ... |
frege64c 43167 | Lemma for ~ frege65c . Pr... |
frege65c 43168 | A kind of Aristotelian inf... |
frege66c 43169 | Swap antecedents of ~ freg... |
frege67c 43170 | Lemma for ~ frege68c . Pr... |
frege68c 43171 | Combination of applying a ... |
dffrege69 43172 | If from the proposition th... |
frege70 43173 | Lemma for ~ frege72 . Pro... |
frege71 43174 | Lemma for ~ frege72 . Pro... |
frege72 43175 | If property ` A ` is hered... |
frege73 43176 | Lemma for ~ frege87 . Pro... |
frege74 43177 | If ` X ` has a property ` ... |
frege75 43178 | If from the proposition th... |
dffrege76 43179 | If from the two propositio... |
frege77 43180 | If ` Y ` follows ` X ` in ... |
frege78 43181 | Commuted form of ~ frege77... |
frege79 43182 | Distributed form of ~ freg... |
frege80 43183 | Add additional condition t... |
frege81 43184 | If ` X ` has a property ` ... |
frege82 43185 | Closed-form deduction base... |
frege83 43186 | Apply commuted form of ~ f... |
frege84 43187 | Commuted form of ~ frege81... |
frege85 43188 | Commuted form of ~ frege77... |
frege86 43189 | Conclusion about element o... |
frege87 43190 | If ` Z ` is a result of an... |
frege88 43191 | Commuted form of ~ frege87... |
frege89 43192 | One direction of ~ dffrege... |
frege90 43193 | Add antecedent to ~ frege8... |
frege91 43194 | Every result of an applica... |
frege92 43195 | Inference from ~ frege91 .... |
frege93 43196 | Necessary condition for tw... |
frege94 43197 | Looking one past a pair re... |
frege95 43198 | Looking one past a pair re... |
frege96 43199 | Every result of an applica... |
frege97 43200 | The property of following ... |
frege98 43201 | If ` Y ` follows ` X ` and... |
dffrege99 43202 | If ` Z ` is identical with... |
frege100 43203 | One direction of ~ dffrege... |
frege101 43204 | Lemma for ~ frege102 . Pr... |
frege102 43205 | If ` Z ` belongs to the ` ... |
frege103 43206 | Proposition 103 of [Frege1... |
frege104 43207 | Proposition 104 of [Frege1... |
frege105 43208 | Proposition 105 of [Frege1... |
frege106 43209 | Whatever follows ` X ` in ... |
frege107 43210 | Proposition 107 of [Frege1... |
frege108 43211 | If ` Y ` belongs to the ` ... |
frege109 43212 | The property of belonging ... |
frege110 43213 | Proposition 110 of [Frege1... |
frege111 43214 | If ` Y ` belongs to the ` ... |
frege112 43215 | Identity implies belonging... |
frege113 43216 | Proposition 113 of [Frege1... |
frege114 43217 | If ` X ` belongs to the ` ... |
dffrege115 43218 | If from the circumstance t... |
frege116 43219 | One direction of ~ dffrege... |
frege117 43220 | Lemma for ~ frege118 . Pr... |
frege118 43221 | Simplified application of ... |
frege119 43222 | Lemma for ~ frege120 . Pr... |
frege120 43223 | Simplified application of ... |
frege121 43224 | Lemma for ~ frege122 . Pr... |
frege122 43225 | If ` X ` is a result of an... |
frege123 43226 | Lemma for ~ frege124 . Pr... |
frege124 43227 | If ` X ` is a result of an... |
frege125 43228 | Lemma for ~ frege126 . Pr... |
frege126 43229 | If ` M ` follows ` Y ` in ... |
frege127 43230 | Communte antecedents of ~ ... |
frege128 43231 | Lemma for ~ frege129 . Pr... |
frege129 43232 | If the procedure ` R ` is ... |
frege130 43233 | Lemma for ~ frege131 . Pr... |
frege131 43234 | If the procedure ` R ` is ... |
frege132 43235 | Lemma for ~ frege133 . Pr... |
frege133 43236 | If the procedure ` R ` is ... |
enrelmap 43237 | The set of all possible re... |
enrelmapr 43238 | The set of all possible re... |
enmappw 43239 | The set of all mappings fr... |
enmappwid 43240 | The set of all mappings fr... |
rfovd 43241 | Value of the operator, ` (... |
rfovfvd 43242 | Value of the operator, ` (... |
rfovfvfvd 43243 | Value of the operator, ` (... |
rfovcnvf1od 43244 | Properties of the operator... |
rfovcnvd 43245 | Value of the converse of t... |
rfovf1od 43246 | The value of the operator,... |
rfovcnvfvd 43247 | Value of the converse of t... |
fsovd 43248 | Value of the operator, ` (... |
fsovrfovd 43249 | The operator which gives a... |
fsovfvd 43250 | Value of the operator, ` (... |
fsovfvfvd 43251 | Value of the operator, ` (... |
fsovfd 43252 | The operator, ` ( A O B ) ... |
fsovcnvlem 43253 | The ` O ` operator, which ... |
fsovcnvd 43254 | The value of the converse ... |
fsovcnvfvd 43255 | The value of the converse ... |
fsovf1od 43256 | The value of ` ( A O B ) `... |
dssmapfvd 43257 | Value of the duality opera... |
dssmapfv2d 43258 | Value of the duality opera... |
dssmapfv3d 43259 | Value of the duality opera... |
dssmapnvod 43260 | For any base set ` B ` the... |
dssmapf1od 43261 | For any base set ` B ` the... |
dssmap2d 43262 | For any base set ` B ` the... |
or3or 43263 | Decompose disjunction into... |
andi3or 43264 | Distribute over triple dis... |
uneqsn 43265 | If a union of classes is e... |
brfvimex 43266 | If a binary relation holds... |
brovmptimex 43267 | If a binary relation holds... |
brovmptimex1 43268 | If a binary relation holds... |
brovmptimex2 43269 | If a binary relation holds... |
brcoffn 43270 | Conditions allowing the de... |
brcofffn 43271 | Conditions allowing the de... |
brco2f1o 43272 | Conditions allowing the de... |
brco3f1o 43273 | Conditions allowing the de... |
ntrclsbex 43274 | If (pseudo-)interior and (... |
ntrclsrcomplex 43275 | The relative complement of... |
neik0imk0p 43276 | Kuratowski's K0 axiom impl... |
ntrk2imkb 43277 | If an interior function is... |
ntrkbimka 43278 | If the interiors of disjoi... |
ntrk0kbimka 43279 | If the interiors of disjoi... |
clsk3nimkb 43280 | If the base set is not emp... |
clsk1indlem0 43281 | The ansatz closure functio... |
clsk1indlem2 43282 | The ansatz closure functio... |
clsk1indlem3 43283 | The ansatz closure functio... |
clsk1indlem4 43284 | The ansatz closure functio... |
clsk1indlem1 43285 | The ansatz closure functio... |
clsk1independent 43286 | For generalized closure fu... |
neik0pk1imk0 43287 | Kuratowski's K0' and K1 ax... |
isotone1 43288 | Two different ways to say ... |
isotone2 43289 | Two different ways to say ... |
ntrk1k3eqk13 43290 | An interior function is bo... |
ntrclsf1o 43291 | If (pseudo-)interior and (... |
ntrclsnvobr 43292 | If (pseudo-)interior and (... |
ntrclsiex 43293 | If (pseudo-)interior and (... |
ntrclskex 43294 | If (pseudo-)interior and (... |
ntrclsfv1 43295 | If (pseudo-)interior and (... |
ntrclsfv2 43296 | If (pseudo-)interior and (... |
ntrclselnel1 43297 | If (pseudo-)interior and (... |
ntrclselnel2 43298 | If (pseudo-)interior and (... |
ntrclsfv 43299 | The value of the interior ... |
ntrclsfveq1 43300 | If interior and closure fu... |
ntrclsfveq2 43301 | If interior and closure fu... |
ntrclsfveq 43302 | If interior and closure fu... |
ntrclsss 43303 | If interior and closure fu... |
ntrclsneine0lem 43304 | If (pseudo-)interior and (... |
ntrclsneine0 43305 | If (pseudo-)interior and (... |
ntrclscls00 43306 | If (pseudo-)interior and (... |
ntrclsiso 43307 | If (pseudo-)interior and (... |
ntrclsk2 43308 | An interior function is co... |
ntrclskb 43309 | The interiors of disjoint ... |
ntrclsk3 43310 | The intersection of interi... |
ntrclsk13 43311 | The interior of the inters... |
ntrclsk4 43312 | Idempotence of the interio... |
ntrneibex 43313 | If (pseudo-)interior and (... |
ntrneircomplex 43314 | The relative complement of... |
ntrneif1o 43315 | If (pseudo-)interior and (... |
ntrneiiex 43316 | If (pseudo-)interior and (... |
ntrneinex 43317 | If (pseudo-)interior and (... |
ntrneicnv 43318 | If (pseudo-)interior and (... |
ntrneifv1 43319 | If (pseudo-)interior and (... |
ntrneifv2 43320 | If (pseudo-)interior and (... |
ntrneiel 43321 | If (pseudo-)interior and (... |
ntrneifv3 43322 | The value of the neighbors... |
ntrneineine0lem 43323 | If (pseudo-)interior and (... |
ntrneineine1lem 43324 | If (pseudo-)interior and (... |
ntrneifv4 43325 | The value of the interior ... |
ntrneiel2 43326 | Membership in iterated int... |
ntrneineine0 43327 | If (pseudo-)interior and (... |
ntrneineine1 43328 | If (pseudo-)interior and (... |
ntrneicls00 43329 | If (pseudo-)interior and (... |
ntrneicls11 43330 | If (pseudo-)interior and (... |
ntrneiiso 43331 | If (pseudo-)interior and (... |
ntrneik2 43332 | An interior function is co... |
ntrneix2 43333 | An interior (closure) func... |
ntrneikb 43334 | The interiors of disjoint ... |
ntrneixb 43335 | The interiors (closures) o... |
ntrneik3 43336 | The intersection of interi... |
ntrneix3 43337 | The closure of the union o... |
ntrneik13 43338 | The interior of the inters... |
ntrneix13 43339 | The closure of the union o... |
ntrneik4w 43340 | Idempotence of the interio... |
ntrneik4 43341 | Idempotence of the interio... |
clsneibex 43342 | If (pseudo-)closure and (p... |
clsneircomplex 43343 | The relative complement of... |
clsneif1o 43344 | If a (pseudo-)closure func... |
clsneicnv 43345 | If a (pseudo-)closure func... |
clsneikex 43346 | If closure and neighborhoo... |
clsneinex 43347 | If closure and neighborhoo... |
clsneiel1 43348 | If a (pseudo-)closure func... |
clsneiel2 43349 | If a (pseudo-)closure func... |
clsneifv3 43350 | Value of the neighborhoods... |
clsneifv4 43351 | Value of the closure (inte... |
neicvgbex 43352 | If (pseudo-)neighborhood a... |
neicvgrcomplex 43353 | The relative complement of... |
neicvgf1o 43354 | If neighborhood and conver... |
neicvgnvo 43355 | If neighborhood and conver... |
neicvgnvor 43356 | If neighborhood and conver... |
neicvgmex 43357 | If the neighborhoods and c... |
neicvgnex 43358 | If the neighborhoods and c... |
neicvgel1 43359 | A subset being an element ... |
neicvgel2 43360 | The complement of a subset... |
neicvgfv 43361 | The value of the neighborh... |
ntrrn 43362 | The range of the interior ... |
ntrf 43363 | The interior function of a... |
ntrf2 43364 | The interior function is a... |
ntrelmap 43365 | The interior function is a... |
clsf2 43366 | The closure function is a ... |
clselmap 43367 | The closure function is a ... |
dssmapntrcls 43368 | The interior and closure o... |
dssmapclsntr 43369 | The closure and interior o... |
gneispa 43370 | Each point ` p ` of the ne... |
gneispb 43371 | Given a neighborhood ` N `... |
gneispace2 43372 | The predicate that ` F ` i... |
gneispace3 43373 | The predicate that ` F ` i... |
gneispace 43374 | The predicate that ` F ` i... |
gneispacef 43375 | A generic neighborhood spa... |
gneispacef2 43376 | A generic neighborhood spa... |
gneispacefun 43377 | A generic neighborhood spa... |
gneispacern 43378 | A generic neighborhood spa... |
gneispacern2 43379 | A generic neighborhood spa... |
gneispace0nelrn 43380 | A generic neighborhood spa... |
gneispace0nelrn2 43381 | A generic neighborhood spa... |
gneispace0nelrn3 43382 | A generic neighborhood spa... |
gneispaceel 43383 | Every neighborhood of a po... |
gneispaceel2 43384 | Every neighborhood of a po... |
gneispacess 43385 | All supersets of a neighbo... |
gneispacess2 43386 | All supersets of a neighbo... |
k0004lem1 43387 | Application of ~ ssin to r... |
k0004lem2 43388 | A mapping with a particula... |
k0004lem3 43389 | When the value of a mappin... |
k0004val 43390 | The topological simplex of... |
k0004ss1 43391 | The topological simplex of... |
k0004ss2 43392 | The topological simplex of... |
k0004ss3 43393 | The topological simplex of... |
k0004val0 43394 | The topological simplex of... |
inductionexd 43395 | Simple induction example. ... |
wwlemuld 43396 | Natural deduction form of ... |
leeq1d 43397 | Specialization of ~ breq1d... |
leeq2d 43398 | Specialization of ~ breq2d... |
absmulrposd 43399 | Specialization of absmuld ... |
imadisjld 43400 | Natural dduction form of o... |
imadisjlnd 43401 | Natural deduction form of ... |
wnefimgd 43402 | The image of a mapping fro... |
fco2d 43403 | Natural deduction form of ... |
wfximgfd 43404 | The value of a function on... |
extoimad 43405 | If |f(x)| <= C for all x t... |
imo72b2lem0 43406 | Lemma for ~ imo72b2 . (Co... |
suprleubrd 43407 | Natural deduction form of ... |
imo72b2lem2 43408 | Lemma for ~ imo72b2 . (Co... |
suprlubrd 43409 | Natural deduction form of ... |
imo72b2lem1 43410 | Lemma for ~ imo72b2 . (Co... |
lemuldiv3d 43411 | 'Less than or equal to' re... |
lemuldiv4d 43412 | 'Less than or equal to' re... |
imo72b2 43413 | IMO 1972 B2. (14th Intern... |
int-addcomd 43414 | AdditionCommutativity gene... |
int-addassocd 43415 | AdditionAssociativity gene... |
int-addsimpd 43416 | AdditionSimplification gen... |
int-mulcomd 43417 | MultiplicationCommutativit... |
int-mulassocd 43418 | MultiplicationAssociativit... |
int-mulsimpd 43419 | MultiplicationSimplificati... |
int-leftdistd 43420 | AdditionMultiplicationLeft... |
int-rightdistd 43421 | AdditionMultiplicationRigh... |
int-sqdefd 43422 | SquareDefinition generator... |
int-mul11d 43423 | First MultiplicationOne ge... |
int-mul12d 43424 | Second MultiplicationOne g... |
int-add01d 43425 | First AdditionZero generat... |
int-add02d 43426 | Second AdditionZero genera... |
int-sqgeq0d 43427 | SquareGEQZero generator ru... |
int-eqprincd 43428 | PrincipleOfEquality genera... |
int-eqtransd 43429 | EqualityTransitivity gener... |
int-eqmvtd 43430 | EquMoveTerm generator rule... |
int-eqineqd 43431 | EquivalenceImpliesDoubleIn... |
int-ineqmvtd 43432 | IneqMoveTerm generator rul... |
int-ineq1stprincd 43433 | FirstPrincipleOfInequality... |
int-ineq2ndprincd 43434 | SecondPrincipleOfInequalit... |
int-ineqtransd 43435 | InequalityTransitivity gen... |
unitadd 43436 | Theorem used in conjunctio... |
gsumws3 43437 | Valuation of a length 3 wo... |
gsumws4 43438 | Valuation of a length 4 wo... |
amgm2d 43439 | Arithmetic-geometric mean ... |
amgm3d 43440 | Arithmetic-geometric mean ... |
amgm4d 43441 | Arithmetic-geometric mean ... |
spALT 43442 | ~ sp can be proven from th... |
elnelneqd 43443 | Two classes are not equal ... |
elnelneq2d 43444 | Two classes are not equal ... |
rr-spce 43445 | Prove an existential. (Co... |
rexlimdvaacbv 43446 | Unpack a restricted existe... |
rexlimddvcbvw 43447 | Unpack a restricted existe... |
rexlimddvcbv 43448 | Unpack a restricted existe... |
rr-elrnmpt3d 43449 | Elementhood in an image se... |
finnzfsuppd 43450 | If a function is zero outs... |
rr-phpd 43451 | Equivalent of ~ php withou... |
suceqd 43452 | Deduction associated with ... |
tfindsd 43453 | Deduction associated with ... |
mnringvald 43456 | Value of the monoid ring f... |
mnringnmulrd 43457 | Components of a monoid rin... |
mnringnmulrdOLD 43458 | Obsolete version of ~ mnri... |
mnringbased 43459 | The base set of a monoid r... |
mnringbasedOLD 43460 | Obsolete version of ~ mnri... |
mnringbaserd 43461 | The base set of a monoid r... |
mnringelbased 43462 | Membership in the base set... |
mnringbasefd 43463 | Elements of a monoid ring ... |
mnringbasefsuppd 43464 | Elements of a monoid ring ... |
mnringaddgd 43465 | The additive operation of ... |
mnringaddgdOLD 43466 | Obsolete version of ~ mnri... |
mnring0gd 43467 | The additive identity of a... |
mnring0g2d 43468 | The additive identity of a... |
mnringmulrd 43469 | The ring product of a mono... |
mnringscad 43470 | The scalar ring of a monoi... |
mnringscadOLD 43471 | Obsolete version of ~ mnri... |
mnringvscad 43472 | The scalar product of a mo... |
mnringvscadOLD 43473 | Obsolete version of ~ mnri... |
mnringlmodd 43474 | Monoid rings are left modu... |
mnringmulrvald 43475 | Value of multiplication in... |
mnringmulrcld 43476 | Monoid rings are closed un... |
gru0eld 43477 | A nonempty Grothendieck un... |
grusucd 43478 | Grothendieck universes are... |
r1rankcld 43479 | Any rank of the cumulative... |
grur1cld 43480 | Grothendieck universes are... |
grurankcld 43481 | Grothendieck universes are... |
grurankrcld 43482 | If a Grothendieck universe... |
scotteqd 43485 | Equality theorem for the S... |
scotteq 43486 | Closed form of ~ scotteqd ... |
nfscott 43487 | Bound-variable hypothesis ... |
scottabf 43488 | Value of the Scott operati... |
scottab 43489 | Value of the Scott operati... |
scottabes 43490 | Value of the Scott operati... |
scottss 43491 | Scott's trick produces a s... |
elscottab 43492 | An element of the output o... |
scottex2 43493 | ~ scottex expressed using ... |
scotteld 43494 | The Scott operation sends ... |
scottelrankd 43495 | Property of a Scott's tric... |
scottrankd 43496 | Rank of a nonempty Scott's... |
gruscottcld 43497 | If a Grothendieck universe... |
dfcoll2 43500 | Alternate definition of th... |
colleq12d 43501 | Equality theorem for the c... |
colleq1 43502 | Equality theorem for the c... |
colleq2 43503 | Equality theorem for the c... |
nfcoll 43504 | Bound-variable hypothesis ... |
collexd 43505 | The output of the collecti... |
cpcolld 43506 | Property of the collection... |
cpcoll2d 43507 | ~ cpcolld with an extra ex... |
grucollcld 43508 | A Grothendieck universe co... |
ismnu 43509 | The hypothesis of this the... |
mnuop123d 43510 | Operations of a minimal un... |
mnussd 43511 | Minimal universes are clos... |
mnuss2d 43512 | ~ mnussd with arguments pr... |
mnu0eld 43513 | A nonempty minimal univers... |
mnuop23d 43514 | Second and third operation... |
mnupwd 43515 | Minimal universes are clos... |
mnusnd 43516 | Minimal universes are clos... |
mnuprssd 43517 | A minimal universe contain... |
mnuprss2d 43518 | Special case of ~ mnuprssd... |
mnuop3d 43519 | Third operation of a minim... |
mnuprdlem1 43520 | Lemma for ~ mnuprd . (Con... |
mnuprdlem2 43521 | Lemma for ~ mnuprd . (Con... |
mnuprdlem3 43522 | Lemma for ~ mnuprd . (Con... |
mnuprdlem4 43523 | Lemma for ~ mnuprd . Gene... |
mnuprd 43524 | Minimal universes are clos... |
mnuunid 43525 | Minimal universes are clos... |
mnuund 43526 | Minimal universes are clos... |
mnutrcld 43527 | Minimal universes contain ... |
mnutrd 43528 | Minimal universes are tran... |
mnurndlem1 43529 | Lemma for ~ mnurnd . (Con... |
mnurndlem2 43530 | Lemma for ~ mnurnd . Dedu... |
mnurnd 43531 | Minimal universes contain ... |
mnugrud 43532 | Minimal universes are Grot... |
grumnudlem 43533 | Lemma for ~ grumnud . (Co... |
grumnud 43534 | Grothendieck universes are... |
grumnueq 43535 | The class of Grothendieck ... |
expandan 43536 | Expand conjunction to prim... |
expandexn 43537 | Expand an existential quan... |
expandral 43538 | Expand a restricted univer... |
expandrexn 43539 | Expand a restricted existe... |
expandrex 43540 | Expand a restricted existe... |
expanduniss 43541 | Expand ` U. A C_ B ` to pr... |
ismnuprim 43542 | Express the predicate on `... |
rr-grothprimbi 43543 | Express "every set is cont... |
inagrud 43544 | Inaccessible levels of the... |
inaex 43545 | Assuming the Tarski-Grothe... |
gruex 43546 | Assuming the Tarski-Grothe... |
rr-groth 43547 | An equivalent of ~ ax-grot... |
rr-grothprim 43548 | An equivalent of ~ ax-grot... |
ismnushort 43549 | Express the predicate on `... |
dfuniv2 43550 | Alternative definition of ... |
rr-grothshortbi 43551 | Express "every set is cont... |
rr-grothshort 43552 | A shorter equivalent of ~ ... |
nanorxor 43553 | 'nand' is equivalent to th... |
undisjrab 43554 | Union of two disjoint rest... |
iso0 43555 | The empty set is an ` R , ... |
ssrecnpr 43556 | ` RR ` is a subset of both... |
seff 43557 | Let set ` S ` be the real ... |
sblpnf 43558 | The infinity ball in the a... |
prmunb2 43559 | The primes are unbounded. ... |
dvgrat 43560 | Ratio test for divergence ... |
cvgdvgrat 43561 | Ratio test for convergence... |
radcnvrat 43562 | Let ` L ` be the limit, if... |
reldvds 43563 | The divides relation is in... |
nznngen 43564 | All positive integers in t... |
nzss 43565 | The set of multiples of _m... |
nzin 43566 | The intersection of the se... |
nzprmdif 43567 | Subtract one prime's multi... |
hashnzfz 43568 | Special case of ~ hashdvds... |
hashnzfz2 43569 | Special case of ~ hashnzfz... |
hashnzfzclim 43570 | As the upper bound ` K ` o... |
caofcan 43571 | Transfer a cancellation la... |
ofsubid 43572 | Function analogue of ~ sub... |
ofmul12 43573 | Function analogue of ~ mul... |
ofdivrec 43574 | Function analogue of ~ div... |
ofdivcan4 43575 | Function analogue of ~ div... |
ofdivdiv2 43576 | Function analogue of ~ div... |
lhe4.4ex1a 43577 | Example of the Fundamental... |
dvsconst 43578 | Derivative of a constant f... |
dvsid 43579 | Derivative of the identity... |
dvsef 43580 | Derivative of the exponent... |
expgrowthi 43581 | Exponential growth and dec... |
dvconstbi 43582 | The derivative of a functi... |
expgrowth 43583 | Exponential growth and dec... |
bccval 43586 | Value of the generalized b... |
bcccl 43587 | Closure of the generalized... |
bcc0 43588 | The generalized binomial c... |
bccp1k 43589 | Generalized binomial coeff... |
bccm1k 43590 | Generalized binomial coeff... |
bccn0 43591 | Generalized binomial coeff... |
bccn1 43592 | Generalized binomial coeff... |
bccbc 43593 | The binomial coefficient a... |
uzmptshftfval 43594 | When ` F ` is a maps-to fu... |
dvradcnv2 43595 | The radius of convergence ... |
binomcxplemwb 43596 | Lemma for ~ binomcxp . Th... |
binomcxplemnn0 43597 | Lemma for ~ binomcxp . Wh... |
binomcxplemrat 43598 | Lemma for ~ binomcxp . As... |
binomcxplemfrat 43599 | Lemma for ~ binomcxp . ~ b... |
binomcxplemradcnv 43600 | Lemma for ~ binomcxp . By... |
binomcxplemdvbinom 43601 | Lemma for ~ binomcxp . By... |
binomcxplemcvg 43602 | Lemma for ~ binomcxp . Th... |
binomcxplemdvsum 43603 | Lemma for ~ binomcxp . Th... |
binomcxplemnotnn0 43604 | Lemma for ~ binomcxp . Wh... |
binomcxp 43605 | Generalize the binomial th... |
pm10.12 43606 | Theorem *10.12 in [Whitehe... |
pm10.14 43607 | Theorem *10.14 in [Whitehe... |
pm10.251 43608 | Theorem *10.251 in [Whiteh... |
pm10.252 43609 | Theorem *10.252 in [Whiteh... |
pm10.253 43610 | Theorem *10.253 in [Whiteh... |
albitr 43611 | Theorem *10.301 in [Whiteh... |
pm10.42 43612 | Theorem *10.42 in [Whitehe... |
pm10.52 43613 | Theorem *10.52 in [Whitehe... |
pm10.53 43614 | Theorem *10.53 in [Whitehe... |
pm10.541 43615 | Theorem *10.541 in [Whiteh... |
pm10.542 43616 | Theorem *10.542 in [Whiteh... |
pm10.55 43617 | Theorem *10.55 in [Whitehe... |
pm10.56 43618 | Theorem *10.56 in [Whitehe... |
pm10.57 43619 | Theorem *10.57 in [Whitehe... |
2alanimi 43620 | Removes two universal quan... |
2al2imi 43621 | Removes two universal quan... |
pm11.11 43622 | Theorem *11.11 in [Whitehe... |
pm11.12 43623 | Theorem *11.12 in [Whitehe... |
19.21vv 43624 | Compare Theorem *11.3 in [... |
2alim 43625 | Theorem *11.32 in [Whitehe... |
2albi 43626 | Theorem *11.33 in [Whitehe... |
2exim 43627 | Theorem *11.34 in [Whitehe... |
2exbi 43628 | Theorem *11.341 in [Whiteh... |
spsbce-2 43629 | Theorem *11.36 in [Whitehe... |
19.33-2 43630 | Theorem *11.421 in [Whiteh... |
19.36vv 43631 | Theorem *11.43 in [Whitehe... |
19.31vv 43632 | Theorem *11.44 in [Whitehe... |
19.37vv 43633 | Theorem *11.46 in [Whitehe... |
19.28vv 43634 | Theorem *11.47 in [Whitehe... |
pm11.52 43635 | Theorem *11.52 in [Whitehe... |
aaanv 43636 | Theorem *11.56 in [Whitehe... |
pm11.57 43637 | Theorem *11.57 in [Whitehe... |
pm11.58 43638 | Theorem *11.58 in [Whitehe... |
pm11.59 43639 | Theorem *11.59 in [Whitehe... |
pm11.6 43640 | Theorem *11.6 in [Whitehea... |
pm11.61 43641 | Theorem *11.61 in [Whitehe... |
pm11.62 43642 | Theorem *11.62 in [Whitehe... |
pm11.63 43643 | Theorem *11.63 in [Whitehe... |
pm11.7 43644 | Theorem *11.7 in [Whitehea... |
pm11.71 43645 | Theorem *11.71 in [Whitehe... |
sbeqal1 43646 | If ` x = y ` always implie... |
sbeqal1i 43647 | Suppose you know ` x = y `... |
sbeqal2i 43648 | If ` x = y ` implies ` x =... |
axc5c4c711 43649 | Proof of a theorem that ca... |
axc5c4c711toc5 43650 | Rederivation of ~ sp from ... |
axc5c4c711toc4 43651 | Rederivation of ~ axc4 fro... |
axc5c4c711toc7 43652 | Rederivation of ~ axc7 fro... |
axc5c4c711to11 43653 | Rederivation of ~ ax-11 fr... |
axc11next 43654 | This theorem shows that, g... |
pm13.13a 43655 | One result of theorem *13.... |
pm13.13b 43656 | Theorem *13.13 in [Whitehe... |
pm13.14 43657 | Theorem *13.14 in [Whitehe... |
pm13.192 43658 | Theorem *13.192 in [Whiteh... |
pm13.193 43659 | Theorem *13.193 in [Whiteh... |
pm13.194 43660 | Theorem *13.194 in [Whiteh... |
pm13.195 43661 | Theorem *13.195 in [Whiteh... |
pm13.196a 43662 | Theorem *13.196 in [Whiteh... |
2sbc6g 43663 | Theorem *13.21 in [Whitehe... |
2sbc5g 43664 | Theorem *13.22 in [Whitehe... |
iotain 43665 | Equivalence between two di... |
iotaexeu 43666 | The iota class exists. Th... |
iotasbc 43667 | Definition *14.01 in [Whit... |
iotasbc2 43668 | Theorem *14.111 in [Whiteh... |
pm14.12 43669 | Theorem *14.12 in [Whitehe... |
pm14.122a 43670 | Theorem *14.122 in [Whiteh... |
pm14.122b 43671 | Theorem *14.122 in [Whiteh... |
pm14.122c 43672 | Theorem *14.122 in [Whiteh... |
pm14.123a 43673 | Theorem *14.123 in [Whiteh... |
pm14.123b 43674 | Theorem *14.123 in [Whiteh... |
pm14.123c 43675 | Theorem *14.123 in [Whiteh... |
pm14.18 43676 | Theorem *14.18 in [Whitehe... |
iotaequ 43677 | Theorem *14.2 in [Whitehea... |
iotavalb 43678 | Theorem *14.202 in [Whiteh... |
iotasbc5 43679 | Theorem *14.205 in [Whiteh... |
pm14.24 43680 | Theorem *14.24 in [Whitehe... |
iotavalsb 43681 | Theorem *14.242 in [Whiteh... |
sbiota1 43682 | Theorem *14.25 in [Whitehe... |
sbaniota 43683 | Theorem *14.26 in [Whitehe... |
eubiOLD 43684 | Obsolete proof of ~ eubi a... |
iotasbcq 43685 | Theorem *14.272 in [Whiteh... |
elnev 43686 | Any set that contains one ... |
rusbcALT 43687 | A version of Russell's par... |
compeq 43688 | Equality between two ways ... |
compne 43689 | The complement of ` A ` is... |
compab 43690 | Two ways of saying "the co... |
conss2 43691 | Contrapositive law for sub... |
conss1 43692 | Contrapositive law for sub... |
ralbidar 43693 | More general form of ~ ral... |
rexbidar 43694 | More general form of ~ rex... |
dropab1 43695 | Theorem to aid use of the ... |
dropab2 43696 | Theorem to aid use of the ... |
ipo0 43697 | If the identity relation p... |
ifr0 43698 | A class that is founded by... |
ordpss 43699 | ~ ordelpss with an anteced... |
fvsb 43700 | Explicit substitution of a... |
fveqsb 43701 | Implicit substitution of a... |
xpexb 43702 | A Cartesian product exists... |
trelpss 43703 | An element of a transitive... |
addcomgi 43704 | Generalization of commutat... |
addrval 43714 | Value of the operation of ... |
subrval 43715 | Value of the operation of ... |
mulvval 43716 | Value of the operation of ... |
addrfv 43717 | Vector addition at a value... |
subrfv 43718 | Vector subtraction at a va... |
mulvfv 43719 | Scalar multiplication at a... |
addrfn 43720 | Vector addition produces a... |
subrfn 43721 | Vector subtraction produce... |
mulvfn 43722 | Scalar multiplication prod... |
addrcom 43723 | Vector addition is commuta... |
idiALT 43727 | Placeholder for ~ idi . T... |
exbir 43728 | Exportation implication al... |
3impexpbicom 43729 | Version of ~ 3impexp where... |
3impexpbicomi 43730 | Inference associated with ... |
bi1imp 43731 | Importation inference simi... |
bi2imp 43732 | Importation inference simi... |
bi3impb 43733 | Similar to ~ 3impb with im... |
bi3impa 43734 | Similar to ~ 3impa with im... |
bi23impib 43735 | ~ 3impib with the inner im... |
bi13impib 43736 | ~ 3impib with the outer im... |
bi123impib 43737 | ~ 3impib with the implicat... |
bi13impia 43738 | ~ 3impia with the outer im... |
bi123impia 43739 | ~ 3impia with the implicat... |
bi33imp12 43740 | ~ 3imp with innermost impl... |
bi23imp13 43741 | ~ 3imp with middle implica... |
bi13imp23 43742 | ~ 3imp with outermost impl... |
bi13imp2 43743 | Similar to ~ 3imp except t... |
bi12imp3 43744 | Similar to ~ 3imp except a... |
bi23imp1 43745 | Similar to ~ 3imp except a... |
bi123imp0 43746 | Similar to ~ 3imp except a... |
4animp1 43747 | A single hypothesis unific... |
4an31 43748 | A rearrangement of conjunc... |
4an4132 43749 | A rearrangement of conjunc... |
expcomdg 43750 | Biconditional form of ~ ex... |
iidn3 43751 | ~ idn3 without virtual ded... |
ee222 43752 | ~ e222 without virtual ded... |
ee3bir 43753 | Right-biconditional form o... |
ee13 43754 | ~ e13 without virtual dedu... |
ee121 43755 | ~ e121 without virtual ded... |
ee122 43756 | ~ e122 without virtual ded... |
ee333 43757 | ~ e333 without virtual ded... |
ee323 43758 | ~ e323 without virtual ded... |
3ornot23 43759 | If the second and third di... |
orbi1r 43760 | ~ orbi1 with order of disj... |
3orbi123 43761 | ~ pm4.39 with a 3-conjunct... |
syl5imp 43762 | Closed form of ~ syl5 . D... |
impexpd 43763 | The following User's Proof... |
com3rgbi 43764 | The following User's Proof... |
impexpdcom 43765 | The following User's Proof... |
ee1111 43766 | Non-virtual deduction form... |
pm2.43bgbi 43767 | Logical equivalence of a 2... |
pm2.43cbi 43768 | Logical equivalence of a 3... |
ee233 43769 | Non-virtual deduction form... |
imbi13 43770 | Join three logical equival... |
ee33 43771 | Non-virtual deduction form... |
con5 43772 | Biconditional contrapositi... |
con5i 43773 | Inference form of ~ con5 .... |
exlimexi 43774 | Inference similar to Theor... |
sb5ALT 43775 | Equivalence for substituti... |
eexinst01 43776 | ~ exinst01 without virtual... |
eexinst11 43777 | ~ exinst11 without virtual... |
vk15.4j 43778 | Excercise 4j of Unit 15 of... |
notnotrALT 43779 | Converse of double negatio... |
con3ALT2 43780 | Contraposition. Alternate... |
ssralv2 43781 | Quantification restricted ... |
sbc3or 43782 | ~ sbcor with a 3-disjuncts... |
alrim3con13v 43783 | Closed form of ~ alrimi wi... |
rspsbc2 43784 | ~ rspsbc with two quantify... |
sbcoreleleq 43785 | Substitution of a setvar v... |
tratrb 43786 | If a class is transitive a... |
ordelordALT 43787 | An element of an ordinal c... |
sbcim2g 43788 | Distribution of class subs... |
sbcbi 43789 | Implication form of ~ sbcb... |
trsbc 43790 | Formula-building inference... |
truniALT 43791 | The union of a class of tr... |
onfrALTlem5 43792 | Lemma for ~ onfrALT . (Co... |
onfrALTlem4 43793 | Lemma for ~ onfrALT . (Co... |
onfrALTlem3 43794 | Lemma for ~ onfrALT . (Co... |
ggen31 43795 | ~ gen31 without virtual de... |
onfrALTlem2 43796 | Lemma for ~ onfrALT . (Co... |
cbvexsv 43797 | A theorem pertaining to th... |
onfrALTlem1 43798 | Lemma for ~ onfrALT . (Co... |
onfrALT 43799 | The membership relation is... |
19.41rg 43800 | Closed form of right-to-le... |
opelopab4 43801 | Ordered pair membership in... |
2pm13.193 43802 | ~ pm13.193 for two variabl... |
hbntal 43803 | A closed form of ~ hbn . ~... |
hbimpg 43804 | A closed form of ~ hbim . ... |
hbalg 43805 | Closed form of ~ hbal . D... |
hbexg 43806 | Closed form of ~ nfex . D... |
ax6e2eq 43807 | Alternate form of ~ ax6e f... |
ax6e2nd 43808 | If at least two sets exist... |
ax6e2ndeq 43809 | "At least two sets exist" ... |
2sb5nd 43810 | Equivalence for double sub... |
2uasbanh 43811 | Distribute the unabbreviat... |
2uasban 43812 | Distribute the unabbreviat... |
e2ebind 43813 | Absorption of an existenti... |
elpwgded 43814 | ~ elpwgdedVD in convention... |
trelded 43815 | Deduction form of ~ trel .... |
jaoded 43816 | Deduction form of ~ jao . ... |
sbtT 43817 | A substitution into a theo... |
not12an2impnot1 43818 | If a double conjunction is... |
in1 43821 | Inference form of ~ df-vd1... |
iin1 43822 | ~ in1 without virtual dedu... |
dfvd1ir 43823 | Inference form of ~ df-vd1... |
idn1 43824 | Virtual deduction identity... |
dfvd1imp 43825 | Left-to-right part of defi... |
dfvd1impr 43826 | Right-to-left part of defi... |
dfvd2 43829 | Definition of a 2-hypothes... |
dfvd2an 43832 | Definition of a 2-hypothes... |
dfvd2ani 43833 | Inference form of ~ dfvd2a... |
dfvd2anir 43834 | Right-to-left inference fo... |
dfvd2i 43835 | Inference form of ~ dfvd2 ... |
dfvd2ir 43836 | Right-to-left inference fo... |
dfvd3 43841 | Definition of a 3-hypothes... |
dfvd3i 43842 | Inference form of ~ dfvd3 ... |
dfvd3ir 43843 | Right-to-left inference fo... |
dfvd3an 43844 | Definition of a 3-hypothes... |
dfvd3ani 43845 | Inference form of ~ dfvd3a... |
dfvd3anir 43846 | Right-to-left inference fo... |
vd01 43847 | A virtual hypothesis virtu... |
vd02 43848 | Two virtual hypotheses vir... |
vd03 43849 | A theorem is virtually inf... |
vd12 43850 | A virtual deduction with 1... |
vd13 43851 | A virtual deduction with 1... |
vd23 43852 | A virtual deduction with 2... |
dfvd2imp 43853 | The virtual deduction form... |
dfvd2impr 43854 | A 2-antecedent nested impl... |
in2 43855 | The virtual deduction intr... |
int2 43856 | The virtual deduction intr... |
iin2 43857 | ~ in2 without virtual dedu... |
in2an 43858 | The virtual deduction intr... |
in3 43859 | The virtual deduction intr... |
iin3 43860 | ~ in3 without virtual dedu... |
in3an 43861 | The virtual deduction intr... |
int3 43862 | The virtual deduction intr... |
idn2 43863 | Virtual deduction identity... |
iden2 43864 | Virtual deduction identity... |
idn3 43865 | Virtual deduction identity... |
gen11 43866 | Virtual deduction generali... |
gen11nv 43867 | Virtual deduction generali... |
gen12 43868 | Virtual deduction generali... |
gen21 43869 | Virtual deduction generali... |
gen21nv 43870 | Virtual deduction form of ... |
gen31 43871 | Virtual deduction generali... |
gen22 43872 | Virtual deduction generali... |
ggen22 43873 | ~ gen22 without virtual de... |
exinst 43874 | Existential Instantiation.... |
exinst01 43875 | Existential Instantiation.... |
exinst11 43876 | Existential Instantiation.... |
e1a 43877 | A Virtual deduction elimin... |
el1 43878 | A Virtual deduction elimin... |
e1bi 43879 | Biconditional form of ~ e1... |
e1bir 43880 | Right biconditional form o... |
e2 43881 | A virtual deduction elimin... |
e2bi 43882 | Biconditional form of ~ e2... |
e2bir 43883 | Right biconditional form o... |
ee223 43884 | ~ e223 without virtual ded... |
e223 43885 | A virtual deduction elimin... |
e222 43886 | A virtual deduction elimin... |
e220 43887 | A virtual deduction elimin... |
ee220 43888 | ~ e220 without virtual ded... |
e202 43889 | A virtual deduction elimin... |
ee202 43890 | ~ e202 without virtual ded... |
e022 43891 | A virtual deduction elimin... |
ee022 43892 | ~ e022 without virtual ded... |
e002 43893 | A virtual deduction elimin... |
ee002 43894 | ~ e002 without virtual ded... |
e020 43895 | A virtual deduction elimin... |
ee020 43896 | ~ e020 without virtual ded... |
e200 43897 | A virtual deduction elimin... |
ee200 43898 | ~ e200 without virtual ded... |
e221 43899 | A virtual deduction elimin... |
ee221 43900 | ~ e221 without virtual ded... |
e212 43901 | A virtual deduction elimin... |
ee212 43902 | ~ e212 without virtual ded... |
e122 43903 | A virtual deduction elimin... |
e112 43904 | A virtual deduction elimin... |
ee112 43905 | ~ e112 without virtual ded... |
e121 43906 | A virtual deduction elimin... |
e211 43907 | A virtual deduction elimin... |
ee211 43908 | ~ e211 without virtual ded... |
e210 43909 | A virtual deduction elimin... |
ee210 43910 | ~ e210 without virtual ded... |
e201 43911 | A virtual deduction elimin... |
ee201 43912 | ~ e201 without virtual ded... |
e120 43913 | A virtual deduction elimin... |
ee120 43914 | Virtual deduction rule ~ e... |
e021 43915 | A virtual deduction elimin... |
ee021 43916 | ~ e021 without virtual ded... |
e012 43917 | A virtual deduction elimin... |
ee012 43918 | ~ e012 without virtual ded... |
e102 43919 | A virtual deduction elimin... |
ee102 43920 | ~ e102 without virtual ded... |
e22 43921 | A virtual deduction elimin... |
e22an 43922 | Conjunction form of ~ e22 ... |
ee22an 43923 | ~ e22an without virtual de... |
e111 43924 | A virtual deduction elimin... |
e1111 43925 | A virtual deduction elimin... |
e110 43926 | A virtual deduction elimin... |
ee110 43927 | ~ e110 without virtual ded... |
e101 43928 | A virtual deduction elimin... |
ee101 43929 | ~ e101 without virtual ded... |
e011 43930 | A virtual deduction elimin... |
ee011 43931 | ~ e011 without virtual ded... |
e100 43932 | A virtual deduction elimin... |
ee100 43933 | ~ e100 without virtual ded... |
e010 43934 | A virtual deduction elimin... |
ee010 43935 | ~ e010 without virtual ded... |
e001 43936 | A virtual deduction elimin... |
ee001 43937 | ~ e001 without virtual ded... |
e11 43938 | A virtual deduction elimin... |
e11an 43939 | Conjunction form of ~ e11 ... |
ee11an 43940 | ~ e11an without virtual de... |
e01 43941 | A virtual deduction elimin... |
e01an 43942 | Conjunction form of ~ e01 ... |
ee01an 43943 | ~ e01an without virtual de... |
e10 43944 | A virtual deduction elimin... |
e10an 43945 | Conjunction form of ~ e10 ... |
ee10an 43946 | ~ e10an without virtual de... |
e02 43947 | A virtual deduction elimin... |
e02an 43948 | Conjunction form of ~ e02 ... |
ee02an 43949 | ~ e02an without virtual de... |
eel021old 43950 | ~ el021old without virtual... |
el021old 43951 | A virtual deduction elimin... |
eel132 43952 | ~ syl2an with antecedents ... |
eel000cT 43953 | An elimination deduction. ... |
eel0TT 43954 | An elimination deduction. ... |
eelT00 43955 | An elimination deduction. ... |
eelTTT 43956 | An elimination deduction. ... |
eelT11 43957 | An elimination deduction. ... |
eelT1 43958 | Syllogism inference combin... |
eelT12 43959 | An elimination deduction. ... |
eelTT1 43960 | An elimination deduction. ... |
eelT01 43961 | An elimination deduction. ... |
eel0T1 43962 | An elimination deduction. ... |
eel12131 43963 | An elimination deduction. ... |
eel2131 43964 | ~ syl2an with antecedents ... |
eel3132 43965 | ~ syl2an with antecedents ... |
eel0321old 43966 | ~ el0321old without virtua... |
el0321old 43967 | A virtual deduction elimin... |
eel2122old 43968 | ~ el2122old without virtua... |
el2122old 43969 | A virtual deduction elimin... |
eel0000 43970 | Elimination rule similar t... |
eel00001 43971 | An elimination deduction. ... |
eel00000 43972 | Elimination rule similar ~... |
eel11111 43973 | Five-hypothesis eliminatio... |
e12 43974 | A virtual deduction elimin... |
e12an 43975 | Conjunction form of ~ e12 ... |
el12 43976 | Virtual deduction form of ... |
e20 43977 | A virtual deduction elimin... |
e20an 43978 | Conjunction form of ~ e20 ... |
ee20an 43979 | ~ e20an without virtual de... |
e21 43980 | A virtual deduction elimin... |
e21an 43981 | Conjunction form of ~ e21 ... |
ee21an 43982 | ~ e21an without virtual de... |
e333 43983 | A virtual deduction elimin... |
e33 43984 | A virtual deduction elimin... |
e33an 43985 | Conjunction form of ~ e33 ... |
ee33an 43986 | ~ e33an without virtual de... |
e3 43987 | Meta-connective form of ~ ... |
e3bi 43988 | Biconditional form of ~ e3... |
e3bir 43989 | Right biconditional form o... |
e03 43990 | A virtual deduction elimin... |
ee03 43991 | ~ e03 without virtual dedu... |
e03an 43992 | Conjunction form of ~ e03 ... |
ee03an 43993 | Conjunction form of ~ ee03... |
e30 43994 | A virtual deduction elimin... |
ee30 43995 | ~ e30 without virtual dedu... |
e30an 43996 | A virtual deduction elimin... |
ee30an 43997 | Conjunction form of ~ ee30... |
e13 43998 | A virtual deduction elimin... |
e13an 43999 | A virtual deduction elimin... |
ee13an 44000 | ~ e13an without virtual de... |
e31 44001 | A virtual deduction elimin... |
ee31 44002 | ~ e31 without virtual dedu... |
e31an 44003 | A virtual deduction elimin... |
ee31an 44004 | ~ e31an without virtual de... |
e23 44005 | A virtual deduction elimin... |
e23an 44006 | A virtual deduction elimin... |
ee23an 44007 | ~ e23an without virtual de... |
e32 44008 | A virtual deduction elimin... |
ee32 44009 | ~ e32 without virtual dedu... |
e32an 44010 | A virtual deduction elimin... |
ee32an 44011 | ~ e33an without virtual de... |
e123 44012 | A virtual deduction elimin... |
ee123 44013 | ~ e123 without virtual ded... |
el123 44014 | A virtual deduction elimin... |
e233 44015 | A virtual deduction elimin... |
e323 44016 | A virtual deduction elimin... |
e000 44017 | A virtual deduction elimin... |
e00 44018 | Elimination rule identical... |
e00an 44019 | Elimination rule identical... |
eel00cT 44020 | An elimination deduction. ... |
eelTT 44021 | An elimination deduction. ... |
e0a 44022 | Elimination rule identical... |
eelT 44023 | An elimination deduction. ... |
eel0cT 44024 | An elimination deduction. ... |
eelT0 44025 | An elimination deduction. ... |
e0bi 44026 | Elimination rule identical... |
e0bir 44027 | Elimination rule identical... |
uun0.1 44028 | Convention notation form o... |
un0.1 44029 | ` T. ` is the constant tru... |
uunT1 44030 | A deduction unionizing a n... |
uunT1p1 44031 | A deduction unionizing a n... |
uunT21 44032 | A deduction unionizing a n... |
uun121 44033 | A deduction unionizing a n... |
uun121p1 44034 | A deduction unionizing a n... |
uun132 44035 | A deduction unionizing a n... |
uun132p1 44036 | A deduction unionizing a n... |
anabss7p1 44037 | A deduction unionizing a n... |
un10 44038 | A unionizing deduction. (... |
un01 44039 | A unionizing deduction. (... |
un2122 44040 | A deduction unionizing a n... |
uun2131 44041 | A deduction unionizing a n... |
uun2131p1 44042 | A deduction unionizing a n... |
uunTT1 44043 | A deduction unionizing a n... |
uunTT1p1 44044 | A deduction unionizing a n... |
uunTT1p2 44045 | A deduction unionizing a n... |
uunT11 44046 | A deduction unionizing a n... |
uunT11p1 44047 | A deduction unionizing a n... |
uunT11p2 44048 | A deduction unionizing a n... |
uunT12 44049 | A deduction unionizing a n... |
uunT12p1 44050 | A deduction unionizing a n... |
uunT12p2 44051 | A deduction unionizing a n... |
uunT12p3 44052 | A deduction unionizing a n... |
uunT12p4 44053 | A deduction unionizing a n... |
uunT12p5 44054 | A deduction unionizing a n... |
uun111 44055 | A deduction unionizing a n... |
3anidm12p1 44056 | A deduction unionizing a n... |
3anidm12p2 44057 | A deduction unionizing a n... |
uun123 44058 | A deduction unionizing a n... |
uun123p1 44059 | A deduction unionizing a n... |
uun123p2 44060 | A deduction unionizing a n... |
uun123p3 44061 | A deduction unionizing a n... |
uun123p4 44062 | A deduction unionizing a n... |
uun2221 44063 | A deduction unionizing a n... |
uun2221p1 44064 | A deduction unionizing a n... |
uun2221p2 44065 | A deduction unionizing a n... |
3impdirp1 44066 | A deduction unionizing a n... |
3impcombi 44067 | A 1-hypothesis proposition... |
trsspwALT 44068 | Virtual deduction proof of... |
trsspwALT2 44069 | Virtual deduction proof of... |
trsspwALT3 44070 | Short predicate calculus p... |
sspwtr 44071 | Virtual deduction proof of... |
sspwtrALT 44072 | Virtual deduction proof of... |
sspwtrALT2 44073 | Short predicate calculus p... |
pwtrVD 44074 | Virtual deduction proof of... |
pwtrrVD 44075 | Virtual deduction proof of... |
suctrALT 44076 | The successor of a transit... |
snssiALTVD 44077 | Virtual deduction proof of... |
snssiALT 44078 | If a class is an element o... |
snsslVD 44079 | Virtual deduction proof of... |
snssl 44080 | If a singleton is a subcla... |
snelpwrVD 44081 | Virtual deduction proof of... |
unipwrVD 44082 | Virtual deduction proof of... |
unipwr 44083 | A class is a subclass of t... |
sstrALT2VD 44084 | Virtual deduction proof of... |
sstrALT2 44085 | Virtual deduction proof of... |
suctrALT2VD 44086 | Virtual deduction proof of... |
suctrALT2 44087 | Virtual deduction proof of... |
elex2VD 44088 | Virtual deduction proof of... |
elex22VD 44089 | Virtual deduction proof of... |
eqsbc2VD 44090 | Virtual deduction proof of... |
zfregs2VD 44091 | Virtual deduction proof of... |
tpid3gVD 44092 | Virtual deduction proof of... |
en3lplem1VD 44093 | Virtual deduction proof of... |
en3lplem2VD 44094 | Virtual deduction proof of... |
en3lpVD 44095 | Virtual deduction proof of... |
simplbi2VD 44096 | Virtual deduction proof of... |
3ornot23VD 44097 | Virtual deduction proof of... |
orbi1rVD 44098 | Virtual deduction proof of... |
bitr3VD 44099 | Virtual deduction proof of... |
3orbi123VD 44100 | Virtual deduction proof of... |
sbc3orgVD 44101 | Virtual deduction proof of... |
19.21a3con13vVD 44102 | Virtual deduction proof of... |
exbirVD 44103 | Virtual deduction proof of... |
exbiriVD 44104 | Virtual deduction proof of... |
rspsbc2VD 44105 | Virtual deduction proof of... |
3impexpVD 44106 | Virtual deduction proof of... |
3impexpbicomVD 44107 | Virtual deduction proof of... |
3impexpbicomiVD 44108 | Virtual deduction proof of... |
sbcoreleleqVD 44109 | Virtual deduction proof of... |
hbra2VD 44110 | Virtual deduction proof of... |
tratrbVD 44111 | Virtual deduction proof of... |
al2imVD 44112 | Virtual deduction proof of... |
syl5impVD 44113 | Virtual deduction proof of... |
idiVD 44114 | Virtual deduction proof of... |
ancomstVD 44115 | Closed form of ~ ancoms . ... |
ssralv2VD 44116 | Quantification restricted ... |
ordelordALTVD 44117 | An element of an ordinal c... |
equncomVD 44118 | If a class equals the unio... |
equncomiVD 44119 | Inference form of ~ equnco... |
sucidALTVD 44120 | A set belongs to its succe... |
sucidALT 44121 | A set belongs to its succe... |
sucidVD 44122 | A set belongs to its succe... |
imbi12VD 44123 | Implication form of ~ imbi... |
imbi13VD 44124 | Join three logical equival... |
sbcim2gVD 44125 | Distribution of class subs... |
sbcbiVD 44126 | Implication form of ~ sbcb... |
trsbcVD 44127 | Formula-building inference... |
truniALTVD 44128 | The union of a class of tr... |
ee33VD 44129 | Non-virtual deduction form... |
trintALTVD 44130 | The intersection of a clas... |
trintALT 44131 | The intersection of a clas... |
undif3VD 44132 | The first equality of Exer... |
sbcssgVD 44133 | Virtual deduction proof of... |
csbingVD 44134 | Virtual deduction proof of... |
onfrALTlem5VD 44135 | Virtual deduction proof of... |
onfrALTlem4VD 44136 | Virtual deduction proof of... |
onfrALTlem3VD 44137 | Virtual deduction proof of... |
simplbi2comtVD 44138 | Virtual deduction proof of... |
onfrALTlem2VD 44139 | Virtual deduction proof of... |
onfrALTlem1VD 44140 | Virtual deduction proof of... |
onfrALTVD 44141 | Virtual deduction proof of... |
csbeq2gVD 44142 | Virtual deduction proof of... |
csbsngVD 44143 | Virtual deduction proof of... |
csbxpgVD 44144 | Virtual deduction proof of... |
csbresgVD 44145 | Virtual deduction proof of... |
csbrngVD 44146 | Virtual deduction proof of... |
csbima12gALTVD 44147 | Virtual deduction proof of... |
csbunigVD 44148 | Virtual deduction proof of... |
csbfv12gALTVD 44149 | Virtual deduction proof of... |
con5VD 44150 | Virtual deduction proof of... |
relopabVD 44151 | Virtual deduction proof of... |
19.41rgVD 44152 | Virtual deduction proof of... |
2pm13.193VD 44153 | Virtual deduction proof of... |
hbimpgVD 44154 | Virtual deduction proof of... |
hbalgVD 44155 | Virtual deduction proof of... |
hbexgVD 44156 | Virtual deduction proof of... |
ax6e2eqVD 44157 | The following User's Proof... |
ax6e2ndVD 44158 | The following User's Proof... |
ax6e2ndeqVD 44159 | The following User's Proof... |
2sb5ndVD 44160 | The following User's Proof... |
2uasbanhVD 44161 | The following User's Proof... |
e2ebindVD 44162 | The following User's Proof... |
sb5ALTVD 44163 | The following User's Proof... |
vk15.4jVD 44164 | The following User's Proof... |
notnotrALTVD 44165 | The following User's Proof... |
con3ALTVD 44166 | The following User's Proof... |
elpwgdedVD 44167 | Membership in a power clas... |
sspwimp 44168 | If a class is a subclass o... |
sspwimpVD 44169 | The following User's Proof... |
sspwimpcf 44170 | If a class is a subclass o... |
sspwimpcfVD 44171 | The following User's Proof... |
suctrALTcf 44172 | The sucessor of a transiti... |
suctrALTcfVD 44173 | The following User's Proof... |
suctrALT3 44174 | The successor of a transit... |
sspwimpALT 44175 | If a class is a subclass o... |
unisnALT 44176 | A set equals the union of ... |
notnotrALT2 44177 | Converse of double negatio... |
sspwimpALT2 44178 | If a class is a subclass o... |
e2ebindALT 44179 | Absorption of an existenti... |
ax6e2ndALT 44180 | If at least two sets exist... |
ax6e2ndeqALT 44181 | "At least two sets exist" ... |
2sb5ndALT 44182 | Equivalence for double sub... |
chordthmALT 44183 | The intersecting chords th... |
isosctrlem1ALT 44184 | Lemma for ~ isosctr . Thi... |
iunconnlem2 44185 | The indexed union of conne... |
iunconnALT 44186 | The indexed union of conne... |
sineq0ALT 44187 | A complex number whose sin... |
evth2f 44188 | A version of ~ evth2 using... |
elunif 44189 | A version of ~ eluni using... |
rzalf 44190 | A version of ~ rzal using ... |
fvelrnbf 44191 | A version of ~ fvelrnb usi... |
rfcnpre1 44192 | If F is a continuous funct... |
ubelsupr 44193 | If U belongs to A and U is... |
fsumcnf 44194 | A finite sum of functions ... |
mulltgt0 44195 | The product of a negative ... |
rspcegf 44196 | A version of ~ rspcev usin... |
rabexgf 44197 | A version of ~ rabexg usin... |
fcnre 44198 | A function continuous with... |
sumsnd 44199 | A sum of a singleton is th... |
evthf 44200 | A version of ~ evth using ... |
cnfex 44201 | The class of continuous fu... |
fnchoice 44202 | For a finite set, a choice... |
refsumcn 44203 | A finite sum of continuous... |
rfcnpre2 44204 | If ` F ` is a continuous f... |
cncmpmax 44205 | When the hypothesis for th... |
rfcnpre3 44206 | If F is a continuous funct... |
rfcnpre4 44207 | If F is a continuous funct... |
sumpair 44208 | Sum of two distinct comple... |
rfcnnnub 44209 | Given a real continuous fu... |
refsum2cnlem1 44210 | This is the core Lemma for... |
refsum2cn 44211 | The sum of two continuus r... |
adantlllr 44212 | Deduction adding a conjunc... |
3adantlr3 44213 | Deduction adding a conjunc... |
3adantll2 44214 | Deduction adding a conjunc... |
3adantll3 44215 | Deduction adding a conjunc... |
ssnel 44216 | If not element of a set, t... |
sncldre 44217 | A singleton is closed w.r.... |
n0p 44218 | A polynomial with a nonzer... |
pm2.65ni 44219 | Inference rule for proof b... |
pwssfi 44220 | Every element of the power... |
iuneq2df 44221 | Equality deduction for ind... |
nnfoctb 44222 | There exists a mapping fro... |
ssinss1d 44223 | Intersection preserves sub... |
elpwinss 44224 | An element of the powerset... |
unidmex 44225 | If ` F ` is a set, then ` ... |
ndisj2 44226 | A non-disjointness conditi... |
zenom 44227 | The set of integer numbers... |
uzwo4 44228 | Well-ordering principle: a... |
unisn0 44229 | The union of the singleton... |
ssin0 44230 | If two classes are disjoin... |
inabs3 44231 | Absorption law for interse... |
pwpwuni 44232 | Relationship between power... |
disjiun2 44233 | In a disjoint collection, ... |
0pwfi 44234 | The empty set is in any po... |
ssinss2d 44235 | Intersection preserves sub... |
zct 44236 | The set of integer numbers... |
pwfin0 44237 | A finite set always belong... |
uzct 44238 | An upper integer set is co... |
iunxsnf 44239 | A singleton index picks ou... |
fiiuncl 44240 | If a set is closed under t... |
iunp1 44241 | The addition of the next s... |
fiunicl 44242 | If a set is closed under t... |
ixpeq2d 44243 | Equality theorem for infin... |
disjxp1 44244 | The sets of a cartesian pr... |
disjsnxp 44245 | The sets in the cartesian ... |
eliind 44246 | Membership in indexed inte... |
rspcef 44247 | Restricted existential spe... |
inn0f 44248 | A nonempty intersection. ... |
ixpssmapc 44249 | An infinite Cartesian prod... |
inn0 44250 | A nonempty intersection. ... |
elintd 44251 | Membership in class inters... |
ssdf 44252 | A sufficient condition for... |
brneqtrd 44253 | Substitution of equal clas... |
ssnct 44254 | A set containing an uncoun... |
ssuniint 44255 | Sufficient condition for b... |
elintdv 44256 | Membership in class inters... |
ssd 44257 | A sufficient condition for... |
ralimralim 44258 | Introducing any antecedent... |
snelmap 44259 | Membership of the element ... |
xrnmnfpnf 44260 | An extended real that is n... |
nelrnmpt 44261 | Non-membership in the rang... |
iuneq1i 44262 | Equality theorem for index... |
nssrex 44263 | Negation of subclass relat... |
ssinc 44264 | Inclusion relation for a m... |
ssdec 44265 | Inclusion relation for a m... |
elixpconstg 44266 | Membership in an infinite ... |
iineq1d 44267 | Equality theorem for index... |
metpsmet 44268 | A metric is a pseudometric... |
ixpssixp 44269 | Subclass theorem for infin... |
ballss3 44270 | A sufficient condition for... |
iunincfi 44271 | Given a sequence of increa... |
nsstr 44272 | If it's not a subclass, it... |
rexanuz3 44273 | Combine two different uppe... |
cbvmpo2 44274 | Rule to change the second ... |
cbvmpo1 44275 | Rule to change the first b... |
eliuniin 44276 | Indexed union of indexed i... |
ssabf 44277 | Subclass of a class abstra... |
pssnssi 44278 | A proper subclass does not... |
rabidim2 44279 | Membership in a restricted... |
eluni2f 44280 | Membership in class union.... |
eliin2f 44281 | Membership in indexed inte... |
nssd 44282 | Negation of subclass relat... |
iineq12dv 44283 | Equality deduction for ind... |
supxrcld 44284 | The supremum of an arbitra... |
elrestd 44285 | A sufficient condition for... |
eliuniincex 44286 | Counterexample to show tha... |
eliincex 44287 | Counterexample to show tha... |
eliinid 44288 | Membership in an indexed i... |
abssf 44289 | Class abstraction in a sub... |
supxrubd 44290 | A member of a set of exten... |
ssrabf 44291 | Subclass of a restricted c... |
ssrabdf 44292 | Subclass of a restricted c... |
eliin2 44293 | Membership in indexed inte... |
ssrab2f 44294 | Subclass relation for a re... |
restuni3 44295 | The underlying set of a su... |
rabssf 44296 | Restricted class abstracti... |
eliuniin2 44297 | Indexed union of indexed i... |
restuni4 44298 | The underlying set of a su... |
restuni6 44299 | The underlying set of a su... |
restuni5 44300 | The underlying set of a su... |
unirestss 44301 | The union of an elementwis... |
iniin1 44302 | Indexed intersection of in... |
iniin2 44303 | Indexed intersection of in... |
cbvrabv2 44304 | A more general version of ... |
cbvrabv2w 44305 | A more general version of ... |
iinssiin 44306 | Subset implication for an ... |
eliind2 44307 | Membership in indexed inte... |
iinssd 44308 | Subset implication for an ... |
rabbida2 44309 | Equivalent wff's yield equ... |
iinexd 44310 | The existence of an indexe... |
rabexf 44311 | Separation Scheme in terms... |
rabbida3 44312 | Equivalent wff's yield equ... |
r19.36vf 44313 | Restricted quantifier vers... |
raleqd 44314 | Equality deduction for res... |
iinssf 44315 | Subset implication for an ... |
iinssdf 44316 | Subset implication for an ... |
resabs2i 44317 | Absorption law for restric... |
ssdf2 44318 | A sufficient condition for... |
rabssd 44319 | Restricted class abstracti... |
rexnegd 44320 | Minus a real number. (Con... |
rexlimd3 44321 | * Inference from Theorem 1... |
resabs1i 44322 | Absorption law for restric... |
nel1nelin 44323 | Membership in an intersect... |
nel2nelin 44324 | Membership in an intersect... |
nel1nelini 44325 | Membership in an intersect... |
nel2nelini 44326 | Membership in an intersect... |
eliunid 44327 | Membership in indexed unio... |
reximddv3 44328 | Deduction from Theorem 19.... |
reximdd 44329 | Deduction from Theorem 19.... |
unfid 44330 | The union of two finite se... |
inopnd 44331 | The intersection of two op... |
ss2rabdf 44332 | Deduction of restricted ab... |
restopn3 44333 | If ` A ` is open, then ` A... |
restopnssd 44334 | A topology restricted to a... |
restsubel 44335 | A subset belongs in the sp... |
toprestsubel 44336 | A subset is open in the to... |
rabidd 44337 | An "identity" law of concr... |
iunssdf 44338 | Subset theorem for an inde... |
iinss2d 44339 | Subset implication for an ... |
r19.3rzf 44340 | Restricted quantification ... |
r19.28zf 44341 | Restricted quantifier vers... |
iindif2f 44342 | Indexed intersection of cl... |
ralfal 44343 | Two ways of expressing emp... |
archd 44344 | Archimedean property of re... |
eliund 44345 | Membership in indexed unio... |
nimnbi 44346 | If an implication is false... |
nimnbi2 44347 | If an implication is false... |
notbicom 44348 | Commutative law for the ne... |
rexeqif 44349 | Equality inference for res... |
rspced 44350 | Restricted existential spe... |
feq1dd 44351 | Equality deduction for fun... |
fnresdmss 44352 | A function does not change... |
fmptsnxp 44353 | Maps-to notation and Carte... |
fvmpt2bd 44354 | Value of a function given ... |
rnmptfi 44355 | The range of a function wi... |
fresin2 44356 | Restriction of a function ... |
ffi 44357 | A function with finite dom... |
suprnmpt 44358 | An explicit bound for the ... |
rnffi 44359 | The range of a function wi... |
mptelpm 44360 | A function in maps-to nota... |
rnmptpr 44361 | Range of a function define... |
resmpti 44362 | Restriction of the mapping... |
founiiun 44363 | Union expressed as an inde... |
rnresun 44364 | Distribution law for range... |
elrnmptf 44365 | The range of a function in... |
rnmptssrn 44366 | Inclusion relation for two... |
disjf1 44367 | A 1 to 1 mapping built fro... |
rnsnf 44368 | The range of a function wh... |
wessf1ornlem 44369 | Given a function ` F ` on ... |
wessf1orn 44370 | Given a function ` F ` on ... |
nelrnres 44371 | If ` A ` is not in the ran... |
disjrnmpt2 44372 | Disjointness of the range ... |
elrnmpt1sf 44373 | Elementhood in an image se... |
founiiun0 44374 | Union expressed as an inde... |
disjf1o 44375 | A bijection built from dis... |
disjinfi 44376 | Only a finite number of di... |
fvovco 44377 | Value of the composition o... |
ssnnf1octb 44378 | There exists a bijection b... |
nnf1oxpnn 44379 | There is a bijection betwe... |
rnmptssd 44380 | The range of a function gi... |
projf1o 44381 | A biijection from a set to... |
fvmap 44382 | Function value for a membe... |
fvixp2 44383 | Projection of a factor of ... |
choicefi 44384 | For a finite set, a choice... |
mpct 44385 | The exponentiation of a co... |
cnmetcoval 44386 | Value of the distance func... |
fcomptss 44387 | Express composition of two... |
elmapsnd 44388 | Membership in a set expone... |
mapss2 44389 | Subset inheritance for set... |
fsneq 44390 | Equality condition for two... |
difmap 44391 | Difference of two sets exp... |
unirnmap 44392 | Given a subset of a set ex... |
inmap 44393 | Intersection of two sets e... |
fcoss 44394 | Composition of two mapping... |
fsneqrn 44395 | Equality condition for two... |
difmapsn 44396 | Difference of two sets exp... |
mapssbi 44397 | Subset inheritance for set... |
unirnmapsn 44398 | Equality theorem for a sub... |
iunmapss 44399 | The indexed union of set e... |
ssmapsn 44400 | A subset ` C ` of a set ex... |
iunmapsn 44401 | The indexed union of set e... |
absfico 44402 | Mapping domain and codomai... |
icof 44403 | The set of left-closed rig... |
elpmrn 44404 | The range of a partial fun... |
imaexi 44405 | The image of a set is a se... |
axccdom 44406 | Relax the constraint on ax... |
dmmptdff 44407 | The domain of the mapping ... |
dmmptdf 44408 | The domain of the mapping ... |
elpmi2 44409 | The domain of a partial fu... |
dmrelrnrel 44410 | A relation preserving func... |
fvcod 44411 | Value of a function compos... |
elrnmpoid 44412 | Membership in the range of... |
axccd 44413 | An alternative version of ... |
axccd2 44414 | An alternative version of ... |
fimassd 44415 | The image of a class is a ... |
feqresmptf 44416 | Express a restricted funct... |
elrnmpt1d 44417 | Elementhood in an image se... |
dmresss 44418 | The domain of a restrictio... |
dmmptssf 44419 | The domain of a mapping is... |
dmmptdf2 44420 | The domain of the mapping ... |
dmuz 44421 | Domain of the upper intege... |
fmptd2f 44422 | Domain and codomain of the... |
mpteq1df 44423 | An equality theorem for th... |
mpteq1dfOLD 44424 | Obsolete version of ~ mpte... |
mptexf 44425 | If the domain of a functio... |
fvmpt4 44426 | Value of a function given ... |
fmptf 44427 | Functionality of the mappi... |
resimass 44428 | The image of a restriction... |
mptssid 44429 | The mapping operation expr... |
mptfnd 44430 | The maps-to notation defin... |
mpteq12daOLD 44431 | Obsolete version of ~ mpte... |
rnmptlb 44432 | Boundness below of the ran... |
rnmptbddlem 44433 | Boundness of the range of ... |
rnmptbdd 44434 | Boundness of the range of ... |
funimaeq 44435 | Membership relation for th... |
rnmptssf 44436 | The range of a function gi... |
rnmptbd2lem 44437 | Boundness below of the ran... |
rnmptbd2 44438 | Boundness below of the ran... |
infnsuprnmpt 44439 | The indexed infimum of rea... |
suprclrnmpt 44440 | Closure of the indexed sup... |
suprubrnmpt2 44441 | A member of a nonempty ind... |
suprubrnmpt 44442 | A member of a nonempty ind... |
rnmptssdf 44443 | The range of a function gi... |
rnmptbdlem 44444 | Boundness above of the ran... |
rnmptbd 44445 | Boundness above of the ran... |
rnmptss2 44446 | The range of a function gi... |
elmptima 44447 | The image of a function in... |
ralrnmpt3 44448 | A restricted quantifier ov... |
fvelima2 44449 | Function value in an image... |
rnmptssbi 44450 | The range of a function gi... |
imass2d 44451 | Subset theorem for image. ... |
imassmpt 44452 | Membership relation for th... |
fpmd 44453 | A total function is a part... |
fconst7 44454 | An alternative way to expr... |
fnmptif 44455 | Functionality and domain o... |
dmmptif 44456 | Domain of the mapping oper... |
mpteq2dfa 44457 | Slightly more general equa... |
dmmpt1 44458 | The domain of the mapping ... |
fmptff 44459 | Functionality of the mappi... |
fvmptelcdmf 44460 | The value of a function at... |
fmptdff 44461 | A version of ~ fmptd using... |
fvmpt2df 44462 | Deduction version of ~ fvm... |
rn1st 44463 | The range of a function wi... |
rnmptssff 44464 | The range of a function gi... |
rnmptssdff 44465 | The range of a function gi... |
fvmpt4d 44466 | Value of a function given ... |
sub2times 44467 | Subtracting from a number,... |
nnxrd 44468 | A natural number is an ext... |
nnxr 44469 | A natural number is an ext... |
abssubrp 44470 | The distance of two distin... |
elfzfzo 44471 | Relationship between membe... |
oddfl 44472 | Odd number representation ... |
abscosbd 44473 | Bound for the absolute val... |
mul13d 44474 | Commutative/associative la... |
negpilt0 44475 | Negative ` _pi ` is negati... |
dstregt0 44476 | A complex number ` A ` tha... |
subadd4b 44477 | Rearrangement of 4 terms i... |
xrlttri5d 44478 | Not equal and not larger i... |
neglt 44479 | The negative of a positive... |
zltlesub 44480 | If an integer ` N ` is les... |
divlt0gt0d 44481 | The ratio of a negative nu... |
subsub23d 44482 | Swap subtrahend and result... |
2timesgt 44483 | Double of a positive real ... |
reopn 44484 | The reals are open with re... |
sub31 44485 | Swap the first and third t... |
nnne1ge2 44486 | A positive integer which i... |
lefldiveq 44487 | A closed enough, smaller r... |
negsubdi3d 44488 | Distribution of negative o... |
ltdiv2dd 44489 | Division of a positive num... |
abssinbd 44490 | Bound for the absolute val... |
halffl 44491 | Floor of ` ( 1 / 2 ) ` . ... |
monoords 44492 | Ordering relation for a st... |
hashssle 44493 | The size of a subset of a ... |
lttri5d 44494 | Not equal and not larger i... |
fzisoeu 44495 | A finite ordered set has a... |
lt3addmuld 44496 | If three real numbers are ... |
absnpncan2d 44497 | Triangular inequality, com... |
fperiodmullem 44498 | A function with period ` T... |
fperiodmul 44499 | A function with period T i... |
upbdrech 44500 | Choice of an upper bound f... |
lt4addmuld 44501 | If four real numbers are l... |
absnpncan3d 44502 | Triangular inequality, com... |
upbdrech2 44503 | Choice of an upper bound f... |
ssfiunibd 44504 | A finite union of bounded ... |
fzdifsuc2 44505 | Remove a successor from th... |
fzsscn 44506 | A finite sequence of integ... |
divcan8d 44507 | A cancellation law for div... |
dmmcand 44508 | Cancellation law for divis... |
fzssre 44509 | A finite sequence of integ... |
bccld 44510 | A binomial coefficient, in... |
leadd12dd 44511 | Addition to both sides of ... |
fzssnn0 44512 | A finite set of sequential... |
xreqle 44513 | Equality implies 'less tha... |
xaddlidd 44514 | ` 0 ` is a left identity f... |
xadd0ge 44515 | A number is less than or e... |
elfzolem1 44516 | A member in a half-open in... |
xrgtned 44517 | 'Greater than' implies not... |
xrleneltd 44518 | 'Less than or equal to' an... |
xaddcomd 44519 | The extended real addition... |
supxrre3 44520 | The supremum of a nonempty... |
uzfissfz 44521 | For any finite subset of t... |
xleadd2d 44522 | Addition of extended reals... |
suprltrp 44523 | The supremum of a nonempty... |
xleadd1d 44524 | Addition of extended reals... |
xreqled 44525 | Equality implies 'less tha... |
xrgepnfd 44526 | An extended real greater t... |
xrge0nemnfd 44527 | A nonnegative extended rea... |
supxrgere 44528 | If a real number can be ap... |
iuneqfzuzlem 44529 | Lemma for ~ iuneqfzuz : he... |
iuneqfzuz 44530 | If two unions indexed by u... |
xle2addd 44531 | Adding both side of two in... |
supxrgelem 44532 | If an extended real number... |
supxrge 44533 | If an extended real number... |
suplesup 44534 | If any element of ` A ` ca... |
infxrglb 44535 | The infimum of a set of ex... |
xadd0ge2 44536 | A number is less than or e... |
nepnfltpnf 44537 | An extended real that is n... |
ltadd12dd 44538 | Addition to both sides of ... |
nemnftgtmnft 44539 | An extended real that is n... |
xrgtso 44540 | 'Greater than' is a strict... |
rpex 44541 | The positive reals form a ... |
xrge0ge0 44542 | A nonnegative extended rea... |
xrssre 44543 | A subset of extended reals... |
ssuzfz 44544 | A finite subset of the upp... |
absfun 44545 | The absolute value is a fu... |
infrpge 44546 | The infimum of a nonempty,... |
xrlexaddrp 44547 | If an extended real number... |
supsubc 44548 | The supremum function dist... |
xralrple2 44549 | Show that ` A ` is less th... |
nnuzdisj 44550 | The first ` N ` elements o... |
ltdivgt1 44551 | Divsion by a number greate... |
xrltned 44552 | 'Less than' implies not eq... |
nnsplit 44553 | Express the set of positiv... |
divdiv3d 44554 | Division into a fraction. ... |
abslt2sqd 44555 | Comparison of the square o... |
qenom 44556 | The set of rational number... |
qct 44557 | The set of rational number... |
xrltnled 44558 | 'Less than' in terms of 'l... |
lenlteq 44559 | 'less than or equal to' bu... |
xrred 44560 | An extended real that is n... |
rr2sscn2 44561 | The cartesian square of ` ... |
infxr 44562 | The infimum of a set of ex... |
infxrunb2 44563 | The infimum of an unbounde... |
infxrbnd2 44564 | The infimum of a bounded-b... |
infleinflem1 44565 | Lemma for ~ infleinf , cas... |
infleinflem2 44566 | Lemma for ~ infleinf , whe... |
infleinf 44567 | If any element of ` B ` ca... |
xralrple4 44568 | Show that ` A ` is less th... |
xralrple3 44569 | Show that ` A ` is less th... |
eluzelzd 44570 | A member of an upper set o... |
suplesup2 44571 | If any element of ` A ` is... |
recnnltrp 44572 | ` N ` is a natural number ... |
nnn0 44573 | The set of positive intege... |
fzct 44574 | A finite set of sequential... |
rpgtrecnn 44575 | Any positive real number i... |
fzossuz 44576 | A half-open integer interv... |
infxrrefi 44577 | The real and extended real... |
xrralrecnnle 44578 | Show that ` A ` is less th... |
fzoct 44579 | A finite set of sequential... |
frexr 44580 | A function taking real val... |
nnrecrp 44581 | The reciprocal of a positi... |
reclt0d 44582 | The reciprocal of a negati... |
lt0neg1dd 44583 | If a number is negative, i... |
infxrcld 44584 | The infimum of an arbitrar... |
xrralrecnnge 44585 | Show that ` A ` is less th... |
reclt0 44586 | The reciprocal of a negati... |
ltmulneg 44587 | Multiplying by a negative ... |
allbutfi 44588 | For all but finitely many.... |
ltdiv23neg 44589 | Swap denominator with othe... |
xreqnltd 44590 | A consequence of trichotom... |
mnfnre2 44591 | Minus infinity is not a re... |
zssxr 44592 | The integers are a subset ... |
fisupclrnmpt 44593 | A nonempty finite indexed ... |
supxrunb3 44594 | The supremum of an unbound... |
elfzod 44595 | Membership in a half-open ... |
fimaxre4 44596 | A nonempty finite set of r... |
ren0 44597 | The set of reals is nonemp... |
eluzelz2 44598 | A member of an upper set o... |
resabs2d 44599 | Absorption law for restric... |
uzid2 44600 | Membership of the least me... |
supxrleubrnmpt 44601 | The supremum of a nonempty... |
uzssre2 44602 | An upper set of integers i... |
uzssd 44603 | Subset relationship for tw... |
eluzd 44604 | Membership in an upper set... |
infxrlbrnmpt2 44605 | A member of a nonempty ind... |
xrre4 44606 | An extended real is real i... |
uz0 44607 | The upper integers functio... |
eluzelz2d 44608 | A member of an upper set o... |
infleinf2 44609 | If any element in ` B ` is... |
unb2ltle 44610 | "Unbounded below" expresse... |
uzidd2 44611 | Membership of the least me... |
uzssd2 44612 | Subset relationship for tw... |
rexabslelem 44613 | An indexed set of absolute... |
rexabsle 44614 | An indexed set of absolute... |
allbutfiinf 44615 | Given a "for all but finit... |
supxrrernmpt 44616 | The real and extended real... |
suprleubrnmpt 44617 | The supremum of a nonempty... |
infrnmptle 44618 | An indexed infimum of exte... |
infxrunb3 44619 | The infimum of an unbounde... |
uzn0d 44620 | The upper integers are all... |
uzssd3 44621 | Subset relationship for tw... |
rexabsle2 44622 | An indexed set of absolute... |
infxrunb3rnmpt 44623 | The infimum of an unbounde... |
supxrre3rnmpt 44624 | The indexed supremum of a ... |
uzublem 44625 | A set of reals, indexed by... |
uzub 44626 | A set of reals, indexed by... |
ssrexr 44627 | A subset of the reals is a... |
supxrmnf2 44628 | Removing minus infinity fr... |
supxrcli 44629 | The supremum of an arbitra... |
uzid3 44630 | Membership of the least me... |
infxrlesupxr 44631 | The supremum of a nonempty... |
xnegeqd 44632 | Equality of two extended n... |
xnegrecl 44633 | The extended real negative... |
xnegnegi 44634 | Extended real version of ~... |
xnegeqi 44635 | Equality of two extended n... |
nfxnegd 44636 | Deduction version of ~ nfx... |
xnegnegd 44637 | Extended real version of ~... |
uzred 44638 | An upper integer is a real... |
xnegcli 44639 | Closure of extended real n... |
supminfrnmpt 44640 | The indexed supremum of a ... |
infxrpnf 44641 | Adding plus infinity to a ... |
infxrrnmptcl 44642 | The infimum of an arbitrar... |
leneg2d 44643 | Negative of one side of 'l... |
supxrltinfxr 44644 | The supremum of the empty ... |
max1d 44645 | A number is less than or e... |
supxrleubrnmptf 44646 | The supremum of a nonempty... |
nleltd 44647 | 'Not less than or equal to... |
zxrd 44648 | An integer is an extended ... |
infxrgelbrnmpt 44649 | The infimum of an indexed ... |
rphalfltd 44650 | Half of a positive real is... |
uzssz2 44651 | An upper set of integers i... |
leneg3d 44652 | Negative of one side of 'l... |
max2d 44653 | A number is less than or e... |
uzn0bi 44654 | The upper integers functio... |
xnegrecl2 44655 | If the extended real negat... |
nfxneg 44656 | Bound-variable hypothesis ... |
uzxrd 44657 | An upper integer is an ext... |
infxrpnf2 44658 | Removing plus infinity fro... |
supminfxr 44659 | The extended real suprema ... |
infrpgernmpt 44660 | The infimum of a nonempty,... |
xnegre 44661 | An extended real is real i... |
xnegrecl2d 44662 | If the extended real negat... |
uzxr 44663 | An upper integer is an ext... |
supminfxr2 44664 | The extended real suprema ... |
xnegred 44665 | An extended real is real i... |
supminfxrrnmpt 44666 | The indexed supremum of a ... |
min1d 44667 | The minimum of two numbers... |
min2d 44668 | The minimum of two numbers... |
pnfged 44669 | Plus infinity is an upper ... |
xrnpnfmnf 44670 | An extended real that is n... |
uzsscn 44671 | An upper set of integers i... |
absimnre 44672 | The absolute value of the ... |
uzsscn2 44673 | An upper set of integers i... |
xrtgcntopre 44674 | The standard topologies on... |
absimlere 44675 | The absolute value of the ... |
rpssxr 44676 | The positive reals are a s... |
monoordxrv 44677 | Ordering relation for a mo... |
monoordxr 44678 | Ordering relation for a mo... |
monoord2xrv 44679 | Ordering relation for a mo... |
monoord2xr 44680 | Ordering relation for a mo... |
xrpnf 44681 | An extended real is plus i... |
xlenegcon1 44682 | Extended real version of ~... |
xlenegcon2 44683 | Extended real version of ~... |
pimxrneun 44684 | The preimage of a set of e... |
caucvgbf 44685 | A function is convergent i... |
cvgcau 44686 | A convergent function is C... |
cvgcaule 44687 | A convergent function is C... |
rexanuz2nf 44688 | A simple counterexample re... |
gtnelioc 44689 | A real number larger than ... |
ioossioc 44690 | An open interval is a subs... |
ioondisj2 44691 | A condition for two open i... |
ioondisj1 44692 | A condition for two open i... |
ioogtlb 44693 | An element of a closed int... |
evthiccabs 44694 | Extreme Value Theorem on y... |
ltnelicc 44695 | A real number smaller than... |
eliood 44696 | Membership in an open real... |
iooabslt 44697 | An upper bound for the dis... |
gtnelicc 44698 | A real number greater than... |
iooinlbub 44699 | An open interval has empty... |
iocgtlb 44700 | An element of a left-open ... |
iocleub 44701 | An element of a left-open ... |
eliccd 44702 | Membership in a closed rea... |
eliccre 44703 | A member of a closed inter... |
eliooshift 44704 | Element of an open interva... |
eliocd 44705 | Membership in a left-open ... |
icoltub 44706 | An element of a left-close... |
eliocre 44707 | A member of a left-open ri... |
iooltub 44708 | An element of an open inte... |
ioontr 44709 | The interior of an interva... |
snunioo1 44710 | The closure of one end of ... |
lbioc 44711 | A left-open right-closed i... |
ioomidp 44712 | The midpoint is an element... |
iccdifioo 44713 | If the open inverval is re... |
iccdifprioo 44714 | An open interval is the cl... |
ioossioobi 44715 | Biconditional form of ~ io... |
iccshift 44716 | A closed interval shifted ... |
iccsuble 44717 | An upper bound to the dist... |
iocopn 44718 | A left-open right-closed i... |
eliccelioc 44719 | Membership in a closed int... |
iooshift 44720 | An open interval shifted b... |
iccintsng 44721 | Intersection of two adiace... |
icoiccdif 44722 | Left-closed right-open int... |
icoopn 44723 | A left-closed right-open i... |
icoub 44724 | A left-closed, right-open ... |
eliccxrd 44725 | Membership in a closed rea... |
pnfel0pnf 44726 | ` +oo ` is a nonnegative e... |
eliccnelico 44727 | An element of a closed int... |
eliccelicod 44728 | A member of a closed inter... |
ge0xrre 44729 | A nonnegative extended rea... |
ge0lere 44730 | A nonnegative extended Rea... |
elicores 44731 | Membership in a left-close... |
inficc 44732 | The infimum of a nonempty ... |
qinioo 44733 | The rational numbers are d... |
lenelioc 44734 | A real number smaller than... |
ioonct 44735 | A nonempty open interval i... |
xrgtnelicc 44736 | A real number greater than... |
iccdificc 44737 | The difference of two clos... |
iocnct 44738 | A nonempty left-open, righ... |
iccnct 44739 | A closed interval, with mo... |
iooiinicc 44740 | A closed interval expresse... |
iccgelbd 44741 | An element of a closed int... |
iooltubd 44742 | An element of an open inte... |
icoltubd 44743 | An element of a left-close... |
qelioo 44744 | The rational numbers are d... |
tgqioo2 44745 | Every open set of reals is... |
iccleubd 44746 | An element of a closed int... |
elioored 44747 | A member of an open interv... |
ioogtlbd 44748 | An element of a closed int... |
ioofun 44749 | ` (,) ` is a function. (C... |
icomnfinre 44750 | A left-closed, right-open,... |
sqrlearg 44751 | The square compared with i... |
ressiocsup 44752 | If the supremum belongs to... |
ressioosup 44753 | If the supremum does not b... |
iooiinioc 44754 | A left-open, right-closed ... |
ressiooinf 44755 | If the infimum does not be... |
icogelbd 44756 | An element of a left-close... |
iocleubd 44757 | An element of a left-open ... |
uzinico 44758 | An upper interval of integ... |
preimaiocmnf 44759 | Preimage of a right-closed... |
uzinico2 44760 | An upper interval of integ... |
uzinico3 44761 | An upper interval of integ... |
icossico2 44762 | Condition for a closed-bel... |
dmico 44763 | The domain of the closed-b... |
ndmico 44764 | The closed-below, open-abo... |
uzubioo 44765 | The upper integers are unb... |
uzubico 44766 | The upper integers are unb... |
uzubioo2 44767 | The upper integers are unb... |
uzubico2 44768 | The upper integers are unb... |
iocgtlbd 44769 | An element of a left-open ... |
xrtgioo2 44770 | The topology on the extend... |
tgioo4 44771 | The standard topology on t... |
fsummulc1f 44772 | Closure of a finite sum of... |
fsumnncl 44773 | Closure of a nonempty, fin... |
fsumge0cl 44774 | The finite sum of nonnegat... |
fsumf1of 44775 | Re-index a finite sum usin... |
fsumiunss 44776 | Sum over a disjoint indexe... |
fsumreclf 44777 | Closure of a finite sum of... |
fsumlessf 44778 | A shorter sum of nonnegati... |
fsumsupp0 44779 | Finite sum of function val... |
fsumsermpt 44780 | A finite sum expressed in ... |
fmul01 44781 | Multiplying a finite numbe... |
fmulcl 44782 | If ' Y ' is closed under t... |
fmuldfeqlem1 44783 | induction step for the pro... |
fmuldfeq 44784 | X and Z are two equivalent... |
fmul01lt1lem1 44785 | Given a finite multiplicat... |
fmul01lt1lem2 44786 | Given a finite multiplicat... |
fmul01lt1 44787 | Given a finite multiplicat... |
cncfmptss 44788 | A continuous complex funct... |
rrpsscn 44789 | The positive reals are a s... |
mulc1cncfg 44790 | A version of ~ mulc1cncf u... |
infrglb 44791 | The infimum of a nonempty ... |
expcnfg 44792 | If ` F ` is a complex cont... |
prodeq2ad 44793 | Equality deduction for pro... |
fprodsplit1 44794 | Separate out a term in a f... |
fprodexp 44795 | Positive integer exponenti... |
fprodabs2 44796 | The absolute value of a fi... |
fprod0 44797 | A finite product with a ze... |
mccllem 44798 | * Induction step for ~ mcc... |
mccl 44799 | A multinomial coefficient,... |
fprodcnlem 44800 | A finite product of functi... |
fprodcn 44801 | A finite product of functi... |
clim1fr1 44802 | A class of sequences of fr... |
isumneg 44803 | Negation of a converging s... |
climrec 44804 | Limit of the reciprocal of... |
climmulf 44805 | A version of ~ climmul usi... |
climexp 44806 | The limit of natural power... |
climinf 44807 | A bounded monotonic noninc... |
climsuselem1 44808 | The subsequence index ` I ... |
climsuse 44809 | A subsequence ` G ` of a c... |
climrecf 44810 | A version of ~ climrec usi... |
climneg 44811 | Complex limit of the negat... |
climinff 44812 | A version of ~ climinf usi... |
climdivf 44813 | Limit of the ratio of two ... |
climreeq 44814 | If ` F ` is a real functio... |
ellimciota 44815 | An explicit value for the ... |
climaddf 44816 | A version of ~ climadd usi... |
mullimc 44817 | Limit of the product of tw... |
ellimcabssub0 44818 | An equivalent condition fo... |
limcdm0 44819 | If a function has empty do... |
islptre 44820 | An equivalence condition f... |
limccog 44821 | Limit of the composition o... |
limciccioolb 44822 | The limit of a function at... |
climf 44823 | Express the predicate: Th... |
mullimcf 44824 | Limit of the multiplicatio... |
constlimc 44825 | Limit of constant function... |
rexlim2d 44826 | Inference removing two res... |
idlimc 44827 | Limit of the identity func... |
divcnvg 44828 | The sequence of reciprocal... |
limcperiod 44829 | If ` F ` is a periodic fun... |
limcrecl 44830 | If ` F ` is a real-valued ... |
sumnnodd 44831 | A series indexed by ` NN `... |
lptioo2 44832 | The upper bound of an open... |
lptioo1 44833 | The lower bound of an open... |
elprn1 44834 | A member of an unordered p... |
elprn2 44835 | A member of an unordered p... |
limcmptdm 44836 | The domain of a maps-to fu... |
clim2f 44837 | Express the predicate: Th... |
limcicciooub 44838 | The limit of a function at... |
ltmod 44839 | A sufficient condition for... |
islpcn 44840 | A characterization for a l... |
lptre2pt 44841 | If a set in the real line ... |
limsupre 44842 | If a sequence is bounded, ... |
limcresiooub 44843 | The left limit doesn't cha... |
limcresioolb 44844 | The right limit doesn't ch... |
limcleqr 44845 | If the left and the right ... |
lptioo2cn 44846 | The upper bound of an open... |
lptioo1cn 44847 | The lower bound of an open... |
neglimc 44848 | Limit of the negative func... |
addlimc 44849 | Sum of two limits. (Contr... |
0ellimcdiv 44850 | If the numerator converges... |
clim2cf 44851 | Express the predicate ` F ... |
limclner 44852 | For a limit point, both fr... |
sublimc 44853 | Subtraction of two limits.... |
reclimc 44854 | Limit of the reciprocal of... |
clim0cf 44855 | Express the predicate ` F ... |
limclr 44856 | For a limit point, both fr... |
divlimc 44857 | Limit of the quotient of t... |
expfac 44858 | Factorial grows faster tha... |
climconstmpt 44859 | A constant sequence conver... |
climresmpt 44860 | A function restricted to u... |
climsubmpt 44861 | Limit of the difference of... |
climsubc2mpt 44862 | Limit of the difference of... |
climsubc1mpt 44863 | Limit of the difference of... |
fnlimfv 44864 | The value of the limit fun... |
climreclf 44865 | The limit of a convergent ... |
climeldmeq 44866 | Two functions that are eve... |
climf2 44867 | Express the predicate: Th... |
fnlimcnv 44868 | The sequence of function v... |
climeldmeqmpt 44869 | Two functions that are eve... |
climfveq 44870 | Two functions that are eve... |
clim2f2 44871 | Express the predicate: Th... |
climfveqmpt 44872 | Two functions that are eve... |
climd 44873 | Express the predicate: Th... |
clim2d 44874 | The limit of complex numbe... |
fnlimfvre 44875 | The limit function of real... |
allbutfifvre 44876 | Given a sequence of real-v... |
climleltrp 44877 | The limit of complex numbe... |
fnlimfvre2 44878 | The limit function of real... |
fnlimf 44879 | The limit function of real... |
fnlimabslt 44880 | A sequence of function val... |
climfveqf 44881 | Two functions that are eve... |
climmptf 44882 | Exhibit a function ` G ` w... |
climfveqmpt3 44883 | Two functions that are eve... |
climeldmeqf 44884 | Two functions that are eve... |
climreclmpt 44885 | The limit of B convergent ... |
limsupref 44886 | If a sequence is bounded, ... |
limsupbnd1f 44887 | If a sequence is eventuall... |
climbddf 44888 | A converging sequence of c... |
climeqf 44889 | Two functions that are eve... |
climeldmeqmpt3 44890 | Two functions that are eve... |
limsupcld 44891 | Closure of the superior li... |
climfv 44892 | The limit of a convergent ... |
limsupval3 44893 | The superior limit of an i... |
climfveqmpt2 44894 | Two functions that are eve... |
limsup0 44895 | The superior limit of the ... |
climeldmeqmpt2 44896 | Two functions that are eve... |
limsupresre 44897 | The supremum limit of a fu... |
climeqmpt 44898 | Two functions that are eve... |
climfvd 44899 | The limit of a convergent ... |
limsuplesup 44900 | An upper bound for the sup... |
limsupresico 44901 | The superior limit doesn't... |
limsuppnfdlem 44902 | If the restriction of a fu... |
limsuppnfd 44903 | If the restriction of a fu... |
limsupresuz 44904 | If the real part of the do... |
limsupub 44905 | If the limsup is not ` +oo... |
limsupres 44906 | The superior limit of a re... |
climinf2lem 44907 | A convergent, nonincreasin... |
climinf2 44908 | A convergent, nonincreasin... |
limsupvaluz 44909 | The superior limit, when t... |
limsupresuz2 44910 | If the domain of a functio... |
limsuppnflem 44911 | If the restriction of a fu... |
limsuppnf 44912 | If the restriction of a fu... |
limsupubuzlem 44913 | If the limsup is not ` +oo... |
limsupubuz 44914 | For a real-valued function... |
climinf2mpt 44915 | A bounded below, monotonic... |
climinfmpt 44916 | A bounded below, monotonic... |
climinf3 44917 | A convergent, nonincreasin... |
limsupvaluzmpt 44918 | The superior limit, when t... |
limsupequzmpt2 44919 | Two functions that are eve... |
limsupubuzmpt 44920 | If the limsup is not ` +oo... |
limsupmnflem 44921 | The superior limit of a fu... |
limsupmnf 44922 | The superior limit of a fu... |
limsupequzlem 44923 | Two functions that are eve... |
limsupequz 44924 | Two functions that are eve... |
limsupre2lem 44925 | Given a function on the ex... |
limsupre2 44926 | Given a function on the ex... |
limsupmnfuzlem 44927 | The superior limit of a fu... |
limsupmnfuz 44928 | The superior limit of a fu... |
limsupequzmptlem 44929 | Two functions that are eve... |
limsupequzmpt 44930 | Two functions that are eve... |
limsupre2mpt 44931 | Given a function on the ex... |
limsupequzmptf 44932 | Two functions that are eve... |
limsupre3lem 44933 | Given a function on the ex... |
limsupre3 44934 | Given a function on the ex... |
limsupre3mpt 44935 | Given a function on the ex... |
limsupre3uzlem 44936 | Given a function on the ex... |
limsupre3uz 44937 | Given a function on the ex... |
limsupreuz 44938 | Given a function on the re... |
limsupvaluz2 44939 | The superior limit, when t... |
limsupreuzmpt 44940 | Given a function on the re... |
supcnvlimsup 44941 | If a function on a set of ... |
supcnvlimsupmpt 44942 | If a function on a set of ... |
0cnv 44943 | If ` (/) ` is a complex nu... |
climuzlem 44944 | Express the predicate: Th... |
climuz 44945 | Express the predicate: Th... |
lmbr3v 44946 | Express the binary relatio... |
climisp 44947 | If a sequence converges to... |
lmbr3 44948 | Express the binary relatio... |
climrescn 44949 | A sequence converging w.r.... |
climxrrelem 44950 | If a sequence ranging over... |
climxrre 44951 | If a sequence ranging over... |
limsuplt2 44954 | The defining property of t... |
liminfgord 44955 | Ordering property of the i... |
limsupvald 44956 | The superior limit of a se... |
limsupresicompt 44957 | The superior limit doesn't... |
limsupcli 44958 | Closure of the superior li... |
liminfgf 44959 | Closure of the inferior li... |
liminfval 44960 | The inferior limit of a se... |
climlimsup 44961 | A sequence of real numbers... |
limsupge 44962 | The defining property of t... |
liminfgval 44963 | Value of the inferior limi... |
liminfcl 44964 | Closure of the inferior li... |
liminfvald 44965 | The inferior limit of a se... |
liminfval5 44966 | The inferior limit of an i... |
limsupresxr 44967 | The superior limit of a fu... |
liminfresxr 44968 | The inferior limit of a fu... |
liminfval2 44969 | The superior limit, relati... |
climlimsupcex 44970 | Counterexample for ~ climl... |
liminfcld 44971 | Closure of the inferior li... |
liminfresico 44972 | The inferior limit doesn't... |
limsup10exlem 44973 | The range of the given fun... |
limsup10ex 44974 | The superior limit of a fu... |
liminf10ex 44975 | The inferior limit of a fu... |
liminflelimsuplem 44976 | The superior limit is grea... |
liminflelimsup 44977 | The superior limit is grea... |
limsupgtlem 44978 | For any positive real, the... |
limsupgt 44979 | Given a sequence of real n... |
liminfresre 44980 | The inferior limit of a fu... |
liminfresicompt 44981 | The inferior limit doesn't... |
liminfltlimsupex 44982 | An example where the ` lim... |
liminfgelimsup 44983 | The inferior limit is grea... |
liminfvalxr 44984 | Alternate definition of ` ... |
liminfresuz 44985 | If the real part of the do... |
liminflelimsupuz 44986 | The superior limit is grea... |
liminfvalxrmpt 44987 | Alternate definition of ` ... |
liminfresuz2 44988 | If the domain of a functio... |
liminfgelimsupuz 44989 | The inferior limit is grea... |
liminfval4 44990 | Alternate definition of ` ... |
liminfval3 44991 | Alternate definition of ` ... |
liminfequzmpt2 44992 | Two functions that are eve... |
liminfvaluz 44993 | Alternate definition of ` ... |
liminf0 44994 | The inferior limit of the ... |
limsupval4 44995 | Alternate definition of ` ... |
liminfvaluz2 44996 | Alternate definition of ` ... |
liminfvaluz3 44997 | Alternate definition of ` ... |
liminflelimsupcex 44998 | A counterexample for ~ lim... |
limsupvaluz3 44999 | Alternate definition of ` ... |
liminfvaluz4 45000 | Alternate definition of ` ... |
limsupvaluz4 45001 | Alternate definition of ` ... |
climliminflimsupd 45002 | If a sequence of real numb... |
liminfreuzlem 45003 | Given a function on the re... |
liminfreuz 45004 | Given a function on the re... |
liminfltlem 45005 | Given a sequence of real n... |
liminflt 45006 | Given a sequence of real n... |
climliminf 45007 | A sequence of real numbers... |
liminflimsupclim 45008 | A sequence of real numbers... |
climliminflimsup 45009 | A sequence of real numbers... |
climliminflimsup2 45010 | A sequence of real numbers... |
climliminflimsup3 45011 | A sequence of real numbers... |
climliminflimsup4 45012 | A sequence of real numbers... |
limsupub2 45013 | A extended real valued fun... |
limsupubuz2 45014 | A sequence with values in ... |
xlimpnfxnegmnf 45015 | A sequence converges to ` ... |
liminflbuz2 45016 | A sequence with values in ... |
liminfpnfuz 45017 | The inferior limit of a fu... |
liminflimsupxrre 45018 | A sequence with values in ... |
xlimrel 45021 | The limit on extended real... |
xlimres 45022 | A function converges iff i... |
xlimcl 45023 | The limit of a sequence of... |
rexlimddv2 45024 | Restricted existential eli... |
xlimclim 45025 | Given a sequence of reals,... |
xlimconst 45026 | A constant sequence conver... |
climxlim 45027 | A converging sequence in t... |
xlimbr 45028 | Express the binary relatio... |
fuzxrpmcn 45029 | A function mapping from an... |
cnrefiisplem 45030 | Lemma for ~ cnrefiisp (som... |
cnrefiisp 45031 | A non-real, complex number... |
xlimxrre 45032 | If a sequence ranging over... |
xlimmnfvlem1 45033 | Lemma for ~ xlimmnfv : the... |
xlimmnfvlem2 45034 | Lemma for ~ xlimmnf : the ... |
xlimmnfv 45035 | A function converges to mi... |
xlimconst2 45036 | A sequence that eventually... |
xlimpnfvlem1 45037 | Lemma for ~ xlimpnfv : the... |
xlimpnfvlem2 45038 | Lemma for ~ xlimpnfv : the... |
xlimpnfv 45039 | A function converges to pl... |
xlimclim2lem 45040 | Lemma for ~ xlimclim2 . H... |
xlimclim2 45041 | Given a sequence of extend... |
xlimmnf 45042 | A function converges to mi... |
xlimpnf 45043 | A function converges to pl... |
xlimmnfmpt 45044 | A function converges to pl... |
xlimpnfmpt 45045 | A function converges to pl... |
climxlim2lem 45046 | In this lemma for ~ climxl... |
climxlim2 45047 | A sequence of extended rea... |
dfxlim2v 45048 | An alternative definition ... |
dfxlim2 45049 | An alternative definition ... |
climresd 45050 | A function restricted to u... |
climresdm 45051 | A real function converges ... |
dmclimxlim 45052 | A real valued sequence tha... |
xlimmnflimsup2 45053 | A sequence of extended rea... |
xlimuni 45054 | An infinite sequence conve... |
xlimclimdm 45055 | A sequence of extended rea... |
xlimfun 45056 | The convergence relation o... |
xlimmnflimsup 45057 | If a sequence of extended ... |
xlimdm 45058 | Two ways to express that a... |
xlimpnfxnegmnf2 45059 | A sequence converges to ` ... |
xlimresdm 45060 | A function converges in th... |
xlimpnfliminf 45061 | If a sequence of extended ... |
xlimpnfliminf2 45062 | A sequence of extended rea... |
xlimliminflimsup 45063 | A sequence of extended rea... |
xlimlimsupleliminf 45064 | A sequence of extended rea... |
coseq0 45065 | A complex number whose cos... |
sinmulcos 45066 | Multiplication formula for... |
coskpi2 45067 | The cosine of an integer m... |
cosnegpi 45068 | The cosine of negative ` _... |
sinaover2ne0 45069 | If ` A ` in ` ( 0 , 2 _pi ... |
cosknegpi 45070 | The cosine of an integer m... |
mulcncff 45071 | The multiplication of two ... |
cncfmptssg 45072 | A continuous complex funct... |
constcncfg 45073 | A constant function is a c... |
idcncfg 45074 | The identity function is a... |
cncfshift 45075 | A periodic continuous func... |
resincncf 45076 | ` sin ` restricted to real... |
addccncf2 45077 | Adding a constant is a con... |
0cnf 45078 | The empty set is a continu... |
fsumcncf 45079 | The finite sum of continuo... |
cncfperiod 45080 | A periodic continuous func... |
subcncff 45081 | The subtraction of two con... |
negcncfg 45082 | The opposite of a continuo... |
cnfdmsn 45083 | A function with a singleto... |
cncfcompt 45084 | Composition of continuous ... |
addcncff 45085 | The sum of two continuous ... |
ioccncflimc 45086 | Limit at the upper bound o... |
cncfuni 45087 | A complex function on a su... |
icccncfext 45088 | A continuous function on a... |
cncficcgt0 45089 | A the absolute value of a ... |
icocncflimc 45090 | Limit at the lower bound, ... |
cncfdmsn 45091 | A complex function with a ... |
divcncff 45092 | The quotient of two contin... |
cncfshiftioo 45093 | A periodic continuous func... |
cncfiooicclem1 45094 | A continuous function ` F ... |
cncfiooicc 45095 | A continuous function ` F ... |
cncfiooiccre 45096 | A continuous function ` F ... |
cncfioobdlem 45097 | ` G ` actually extends ` F... |
cncfioobd 45098 | A continuous function ` F ... |
jumpncnp 45099 | Jump discontinuity or disc... |
cxpcncf2 45100 | The complex power function... |
fprodcncf 45101 | The finite product of cont... |
add1cncf 45102 | Addition to a constant is ... |
add2cncf 45103 | Addition to a constant is ... |
sub1cncfd 45104 | Subtracting a constant is ... |
sub2cncfd 45105 | Subtraction from a constan... |
fprodsub2cncf 45106 | ` F ` is continuous. (Con... |
fprodadd2cncf 45107 | ` F ` is continuous. (Con... |
fprodsubrecnncnvlem 45108 | The sequence ` S ` of fini... |
fprodsubrecnncnv 45109 | The sequence ` S ` of fini... |
fprodaddrecnncnvlem 45110 | The sequence ` S ` of fini... |
fprodaddrecnncnv 45111 | The sequence ` S ` of fini... |
dvsinexp 45112 | The derivative of sin^N . ... |
dvcosre 45113 | The real derivative of the... |
dvsinax 45114 | Derivative exercise: the d... |
dvsubf 45115 | The subtraction rule for e... |
dvmptconst 45116 | Function-builder for deriv... |
dvcnre 45117 | From complex differentiati... |
dvmptidg 45118 | Function-builder for deriv... |
dvresntr 45119 | Function-builder for deriv... |
fperdvper 45120 | The derivative of a period... |
dvasinbx 45121 | Derivative exercise: the d... |
dvresioo 45122 | Restriction of a derivativ... |
dvdivf 45123 | The quotient rule for ever... |
dvdivbd 45124 | A sufficient condition for... |
dvsubcncf 45125 | A sufficient condition for... |
dvmulcncf 45126 | A sufficient condition for... |
dvcosax 45127 | Derivative exercise: the d... |
dvdivcncf 45128 | A sufficient condition for... |
dvbdfbdioolem1 45129 | Given a function with boun... |
dvbdfbdioolem2 45130 | A function on an open inte... |
dvbdfbdioo 45131 | A function on an open inte... |
ioodvbdlimc1lem1 45132 | If ` F ` has bounded deriv... |
ioodvbdlimc1lem2 45133 | Limit at the lower bound o... |
ioodvbdlimc1 45134 | A real function with bound... |
ioodvbdlimc2lem 45135 | Limit at the upper bound o... |
ioodvbdlimc2 45136 | A real function with bound... |
dvdmsscn 45137 | ` X ` is a subset of ` CC ... |
dvmptmulf 45138 | Function-builder for deriv... |
dvnmptdivc 45139 | Function-builder for itera... |
dvdsn1add 45140 | If ` K ` divides ` N ` but... |
dvxpaek 45141 | Derivative of the polynomi... |
dvnmptconst 45142 | The ` N ` -th derivative o... |
dvnxpaek 45143 | The ` n ` -th derivative o... |
dvnmul 45144 | Function-builder for the `... |
dvmptfprodlem 45145 | Induction step for ~ dvmpt... |
dvmptfprod 45146 | Function-builder for deriv... |
dvnprodlem1 45147 | ` D ` is bijective. (Cont... |
dvnprodlem2 45148 | Induction step for ~ dvnpr... |
dvnprodlem3 45149 | The multinomial formula fo... |
dvnprod 45150 | The multinomial formula fo... |
itgsin0pilem1 45151 | Calculation of the integra... |
ibliccsinexp 45152 | sin^n on a closed interval... |
itgsin0pi 45153 | Calculation of the integra... |
iblioosinexp 45154 | sin^n on an open integral ... |
itgsinexplem1 45155 | Integration by parts is ap... |
itgsinexp 45156 | A recursive formula for th... |
iblconstmpt 45157 | A constant function is int... |
itgeq1d 45158 | Equality theorem for an in... |
mbfres2cn 45159 | Measurability of a piecewi... |
vol0 45160 | The measure of the empty s... |
ditgeqiooicc 45161 | A function ` F ` on an ope... |
volge0 45162 | The volume of a set is alw... |
cnbdibl 45163 | A continuous bounded funct... |
snmbl 45164 | A singleton is measurable.... |
ditgeq3d 45165 | Equality theorem for the d... |
iblempty 45166 | The empty function is inte... |
iblsplit 45167 | The union of two integrabl... |
volsn 45168 | A singleton has 0 Lebesgue... |
itgvol0 45169 | If the domani is negligibl... |
itgcoscmulx 45170 | Exercise: the integral of ... |
iblsplitf 45171 | A version of ~ iblsplit us... |
ibliooicc 45172 | If a function is integrabl... |
volioc 45173 | The measure of a left-open... |
iblspltprt 45174 | If a function is integrabl... |
itgsincmulx 45175 | Exercise: the integral of ... |
itgsubsticclem 45176 | lemma for ~ itgsubsticc . ... |
itgsubsticc 45177 | Integration by u-substitut... |
itgioocnicc 45178 | The integral of a piecewis... |
iblcncfioo 45179 | A continuous function ` F ... |
itgspltprt 45180 | The ` S. ` integral splits... |
itgiccshift 45181 | The integral of a function... |
itgperiod 45182 | The integral of a periodic... |
itgsbtaddcnst 45183 | Integral substitution, add... |
volico 45184 | The measure of left-closed... |
sublevolico 45185 | The Lebesgue measure of a ... |
dmvolss 45186 | Lebesgue measurable sets a... |
ismbl3 45187 | The predicate " ` A ` is L... |
volioof 45188 | The function that assigns ... |
ovolsplit 45189 | The Lebesgue outer measure... |
fvvolioof 45190 | The function value of the ... |
volioore 45191 | The measure of an open int... |
fvvolicof 45192 | The function value of the ... |
voliooico 45193 | An open interval and a lef... |
ismbl4 45194 | The predicate " ` A ` is L... |
volioofmpt 45195 | ` ( ( vol o. (,) ) o. F ) ... |
volicoff 45196 | ` ( ( vol o. [,) ) o. F ) ... |
voliooicof 45197 | The Lebesgue measure of op... |
volicofmpt 45198 | ` ( ( vol o. [,) ) o. F ) ... |
volicc 45199 | The Lebesgue measure of a ... |
voliccico 45200 | A closed interval and a le... |
mbfdmssre 45201 | The domain of a measurable... |
stoweidlem1 45202 | Lemma for ~ stoweid . Thi... |
stoweidlem2 45203 | lemma for ~ stoweid : here... |
stoweidlem3 45204 | Lemma for ~ stoweid : if `... |
stoweidlem4 45205 | Lemma for ~ stoweid : a cl... |
stoweidlem5 45206 | There exists a δ as ... |
stoweidlem6 45207 | Lemma for ~ stoweid : two ... |
stoweidlem7 45208 | This lemma is used to prov... |
stoweidlem8 45209 | Lemma for ~ stoweid : two ... |
stoweidlem9 45210 | Lemma for ~ stoweid : here... |
stoweidlem10 45211 | Lemma for ~ stoweid . Thi... |
stoweidlem11 45212 | This lemma is used to prov... |
stoweidlem12 45213 | Lemma for ~ stoweid . Thi... |
stoweidlem13 45214 | Lemma for ~ stoweid . Thi... |
stoweidlem14 45215 | There exists a ` k ` as in... |
stoweidlem15 45216 | This lemma is used to prov... |
stoweidlem16 45217 | Lemma for ~ stoweid . The... |
stoweidlem17 45218 | This lemma proves that the... |
stoweidlem18 45219 | This theorem proves Lemma ... |
stoweidlem19 45220 | If a set of real functions... |
stoweidlem20 45221 | If a set A of real functio... |
stoweidlem21 45222 | Once the Stone Weierstrass... |
stoweidlem22 45223 | If a set of real functions... |
stoweidlem23 45224 | This lemma is used to prov... |
stoweidlem24 45225 | This lemma proves that for... |
stoweidlem25 45226 | This lemma proves that for... |
stoweidlem26 45227 | This lemma is used to prov... |
stoweidlem27 45228 | This lemma is used to prov... |
stoweidlem28 45229 | There exists a δ as ... |
stoweidlem29 45230 | When the hypothesis for th... |
stoweidlem30 45231 | This lemma is used to prov... |
stoweidlem31 45232 | This lemma is used to prov... |
stoweidlem32 45233 | If a set A of real functio... |
stoweidlem33 45234 | If a set of real functions... |
stoweidlem34 45235 | This lemma proves that for... |
stoweidlem35 45236 | This lemma is used to prov... |
stoweidlem36 45237 | This lemma is used to prov... |
stoweidlem37 45238 | This lemma is used to prov... |
stoweidlem38 45239 | This lemma is used to prov... |
stoweidlem39 45240 | This lemma is used to prov... |
stoweidlem40 45241 | This lemma proves that q_n... |
stoweidlem41 45242 | This lemma is used to prov... |
stoweidlem42 45243 | This lemma is used to prov... |
stoweidlem43 45244 | This lemma is used to prov... |
stoweidlem44 45245 | This lemma is used to prov... |
stoweidlem45 45246 | This lemma proves that, gi... |
stoweidlem46 45247 | This lemma proves that set... |
stoweidlem47 45248 | Subtracting a constant fro... |
stoweidlem48 45249 | This lemma is used to prov... |
stoweidlem49 45250 | There exists a function q_... |
stoweidlem50 45251 | This lemma proves that set... |
stoweidlem51 45252 | There exists a function x ... |
stoweidlem52 45253 | There exists a neighborhoo... |
stoweidlem53 45254 | This lemma is used to prov... |
stoweidlem54 45255 | There exists a function ` ... |
stoweidlem55 45256 | This lemma proves the exis... |
stoweidlem56 45257 | This theorem proves Lemma ... |
stoweidlem57 45258 | There exists a function x ... |
stoweidlem58 45259 | This theorem proves Lemma ... |
stoweidlem59 45260 | This lemma proves that the... |
stoweidlem60 45261 | This lemma proves that the... |
stoweidlem61 45262 | This lemma proves that the... |
stoweidlem62 45263 | This theorem proves the St... |
stoweid 45264 | This theorem proves the St... |
stowei 45265 | This theorem proves the St... |
wallispilem1 45266 | ` I ` is monotone: increas... |
wallispilem2 45267 | A first set of properties ... |
wallispilem3 45268 | I maps to real values. (C... |
wallispilem4 45269 | ` F ` maps to explicit exp... |
wallispilem5 45270 | The sequence ` H ` converg... |
wallispi 45271 | Wallis' formula for π :... |
wallispi2lem1 45272 | An intermediate step betwe... |
wallispi2lem2 45273 | Two expressions are proven... |
wallispi2 45274 | An alternative version of ... |
stirlinglem1 45275 | A simple limit of fraction... |
stirlinglem2 45276 | ` A ` maps to positive rea... |
stirlinglem3 45277 | Long but simple algebraic ... |
stirlinglem4 45278 | Algebraic manipulation of ... |
stirlinglem5 45279 | If ` T ` is between ` 0 ` ... |
stirlinglem6 45280 | A series that converges to... |
stirlinglem7 45281 | Algebraic manipulation of ... |
stirlinglem8 45282 | If ` A ` converges to ` C ... |
stirlinglem9 45283 | ` ( ( B `` N ) - ( B `` ( ... |
stirlinglem10 45284 | A bound for any B(N)-B(N +... |
stirlinglem11 45285 | ` B ` is decreasing. (Con... |
stirlinglem12 45286 | The sequence ` B ` is boun... |
stirlinglem13 45287 | ` B ` is decreasing and ha... |
stirlinglem14 45288 | The sequence ` A ` converg... |
stirlinglem15 45289 | The Stirling's formula is ... |
stirling 45290 | Stirling's approximation f... |
stirlingr 45291 | Stirling's approximation f... |
dirkerval 45292 | The N_th Dirichlet Kernel.... |
dirker2re 45293 | The Dirichlet Kernel value... |
dirkerdenne0 45294 | The Dirichlet Kernel denom... |
dirkerval2 45295 | The N_th Dirichlet Kernel ... |
dirkerre 45296 | The Dirichlet Kernel at an... |
dirkerper 45297 | the Dirichlet Kernel has p... |
dirkerf 45298 | For any natural number ` N... |
dirkertrigeqlem1 45299 | Sum of an even number of a... |
dirkertrigeqlem2 45300 | Trigonomic equality lemma ... |
dirkertrigeqlem3 45301 | Trigonometric equality lem... |
dirkertrigeq 45302 | Trigonometric equality for... |
dirkeritg 45303 | The definite integral of t... |
dirkercncflem1 45304 | If ` Y ` is a multiple of ... |
dirkercncflem2 45305 | Lemma used to prove that t... |
dirkercncflem3 45306 | The Dirichlet Kernel is co... |
dirkercncflem4 45307 | The Dirichlet Kernel is co... |
dirkercncf 45308 | For any natural number ` N... |
fourierdlem1 45309 | A partition interval is a ... |
fourierdlem2 45310 | Membership in a partition.... |
fourierdlem3 45311 | Membership in a partition.... |
fourierdlem4 45312 | ` E ` is a function that m... |
fourierdlem5 45313 | ` S ` is a function. (Con... |
fourierdlem6 45314 | ` X ` is in the periodic p... |
fourierdlem7 45315 | The difference between the... |
fourierdlem8 45316 | A partition interval is a ... |
fourierdlem9 45317 | ` H ` is a complex functio... |
fourierdlem10 45318 | Condition on the bounds of... |
fourierdlem11 45319 | If there is a partition, t... |
fourierdlem12 45320 | A point of a partition is ... |
fourierdlem13 45321 | Value of ` V ` in terms of... |
fourierdlem14 45322 | Given the partition ` V ` ... |
fourierdlem15 45323 | The range of the partition... |
fourierdlem16 45324 | The coefficients of the fo... |
fourierdlem17 45325 | The defined ` L ` is actua... |
fourierdlem18 45326 | The function ` S ` is cont... |
fourierdlem19 45327 | If two elements of ` D ` h... |
fourierdlem20 45328 | Every interval in the part... |
fourierdlem21 45329 | The coefficients of the fo... |
fourierdlem22 45330 | The coefficients of the fo... |
fourierdlem23 45331 | If ` F ` is continuous and... |
fourierdlem24 45332 | A sufficient condition for... |
fourierdlem25 45333 | If ` C ` is not in the ran... |
fourierdlem26 45334 | Periodic image of a point ... |
fourierdlem27 45335 | A partition open interval ... |
fourierdlem28 45336 | Derivative of ` ( F `` ( X... |
fourierdlem29 45337 | Explicit function value fo... |
fourierdlem30 45338 | Sum of three small pieces ... |
fourierdlem31 45339 | If ` A ` is finite and for... |
fourierdlem32 45340 | Limit of a continuous func... |
fourierdlem33 45341 | Limit of a continuous func... |
fourierdlem34 45342 | A partition is one to one.... |
fourierdlem35 45343 | There is a single point in... |
fourierdlem36 45344 | ` F ` is an isomorphism. ... |
fourierdlem37 45345 | ` I ` is a function that m... |
fourierdlem38 45346 | The function ` F ` is cont... |
fourierdlem39 45347 | Integration by parts of ... |
fourierdlem40 45348 | ` H ` is a continuous func... |
fourierdlem41 45349 | Lemma used to prove that e... |
fourierdlem42 45350 | The set of points in a mov... |
fourierdlem43 45351 | ` K ` is a real function. ... |
fourierdlem44 45352 | A condition for having ` (... |
fourierdlem46 45353 | The function ` F ` has a l... |
fourierdlem47 45354 | For ` r ` large enough, th... |
fourierdlem48 45355 | The given periodic functio... |
fourierdlem49 45356 | The given periodic functio... |
fourierdlem50 45357 | Continuity of ` O ` and it... |
fourierdlem51 45358 | ` X ` is in the periodic p... |
fourierdlem52 45359 | d16:d17,d18:jca |- ( ph ->... |
fourierdlem53 45360 | The limit of ` F ( s ) ` a... |
fourierdlem54 45361 | Given a partition ` Q ` an... |
fourierdlem55 45362 | ` U ` is a real function. ... |
fourierdlem56 45363 | Derivative of the ` K ` fu... |
fourierdlem57 45364 | The derivative of ` O ` . ... |
fourierdlem58 45365 | The derivative of ` K ` is... |
fourierdlem59 45366 | The derivative of ` H ` is... |
fourierdlem60 45367 | Given a differentiable fun... |
fourierdlem61 45368 | Given a differentiable fun... |
fourierdlem62 45369 | The function ` K ` is cont... |
fourierdlem63 45370 | The upper bound of interva... |
fourierdlem64 45371 | The partition ` V ` is fin... |
fourierdlem65 45372 | The distance of two adjace... |
fourierdlem66 45373 | Value of the ` G ` functio... |
fourierdlem67 45374 | ` G ` is a function. (Con... |
fourierdlem68 45375 | The derivative of ` O ` is... |
fourierdlem69 45376 | A piecewise continuous fun... |
fourierdlem70 45377 | A piecewise continuous fun... |
fourierdlem71 45378 | A periodic piecewise conti... |
fourierdlem72 45379 | The derivative of ` O ` is... |
fourierdlem73 45380 | A version of the Riemann L... |
fourierdlem74 45381 | Given a piecewise smooth f... |
fourierdlem75 45382 | Given a piecewise smooth f... |
fourierdlem76 45383 | Continuity of ` O ` and it... |
fourierdlem77 45384 | If ` H ` is bounded, then ... |
fourierdlem78 45385 | ` G ` is continuous when r... |
fourierdlem79 45386 | ` E ` projects every inter... |
fourierdlem80 45387 | The derivative of ` O ` is... |
fourierdlem81 45388 | The integral of a piecewis... |
fourierdlem82 45389 | Integral by substitution, ... |
fourierdlem83 45390 | The fourier partial sum fo... |
fourierdlem84 45391 | If ` F ` is piecewise coni... |
fourierdlem85 45392 | Limit of the function ` G ... |
fourierdlem86 45393 | Continuity of ` O ` and it... |
fourierdlem87 45394 | The integral of ` G ` goes... |
fourierdlem88 45395 | Given a piecewise continuo... |
fourierdlem89 45396 | Given a piecewise continuo... |
fourierdlem90 45397 | Given a piecewise continuo... |
fourierdlem91 45398 | Given a piecewise continuo... |
fourierdlem92 45399 | The integral of a piecewis... |
fourierdlem93 45400 | Integral by substitution (... |
fourierdlem94 45401 | For a piecewise smooth fun... |
fourierdlem95 45402 | Algebraic manipulation of ... |
fourierdlem96 45403 | limit for ` F ` at the low... |
fourierdlem97 45404 | ` F ` is continuous on the... |
fourierdlem98 45405 | ` F ` is continuous on the... |
fourierdlem99 45406 | limit for ` F ` at the upp... |
fourierdlem100 45407 | A piecewise continuous fun... |
fourierdlem101 45408 | Integral by substitution f... |
fourierdlem102 45409 | For a piecewise smooth fun... |
fourierdlem103 45410 | The half lower part of the... |
fourierdlem104 45411 | The half upper part of the... |
fourierdlem105 45412 | A piecewise continuous fun... |
fourierdlem106 45413 | For a piecewise smooth fun... |
fourierdlem107 45414 | The integral of a piecewis... |
fourierdlem108 45415 | The integral of a piecewis... |
fourierdlem109 45416 | The integral of a piecewis... |
fourierdlem110 45417 | The integral of a piecewis... |
fourierdlem111 45418 | The fourier partial sum fo... |
fourierdlem112 45419 | Here abbreviations (local ... |
fourierdlem113 45420 | Fourier series convergence... |
fourierdlem114 45421 | Fourier series convergence... |
fourierdlem115 45422 | Fourier serier convergence... |
fourierd 45423 | Fourier series convergence... |
fourierclimd 45424 | Fourier series convergence... |
fourierclim 45425 | Fourier series convergence... |
fourier 45426 | Fourier series convergence... |
fouriercnp 45427 | If ` F ` is continuous at ... |
fourier2 45428 | Fourier series convergence... |
sqwvfoura 45429 | Fourier coefficients for t... |
sqwvfourb 45430 | Fourier series ` B ` coeff... |
fourierswlem 45431 | The Fourier series for the... |
fouriersw 45432 | Fourier series convergence... |
fouriercn 45433 | If the derivative of ` F `... |
elaa2lem 45434 | Elementhood in the set of ... |
elaa2 45435 | Elementhood in the set of ... |
etransclem1 45436 | ` H ` is a function. (Con... |
etransclem2 45437 | Derivative of ` G ` . (Co... |
etransclem3 45438 | The given ` if ` term is a... |
etransclem4 45439 | ` F ` expressed as a finit... |
etransclem5 45440 | A change of bound variable... |
etransclem6 45441 | A change of bound variable... |
etransclem7 45442 | The given product is an in... |
etransclem8 45443 | ` F ` is a function. (Con... |
etransclem9 45444 | If ` K ` divides ` N ` but... |
etransclem10 45445 | The given ` if ` term is a... |
etransclem11 45446 | A change of bound variable... |
etransclem12 45447 | ` C ` applied to ` N ` . ... |
etransclem13 45448 | ` F ` applied to ` Y ` . ... |
etransclem14 45449 | Value of the term ` T ` , ... |
etransclem15 45450 | Value of the term ` T ` , ... |
etransclem16 45451 | Every element in the range... |
etransclem17 45452 | The ` N ` -th derivative o... |
etransclem18 45453 | The given function is inte... |
etransclem19 45454 | The ` N ` -th derivative o... |
etransclem20 45455 | ` H ` is smooth. (Contrib... |
etransclem21 45456 | The ` N ` -th derivative o... |
etransclem22 45457 | The ` N ` -th derivative o... |
etransclem23 45458 | This is the claim proof in... |
etransclem24 45459 | ` P ` divides the I -th de... |
etransclem25 45460 | ` P ` factorial divides th... |
etransclem26 45461 | Every term in the sum of t... |
etransclem27 45462 | The ` N ` -th derivative o... |
etransclem28 45463 | ` ( P - 1 ) ` factorial di... |
etransclem29 45464 | The ` N ` -th derivative o... |
etransclem30 45465 | The ` N ` -th derivative o... |
etransclem31 45466 | The ` N ` -th derivative o... |
etransclem32 45467 | This is the proof for the ... |
etransclem33 45468 | ` F ` is smooth. (Contrib... |
etransclem34 45469 | The ` N ` -th derivative o... |
etransclem35 45470 | ` P ` does not divide the ... |
etransclem36 45471 | The ` N ` -th derivative o... |
etransclem37 45472 | ` ( P - 1 ) ` factorial di... |
etransclem38 45473 | ` P ` divides the I -th de... |
etransclem39 45474 | ` G ` is a function. (Con... |
etransclem40 45475 | The ` N ` -th derivative o... |
etransclem41 45476 | ` P ` does not divide the ... |
etransclem42 45477 | The ` N ` -th derivative o... |
etransclem43 45478 | ` G ` is a continuous func... |
etransclem44 45479 | The given finite sum is no... |
etransclem45 45480 | ` K ` is an integer. (Con... |
etransclem46 45481 | This is the proof for equa... |
etransclem47 45482 | ` _e ` is transcendental. ... |
etransclem48 45483 | ` _e ` is transcendental. ... |
etransc 45484 | ` _e ` is transcendental. ... |
rrxtopn 45485 | The topology of the genera... |
rrxngp 45486 | Generalized Euclidean real... |
rrxtps 45487 | Generalized Euclidean real... |
rrxtopnfi 45488 | The topology of the n-dime... |
rrxtopon 45489 | The topology on generalize... |
rrxtop 45490 | The topology on generalize... |
rrndistlt 45491 | Given two points in the sp... |
rrxtoponfi 45492 | The topology on n-dimensio... |
rrxunitopnfi 45493 | The base set of the standa... |
rrxtopn0 45494 | The topology of the zero-d... |
qndenserrnbllem 45495 | n-dimensional rational num... |
qndenserrnbl 45496 | n-dimensional rational num... |
rrxtopn0b 45497 | The topology of the zero-d... |
qndenserrnopnlem 45498 | n-dimensional rational num... |
qndenserrnopn 45499 | n-dimensional rational num... |
qndenserrn 45500 | n-dimensional rational num... |
rrxsnicc 45501 | A multidimensional singlet... |
rrnprjdstle 45502 | The distance between two p... |
rrndsmet 45503 | ` D ` is a metric for the ... |
rrndsxmet 45504 | ` D ` is an extended metri... |
ioorrnopnlem 45505 | The a point in an indexed ... |
ioorrnopn 45506 | The indexed product of ope... |
ioorrnopnxrlem 45507 | Given a point ` F ` that b... |
ioorrnopnxr 45508 | The indexed product of ope... |
issal 45515 | Express the predicate " ` ... |
pwsal 45516 | The power set of a given s... |
salunicl 45517 | SAlg sigma-algebra is clos... |
saluncl 45518 | The union of two sets in a... |
prsal 45519 | The pair of the empty set ... |
saldifcl 45520 | The complement of an eleme... |
0sal 45521 | The empty set belongs to e... |
salgenval 45522 | The sigma-algebra generate... |
saliunclf 45523 | SAlg sigma-algebra is clos... |
saliuncl 45524 | SAlg sigma-algebra is clos... |
salincl 45525 | The intersection of two se... |
saluni 45526 | A set is an element of any... |
saliinclf 45527 | SAlg sigma-algebra is clos... |
saliincl 45528 | SAlg sigma-algebra is clos... |
saldifcl2 45529 | The difference of two elem... |
intsaluni 45530 | The union of an arbitrary ... |
intsal 45531 | The arbitrary intersection... |
salgenn0 45532 | The set used in the defini... |
salgencl 45533 | ` SalGen ` actually genera... |
issald 45534 | Sufficient condition to pr... |
salexct 45535 | An example of nontrivial s... |
sssalgen 45536 | A set is a subset of the s... |
salgenss 45537 | The sigma-algebra generate... |
salgenuni 45538 | The base set of the sigma-... |
issalgend 45539 | One side of ~ dfsalgen2 . ... |
salexct2 45540 | An example of a subset tha... |
unisalgen 45541 | The union of a set belongs... |
dfsalgen2 45542 | Alternate characterization... |
salexct3 45543 | An example of a sigma-alge... |
salgencntex 45544 | This counterexample shows ... |
salgensscntex 45545 | This counterexample shows ... |
issalnnd 45546 | Sufficient condition to pr... |
dmvolsal 45547 | Lebesgue measurable sets f... |
saldifcld 45548 | The complement of an eleme... |
saluncld 45549 | The union of two sets in a... |
salgencld 45550 | ` SalGen ` actually genera... |
0sald 45551 | The empty set belongs to e... |
iooborel 45552 | An open interval is a Bore... |
salincld 45553 | The intersection of two se... |
salunid 45554 | A set is an element of any... |
unisalgen2 45555 | The union of a set belongs... |
bor1sal 45556 | The Borel sigma-algebra on... |
iocborel 45557 | A left-open, right-closed ... |
subsaliuncllem 45558 | A subspace sigma-algebra i... |
subsaliuncl 45559 | A subspace sigma-algebra i... |
subsalsal 45560 | A subspace sigma-algebra i... |
subsaluni 45561 | A set belongs to the subsp... |
salrestss 45562 | A sigma-algebra restricted... |
sge0rnre 45565 | When ` sum^ ` is applied t... |
fge0icoicc 45566 | If ` F ` maps to nonnegati... |
sge0val 45567 | The value of the sum of no... |
fge0npnf 45568 | If ` F ` maps to nonnegati... |
sge0rnn0 45569 | The range used in the defi... |
sge0vald 45570 | The value of the sum of no... |
fge0iccico 45571 | A range of nonnegative ext... |
gsumge0cl 45572 | Closure of group sum, for ... |
sge0reval 45573 | Value of the sum of nonneg... |
sge0pnfval 45574 | If a term in the sum of no... |
fge0iccre 45575 | A range of nonnegative ext... |
sge0z 45576 | Any nonnegative extended s... |
sge00 45577 | The sum of nonnegative ext... |
fsumlesge0 45578 | Every finite subsum of non... |
sge0revalmpt 45579 | Value of the sum of nonneg... |
sge0sn 45580 | A sum of a nonnegative ext... |
sge0tsms 45581 | ` sum^ ` applied to a nonn... |
sge0cl 45582 | The arbitrary sum of nonne... |
sge0f1o 45583 | Re-index a nonnegative ext... |
sge0snmpt 45584 | A sum of a nonnegative ext... |
sge0ge0 45585 | The sum of nonnegative ext... |
sge0xrcl 45586 | The arbitrary sum of nonne... |
sge0repnf 45587 | The of nonnegative extende... |
sge0fsum 45588 | The arbitrary sum of a fin... |
sge0rern 45589 | If the sum of nonnegative ... |
sge0supre 45590 | If the arbitrary sum of no... |
sge0fsummpt 45591 | The arbitrary sum of a fin... |
sge0sup 45592 | The arbitrary sum of nonne... |
sge0less 45593 | A shorter sum of nonnegati... |
sge0rnbnd 45594 | The range used in the defi... |
sge0pr 45595 | Sum of a pair of nonnegati... |
sge0gerp 45596 | The arbitrary sum of nonne... |
sge0pnffigt 45597 | If the sum of nonnegative ... |
sge0ssre 45598 | If a sum of nonnegative ex... |
sge0lefi 45599 | A sum of nonnegative exten... |
sge0lessmpt 45600 | A shorter sum of nonnegati... |
sge0ltfirp 45601 | If the sum of nonnegative ... |
sge0prle 45602 | The sum of a pair of nonne... |
sge0gerpmpt 45603 | The arbitrary sum of nonne... |
sge0resrnlem 45604 | The sum of nonnegative ext... |
sge0resrn 45605 | The sum of nonnegative ext... |
sge0ssrempt 45606 | If a sum of nonnegative ex... |
sge0resplit 45607 | ` sum^ ` splits into two p... |
sge0le 45608 | If all of the terms of sum... |
sge0ltfirpmpt 45609 | If the extended sum of non... |
sge0split 45610 | Split a sum of nonnegative... |
sge0lempt 45611 | If all of the terms of sum... |
sge0splitmpt 45612 | Split a sum of nonnegative... |
sge0ss 45613 | Change the index set to a ... |
sge0iunmptlemfi 45614 | Sum of nonnegative extende... |
sge0p1 45615 | The addition of the next t... |
sge0iunmptlemre 45616 | Sum of nonnegative extende... |
sge0fodjrnlem 45617 | Re-index a nonnegative ext... |
sge0fodjrn 45618 | Re-index a nonnegative ext... |
sge0iunmpt 45619 | Sum of nonnegative extende... |
sge0iun 45620 | Sum of nonnegative extende... |
sge0nemnf 45621 | The generalized sum of non... |
sge0rpcpnf 45622 | The sum of an infinite num... |
sge0rernmpt 45623 | If the sum of nonnegative ... |
sge0lefimpt 45624 | A sum of nonnegative exten... |
nn0ssge0 45625 | Nonnegative integers are n... |
sge0clmpt 45626 | The generalized sum of non... |
sge0ltfirpmpt2 45627 | If the extended sum of non... |
sge0isum 45628 | If a series of nonnegative... |
sge0xrclmpt 45629 | The generalized sum of non... |
sge0xp 45630 | Combine two generalized su... |
sge0isummpt 45631 | If a series of nonnegative... |
sge0ad2en 45632 | The value of the infinite ... |
sge0isummpt2 45633 | If a series of nonnegative... |
sge0xaddlem1 45634 | The extended addition of t... |
sge0xaddlem2 45635 | The extended addition of t... |
sge0xadd 45636 | The extended addition of t... |
sge0fsummptf 45637 | The generalized sum of a f... |
sge0snmptf 45638 | A sum of a nonnegative ext... |
sge0ge0mpt 45639 | The sum of nonnegative ext... |
sge0repnfmpt 45640 | The of nonnegative extende... |
sge0pnffigtmpt 45641 | If the generalized sum of ... |
sge0splitsn 45642 | Separate out a term in a g... |
sge0pnffsumgt 45643 | If the sum of nonnegative ... |
sge0gtfsumgt 45644 | If the generalized sum of ... |
sge0uzfsumgt 45645 | If a real number is smalle... |
sge0pnfmpt 45646 | If a term in the sum of no... |
sge0seq 45647 | A series of nonnegative re... |
sge0reuz 45648 | Value of the generalized s... |
sge0reuzb 45649 | Value of the generalized s... |
ismea 45652 | Express the predicate " ` ... |
dmmeasal 45653 | The domain of a measure is... |
meaf 45654 | A measure is a function th... |
mea0 45655 | The measure of the empty s... |
nnfoctbdjlem 45656 | There exists a mapping fro... |
nnfoctbdj 45657 | There exists a mapping fro... |
meadjuni 45658 | The measure of the disjoin... |
meacl 45659 | The measure of a set is a ... |
iundjiunlem 45660 | The sets in the sequence `... |
iundjiun 45661 | Given a sequence ` E ` of ... |
meaxrcl 45662 | The measure of a set is an... |
meadjun 45663 | The measure of the union o... |
meassle 45664 | The measure of a set is gr... |
meaunle 45665 | The measure of the union o... |
meadjiunlem 45666 | The sum of nonnegative ext... |
meadjiun 45667 | The measure of the disjoin... |
ismeannd 45668 | Sufficient condition to pr... |
meaiunlelem 45669 | The measure of the union o... |
meaiunle 45670 | The measure of the union o... |
psmeasurelem 45671 | ` M ` applied to a disjoin... |
psmeasure 45672 | Point supported measure, R... |
voliunsge0lem 45673 | The Lebesgue measure funct... |
voliunsge0 45674 | The Lebesgue measure funct... |
volmea 45675 | The Lebesgue measure on th... |
meage0 45676 | If the measure of a measur... |
meadjunre 45677 | The measure of the union o... |
meassre 45678 | If the measure of a measur... |
meale0eq0 45679 | A measure that is less tha... |
meadif 45680 | The measure of the differe... |
meaiuninclem 45681 | Measures are continuous fr... |
meaiuninc 45682 | Measures are continuous fr... |
meaiuninc2 45683 | Measures are continuous fr... |
meaiunincf 45684 | Measures are continuous fr... |
meaiuninc3v 45685 | Measures are continuous fr... |
meaiuninc3 45686 | Measures are continuous fr... |
meaiininclem 45687 | Measures are continuous fr... |
meaiininc 45688 | Measures are continuous fr... |
meaiininc2 45689 | Measures are continuous fr... |
caragenval 45694 | The sigma-algebra generate... |
isome 45695 | Express the predicate " ` ... |
caragenel 45696 | Membership in the Caratheo... |
omef 45697 | An outer measure is a func... |
ome0 45698 | The outer measure of the e... |
omessle 45699 | The outer measure of a set... |
omedm 45700 | The domain of an outer mea... |
caragensplit 45701 | If ` E ` is in the set gen... |
caragenelss 45702 | An element of the Caratheo... |
carageneld 45703 | Membership in the Caratheo... |
omecl 45704 | The outer measure of a set... |
caragenss 45705 | The sigma-algebra generate... |
omeunile 45706 | The outer measure of the u... |
caragen0 45707 | The empty set belongs to a... |
omexrcl 45708 | The outer measure of a set... |
caragenunidm 45709 | The base set of an outer m... |
caragensspw 45710 | The sigma-algebra generate... |
omessre 45711 | If the outer measure of a ... |
caragenuni 45712 | The base set of the sigma-... |
caragenuncllem 45713 | The Caratheodory's constru... |
caragenuncl 45714 | The Caratheodory's constru... |
caragendifcl 45715 | The Caratheodory's constru... |
caragenfiiuncl 45716 | The Caratheodory's constru... |
omeunle 45717 | The outer measure of the u... |
omeiunle 45718 | The outer measure of the i... |
omelesplit 45719 | The outer measure of a set... |
omeiunltfirp 45720 | If the outer measure of a ... |
omeiunlempt 45721 | The outer measure of the i... |
carageniuncllem1 45722 | The outer measure of ` A i... |
carageniuncllem2 45723 | The Caratheodory's constru... |
carageniuncl 45724 | The Caratheodory's constru... |
caragenunicl 45725 | The Caratheodory's constru... |
caragensal 45726 | Caratheodory's method gene... |
caratheodorylem1 45727 | Lemma used to prove that C... |
caratheodorylem2 45728 | Caratheodory's constructio... |
caratheodory 45729 | Caratheodory's constructio... |
0ome 45730 | The map that assigns 0 to ... |
isomenndlem 45731 | ` O ` is sub-additive w.r.... |
isomennd 45732 | Sufficient condition to pr... |
caragenel2d 45733 | Membership in the Caratheo... |
omege0 45734 | If the outer measure of a ... |
omess0 45735 | If the outer measure of a ... |
caragencmpl 45736 | A measure built with the C... |
vonval 45741 | Value of the Lebesgue meas... |
ovnval 45742 | Value of the Lebesgue oute... |
elhoi 45743 | Membership in a multidimen... |
icoresmbl 45744 | A closed-below, open-above... |
hoissre 45745 | The projection of a half-o... |
ovnval2 45746 | Value of the Lebesgue oute... |
volicorecl 45747 | The Lebesgue measure of a ... |
hoiprodcl 45748 | The pre-measure of half-op... |
hoicvr 45749 | ` I ` is a countable set o... |
hoissrrn 45750 | A half-open interval is a ... |
ovn0val 45751 | The Lebesgue outer measure... |
ovnn0val 45752 | The value of a (multidimen... |
ovnval2b 45753 | Value of the Lebesgue oute... |
volicorescl 45754 | The Lebesgue measure of a ... |
ovnprodcl 45755 | The product used in the de... |
hoiprodcl2 45756 | The pre-measure of half-op... |
hoicvrrex 45757 | Any subset of the multidim... |
ovnsupge0 45758 | The set used in the defini... |
ovnlecvr 45759 | Given a subset of multidim... |
ovnpnfelsup 45760 | ` +oo ` is an element of t... |
ovnsslelem 45761 | The (multidimensional, non... |
ovnssle 45762 | The (multidimensional) Leb... |
ovnlerp 45763 | The Lebesgue outer measure... |
ovnf 45764 | The Lebesgue outer measure... |
ovncvrrp 45765 | The Lebesgue outer measure... |
ovn0lem 45766 | For any finite dimension, ... |
ovn0 45767 | For any finite dimension, ... |
ovncl 45768 | The Lebesgue outer measure... |
ovn02 45769 | For the zero-dimensional s... |
ovnxrcl 45770 | The Lebesgue outer measure... |
ovnsubaddlem1 45771 | The Lebesgue outer measure... |
ovnsubaddlem2 45772 | ` ( voln* `` X ) ` is suba... |
ovnsubadd 45773 | ` ( voln* `` X ) ` is suba... |
ovnome 45774 | ` ( voln* `` X ) ` is an o... |
vonmea 45775 | ` ( voln `` X ) ` is a mea... |
volicon0 45776 | The measure of a nonempty ... |
hsphoif 45777 | ` H ` is a function (that ... |
hoidmvval 45778 | The dimensional volume of ... |
hoissrrn2 45779 | A half-open interval is a ... |
hsphoival 45780 | ` H ` is a function (that ... |
hoiprodcl3 45781 | The pre-measure of half-op... |
volicore 45782 | The Lebesgue measure of a ... |
hoidmvcl 45783 | The dimensional volume of ... |
hoidmv0val 45784 | The dimensional volume of ... |
hoidmvn0val 45785 | The dimensional volume of ... |
hsphoidmvle2 45786 | The dimensional volume of ... |
hsphoidmvle 45787 | The dimensional volume of ... |
hoidmvval0 45788 | The dimensional volume of ... |
hoiprodp1 45789 | The dimensional volume of ... |
sge0hsphoire 45790 | If the generalized sum of ... |
hoidmvval0b 45791 | The dimensional volume of ... |
hoidmv1lelem1 45792 | The supremum of ` U ` belo... |
hoidmv1lelem2 45793 | This is the contradiction ... |
hoidmv1lelem3 45794 | The dimensional volume of ... |
hoidmv1le 45795 | The dimensional volume of ... |
hoidmvlelem1 45796 | The supremum of ` U ` belo... |
hoidmvlelem2 45797 | This is the contradiction ... |
hoidmvlelem3 45798 | This is the contradiction ... |
hoidmvlelem4 45799 | The dimensional volume of ... |
hoidmvlelem5 45800 | The dimensional volume of ... |
hoidmvle 45801 | The dimensional volume of ... |
ovnhoilem1 45802 | The Lebesgue outer measure... |
ovnhoilem2 45803 | The Lebesgue outer measure... |
ovnhoi 45804 | The Lebesgue outer measure... |
dmovn 45805 | The domain of the Lebesgue... |
hoicoto2 45806 | The half-open interval exp... |
dmvon 45807 | Lebesgue measurable n-dime... |
hoi2toco 45808 | The half-open interval exp... |
hoidifhspval 45809 | ` D ` is a function that r... |
hspval 45810 | The value of the half-spac... |
ovnlecvr2 45811 | Given a subset of multidim... |
ovncvr2 45812 | ` B ` and ` T ` are the le... |
dmovnsal 45813 | The domain of the Lebesgue... |
unidmovn 45814 | Base set of the n-dimensio... |
rrnmbl 45815 | The set of n-dimensional R... |
hoidifhspval2 45816 | ` D ` is a function that r... |
hspdifhsp 45817 | A n-dimensional half-open ... |
unidmvon 45818 | Base set of the n-dimensio... |
hoidifhspf 45819 | ` D ` is a function that r... |
hoidifhspval3 45820 | ` D ` is a function that r... |
hoidifhspdmvle 45821 | The dimensional volume of ... |
voncmpl 45822 | The Lebesgue measure is co... |
hoiqssbllem1 45823 | The center of the n-dimens... |
hoiqssbllem2 45824 | The center of the n-dimens... |
hoiqssbllem3 45825 | A n-dimensional ball conta... |
hoiqssbl 45826 | A n-dimensional ball conta... |
hspmbllem1 45827 | Any half-space of the n-di... |
hspmbllem2 45828 | Any half-space of the n-di... |
hspmbllem3 45829 | Any half-space of the n-di... |
hspmbl 45830 | Any half-space of the n-di... |
hoimbllem 45831 | Any n-dimensional half-ope... |
hoimbl 45832 | Any n-dimensional half-ope... |
opnvonmbllem1 45833 | The half-open interval exp... |
opnvonmbllem2 45834 | An open subset of the n-di... |
opnvonmbl 45835 | An open subset of the n-di... |
opnssborel 45836 | Open sets of a generalized... |
borelmbl 45837 | All Borel subsets of the n... |
volicorege0 45838 | The Lebesgue measure of a ... |
isvonmbl 45839 | The predicate " ` A ` is m... |
mblvon 45840 | The n-dimensional Lebesgue... |
vonmblss 45841 | n-dimensional Lebesgue mea... |
volico2 45842 | The measure of left-closed... |
vonmblss2 45843 | n-dimensional Lebesgue mea... |
ovolval2lem 45844 | The value of the Lebesgue ... |
ovolval2 45845 | The value of the Lebesgue ... |
ovnsubadd2lem 45846 | ` ( voln* `` X ) ` is suba... |
ovnsubadd2 45847 | ` ( voln* `` X ) ` is suba... |
ovolval3 45848 | The value of the Lebesgue ... |
ovnsplit 45849 | The n-dimensional Lebesgue... |
ovolval4lem1 45850 | |- ( ( ph /\ n e. A ) -> ... |
ovolval4lem2 45851 | The value of the Lebesgue ... |
ovolval4 45852 | The value of the Lebesgue ... |
ovolval5lem1 45853 | ` |- ( ph -> ( sum^ `` ( n... |
ovolval5lem2 45854 | ` |- ( ( ph /\ n e. NN ) -... |
ovolval5lem3 45855 | The value of the Lebesgue ... |
ovolval5 45856 | The value of the Lebesgue ... |
ovnovollem1 45857 | if ` F ` is a cover of ` B... |
ovnovollem2 45858 | if ` I ` is a cover of ` (... |
ovnovollem3 45859 | The 1-dimensional Lebesgue... |
ovnovol 45860 | The 1-dimensional Lebesgue... |
vonvolmbllem 45861 | If a subset ` B ` of real ... |
vonvolmbl 45862 | A subset of Real numbers i... |
vonvol 45863 | The 1-dimensional Lebesgue... |
vonvolmbl2 45864 | A subset ` X ` of the spac... |
vonvol2 45865 | The 1-dimensional Lebesgue... |
hoimbl2 45866 | Any n-dimensional half-ope... |
voncl 45867 | The Lebesgue measure of a ... |
vonhoi 45868 | The Lebesgue outer measure... |
vonxrcl 45869 | The Lebesgue measure of a ... |
ioosshoi 45870 | A n-dimensional open inter... |
vonn0hoi 45871 | The Lebesgue outer measure... |
von0val 45872 | The Lebesgue measure (for ... |
vonhoire 45873 | The Lebesgue measure of a ... |
iinhoiicclem 45874 | A n-dimensional closed int... |
iinhoiicc 45875 | A n-dimensional closed int... |
iunhoiioolem 45876 | A n-dimensional open inter... |
iunhoiioo 45877 | A n-dimensional open inter... |
ioovonmbl 45878 | Any n-dimensional open int... |
iccvonmbllem 45879 | Any n-dimensional closed i... |
iccvonmbl 45880 | Any n-dimensional closed i... |
vonioolem1 45881 | The sequence of the measur... |
vonioolem2 45882 | The n-dimensional Lebesgue... |
vonioo 45883 | The n-dimensional Lebesgue... |
vonicclem1 45884 | The sequence of the measur... |
vonicclem2 45885 | The n-dimensional Lebesgue... |
vonicc 45886 | The n-dimensional Lebesgue... |
snvonmbl 45887 | A n-dimensional singleton ... |
vonn0ioo 45888 | The n-dimensional Lebesgue... |
vonn0icc 45889 | The n-dimensional Lebesgue... |
ctvonmbl 45890 | Any n-dimensional countabl... |
vonn0ioo2 45891 | The n-dimensional Lebesgue... |
vonsn 45892 | The n-dimensional Lebesgue... |
vonn0icc2 45893 | The n-dimensional Lebesgue... |
vonct 45894 | The n-dimensional Lebesgue... |
vitali2 45895 | There are non-measurable s... |
pimltmnf2f 45898 | Given a real-valued functi... |
pimltmnf2 45899 | Given a real-valued functi... |
preimagelt 45900 | The preimage of a right-op... |
preimalegt 45901 | The preimage of a left-ope... |
pimconstlt0 45902 | Given a constant function,... |
pimconstlt1 45903 | Given a constant function,... |
pimltpnff 45904 | Given a real-valued functi... |
pimltpnf 45905 | Given a real-valued functi... |
pimgtpnf2f 45906 | Given a real-valued functi... |
pimgtpnf2 45907 | Given a real-valued functi... |
salpreimagelt 45908 | If all the preimages of le... |
pimrecltpos 45909 | The preimage of an unbound... |
salpreimalegt 45910 | If all the preimages of ri... |
pimiooltgt 45911 | The preimage of an open in... |
preimaicomnf 45912 | Preimage of an open interv... |
pimltpnf2f 45913 | Given a real-valued functi... |
pimltpnf2 45914 | Given a real-valued functi... |
pimgtmnf2 45915 | Given a real-valued functi... |
pimdecfgtioc 45916 | Given a nonincreasing func... |
pimincfltioc 45917 | Given a nondecreasing func... |
pimdecfgtioo 45918 | Given a nondecreasing func... |
pimincfltioo 45919 | Given a nondecreasing func... |
preimaioomnf 45920 | Preimage of an open interv... |
preimageiingt 45921 | A preimage of a left-close... |
preimaleiinlt 45922 | A preimage of a left-open,... |
pimgtmnff 45923 | Given a real-valued functi... |
pimgtmnf 45924 | Given a real-valued functi... |
pimrecltneg 45925 | The preimage of an unbound... |
salpreimagtge 45926 | If all the preimages of le... |
salpreimaltle 45927 | If all the preimages of ri... |
issmflem 45928 | The predicate " ` F ` is a... |
issmf 45929 | The predicate " ` F ` is a... |
salpreimalelt 45930 | If all the preimages of ri... |
salpreimagtlt 45931 | If all the preimages of le... |
smfpreimalt 45932 | Given a function measurabl... |
smff 45933 | A function measurable w.r.... |
smfdmss 45934 | The domain of a function m... |
issmff 45935 | The predicate " ` F ` is a... |
issmfd 45936 | A sufficient condition for... |
smfpreimaltf 45937 | Given a function measurabl... |
issmfdf 45938 | A sufficient condition for... |
sssmf 45939 | The restriction of a sigma... |
mbfresmf 45940 | A real-valued measurable f... |
cnfsmf 45941 | A continuous function is m... |
incsmflem 45942 | A nondecreasing function i... |
incsmf 45943 | A real-valued, nondecreasi... |
smfsssmf 45944 | If a function is measurabl... |
issmflelem 45945 | The predicate " ` F ` is a... |
issmfle 45946 | The predicate " ` F ` is a... |
smfpimltmpt 45947 | Given a function measurabl... |
smfpimltxr 45948 | Given a function measurabl... |
issmfdmpt 45949 | A sufficient condition for... |
smfconst 45950 | Given a sigma-algebra over... |
sssmfmpt 45951 | The restriction of a sigma... |
cnfrrnsmf 45952 | A function, continuous fro... |
smfid 45953 | The identity function is B... |
bormflebmf 45954 | A Borel measurable functio... |
smfpreimale 45955 | Given a function measurabl... |
issmfgtlem 45956 | The predicate " ` F ` is a... |
issmfgt 45957 | The predicate " ` F ` is a... |
issmfled 45958 | A sufficient condition for... |
smfpimltxrmptf 45959 | Given a function measurabl... |
smfpimltxrmpt 45960 | Given a function measurabl... |
smfmbfcex 45961 | A constant function, with ... |
issmfgtd 45962 | A sufficient condition for... |
smfpreimagt 45963 | Given a function measurabl... |
smfaddlem1 45964 | Given the sum of two funct... |
smfaddlem2 45965 | The sum of two sigma-measu... |
smfadd 45966 | The sum of two sigma-measu... |
decsmflem 45967 | A nonincreasing function i... |
decsmf 45968 | A real-valued, nonincreasi... |
smfpreimagtf 45969 | Given a function measurabl... |
issmfgelem 45970 | The predicate " ` F ` is a... |
issmfge 45971 | The predicate " ` F ` is a... |
smflimlem1 45972 | Lemma for the proof that t... |
smflimlem2 45973 | Lemma for the proof that t... |
smflimlem3 45974 | The limit of sigma-measura... |
smflimlem4 45975 | Lemma for the proof that t... |
smflimlem5 45976 | Lemma for the proof that t... |
smflimlem6 45977 | Lemma for the proof that t... |
smflim 45978 | The limit of sigma-measura... |
nsssmfmbflem 45979 | The sigma-measurable funct... |
nsssmfmbf 45980 | The sigma-measurable funct... |
smfpimgtxr 45981 | Given a function measurabl... |
smfpimgtmpt 45982 | Given a function measurabl... |
smfpreimage 45983 | Given a function measurabl... |
mbfpsssmf 45984 | Real-valued measurable fun... |
smfpimgtxrmptf 45985 | Given a function measurabl... |
smfpimgtxrmpt 45986 | Given a function measurabl... |
smfpimioompt 45987 | Given a function measurabl... |
smfpimioo 45988 | Given a function measurabl... |
smfresal 45989 | Given a sigma-measurable f... |
smfrec 45990 | The reciprocal of a sigma-... |
smfres 45991 | The restriction of sigma-m... |
smfmullem1 45992 | The multiplication of two ... |
smfmullem2 45993 | The multiplication of two ... |
smfmullem3 45994 | The multiplication of two ... |
smfmullem4 45995 | The multiplication of two ... |
smfmul 45996 | The multiplication of two ... |
smfmulc1 45997 | A sigma-measurable functio... |
smfdiv 45998 | The fraction of two sigma-... |
smfpimbor1lem1 45999 | Every open set belongs to ... |
smfpimbor1lem2 46000 | Given a sigma-measurable f... |
smfpimbor1 46001 | Given a sigma-measurable f... |
smf2id 46002 | Twice the identity functio... |
smfco 46003 | The composition of a Borel... |
smfneg 46004 | The negative of a sigma-me... |
smffmptf 46005 | A function measurable w.r.... |
smffmpt 46006 | A function measurable w.r.... |
smflim2 46007 | The limit of a sequence of... |
smfpimcclem 46008 | Lemma for ~ smfpimcc given... |
smfpimcc 46009 | Given a countable set of s... |
issmfle2d 46010 | A sufficient condition for... |
smflimmpt 46011 | The limit of a sequence of... |
smfsuplem1 46012 | The supremum of a countabl... |
smfsuplem2 46013 | The supremum of a countabl... |
smfsuplem3 46014 | The supremum of a countabl... |
smfsup 46015 | The supremum of a countabl... |
smfsupmpt 46016 | The supremum of a countabl... |
smfsupxr 46017 | The supremum of a countabl... |
smfinflem 46018 | The infimum of a countable... |
smfinf 46019 | The infimum of a countable... |
smfinfmpt 46020 | The infimum of a countable... |
smflimsuplem1 46021 | If ` H ` converges, the ` ... |
smflimsuplem2 46022 | The superior limit of a se... |
smflimsuplem3 46023 | The limit of the ` ( H `` ... |
smflimsuplem4 46024 | If ` H ` converges, the ` ... |
smflimsuplem5 46025 | ` H ` converges to the sup... |
smflimsuplem6 46026 | The superior limit of a se... |
smflimsuplem7 46027 | The superior limit of a se... |
smflimsuplem8 46028 | The superior limit of a se... |
smflimsup 46029 | The superior limit of a se... |
smflimsupmpt 46030 | The superior limit of a se... |
smfliminflem 46031 | The inferior limit of a co... |
smfliminf 46032 | The inferior limit of a co... |
smfliminfmpt 46033 | The inferior limit of a co... |
adddmmbl 46034 | If two functions have doma... |
adddmmbl2 46035 | If two functions have doma... |
muldmmbl 46036 | If two functions have doma... |
muldmmbl2 46037 | If two functions have doma... |
smfdmmblpimne 46038 | If a measurable function w... |
smfdivdmmbl 46039 | If a functions and a sigma... |
smfpimne 46040 | Given a function measurabl... |
smfpimne2 46041 | Given a function measurabl... |
smfdivdmmbl2 46042 | If a functions and a sigma... |
fsupdm 46043 | The domain of the sup func... |
fsupdm2 46044 | The domain of the sup func... |
smfsupdmmbllem 46045 | If a countable set of sigm... |
smfsupdmmbl 46046 | If a countable set of sigm... |
finfdm 46047 | The domain of the inf func... |
finfdm2 46048 | The domain of the inf func... |
smfinfdmmbllem 46049 | If a countable set of sigm... |
smfinfdmmbl 46050 | If a countable set of sigm... |
sigarval 46051 | Define the signed area by ... |
sigarim 46052 | Signed area takes value in... |
sigarac 46053 | Signed area is anticommuta... |
sigaraf 46054 | Signed area is additive by... |
sigarmf 46055 | Signed area is additive (w... |
sigaras 46056 | Signed area is additive by... |
sigarms 46057 | Signed area is additive (w... |
sigarls 46058 | Signed area is linear by t... |
sigarid 46059 | Signed area of a flat para... |
sigarexp 46060 | Expand the signed area for... |
sigarperm 46061 | Signed area ` ( A - C ) G ... |
sigardiv 46062 | If signed area between vec... |
sigarimcd 46063 | Signed area takes value in... |
sigariz 46064 | If signed area is zero, th... |
sigarcol 46065 | Given three points ` A ` ,... |
sharhght 46066 | Let ` A B C ` be a triangl... |
sigaradd 46067 | Subtracting (double) area ... |
cevathlem1 46068 | Ceva's theorem first lemma... |
cevathlem2 46069 | Ceva's theorem second lemm... |
cevath 46070 | Ceva's theorem. Let ` A B... |
simpcntrab 46071 | The center of a simple gro... |
et-ltneverrefl 46072 | Less-than class is never r... |
et-equeucl 46073 | Alternative proof that equ... |
et-sqrtnegnre 46074 | The square root of a negat... |
natlocalincr 46075 | Global monotonicity on hal... |
natglobalincr 46076 | Local monotonicity on half... |
upwordnul 46079 | Empty set is an increasing... |
upwordisword 46080 | Any increasing sequence is... |
singoutnword 46081 | Singleton with character o... |
singoutnupword 46082 | Singleton with character o... |
upwordsing 46083 | Singleton is an increasing... |
upwordsseti 46084 | Strictly increasing sequen... |
tworepnotupword 46085 | Concatenation of identical... |
upwrdfi 46086 | There is a finite number o... |
hirstL-ax3 46087 | The third axiom of a syste... |
ax3h 46088 | Recover ~ ax-3 from ~ hirs... |
aibandbiaiffaiffb 46089 | A closed form showing (a i... |
aibandbiaiaiffb 46090 | A closed form showing (a i... |
notatnand 46091 | Do not use. Use intnanr i... |
aistia 46092 | Given a is equivalent to `... |
aisfina 46093 | Given a is equivalent to `... |
bothtbothsame 46094 | Given both a, b are equiva... |
bothfbothsame 46095 | Given both a, b are equiva... |
aiffbbtat 46096 | Given a is equivalent to b... |
aisbbisfaisf 46097 | Given a is equivalent to b... |
axorbtnotaiffb 46098 | Given a is exclusive to b,... |
aiffnbandciffatnotciffb 46099 | Given a is equivalent to (... |
axorbciffatcxorb 46100 | Given a is equivalent to (... |
aibnbna 46101 | Given a implies b, (not b)... |
aibnbaif 46102 | Given a implies b, not b, ... |
aiffbtbat 46103 | Given a is equivalent to b... |
astbstanbst 46104 | Given a is equivalent to T... |
aistbistaandb 46105 | Given a is equivalent to T... |
aisbnaxb 46106 | Given a is equivalent to b... |
atbiffatnnb 46107 | If a implies b, then a imp... |
bisaiaisb 46108 | Application of bicom1 with... |
atbiffatnnbalt 46109 | If a implies b, then a imp... |
abnotbtaxb 46110 | Assuming a, not b, there e... |
abnotataxb 46111 | Assuming not a, b, there e... |
conimpf 46112 | Assuming a, not b, and a i... |
conimpfalt 46113 | Assuming a, not b, and a i... |
aistbisfiaxb 46114 | Given a is equivalent to T... |
aisfbistiaxb 46115 | Given a is equivalent to F... |
aifftbifffaibif 46116 | Given a is equivalent to T... |
aifftbifffaibifff 46117 | Given a is equivalent to T... |
atnaiana 46118 | Given a, it is not the cas... |
ainaiaandna 46119 | Given a, a implies it is n... |
abcdta 46120 | Given (((a and b) and c) a... |
abcdtb 46121 | Given (((a and b) and c) a... |
abcdtc 46122 | Given (((a and b) and c) a... |
abcdtd 46123 | Given (((a and b) and c) a... |
abciffcbatnabciffncba 46124 | Operands in a biconditiona... |
abciffcbatnabciffncbai 46125 | Operands in a biconditiona... |
nabctnabc 46126 | not ( a -> ( b /\ c ) ) we... |
jabtaib 46127 | For when pm3.4 lacks a pm3... |
onenotinotbothi 46128 | From one negated implicati... |
twonotinotbothi 46129 | From these two negated imp... |
clifte 46130 | show d is the same as an i... |
cliftet 46131 | show d is the same as an i... |
clifteta 46132 | show d is the same as an i... |
cliftetb 46133 | show d is the same as an i... |
confun 46134 | Given the hypotheses there... |
confun2 46135 | Confun simplified to two p... |
confun3 46136 | Confun's more complex form... |
confun4 46137 | An attempt at derivative. ... |
confun5 46138 | An attempt at derivative. ... |
plcofph 46139 | Given, a,b and a "definiti... |
pldofph 46140 | Given, a,b c, d, "definiti... |
plvcofph 46141 | Given, a,b,d, and "definit... |
plvcofphax 46142 | Given, a,b,d, and "definit... |
plvofpos 46143 | rh is derivable because ON... |
mdandyv0 46144 | Given the equivalences set... |
mdandyv1 46145 | Given the equivalences set... |
mdandyv2 46146 | Given the equivalences set... |
mdandyv3 46147 | Given the equivalences set... |
mdandyv4 46148 | Given the equivalences set... |
mdandyv5 46149 | Given the equivalences set... |
mdandyv6 46150 | Given the equivalences set... |
mdandyv7 46151 | Given the equivalences set... |
mdandyv8 46152 | Given the equivalences set... |
mdandyv9 46153 | Given the equivalences set... |
mdandyv10 46154 | Given the equivalences set... |
mdandyv11 46155 | Given the equivalences set... |
mdandyv12 46156 | Given the equivalences set... |
mdandyv13 46157 | Given the equivalences set... |
mdandyv14 46158 | Given the equivalences set... |
mdandyv15 46159 | Given the equivalences set... |
mdandyvr0 46160 | Given the equivalences set... |
mdandyvr1 46161 | Given the equivalences set... |
mdandyvr2 46162 | Given the equivalences set... |
mdandyvr3 46163 | Given the equivalences set... |
mdandyvr4 46164 | Given the equivalences set... |
mdandyvr5 46165 | Given the equivalences set... |
mdandyvr6 46166 | Given the equivalences set... |
mdandyvr7 46167 | Given the equivalences set... |
mdandyvr8 46168 | Given the equivalences set... |
mdandyvr9 46169 | Given the equivalences set... |
mdandyvr10 46170 | Given the equivalences set... |
mdandyvr11 46171 | Given the equivalences set... |
mdandyvr12 46172 | Given the equivalences set... |
mdandyvr13 46173 | Given the equivalences set... |
mdandyvr14 46174 | Given the equivalences set... |
mdandyvr15 46175 | Given the equivalences set... |
mdandyvrx0 46176 | Given the exclusivities se... |
mdandyvrx1 46177 | Given the exclusivities se... |
mdandyvrx2 46178 | Given the exclusivities se... |
mdandyvrx3 46179 | Given the exclusivities se... |
mdandyvrx4 46180 | Given the exclusivities se... |
mdandyvrx5 46181 | Given the exclusivities se... |
mdandyvrx6 46182 | Given the exclusivities se... |
mdandyvrx7 46183 | Given the exclusivities se... |
mdandyvrx8 46184 | Given the exclusivities se... |
mdandyvrx9 46185 | Given the exclusivities se... |
mdandyvrx10 46186 | Given the exclusivities se... |
mdandyvrx11 46187 | Given the exclusivities se... |
mdandyvrx12 46188 | Given the exclusivities se... |
mdandyvrx13 46189 | Given the exclusivities se... |
mdandyvrx14 46190 | Given the exclusivities se... |
mdandyvrx15 46191 | Given the exclusivities se... |
H15NH16TH15IH16 46192 | Given 15 hypotheses and a ... |
dandysum2p2e4 46193 | CONTRADICTION PROVED AT 1 ... |
mdandysum2p2e4 46194 | CONTRADICTION PROVED AT 1 ... |
adh-jarrsc 46195 | Replacement of a nested an... |
adh-minim 46196 | A single axiom for minimal... |
adh-minim-ax1-ax2-lem1 46197 | First lemma for the deriva... |
adh-minim-ax1-ax2-lem2 46198 | Second lemma for the deriv... |
adh-minim-ax1-ax2-lem3 46199 | Third lemma for the deriva... |
adh-minim-ax1-ax2-lem4 46200 | Fourth lemma for the deriv... |
adh-minim-ax1 46201 | Derivation of ~ ax-1 from ... |
adh-minim-ax2-lem5 46202 | Fifth lemma for the deriva... |
adh-minim-ax2-lem6 46203 | Sixth lemma for the deriva... |
adh-minim-ax2c 46204 | Derivation of a commuted f... |
adh-minim-ax2 46205 | Derivation of ~ ax-2 from ... |
adh-minim-idALT 46206 | Derivation of ~ id (reflex... |
adh-minim-pm2.43 46207 | Derivation of ~ pm2.43 Whi... |
adh-minimp 46208 | Another single axiom for m... |
adh-minimp-jarr-imim1-ax2c-lem1 46209 | First lemma for the deriva... |
adh-minimp-jarr-lem2 46210 | Second lemma for the deriv... |
adh-minimp-jarr-ax2c-lem3 46211 | Third lemma for the deriva... |
adh-minimp-sylsimp 46212 | Derivation of ~ jarr (also... |
adh-minimp-ax1 46213 | Derivation of ~ ax-1 from ... |
adh-minimp-imim1 46214 | Derivation of ~ imim1 ("le... |
adh-minimp-ax2c 46215 | Derivation of a commuted f... |
adh-minimp-ax2-lem4 46216 | Fourth lemma for the deriv... |
adh-minimp-ax2 46217 | Derivation of ~ ax-2 from ... |
adh-minimp-idALT 46218 | Derivation of ~ id (reflex... |
adh-minimp-pm2.43 46219 | Derivation of ~ pm2.43 Whi... |
n0nsn2el 46220 | If a class with one elemen... |
eusnsn 46221 | There is a unique element ... |
absnsb 46222 | If the class abstraction `... |
euabsneu 46223 | Another way to express exi... |
elprneb 46224 | An element of a proper uno... |
oppr 46225 | Equality for ordered pairs... |
opprb 46226 | Equality for unordered pai... |
or2expropbilem1 46227 | Lemma 1 for ~ or2expropbi ... |
or2expropbilem2 46228 | Lemma 2 for ~ or2expropbi ... |
or2expropbi 46229 | If two classes are strictl... |
eubrv 46230 | If there is a unique set w... |
eubrdm 46231 | If there is a unique set w... |
eldmressn 46232 | Element of the domain of a... |
iota0def 46233 | Example for a defined iota... |
iota0ndef 46234 | Example for an undefined i... |
fveqvfvv 46235 | If a function's value at a... |
fnresfnco 46236 | Composition of two functio... |
funcoressn 46237 | A composition restricted t... |
funressnfv 46238 | A restriction to a singlet... |
funressndmfvrn 46239 | The value of a function ` ... |
funressnvmo 46240 | A function restricted to a... |
funressnmo 46241 | A function restricted to a... |
funressneu 46242 | There is exactly one value... |
fresfo 46243 | Conditions for a restricti... |
fsetsniunop 46244 | The class of all functions... |
fsetabsnop 46245 | The class of all functions... |
fsetsnf 46246 | The mapping of an element ... |
fsetsnf1 46247 | The mapping of an element ... |
fsetsnfo 46248 | The mapping of an element ... |
fsetsnf1o 46249 | The mapping of an element ... |
fsetsnprcnex 46250 | The class of all functions... |
cfsetssfset 46251 | The class of constant func... |
cfsetsnfsetfv 46252 | The function value of the ... |
cfsetsnfsetf 46253 | The mapping of the class o... |
cfsetsnfsetf1 46254 | The mapping of the class o... |
cfsetsnfsetfo 46255 | The mapping of the class o... |
cfsetsnfsetf1o 46256 | The mapping of the class o... |
fsetprcnexALT 46257 | First version of proof for... |
fcoreslem1 46258 | Lemma 1 for ~ fcores . (C... |
fcoreslem2 46259 | Lemma 2 for ~ fcores . (C... |
fcoreslem3 46260 | Lemma 3 for ~ fcores . (C... |
fcoreslem4 46261 | Lemma 4 for ~ fcores . (C... |
fcores 46262 | Every composite function `... |
fcoresf1lem 46263 | Lemma for ~ fcoresf1 . (C... |
fcoresf1 46264 | If a composition is inject... |
fcoresf1b 46265 | A composition is injective... |
fcoresfo 46266 | If a composition is surjec... |
fcoresfob 46267 | A composition is surjectiv... |
fcoresf1ob 46268 | A composition is bijective... |
f1cof1blem 46269 | Lemma for ~ f1cof1b and ~ ... |
f1cof1b 46270 | If the range of ` F ` equa... |
funfocofob 46271 | If the domain of a functio... |
fnfocofob 46272 | If the domain of a functio... |
focofob 46273 | If the domain of a functio... |
f1ocof1ob 46274 | If the range of ` F ` equa... |
f1ocof1ob2 46275 | If the range of ` F ` equa... |
aiotajust 46277 | Soundness justification th... |
dfaiota2 46279 | Alternate definition of th... |
reuabaiotaiota 46280 | The iota and the alternate... |
reuaiotaiota 46281 | The iota and the alternate... |
aiotaexb 46282 | The alternate iota over a ... |
aiotavb 46283 | The alternate iota over a ... |
aiotaint 46284 | This is to ~ df-aiota what... |
dfaiota3 46285 | Alternate definition of ` ... |
iotan0aiotaex 46286 | If the iota over a wff ` p... |
aiotaexaiotaiota 46287 | The alternate iota over a ... |
aiotaval 46288 | Theorem 8.19 in [Quine] p.... |
aiota0def 46289 | Example for a defined alte... |
aiota0ndef 46290 | Example for an undefined a... |
r19.32 46291 | Theorem 19.32 of [Margaris... |
rexsb 46292 | An equivalent expression f... |
rexrsb 46293 | An equivalent expression f... |
2rexsb 46294 | An equivalent expression f... |
2rexrsb 46295 | An equivalent expression f... |
cbvral2 46296 | Change bound variables of ... |
cbvrex2 46297 | Change bound variables of ... |
ralndv1 46298 | Example for a theorem abou... |
ralndv2 46299 | Second example for a theor... |
reuf1odnf 46300 | There is exactly one eleme... |
reuf1od 46301 | There is exactly one eleme... |
euoreqb 46302 | There is a set which is eq... |
2reu3 46303 | Double restricted existent... |
2reu7 46304 | Two equivalent expressions... |
2reu8 46305 | Two equivalent expressions... |
2reu8i 46306 | Implication of a double re... |
2reuimp0 46307 | Implication of a double re... |
2reuimp 46308 | Implication of a double re... |
ralbinrald 46315 | Elemination of a restricte... |
nvelim 46316 | If a class is the universa... |
alneu 46317 | If a statement holds for a... |
eu2ndop1stv 46318 | If there is a unique secon... |
dfateq12d 46319 | Equality deduction for "de... |
nfdfat 46320 | Bound-variable hypothesis ... |
dfdfat2 46321 | Alternate definition of th... |
fundmdfat 46322 | A function is defined at a... |
dfatprc 46323 | A function is not defined ... |
dfatelrn 46324 | The value of a function ` ... |
dfafv2 46325 | Alternative definition of ... |
afveq12d 46326 | Equality deduction for fun... |
afveq1 46327 | Equality theorem for funct... |
afveq2 46328 | Equality theorem for funct... |
nfafv 46329 | Bound-variable hypothesis ... |
csbafv12g 46330 | Move class substitution in... |
afvfundmfveq 46331 | If a class is a function r... |
afvnfundmuv 46332 | If a set is not in the dom... |
ndmafv 46333 | The value of a class outsi... |
afvvdm 46334 | If the function value of a... |
nfunsnafv 46335 | If the restriction of a cl... |
afvvfunressn 46336 | If the function value of a... |
afvprc 46337 | A function's value at a pr... |
afvvv 46338 | If a function's value at a... |
afvpcfv0 46339 | If the value of the altern... |
afvnufveq 46340 | The value of the alternati... |
afvvfveq 46341 | The value of the alternati... |
afv0fv0 46342 | If the value of the altern... |
afvfvn0fveq 46343 | If the function's value at... |
afv0nbfvbi 46344 | The function's value at an... |
afvfv0bi 46345 | The function's value at an... |
afveu 46346 | The value of a function at... |
fnbrafvb 46347 | Equivalence of function va... |
fnopafvb 46348 | Equivalence of function va... |
funbrafvb 46349 | Equivalence of function va... |
funopafvb 46350 | Equivalence of function va... |
funbrafv 46351 | The second argument of a b... |
funbrafv2b 46352 | Function value in terms of... |
dfafn5a 46353 | Representation of a functi... |
dfafn5b 46354 | Representation of a functi... |
fnrnafv 46355 | The range of a function ex... |
afvelrnb 46356 | A member of a function's r... |
afvelrnb0 46357 | A member of a function's r... |
dfaimafn 46358 | Alternate definition of th... |
dfaimafn2 46359 | Alternate definition of th... |
afvelima 46360 | Function value in an image... |
afvelrn 46361 | A function's value belongs... |
fnafvelrn 46362 | A function's value belongs... |
fafvelcdm 46363 | A function's value belongs... |
ffnafv 46364 | A function maps to a class... |
afvres 46365 | The value of a restricted ... |
tz6.12-afv 46366 | Function value. Theorem 6... |
tz6.12-1-afv 46367 | Function value (Theorem 6.... |
dmfcoafv 46368 | Domains of a function comp... |
afvco2 46369 | Value of a function compos... |
rlimdmafv 46370 | Two ways to express that a... |
aoveq123d 46371 | Equality deduction for ope... |
nfaov 46372 | Bound-variable hypothesis ... |
csbaovg 46373 | Move class substitution in... |
aovfundmoveq 46374 | If a class is a function r... |
aovnfundmuv 46375 | If an ordered pair is not ... |
ndmaov 46376 | The value of an operation ... |
ndmaovg 46377 | The value of an operation ... |
aovvdm 46378 | If the operation value of ... |
nfunsnaov 46379 | If the restriction of a cl... |
aovvfunressn 46380 | If the operation value of ... |
aovprc 46381 | The value of an operation ... |
aovrcl 46382 | Reverse closure for an ope... |
aovpcov0 46383 | If the alternative value o... |
aovnuoveq 46384 | The alternative value of t... |
aovvoveq 46385 | The alternative value of t... |
aov0ov0 46386 | If the alternative value o... |
aovovn0oveq 46387 | If the operation's value a... |
aov0nbovbi 46388 | The operation's value on a... |
aovov0bi 46389 | The operation's value on a... |
rspceaov 46390 | A frequently used special ... |
fnotaovb 46391 | Equivalence of operation v... |
ffnaov 46392 | An operation maps to a cla... |
faovcl 46393 | Closure law for an operati... |
aovmpt4g 46394 | Value of a function given ... |
aoprssdm 46395 | Domain of closure of an op... |
ndmaovcl 46396 | The "closure" of an operat... |
ndmaovrcl 46397 | Reverse closure law, in co... |
ndmaovcom 46398 | Any operation is commutati... |
ndmaovass 46399 | Any operation is associati... |
ndmaovdistr 46400 | Any operation is distribut... |
dfatafv2iota 46403 | If a function is defined a... |
ndfatafv2 46404 | The alternate function val... |
ndfatafv2undef 46405 | The alternate function val... |
dfatafv2ex 46406 | The alternate function val... |
afv2ex 46407 | The alternate function val... |
afv2eq12d 46408 | Equality deduction for fun... |
afv2eq1 46409 | Equality theorem for funct... |
afv2eq2 46410 | Equality theorem for funct... |
nfafv2 46411 | Bound-variable hypothesis ... |
csbafv212g 46412 | Move class substitution in... |
fexafv2ex 46413 | The alternate function val... |
ndfatafv2nrn 46414 | The alternate function val... |
ndmafv2nrn 46415 | The value of a class outsi... |
funressndmafv2rn 46416 | The alternate function val... |
afv2ndefb 46417 | Two ways to say that an al... |
nfunsnafv2 46418 | If the restriction of a cl... |
afv2prc 46419 | A function's value at a pr... |
dfatafv2rnb 46420 | The alternate function val... |
afv2orxorb 46421 | If a set is in the range o... |
dmafv2rnb 46422 | The alternate function val... |
fundmafv2rnb 46423 | The alternate function val... |
afv2elrn 46424 | An alternate function valu... |
afv20defat 46425 | If the alternate function ... |
fnafv2elrn 46426 | An alternate function valu... |
fafv2elcdm 46427 | An alternate function valu... |
fafv2elrnb 46428 | An alternate function valu... |
fcdmvafv2v 46429 | If the codomain of a funct... |
tz6.12-2-afv2 46430 | Function value when ` F ` ... |
afv2eu 46431 | The value of a function at... |
afv2res 46432 | The value of a restricted ... |
tz6.12-afv2 46433 | Function value (Theorem 6.... |
tz6.12-1-afv2 46434 | Function value (Theorem 6.... |
tz6.12c-afv2 46435 | Corollary of Theorem 6.12(... |
tz6.12i-afv2 46436 | Corollary of Theorem 6.12(... |
funressnbrafv2 46437 | The second argument of a b... |
dfatbrafv2b 46438 | Equivalence of function va... |
dfatopafv2b 46439 | Equivalence of function va... |
funbrafv2 46440 | The second argument of a b... |
fnbrafv2b 46441 | Equivalence of function va... |
fnopafv2b 46442 | Equivalence of function va... |
funbrafv22b 46443 | Equivalence of function va... |
funopafv2b 46444 | Equivalence of function va... |
dfatsnafv2 46445 | Singleton of function valu... |
dfafv23 46446 | A definition of function v... |
dfatdmfcoafv2 46447 | Domain of a function compo... |
dfatcolem 46448 | Lemma for ~ dfatco . (Con... |
dfatco 46449 | The predicate "defined at"... |
afv2co2 46450 | Value of a function compos... |
rlimdmafv2 46451 | Two ways to express that a... |
dfafv22 46452 | Alternate definition of ` ... |
afv2ndeffv0 46453 | If the alternate function ... |
dfatafv2eqfv 46454 | If a function is defined a... |
afv2rnfveq 46455 | If the alternate function ... |
afv20fv0 46456 | If the alternate function ... |
afv2fvn0fveq 46457 | If the function's value at... |
afv2fv0 46458 | If the function's value at... |
afv2fv0b 46459 | The function's value at an... |
afv2fv0xorb 46460 | If a set is in the range o... |
an4com24 46461 | Rearrangement of 4 conjunc... |
3an4ancom24 46462 | Commutative law for a conj... |
4an21 46463 | Rearrangement of 4 conjunc... |
dfnelbr2 46466 | Alternate definition of th... |
nelbr 46467 | The binary relation of a s... |
nelbrim 46468 | If a set is related to ano... |
nelbrnel 46469 | A set is related to anothe... |
nelbrnelim 46470 | If a set is related to ano... |
ralralimp 46471 | Selecting one of two alter... |
otiunsndisjX 46472 | The union of singletons co... |
fvifeq 46473 | Equality of function value... |
rnfdmpr 46474 | The range of a one-to-one ... |
imarnf1pr 46475 | The image of the range of ... |
funop1 46476 | A function is an ordered p... |
fun2dmnopgexmpl 46477 | A function with a domain c... |
opabresex0d 46478 | A collection of ordered pa... |
opabbrfex0d 46479 | A collection of ordered pa... |
opabresexd 46480 | A collection of ordered pa... |
opabbrfexd 46481 | A collection of ordered pa... |
f1oresf1orab 46482 | Build a bijection by restr... |
f1oresf1o 46483 | Build a bijection by restr... |
f1oresf1o2 46484 | Build a bijection by restr... |
fvmptrab 46485 | Value of a function mappin... |
fvmptrabdm 46486 | Value of a function mappin... |
cnambpcma 46487 | ((a-b)+c)-a = c-a holds fo... |
cnapbmcpd 46488 | ((a+b)-c)+d = ((a+d)+b)-c ... |
addsubeq0 46489 | The sum of two complex num... |
leaddsuble 46490 | Addition and subtraction o... |
2leaddle2 46491 | If two real numbers are le... |
ltnltne 46492 | Variant of trichotomy law ... |
p1lep2 46493 | A real number increasd by ... |
ltsubsubaddltsub 46494 | If the result of subtracti... |
zm1nn 46495 | An integer minus 1 is posi... |
readdcnnred 46496 | The sum of a real number a... |
resubcnnred 46497 | The difference of a real n... |
recnmulnred 46498 | The product of a real numb... |
cndivrenred 46499 | The quotient of an imagina... |
sqrtnegnre 46500 | The square root of a negat... |
nn0resubcl 46501 | Closure law for subtractio... |
zgeltp1eq 46502 | If an integer is between a... |
1t10e1p1e11 46503 | 11 is 1 times 10 to the po... |
deccarry 46504 | Add 1 to a 2 digit number ... |
eluzge0nn0 46505 | If an integer is greater t... |
nltle2tri 46506 | Negated extended trichotom... |
ssfz12 46507 | Subset relationship for fi... |
elfz2z 46508 | Membership of an integer i... |
2elfz3nn0 46509 | If there are two elements ... |
fz0addcom 46510 | The addition of two member... |
2elfz2melfz 46511 | If the sum of two integers... |
fz0addge0 46512 | The sum of two integers in... |
elfzlble 46513 | Membership of an integer i... |
elfzelfzlble 46514 | Membership of an element o... |
fzopred 46515 | Join a predecessor to the ... |
fzopredsuc 46516 | Join a predecessor and a s... |
1fzopredsuc 46517 | Join 0 and a successor to ... |
el1fzopredsuc 46518 | An element of an open inte... |
subsubelfzo0 46519 | Subtracting a difference f... |
fzoopth 46520 | A half-open integer range ... |
2ffzoeq 46521 | Two functions over a half-... |
m1mod0mod1 46522 | An integer decreased by 1 ... |
elmod2 46523 | An integer modulo 2 is eit... |
smonoord 46524 | Ordering relation for a st... |
fsummsndifre 46525 | A finite sum with one of i... |
fsumsplitsndif 46526 | Separate out a term in a f... |
fsummmodsndifre 46527 | A finite sum of summands m... |
fsummmodsnunz 46528 | A finite sum of summands m... |
setsidel 46529 | The injected slot is an el... |
setsnidel 46530 | The injected slot is an el... |
setsv 46531 | The value of the structure... |
preimafvsnel 46532 | The preimage of a function... |
preimafvn0 46533 | The preimage of a function... |
uniimafveqt 46534 | The union of the image of ... |
uniimaprimaeqfv 46535 | The union of the image of ... |
setpreimafvex 46536 | The class ` P ` of all pre... |
elsetpreimafvb 46537 | The characterization of an... |
elsetpreimafv 46538 | An element of the class ` ... |
elsetpreimafvssdm 46539 | An element of the class ` ... |
fvelsetpreimafv 46540 | There is an element in a p... |
preimafvelsetpreimafv 46541 | The preimage of a function... |
preimafvsspwdm 46542 | The class ` P ` of all pre... |
0nelsetpreimafv 46543 | The empty set is not an el... |
elsetpreimafvbi 46544 | An element of the preimage... |
elsetpreimafveqfv 46545 | The elements of the preima... |
eqfvelsetpreimafv 46546 | If an element of the domai... |
elsetpreimafvrab 46547 | An element of the preimage... |
imaelsetpreimafv 46548 | The image of an element of... |
uniimaelsetpreimafv 46549 | The union of the image of ... |
elsetpreimafveq 46550 | If two preimages of functi... |
fundcmpsurinjlem1 46551 | Lemma 1 for ~ fundcmpsurin... |
fundcmpsurinjlem2 46552 | Lemma 2 for ~ fundcmpsurin... |
fundcmpsurinjlem3 46553 | Lemma 3 for ~ fundcmpsurin... |
imasetpreimafvbijlemf 46554 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbijlemfv 46555 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbijlemfv1 46556 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbijlemf1 46557 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbijlemfo 46558 | Lemma for ~ imasetpreimafv... |
imasetpreimafvbij 46559 | The mapping ` H ` is a bij... |
fundcmpsurbijinjpreimafv 46560 | Every function ` F : A -->... |
fundcmpsurinjpreimafv 46561 | Every function ` F : A -->... |
fundcmpsurinj 46562 | Every function ` F : A -->... |
fundcmpsurbijinj 46563 | Every function ` F : A -->... |
fundcmpsurinjimaid 46564 | Every function ` F : A -->... |
fundcmpsurinjALT 46565 | Alternate proof of ~ fundc... |
iccpval 46568 | Partition consisting of a ... |
iccpart 46569 | A special partition. Corr... |
iccpartimp 46570 | Implications for a class b... |
iccpartres 46571 | The restriction of a parti... |
iccpartxr 46572 | If there is a partition, t... |
iccpartgtprec 46573 | If there is a partition, t... |
iccpartipre 46574 | If there is a partition, t... |
iccpartiltu 46575 | If there is a partition, t... |
iccpartigtl 46576 | If there is a partition, t... |
iccpartlt 46577 | If there is a partition, t... |
iccpartltu 46578 | If there is a partition, t... |
iccpartgtl 46579 | If there is a partition, t... |
iccpartgt 46580 | If there is a partition, t... |
iccpartleu 46581 | If there is a partition, t... |
iccpartgel 46582 | If there is a partition, t... |
iccpartrn 46583 | If there is a partition, t... |
iccpartf 46584 | The range of the partition... |
iccpartel 46585 | If there is a partition, t... |
iccelpart 46586 | An element of any partitio... |
iccpartiun 46587 | A half-open interval of ex... |
icceuelpartlem 46588 | Lemma for ~ icceuelpart . ... |
icceuelpart 46589 | An element of a partitione... |
iccpartdisj 46590 | The segments of a partitio... |
iccpartnel 46591 | A point of a partition is ... |
fargshiftfv 46592 | If a class is a function, ... |
fargshiftf 46593 | If a class is a function, ... |
fargshiftf1 46594 | If a function is 1-1, then... |
fargshiftfo 46595 | If a function is onto, the... |
fargshiftfva 46596 | The values of a shifted fu... |
lswn0 46597 | The last symbol of a not e... |
nfich1 46600 | The first interchangeable ... |
nfich2 46601 | The second interchangeable... |
ichv 46602 | Setvar variables are inter... |
ichf 46603 | Setvar variables are inter... |
ichid 46604 | A setvar variable is alway... |
icht 46605 | A theorem is interchangeab... |
ichbidv 46606 | Formula building rule for ... |
ichcircshi 46607 | The setvar variables are i... |
ichan 46608 | If two setvar variables ar... |
ichn 46609 | Negation does not affect i... |
ichim 46610 | Formula building rule for ... |
dfich2 46611 | Alternate definition of th... |
ichcom 46612 | The interchangeability of ... |
ichbi12i 46613 | Equivalence for interchang... |
icheqid 46614 | In an equality for the sam... |
icheq 46615 | In an equality of setvar v... |
ichnfimlem 46616 | Lemma for ~ ichnfim : A s... |
ichnfim 46617 | If in an interchangeabilit... |
ichnfb 46618 | If ` x ` and ` y ` are int... |
ichal 46619 | Move a universal quantifie... |
ich2al 46620 | Two setvar variables are a... |
ich2ex 46621 | Two setvar variables are a... |
ichexmpl1 46622 | Example for interchangeabl... |
ichexmpl2 46623 | Example for interchangeabl... |
ich2exprop 46624 | If the setvar variables ar... |
ichnreuop 46625 | If the setvar variables ar... |
ichreuopeq 46626 | If the setvar variables ar... |
sprid 46627 | Two identical representati... |
elsprel 46628 | An unordered pair is an el... |
spr0nelg 46629 | The empty set is not an el... |
sprval 46632 | The set of all unordered p... |
sprvalpw 46633 | The set of all unordered p... |
sprssspr 46634 | The set of all unordered p... |
spr0el 46635 | The empty set is not an un... |
sprvalpwn0 46636 | The set of all unordered p... |
sprel 46637 | An element of the set of a... |
prssspr 46638 | An element of a subset of ... |
prelspr 46639 | An unordered pair of eleme... |
prsprel 46640 | The elements of a pair fro... |
prsssprel 46641 | The elements of a pair fro... |
sprvalpwle2 46642 | The set of all unordered p... |
sprsymrelfvlem 46643 | Lemma for ~ sprsymrelf and... |
sprsymrelf1lem 46644 | Lemma for ~ sprsymrelf1 . ... |
sprsymrelfolem1 46645 | Lemma 1 for ~ sprsymrelfo ... |
sprsymrelfolem2 46646 | Lemma 2 for ~ sprsymrelfo ... |
sprsymrelfv 46647 | The value of the function ... |
sprsymrelf 46648 | The mapping ` F ` is a fun... |
sprsymrelf1 46649 | The mapping ` F ` is a one... |
sprsymrelfo 46650 | The mapping ` F ` is a fun... |
sprsymrelf1o 46651 | The mapping ` F ` is a bij... |
sprbisymrel 46652 | There is a bijection betwe... |
sprsymrelen 46653 | The class ` P ` of subsets... |
prpair 46654 | Characterization of a prop... |
prproropf1olem0 46655 | Lemma 0 for ~ prproropf1o ... |
prproropf1olem1 46656 | Lemma 1 for ~ prproropf1o ... |
prproropf1olem2 46657 | Lemma 2 for ~ prproropf1o ... |
prproropf1olem3 46658 | Lemma 3 for ~ prproropf1o ... |
prproropf1olem4 46659 | Lemma 4 for ~ prproropf1o ... |
prproropf1o 46660 | There is a bijection betwe... |
prproropen 46661 | The set of proper pairs an... |
prproropreud 46662 | There is exactly one order... |
pairreueq 46663 | Two equivalent representat... |
paireqne 46664 | Two sets are not equal iff... |
prprval 46667 | The set of all proper unor... |
prprvalpw 46668 | The set of all proper unor... |
prprelb 46669 | An element of the set of a... |
prprelprb 46670 | A set is an element of the... |
prprspr2 46671 | The set of all proper unor... |
prprsprreu 46672 | There is a unique proper u... |
prprreueq 46673 | There is a unique proper u... |
sbcpr 46674 | The proper substitution of... |
reupr 46675 | There is a unique unordere... |
reuprpr 46676 | There is a unique proper u... |
poprelb 46677 | Equality for unordered pai... |
2exopprim 46678 | The existence of an ordere... |
reuopreuprim 46679 | There is a unique unordere... |
fmtno 46682 | The ` N ` th Fermat number... |
fmtnoge3 46683 | Each Fermat number is grea... |
fmtnonn 46684 | Each Fermat number is a po... |
fmtnom1nn 46685 | A Fermat number minus one ... |
fmtnoodd 46686 | Each Fermat number is odd.... |
fmtnorn 46687 | A Fermat number is a funct... |
fmtnof1 46688 | The enumeration of the Fer... |
fmtnoinf 46689 | The set of Fermat numbers ... |
fmtnorec1 46690 | The first recurrence relat... |
sqrtpwpw2p 46691 | The floor of the square ro... |
fmtnosqrt 46692 | The floor of the square ro... |
fmtno0 46693 | The ` 0 ` th Fermat number... |
fmtno1 46694 | The ` 1 ` st Fermat number... |
fmtnorec2lem 46695 | Lemma for ~ fmtnorec2 (ind... |
fmtnorec2 46696 | The second recurrence rela... |
fmtnodvds 46697 | Any Fermat number divides ... |
goldbachthlem1 46698 | Lemma 1 for ~ goldbachth .... |
goldbachthlem2 46699 | Lemma 2 for ~ goldbachth .... |
goldbachth 46700 | Goldbach's theorem: Two d... |
fmtnorec3 46701 | The third recurrence relat... |
fmtnorec4 46702 | The fourth recurrence rela... |
fmtno2 46703 | The ` 2 ` nd Fermat number... |
fmtno3 46704 | The ` 3 ` rd Fermat number... |
fmtno4 46705 | The ` 4 ` th Fermat number... |
fmtno5lem1 46706 | Lemma 1 for ~ fmtno5 . (C... |
fmtno5lem2 46707 | Lemma 2 for ~ fmtno5 . (C... |
fmtno5lem3 46708 | Lemma 3 for ~ fmtno5 . (C... |
fmtno5lem4 46709 | Lemma 4 for ~ fmtno5 . (C... |
fmtno5 46710 | The ` 5 ` th Fermat number... |
fmtno0prm 46711 | The ` 0 ` th Fermat number... |
fmtno1prm 46712 | The ` 1 ` st Fermat number... |
fmtno2prm 46713 | The ` 2 ` nd Fermat number... |
257prm 46714 | 257 is a prime number (the... |
fmtno3prm 46715 | The ` 3 ` rd Fermat number... |
odz2prm2pw 46716 | Any power of two is coprim... |
fmtnoprmfac1lem 46717 | Lemma for ~ fmtnoprmfac1 :... |
fmtnoprmfac1 46718 | Divisor of Fermat number (... |
fmtnoprmfac2lem1 46719 | Lemma for ~ fmtnoprmfac2 .... |
fmtnoprmfac2 46720 | Divisor of Fermat number (... |
fmtnofac2lem 46721 | Lemma for ~ fmtnofac2 (Ind... |
fmtnofac2 46722 | Divisor of Fermat number (... |
fmtnofac1 46723 | Divisor of Fermat number (... |
fmtno4sqrt 46724 | The floor of the square ro... |
fmtno4prmfac 46725 | If P was a (prime) factor ... |
fmtno4prmfac193 46726 | If P was a (prime) factor ... |
fmtno4nprmfac193 46727 | 193 is not a (prime) facto... |
fmtno4prm 46728 | The ` 4 `-th Fermat number... |
65537prm 46729 | 65537 is a prime number (t... |
fmtnofz04prm 46730 | The first five Fermat numb... |
fmtnole4prm 46731 | The first five Fermat numb... |
fmtno5faclem1 46732 | Lemma 1 for ~ fmtno5fac . ... |
fmtno5faclem2 46733 | Lemma 2 for ~ fmtno5fac . ... |
fmtno5faclem3 46734 | Lemma 3 for ~ fmtno5fac . ... |
fmtno5fac 46735 | The factorisation of the `... |
fmtno5nprm 46736 | The ` 5 ` th Fermat number... |
prmdvdsfmtnof1lem1 46737 | Lemma 1 for ~ prmdvdsfmtno... |
prmdvdsfmtnof1lem2 46738 | Lemma 2 for ~ prmdvdsfmtno... |
prmdvdsfmtnof 46739 | The mapping of a Fermat nu... |
prmdvdsfmtnof1 46740 | The mapping of a Fermat nu... |
prminf2 46741 | The set of prime numbers i... |
2pwp1prm 46742 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
2pwp1prmfmtno 46743 | Every prime number of the ... |
m2prm 46744 | The second Mersenne number... |
m3prm 46745 | The third Mersenne number ... |
flsqrt 46746 | A condition equivalent to ... |
flsqrt5 46747 | The floor of the square ro... |
3ndvds4 46748 | 3 does not divide 4. (Con... |
139prmALT 46749 | 139 is a prime number. In... |
31prm 46750 | 31 is a prime number. In ... |
m5prm 46751 | The fifth Mersenne number ... |
127prm 46752 | 127 is a prime number. (C... |
m7prm 46753 | The seventh Mersenne numbe... |
m11nprm 46754 | The eleventh Mersenne numb... |
mod42tp1mod8 46755 | If a number is ` 3 ` modul... |
sfprmdvdsmersenne 46756 | If ` Q ` is a safe prime (... |
sgprmdvdsmersenne 46757 | If ` P ` is a Sophie Germa... |
lighneallem1 46758 | Lemma 1 for ~ lighneal . ... |
lighneallem2 46759 | Lemma 2 for ~ lighneal . ... |
lighneallem3 46760 | Lemma 3 for ~ lighneal . ... |
lighneallem4a 46761 | Lemma 1 for ~ lighneallem4... |
lighneallem4b 46762 | Lemma 2 for ~ lighneallem4... |
lighneallem4 46763 | Lemma 3 for ~ lighneal . ... |
lighneal 46764 | If a power of a prime ` P ... |
modexp2m1d 46765 | The square of an integer w... |
proththdlem 46766 | Lemma for ~ proththd . (C... |
proththd 46767 | Proth's theorem (1878). I... |
5tcu2e40 46768 | 5 times the cube of 2 is 4... |
3exp4mod41 46769 | 3 to the fourth power is -... |
41prothprmlem1 46770 | Lemma 1 for ~ 41prothprm .... |
41prothprmlem2 46771 | Lemma 2 for ~ 41prothprm .... |
41prothprm 46772 | 41 is a _Proth prime_. (C... |
quad1 46773 | A condition for a quadrati... |
requad01 46774 | A condition for a quadrati... |
requad1 46775 | A condition for a quadrati... |
requad2 46776 | A condition for a quadrati... |
iseven 46781 | The predicate "is an even ... |
isodd 46782 | The predicate "is an odd n... |
evenz 46783 | An even number is an integ... |
oddz 46784 | An odd number is an intege... |
evendiv2z 46785 | The result of dividing an ... |
oddp1div2z 46786 | The result of dividing an ... |
oddm1div2z 46787 | The result of dividing an ... |
isodd2 46788 | The predicate "is an odd n... |
dfodd2 46789 | Alternate definition for o... |
dfodd6 46790 | Alternate definition for o... |
dfeven4 46791 | Alternate definition for e... |
evenm1odd 46792 | The predecessor of an even... |
evenp1odd 46793 | The successor of an even n... |
oddp1eveni 46794 | The successor of an odd nu... |
oddm1eveni 46795 | The predecessor of an odd ... |
evennodd 46796 | An even number is not an o... |
oddneven 46797 | An odd number is not an ev... |
enege 46798 | The negative of an even nu... |
onego 46799 | The negative of an odd num... |
m1expevenALTV 46800 | Exponentiation of -1 by an... |
m1expoddALTV 46801 | Exponentiation of -1 by an... |
dfeven2 46802 | Alternate definition for e... |
dfodd3 46803 | Alternate definition for o... |
iseven2 46804 | The predicate "is an even ... |
isodd3 46805 | The predicate "is an odd n... |
2dvdseven 46806 | 2 divides an even number. ... |
m2even 46807 | A multiple of 2 is an even... |
2ndvdsodd 46808 | 2 does not divide an odd n... |
2dvdsoddp1 46809 | 2 divides an odd number in... |
2dvdsoddm1 46810 | 2 divides an odd number de... |
dfeven3 46811 | Alternate definition for e... |
dfodd4 46812 | Alternate definition for o... |
dfodd5 46813 | Alternate definition for o... |
zefldiv2ALTV 46814 | The floor of an even numbe... |
zofldiv2ALTV 46815 | The floor of an odd numer ... |
oddflALTV 46816 | Odd number representation ... |
iseven5 46817 | The predicate "is an even ... |
isodd7 46818 | The predicate "is an odd n... |
dfeven5 46819 | Alternate definition for e... |
dfodd7 46820 | Alternate definition for o... |
gcd2odd1 46821 | The greatest common diviso... |
zneoALTV 46822 | No even integer equals an ... |
zeoALTV 46823 | An integer is even or odd.... |
zeo2ALTV 46824 | An integer is even or odd ... |
nneoALTV 46825 | A positive integer is even... |
nneoiALTV 46826 | A positive integer is even... |
odd2np1ALTV 46827 | An integer is odd iff it i... |
oddm1evenALTV 46828 | An integer is odd iff its ... |
oddp1evenALTV 46829 | An integer is odd iff its ... |
oexpnegALTV 46830 | The exponential of the neg... |
oexpnegnz 46831 | The exponential of the neg... |
bits0ALTV 46832 | Value of the zeroth bit. ... |
bits0eALTV 46833 | The zeroth bit of an even ... |
bits0oALTV 46834 | The zeroth bit of an odd n... |
divgcdoddALTV 46835 | Either ` A / ( A gcd B ) `... |
opoeALTV 46836 | The sum of two odds is eve... |
opeoALTV 46837 | The sum of an odd and an e... |
omoeALTV 46838 | The difference of two odds... |
omeoALTV 46839 | The difference of an odd a... |
oddprmALTV 46840 | A prime not equal to ` 2 `... |
0evenALTV 46841 | 0 is an even number. (Con... |
0noddALTV 46842 | 0 is not an odd number. (... |
1oddALTV 46843 | 1 is an odd number. (Cont... |
1nevenALTV 46844 | 1 is not an even number. ... |
2evenALTV 46845 | 2 is an even number. (Con... |
2noddALTV 46846 | 2 is not an odd number. (... |
nn0o1gt2ALTV 46847 | An odd nonnegative integer... |
nnoALTV 46848 | An alternate characterizat... |
nn0oALTV 46849 | An alternate characterizat... |
nn0e 46850 | An alternate characterizat... |
nneven 46851 | An alternate characterizat... |
nn0onn0exALTV 46852 | For each odd nonnegative i... |
nn0enn0exALTV 46853 | For each even nonnegative ... |
nnennexALTV 46854 | For each even positive int... |
nnpw2evenALTV 46855 | 2 to the power of a positi... |
epoo 46856 | The sum of an even and an ... |
emoo 46857 | The difference of an even ... |
epee 46858 | The sum of two even number... |
emee 46859 | The difference of two even... |
evensumeven 46860 | If a summand is even, the ... |
3odd 46861 | 3 is an odd number. (Cont... |
4even 46862 | 4 is an even number. (Con... |
5odd 46863 | 5 is an odd number. (Cont... |
6even 46864 | 6 is an even number. (Con... |
7odd 46865 | 7 is an odd number. (Cont... |
8even 46866 | 8 is an even number. (Con... |
evenprm2 46867 | A prime number is even iff... |
oddprmne2 46868 | Every prime number not bei... |
oddprmuzge3 46869 | A prime number which is od... |
evenltle 46870 | If an even number is great... |
odd2prm2 46871 | If an odd number is the su... |
even3prm2 46872 | If an even number is the s... |
mogoldbblem 46873 | Lemma for ~ mogoldbb . (C... |
perfectALTVlem1 46874 | Lemma for ~ perfectALTV . ... |
perfectALTVlem2 46875 | Lemma for ~ perfectALTV . ... |
perfectALTV 46876 | The Euclid-Euler theorem, ... |
fppr 46879 | The set of Fermat pseudopr... |
fpprmod 46880 | The set of Fermat pseudopr... |
fpprel 46881 | A Fermat pseudoprime to th... |
fpprbasnn 46882 | The base of a Fermat pseud... |
fpprnn 46883 | A Fermat pseudoprime to th... |
fppr2odd 46884 | A Fermat pseudoprime to th... |
11t31e341 46885 | 341 is the product of 11 a... |
2exp340mod341 46886 | Eight to the eighth power ... |
341fppr2 46887 | 341 is the (smallest) _Pou... |
4fppr1 46888 | 4 is the (smallest) Fermat... |
8exp8mod9 46889 | Eight to the eighth power ... |
9fppr8 46890 | 9 is the (smallest) Fermat... |
dfwppr 46891 | Alternate definition of a ... |
fpprwppr 46892 | A Fermat pseudoprime to th... |
fpprwpprb 46893 | An integer ` X ` which is ... |
fpprel2 46894 | An alternate definition fo... |
nfermltl8rev 46895 | Fermat's little theorem wi... |
nfermltl2rev 46896 | Fermat's little theorem wi... |
nfermltlrev 46897 | Fermat's little theorem re... |
isgbe 46904 | The predicate "is an even ... |
isgbow 46905 | The predicate "is a weak o... |
isgbo 46906 | The predicate "is an odd G... |
gbeeven 46907 | An even Goldbach number is... |
gbowodd 46908 | A weak odd Goldbach number... |
gbogbow 46909 | A (strong) odd Goldbach nu... |
gboodd 46910 | An odd Goldbach number is ... |
gbepos 46911 | Any even Goldbach number i... |
gbowpos 46912 | Any weak odd Goldbach numb... |
gbopos 46913 | Any odd Goldbach number is... |
gbegt5 46914 | Any even Goldbach number i... |
gbowgt5 46915 | Any weak odd Goldbach numb... |
gbowge7 46916 | Any weak odd Goldbach numb... |
gboge9 46917 | Any odd Goldbach number is... |
gbege6 46918 | Any even Goldbach number i... |
gbpart6 46919 | The Goldbach partition of ... |
gbpart7 46920 | The (weak) Goldbach partit... |
gbpart8 46921 | The Goldbach partition of ... |
gbpart9 46922 | The (strong) Goldbach part... |
gbpart11 46923 | The (strong) Goldbach part... |
6gbe 46924 | 6 is an even Goldbach numb... |
7gbow 46925 | 7 is a weak odd Goldbach n... |
8gbe 46926 | 8 is an even Goldbach numb... |
9gbo 46927 | 9 is an odd Goldbach numbe... |
11gbo 46928 | 11 is an odd Goldbach numb... |
stgoldbwt 46929 | If the strong ternary Gold... |
sbgoldbwt 46930 | If the strong binary Goldb... |
sbgoldbst 46931 | If the strong binary Goldb... |
sbgoldbaltlem1 46932 | Lemma 1 for ~ sbgoldbalt :... |
sbgoldbaltlem2 46933 | Lemma 2 for ~ sbgoldbalt :... |
sbgoldbalt 46934 | An alternate (related to t... |
sbgoldbb 46935 | If the strong binary Goldb... |
sgoldbeven3prm 46936 | If the binary Goldbach con... |
sbgoldbm 46937 | If the strong binary Goldb... |
mogoldbb 46938 | If the modern version of t... |
sbgoldbmb 46939 | The strong binary Goldbach... |
sbgoldbo 46940 | If the strong binary Goldb... |
nnsum3primes4 46941 | 4 is the sum of at most 3 ... |
nnsum4primes4 46942 | 4 is the sum of at most 4 ... |
nnsum3primesprm 46943 | Every prime is "the sum of... |
nnsum4primesprm 46944 | Every prime is "the sum of... |
nnsum3primesgbe 46945 | Any even Goldbach number i... |
nnsum4primesgbe 46946 | Any even Goldbach number i... |
nnsum3primesle9 46947 | Every integer greater than... |
nnsum4primesle9 46948 | Every integer greater than... |
nnsum4primesodd 46949 | If the (weak) ternary Gold... |
nnsum4primesoddALTV 46950 | If the (strong) ternary Go... |
evengpop3 46951 | If the (weak) ternary Gold... |
evengpoap3 46952 | If the (strong) ternary Go... |
nnsum4primeseven 46953 | If the (weak) ternary Gold... |
nnsum4primesevenALTV 46954 | If the (strong) ternary Go... |
wtgoldbnnsum4prm 46955 | If the (weak) ternary Gold... |
stgoldbnnsum4prm 46956 | If the (strong) ternary Go... |
bgoldbnnsum3prm 46957 | If the binary Goldbach con... |
bgoldbtbndlem1 46958 | Lemma 1 for ~ bgoldbtbnd :... |
bgoldbtbndlem2 46959 | Lemma 2 for ~ bgoldbtbnd .... |
bgoldbtbndlem3 46960 | Lemma 3 for ~ bgoldbtbnd .... |
bgoldbtbndlem4 46961 | Lemma 4 for ~ bgoldbtbnd .... |
bgoldbtbnd 46962 | If the binary Goldbach con... |
tgoldbachgtALTV 46965 | Variant of Thierry Arnoux'... |
bgoldbachlt 46966 | The binary Goldbach conjec... |
tgblthelfgott 46968 | The ternary Goldbach conje... |
tgoldbachlt 46969 | The ternary Goldbach conje... |
tgoldbach 46970 | The ternary Goldbach conje... |
isomgrrel 46975 | The isomorphy relation for... |
isomgr 46976 | The isomorphy relation for... |
isisomgr 46977 | Implications of two graphs... |
isomgreqve 46978 | A set is isomorphic to a h... |
isomushgr 46979 | The isomorphy relation for... |
isomuspgrlem1 46980 | Lemma 1 for ~ isomuspgr . ... |
isomuspgrlem2a 46981 | Lemma 1 for ~ isomuspgrlem... |
isomuspgrlem2b 46982 | Lemma 2 for ~ isomuspgrlem... |
isomuspgrlem2c 46983 | Lemma 3 for ~ isomuspgrlem... |
isomuspgrlem2d 46984 | Lemma 4 for ~ isomuspgrlem... |
isomuspgrlem2e 46985 | Lemma 5 for ~ isomuspgrlem... |
isomuspgrlem2 46986 | Lemma 2 for ~ isomuspgr . ... |
isomuspgr 46987 | The isomorphy relation for... |
isomgrref 46988 | The isomorphy relation is ... |
isomgrsym 46989 | The isomorphy relation is ... |
isomgrsymb 46990 | The isomorphy relation is ... |
isomgrtrlem 46991 | Lemma for ~ isomgrtr . (C... |
isomgrtr 46992 | The isomorphy relation is ... |
strisomgrop 46993 | A graph represented as an ... |
ushrisomgr 46994 | A simple hypergraph (with ... |
1hegrlfgr 46995 | A graph ` G ` with one hyp... |
upwlksfval 46998 | The set of simple walks (i... |
isupwlk 46999 | Properties of a pair of fu... |
isupwlkg 47000 | Generalization of ~ isupwl... |
upwlkbprop 47001 | Basic properties of a simp... |
upwlkwlk 47002 | A simple walk is a walk. ... |
upgrwlkupwlk 47003 | In a pseudograph, a walk i... |
upgrwlkupwlkb 47004 | In a pseudograph, the defi... |
upgrisupwlkALT 47005 | Alternate proof of ~ upgri... |
upgredgssspr 47006 | The set of edges of a pseu... |
uspgropssxp 47007 | The set ` G ` of "simple p... |
uspgrsprfv 47008 | The value of the function ... |
uspgrsprf 47009 | The mapping ` F ` is a fun... |
uspgrsprf1 47010 | The mapping ` F ` is a one... |
uspgrsprfo 47011 | The mapping ` F ` is a fun... |
uspgrsprf1o 47012 | The mapping ` F ` is a bij... |
uspgrex 47013 | The class ` G ` of all "si... |
uspgrbispr 47014 | There is a bijection betwe... |
uspgrspren 47015 | The set ` G ` of the "simp... |
uspgrymrelen 47016 | The set ` G ` of the "simp... |
uspgrbisymrel 47017 | There is a bijection betwe... |
uspgrbisymrelALT 47018 | Alternate proof of ~ uspgr... |
ovn0dmfun 47019 | If a class operation value... |
xpsnopab 47020 | A Cartesian product with a... |
xpiun 47021 | A Cartesian product expres... |
ovn0ssdmfun 47022 | If a class' operation valu... |
fnxpdmdm 47023 | The domain of the domain o... |
cnfldsrngbas 47024 | The base set of a subring ... |
cnfldsrngadd 47025 | The group addition operati... |
cnfldsrngmul 47026 | The ring multiplication op... |
plusfreseq 47027 | If the empty set is not co... |
mgmplusfreseq 47028 | If the empty set is not co... |
0mgm 47029 | A set with an empty base s... |
opmpoismgm 47030 | A structure with a group a... |
copissgrp 47031 | A structure with a constan... |
copisnmnd 47032 | A structure with a constan... |
0nodd 47033 | 0 is not an odd integer. ... |
1odd 47034 | 1 is an odd integer. (Con... |
2nodd 47035 | 2 is not an odd integer. ... |
oddibas 47036 | Lemma 1 for ~ oddinmgm : ... |
oddiadd 47037 | Lemma 2 for ~ oddinmgm : ... |
oddinmgm 47038 | The structure of all odd i... |
nnsgrpmgm 47039 | The structure of positive ... |
nnsgrp 47040 | The structure of positive ... |
nnsgrpnmnd 47041 | The structure of positive ... |
nn0mnd 47042 | The set of nonnegative int... |
gsumsplit2f 47043 | Split a group sum into two... |
gsumdifsndf 47044 | Extract a summand from a f... |
gsumfsupp 47045 | A group sum of a family ca... |
iscllaw 47052 | The predicate "is a closed... |
iscomlaw 47053 | The predicate "is a commut... |
clcllaw 47054 | Closure of a closed operat... |
isasslaw 47055 | The predicate "is an assoc... |
asslawass 47056 | Associativity of an associ... |
mgmplusgiopALT 47057 | Slot 2 (group operation) o... |
sgrpplusgaopALT 47058 | Slot 2 (group operation) o... |
intopval 47065 | The internal (binary) oper... |
intop 47066 | An internal (binary) opera... |
clintopval 47067 | The closed (internal binar... |
assintopval 47068 | The associative (closed in... |
assintopmap 47069 | The associative (closed in... |
isclintop 47070 | The predicate "is a closed... |
clintop 47071 | A closed (internal binary)... |
assintop 47072 | An associative (closed int... |
isassintop 47073 | The predicate "is an assoc... |
clintopcllaw 47074 | The closure law holds for ... |
assintopcllaw 47075 | The closure low holds for ... |
assintopasslaw 47076 | The associative low holds ... |
assintopass 47077 | An associative (closed int... |
ismgmALT 47086 | The predicate "is a magma"... |
iscmgmALT 47087 | The predicate "is a commut... |
issgrpALT 47088 | The predicate "is a semigr... |
iscsgrpALT 47089 | The predicate "is a commut... |
mgm2mgm 47090 | Equivalence of the two def... |
sgrp2sgrp 47091 | Equivalence of the two def... |
lmod0rng 47092 | If the scalar ring of a mo... |
nzrneg1ne0 47093 | The additive inverse of th... |
lidldomn1 47094 | If a (left) ideal (which i... |
lidlabl 47095 | A (left) ideal of a ring i... |
lidlrng 47096 | A (left) ideal of a ring i... |
zlidlring 47097 | The zero (left) ideal of a... |
uzlidlring 47098 | Only the zero (left) ideal... |
lidldomnnring 47099 | A (left) ideal of a domain... |
0even 47100 | 0 is an even integer. (Co... |
1neven 47101 | 1 is not an even integer. ... |
2even 47102 | 2 is an even integer. (Co... |
2zlidl 47103 | The even integers are a (l... |
2zrng 47104 | The ring of integers restr... |
2zrngbas 47105 | The base set of R is the s... |
2zrngadd 47106 | The group addition operati... |
2zrng0 47107 | The additive identity of R... |
2zrngamgm 47108 | R is an (additive) magma. ... |
2zrngasgrp 47109 | R is an (additive) semigro... |
2zrngamnd 47110 | R is an (additive) monoid.... |
2zrngacmnd 47111 | R is a commutative (additi... |
2zrngagrp 47112 | R is an (additive) group. ... |
2zrngaabl 47113 | R is an (additive) abelian... |
2zrngmul 47114 | The ring multiplication op... |
2zrngmmgm 47115 | R is a (multiplicative) ma... |
2zrngmsgrp 47116 | R is a (multiplicative) se... |
2zrngALT 47117 | The ring of integers restr... |
2zrngnmlid 47118 | R has no multiplicative (l... |
2zrngnmrid 47119 | R has no multiplicative (r... |
2zrngnmlid2 47120 | R has no multiplicative (l... |
2zrngnring 47121 | R is not a unital ring. (... |
cznrnglem 47122 | Lemma for ~ cznrng : The ... |
cznabel 47123 | The ring constructed from ... |
cznrng 47124 | The ring constructed from ... |
cznnring 47125 | The ring constructed from ... |
rngcvalALTV 47128 | Value of the category of n... |
rngcbasALTV 47129 | Set of objects of the cate... |
rngchomfvalALTV 47130 | Set of arrows of the categ... |
rngchomALTV 47131 | Set of arrows of the categ... |
elrngchomALTV 47132 | A morphism of non-unital r... |
rngccofvalALTV 47133 | Composition in the categor... |
rngccoALTV 47134 | Composition in the categor... |
rngccatidALTV 47135 | Lemma for ~ rngccatALTV . ... |
rngccatALTV 47136 | The category of non-unital... |
rngcidALTV 47137 | The identity arrow in the ... |
rngcsectALTV 47138 | A section in the category ... |
rngcinvALTV 47139 | An inverse in the category... |
rngcisoALTV 47140 | An isomorphism in the cate... |
rngchomffvalALTV 47141 | The value of the functiona... |
rngchomrnghmresALTV 47142 | The value of the functiona... |
rngcrescrhmALTV 47143 | The category of non-unital... |
rhmsubcALTVlem1 47144 | Lemma 1 for ~ rhmsubcALTV ... |
rhmsubcALTVlem2 47145 | Lemma 2 for ~ rhmsubcALTV ... |
rhmsubcALTVlem3 47146 | Lemma 3 for ~ rhmsubcALTV ... |
rhmsubcALTVlem4 47147 | Lemma 4 for ~ rhmsubcALTV ... |
rhmsubcALTV 47148 | According to ~ df-subc , t... |
rhmsubcALTVcat 47149 | The restriction of the cat... |
ringcvalALTV 47152 | Value of the category of r... |
funcringcsetcALTV2lem1 47153 | Lemma 1 for ~ funcringcset... |
funcringcsetcALTV2lem2 47154 | Lemma 2 for ~ funcringcset... |
funcringcsetcALTV2lem3 47155 | Lemma 3 for ~ funcringcset... |
funcringcsetcALTV2lem4 47156 | Lemma 4 for ~ funcringcset... |
funcringcsetcALTV2lem5 47157 | Lemma 5 for ~ funcringcset... |
funcringcsetcALTV2lem6 47158 | Lemma 6 for ~ funcringcset... |
funcringcsetcALTV2lem7 47159 | Lemma 7 for ~ funcringcset... |
funcringcsetcALTV2lem8 47160 | Lemma 8 for ~ funcringcset... |
funcringcsetcALTV2lem9 47161 | Lemma 9 for ~ funcringcset... |
funcringcsetcALTV2 47162 | The "natural forgetful fun... |
ringcbasALTV 47163 | Set of objects of the cate... |
ringchomfvalALTV 47164 | Set of arrows of the categ... |
ringchomALTV 47165 | Set of arrows of the categ... |
elringchomALTV 47166 | A morphism of rings is a f... |
ringccofvalALTV 47167 | Composition in the categor... |
ringccoALTV 47168 | Composition in the categor... |
ringccatidALTV 47169 | Lemma for ~ ringccatALTV .... |
ringccatALTV 47170 | The category of rings is a... |
ringcidALTV 47171 | The identity arrow in the ... |
ringcsectALTV 47172 | A section in the category ... |
ringcinvALTV 47173 | An inverse in the category... |
ringcisoALTV 47174 | An isomorphism in the cate... |
ringcbasbasALTV 47175 | An element of the base set... |
funcringcsetclem1ALTV 47176 | Lemma 1 for ~ funcringcset... |
funcringcsetclem2ALTV 47177 | Lemma 2 for ~ funcringcset... |
funcringcsetclem3ALTV 47178 | Lemma 3 for ~ funcringcset... |
funcringcsetclem4ALTV 47179 | Lemma 4 for ~ funcringcset... |
funcringcsetclem5ALTV 47180 | Lemma 5 for ~ funcringcset... |
funcringcsetclem6ALTV 47181 | Lemma 6 for ~ funcringcset... |
funcringcsetclem7ALTV 47182 | Lemma 7 for ~ funcringcset... |
funcringcsetclem8ALTV 47183 | Lemma 8 for ~ funcringcset... |
funcringcsetclem9ALTV 47184 | Lemma 9 for ~ funcringcset... |
funcringcsetcALTV 47185 | The "natural forgetful fun... |
srhmsubcALTVlem1 47186 | Lemma 1 for ~ srhmsubcALTV... |
srhmsubcALTVlem2 47187 | Lemma 2 for ~ srhmsubcALTV... |
srhmsubcALTV 47188 | According to ~ df-subc , t... |
sringcatALTV 47189 | The restriction of the cat... |
crhmsubcALTV 47190 | According to ~ df-subc , t... |
cringcatALTV 47191 | The restriction of the cat... |
drhmsubcALTV 47192 | According to ~ df-subc , t... |
drngcatALTV 47193 | The restriction of the cat... |
fldcatALTV 47194 | The restriction of the cat... |
fldcALTV 47195 | The restriction of the cat... |
fldhmsubcALTV 47196 | According to ~ df-subc , t... |
opeliun2xp 47197 | Membership of an ordered p... |
eliunxp2 47198 | Membership in a union of C... |
mpomptx2 47199 | Express a two-argument fun... |
cbvmpox2 47200 | Rule to change the bound v... |
dmmpossx2 47201 | The domain of a mapping is... |
mpoexxg2 47202 | Existence of an operation ... |
ovmpordxf 47203 | Value of an operation give... |
ovmpordx 47204 | Value of an operation give... |
ovmpox2 47205 | The value of an operation ... |
fdmdifeqresdif 47206 | The restriction of a condi... |
offvalfv 47207 | The function operation exp... |
ofaddmndmap 47208 | The function operation app... |
mapsnop 47209 | A singleton of an ordered ... |
fprmappr 47210 | A function with a domain o... |
mapprop 47211 | An unordered pair containi... |
ztprmneprm 47212 | A prime is not an integer ... |
2t6m3t4e0 47213 | 2 times 6 minus 3 times 4 ... |
ssnn0ssfz 47214 | For any finite subset of `... |
nn0sumltlt 47215 | If the sum of two nonnegat... |
bcpascm1 47216 | Pascal's rule for the bino... |
altgsumbc 47217 | The sum of binomial coeffi... |
altgsumbcALT 47218 | Alternate proof of ~ altgs... |
zlmodzxzlmod 47219 | The ` ZZ `-module ` ZZ X. ... |
zlmodzxzel 47220 | An element of the (base se... |
zlmodzxz0 47221 | The ` 0 ` of the ` ZZ `-mo... |
zlmodzxzscm 47222 | The scalar multiplication ... |
zlmodzxzadd 47223 | The addition of the ` ZZ `... |
zlmodzxzsubm 47224 | The subtraction of the ` Z... |
zlmodzxzsub 47225 | The subtraction of the ` Z... |
mgpsumunsn 47226 | Extract a summand/factor f... |
mgpsumz 47227 | If the group sum for the m... |
mgpsumn 47228 | If the group sum for the m... |
exple2lt6 47229 | A nonnegative integer to t... |
pgrple2abl 47230 | Every symmetric group on a... |
pgrpgt2nabl 47231 | Every symmetric group on a... |
invginvrid 47232 | Identity for a multiplicat... |
rmsupp0 47233 | The support of a mapping o... |
domnmsuppn0 47234 | The support of a mapping o... |
rmsuppss 47235 | The support of a mapping o... |
mndpsuppss 47236 | The support of a mapping o... |
scmsuppss 47237 | The support of a mapping o... |
rmsuppfi 47238 | The support of a mapping o... |
rmfsupp 47239 | A mapping of a multiplicat... |
mndpsuppfi 47240 | The support of a mapping o... |
mndpfsupp 47241 | A mapping of a scalar mult... |
scmsuppfi 47242 | The support of a mapping o... |
scmfsupp 47243 | A mapping of a scalar mult... |
suppmptcfin 47244 | The support of a mapping w... |
mptcfsupp 47245 | A mapping with value 0 exc... |
fsuppmptdmf 47246 | A mapping with a finite do... |
lmodvsmdi 47247 | Multiple distributive law ... |
gsumlsscl 47248 | Closure of a group sum in ... |
assaascl0 47249 | The scalar 0 embedded into... |
assaascl1 47250 | The scalar 1 embedded into... |
ply1vr1smo 47251 | The variable in a polynomi... |
ply1sclrmsm 47252 | The ring multiplication of... |
coe1id 47253 | Coefficient vector of the ... |
coe1sclmulval 47254 | The value of the coefficie... |
ply1mulgsumlem1 47255 | Lemma 1 for ~ ply1mulgsum ... |
ply1mulgsumlem2 47256 | Lemma 2 for ~ ply1mulgsum ... |
ply1mulgsumlem3 47257 | Lemma 3 for ~ ply1mulgsum ... |
ply1mulgsumlem4 47258 | Lemma 4 for ~ ply1mulgsum ... |
ply1mulgsum 47259 | The product of two polynom... |
evl1at0 47260 | Polynomial evaluation for ... |
evl1at1 47261 | Polynomial evaluation for ... |
linply1 47262 | A term of the form ` x - C... |
lineval 47263 | A term of the form ` x - C... |
linevalexample 47264 | The polynomial ` x - 3 ` o... |
dmatALTval 47269 | The algebra of ` N ` x ` N... |
dmatALTbas 47270 | The base set of the algebr... |
dmatALTbasel 47271 | An element of the base set... |
dmatbas 47272 | The set of all ` N ` x ` N... |
lincop 47277 | A linear combination as op... |
lincval 47278 | The value of a linear comb... |
dflinc2 47279 | Alternative definition of ... |
lcoop 47280 | A linear combination as op... |
lcoval 47281 | The value of a linear comb... |
lincfsuppcl 47282 | A linear combination of ve... |
linccl 47283 | A linear combination of ve... |
lincval0 47284 | The value of an empty line... |
lincvalsng 47285 | The linear combination ove... |
lincvalsn 47286 | The linear combination ove... |
lincvalpr 47287 | The linear combination ove... |
lincval1 47288 | The linear combination ove... |
lcosn0 47289 | Properties of a linear com... |
lincvalsc0 47290 | The linear combination whe... |
lcoc0 47291 | Properties of a linear com... |
linc0scn0 47292 | If a set contains the zero... |
lincdifsn 47293 | A vector is a linear combi... |
linc1 47294 | A vector is a linear combi... |
lincellss 47295 | A linear combination of a ... |
lco0 47296 | The set of empty linear co... |
lcoel0 47297 | The zero vector is always ... |
lincsum 47298 | The sum of two linear comb... |
lincscm 47299 | A linear combinations mult... |
lincsumcl 47300 | The sum of two linear comb... |
lincscmcl 47301 | The multiplication of a li... |
lincsumscmcl 47302 | The sum of a linear combin... |
lincolss 47303 | According to the statement... |
ellcoellss 47304 | Every linear combination o... |
lcoss 47305 | A set of vectors of a modu... |
lspsslco 47306 | Lemma for ~ lspeqlco . (C... |
lcosslsp 47307 | Lemma for ~ lspeqlco . (C... |
lspeqlco 47308 | Equivalence of a _span_ of... |
rellininds 47312 | The class defining the rel... |
linindsv 47314 | The classes of the module ... |
islininds 47315 | The property of being a li... |
linindsi 47316 | The implications of being ... |
linindslinci 47317 | The implications of being ... |
islinindfis 47318 | The property of being a li... |
islinindfiss 47319 | The property of being a li... |
linindscl 47320 | A linearly independent set... |
lindepsnlininds 47321 | A linearly dependent subse... |
islindeps 47322 | The property of being a li... |
lincext1 47323 | Property 1 of an extension... |
lincext2 47324 | Property 2 of an extension... |
lincext3 47325 | Property 3 of an extension... |
lindslinindsimp1 47326 | Implication 1 for ~ lindsl... |
lindslinindimp2lem1 47327 | Lemma 1 for ~ lindslininds... |
lindslinindimp2lem2 47328 | Lemma 2 for ~ lindslininds... |
lindslinindimp2lem3 47329 | Lemma 3 for ~ lindslininds... |
lindslinindimp2lem4 47330 | Lemma 4 for ~ lindslininds... |
lindslinindsimp2lem5 47331 | Lemma 5 for ~ lindslininds... |
lindslinindsimp2 47332 | Implication 2 for ~ lindsl... |
lindslininds 47333 | Equivalence of definitions... |
linds0 47334 | The empty set is always a ... |
el0ldep 47335 | A set containing the zero ... |
el0ldepsnzr 47336 | A set containing the zero ... |
lindsrng01 47337 | Any subset of a module is ... |
lindszr 47338 | Any subset of a module ove... |
snlindsntorlem 47339 | Lemma for ~ snlindsntor . ... |
snlindsntor 47340 | A singleton is linearly in... |
ldepsprlem 47341 | Lemma for ~ ldepspr . (Co... |
ldepspr 47342 | If a vector is a scalar mu... |
lincresunit3lem3 47343 | Lemma 3 for ~ lincresunit3... |
lincresunitlem1 47344 | Lemma 1 for properties of ... |
lincresunitlem2 47345 | Lemma for properties of a ... |
lincresunit1 47346 | Property 1 of a specially ... |
lincresunit2 47347 | Property 2 of a specially ... |
lincresunit3lem1 47348 | Lemma 1 for ~ lincresunit3... |
lincresunit3lem2 47349 | Lemma 2 for ~ lincresunit3... |
lincresunit3 47350 | Property 3 of a specially ... |
lincreslvec3 47351 | Property 3 of a specially ... |
islindeps2 47352 | Conditions for being a lin... |
islininds2 47353 | Implication of being a lin... |
isldepslvec2 47354 | Alternative definition of ... |
lindssnlvec 47355 | A singleton not containing... |
lmod1lem1 47356 | Lemma 1 for ~ lmod1 . (Co... |
lmod1lem2 47357 | Lemma 2 for ~ lmod1 . (Co... |
lmod1lem3 47358 | Lemma 3 for ~ lmod1 . (Co... |
lmod1lem4 47359 | Lemma 4 for ~ lmod1 . (Co... |
lmod1lem5 47360 | Lemma 5 for ~ lmod1 . (Co... |
lmod1 47361 | The (smallest) structure r... |
lmod1zr 47362 | The (smallest) structure r... |
lmod1zrnlvec 47363 | There is a (left) module (... |
lmodn0 47364 | Left modules exist. (Cont... |
zlmodzxzequa 47365 | Example of an equation wit... |
zlmodzxznm 47366 | Example of a linearly depe... |
zlmodzxzldeplem 47367 | A and B are not equal. (C... |
zlmodzxzequap 47368 | Example of an equation wit... |
zlmodzxzldeplem1 47369 | Lemma 1 for ~ zlmodzxzldep... |
zlmodzxzldeplem2 47370 | Lemma 2 for ~ zlmodzxzldep... |
zlmodzxzldeplem3 47371 | Lemma 3 for ~ zlmodzxzldep... |
zlmodzxzldeplem4 47372 | Lemma 4 for ~ zlmodzxzldep... |
zlmodzxzldep 47373 | { A , B } is a linearly de... |
ldepsnlinclem1 47374 | Lemma 1 for ~ ldepsnlinc .... |
ldepsnlinclem2 47375 | Lemma 2 for ~ ldepsnlinc .... |
lvecpsslmod 47376 | The class of all (left) ve... |
ldepsnlinc 47377 | The reverse implication of... |
ldepslinc 47378 | For (left) vector spaces, ... |
suppdm 47379 | If the range of a function... |
eluz2cnn0n1 47380 | An integer greater than 1 ... |
divge1b 47381 | The ratio of a real number... |
divgt1b 47382 | The ratio of a real number... |
ltsubaddb 47383 | Equivalence for the "less ... |
ltsubsubb 47384 | Equivalence for the "less ... |
ltsubadd2b 47385 | Equivalence for the "less ... |
divsub1dir 47386 | Distribution of division o... |
expnegico01 47387 | An integer greater than 1 ... |
elfzolborelfzop1 47388 | An element of a half-open ... |
pw2m1lepw2m1 47389 | 2 to the power of a positi... |
zgtp1leeq 47390 | If an integer is between a... |
flsubz 47391 | An integer can be moved in... |
fldivmod 47392 | Expressing the floor of a ... |
mod0mul 47393 | If an integer is 0 modulo ... |
modn0mul 47394 | If an integer is not 0 mod... |
m1modmmod 47395 | An integer decreased by 1 ... |
difmodm1lt 47396 | The difference between an ... |
nn0onn0ex 47397 | For each odd nonnegative i... |
nn0enn0ex 47398 | For each even nonnegative ... |
nnennex 47399 | For each even positive int... |
nneop 47400 | A positive integer is even... |
nneom 47401 | A positive integer is even... |
nn0eo 47402 | A nonnegative integer is e... |
nnpw2even 47403 | 2 to the power of a positi... |
zefldiv2 47404 | The floor of an even integ... |
zofldiv2 47405 | The floor of an odd intege... |
nn0ofldiv2 47406 | The floor of an odd nonneg... |
flnn0div2ge 47407 | The floor of a positive in... |
flnn0ohalf 47408 | The floor of the half of a... |
logcxp0 47409 | Logarithm of a complex pow... |
regt1loggt0 47410 | The natural logarithm for ... |
fdivval 47413 | The quotient of two functi... |
fdivmpt 47414 | The quotient of two functi... |
fdivmptf 47415 | The quotient of two functi... |
refdivmptf 47416 | The quotient of two functi... |
fdivpm 47417 | The quotient of two functi... |
refdivpm 47418 | The quotient of two functi... |
fdivmptfv 47419 | The function value of a qu... |
refdivmptfv 47420 | The function value of a qu... |
bigoval 47423 | Set of functions of order ... |
elbigofrcl 47424 | Reverse closure of the "bi... |
elbigo 47425 | Properties of a function o... |
elbigo2 47426 | Properties of a function o... |
elbigo2r 47427 | Sufficient condition for a... |
elbigof 47428 | A function of order G(x) i... |
elbigodm 47429 | The domain of a function o... |
elbigoimp 47430 | The defining property of a... |
elbigolo1 47431 | A function (into the posit... |
rege1logbrege0 47432 | The general logarithm, wit... |
rege1logbzge0 47433 | The general logarithm, wit... |
fllogbd 47434 | A real number is between t... |
relogbmulbexp 47435 | The logarithm of the produ... |
relogbdivb 47436 | The logarithm of the quoti... |
logbge0b 47437 | The logarithm of a number ... |
logblt1b 47438 | The logarithm of a number ... |
fldivexpfllog2 47439 | The floor of a positive re... |
nnlog2ge0lt1 47440 | A positive integer is 1 if... |
logbpw2m1 47441 | The floor of the binary lo... |
fllog2 47442 | The floor of the binary lo... |
blenval 47445 | The binary length of an in... |
blen0 47446 | The binary length of 0. (... |
blenn0 47447 | The binary length of a "nu... |
blenre 47448 | The binary length of a pos... |
blennn 47449 | The binary length of a pos... |
blennnelnn 47450 | The binary length of a pos... |
blennn0elnn 47451 | The binary length of a non... |
blenpw2 47452 | The binary length of a pow... |
blenpw2m1 47453 | The binary length of a pow... |
nnpw2blen 47454 | A positive integer is betw... |
nnpw2blenfzo 47455 | A positive integer is betw... |
nnpw2blenfzo2 47456 | A positive integer is eith... |
nnpw2pmod 47457 | Every positive integer can... |
blen1 47458 | The binary length of 1. (... |
blen2 47459 | The binary length of 2. (... |
nnpw2p 47460 | Every positive integer can... |
nnpw2pb 47461 | A number is a positive int... |
blen1b 47462 | The binary length of a non... |
blennnt2 47463 | The binary length of a pos... |
nnolog2flm1 47464 | The floor of the binary lo... |
blennn0em1 47465 | The binary length of the h... |
blennngt2o2 47466 | The binary length of an od... |
blengt1fldiv2p1 47467 | The binary length of an in... |
blennn0e2 47468 | The binary length of an ev... |
digfval 47471 | Operation to obtain the ` ... |
digval 47472 | The ` K ` th digit of a no... |
digvalnn0 47473 | The ` K ` th digit of a no... |
nn0digval 47474 | The ` K ` th digit of a no... |
dignn0fr 47475 | The digits of the fraction... |
dignn0ldlem 47476 | Lemma for ~ dignnld . (Co... |
dignnld 47477 | The leading digits of a po... |
dig2nn0ld 47478 | The leading digits of a po... |
dig2nn1st 47479 | The first (relevant) digit... |
dig0 47480 | All digits of 0 are 0. (C... |
digexp 47481 | The ` K ` th digit of a po... |
dig1 47482 | All but one digits of 1 ar... |
0dig1 47483 | The ` 0 ` th digit of 1 is... |
0dig2pr01 47484 | The integers 0 and 1 corre... |
dig2nn0 47485 | A digit of a nonnegative i... |
0dig2nn0e 47486 | The last bit of an even in... |
0dig2nn0o 47487 | The last bit of an odd int... |
dig2bits 47488 | The ` K ` th digit of a no... |
dignn0flhalflem1 47489 | Lemma 1 for ~ dignn0flhalf... |
dignn0flhalflem2 47490 | Lemma 2 for ~ dignn0flhalf... |
dignn0ehalf 47491 | The digits of the half of ... |
dignn0flhalf 47492 | The digits of the rounded ... |
nn0sumshdiglemA 47493 | Lemma for ~ nn0sumshdig (i... |
nn0sumshdiglemB 47494 | Lemma for ~ nn0sumshdig (i... |
nn0sumshdiglem1 47495 | Lemma 1 for ~ nn0sumshdig ... |
nn0sumshdiglem2 47496 | Lemma 2 for ~ nn0sumshdig ... |
nn0sumshdig 47497 | A nonnegative integer can ... |
nn0mulfsum 47498 | Trivial algorithm to calcu... |
nn0mullong 47499 | Standard algorithm (also k... |
naryfval 47502 | The set of the n-ary (endo... |
naryfvalixp 47503 | The set of the n-ary (endo... |
naryfvalel 47504 | An n-ary (endo)function on... |
naryrcl 47505 | Reverse closure for n-ary ... |
naryfvalelfv 47506 | The value of an n-ary (end... |
naryfvalelwrdf 47507 | An n-ary (endo)function on... |
0aryfvalel 47508 | A nullary (endo)function o... |
0aryfvalelfv 47509 | The value of a nullary (en... |
1aryfvalel 47510 | A unary (endo)function on ... |
fv1arycl 47511 | Closure of a unary (endo)f... |
1arympt1 47512 | A unary (endo)function in ... |
1arympt1fv 47513 | The value of a unary (endo... |
1arymaptfv 47514 | The value of the mapping o... |
1arymaptf 47515 | The mapping of unary (endo... |
1arymaptf1 47516 | The mapping of unary (endo... |
1arymaptfo 47517 | The mapping of unary (endo... |
1arymaptf1o 47518 | The mapping of unary (endo... |
1aryenef 47519 | The set of unary (endo)fun... |
1aryenefmnd 47520 | The set of unary (endo)fun... |
2aryfvalel 47521 | A binary (endo)function on... |
fv2arycl 47522 | Closure of a binary (endo)... |
2arympt 47523 | A binary (endo)function in... |
2arymptfv 47524 | The value of a binary (end... |
2arymaptfv 47525 | The value of the mapping o... |
2arymaptf 47526 | The mapping of binary (end... |
2arymaptf1 47527 | The mapping of binary (end... |
2arymaptfo 47528 | The mapping of binary (end... |
2arymaptf1o 47529 | The mapping of binary (end... |
2aryenef 47530 | The set of binary (endo)fu... |
itcoval 47535 | The value of the function ... |
itcoval0 47536 | A function iterated zero t... |
itcoval1 47537 | A function iterated once. ... |
itcoval2 47538 | A function iterated twice.... |
itcoval3 47539 | A function iterated three ... |
itcoval0mpt 47540 | A mapping iterated zero ti... |
itcovalsuc 47541 | The value of the function ... |
itcovalsucov 47542 | The value of the function ... |
itcovalendof 47543 | The n-th iterate of an end... |
itcovalpclem1 47544 | Lemma 1 for ~ itcovalpc : ... |
itcovalpclem2 47545 | Lemma 2 for ~ itcovalpc : ... |
itcovalpc 47546 | The value of the function ... |
itcovalt2lem2lem1 47547 | Lemma 1 for ~ itcovalt2lem... |
itcovalt2lem2lem2 47548 | Lemma 2 for ~ itcovalt2lem... |
itcovalt2lem1 47549 | Lemma 1 for ~ itcovalt2 : ... |
itcovalt2lem2 47550 | Lemma 2 for ~ itcovalt2 : ... |
itcovalt2 47551 | The value of the function ... |
ackvalsuc1mpt 47552 | The Ackermann function at ... |
ackvalsuc1 47553 | The Ackermann function at ... |
ackval0 47554 | The Ackermann function at ... |
ackval1 47555 | The Ackermann function at ... |
ackval2 47556 | The Ackermann function at ... |
ackval3 47557 | The Ackermann function at ... |
ackendofnn0 47558 | The Ackermann function at ... |
ackfnnn0 47559 | The Ackermann function at ... |
ackval0val 47560 | The Ackermann function at ... |
ackvalsuc0val 47561 | The Ackermann function at ... |
ackvalsucsucval 47562 | The Ackermann function at ... |
ackval0012 47563 | The Ackermann function at ... |
ackval1012 47564 | The Ackermann function at ... |
ackval2012 47565 | The Ackermann function at ... |
ackval3012 47566 | The Ackermann function at ... |
ackval40 47567 | The Ackermann function at ... |
ackval41a 47568 | The Ackermann function at ... |
ackval41 47569 | The Ackermann function at ... |
ackval42 47570 | The Ackermann function at ... |
ackval42a 47571 | The Ackermann function at ... |
ackval50 47572 | The Ackermann function at ... |
fv1prop 47573 | The function value of unor... |
fv2prop 47574 | The function value of unor... |
submuladdmuld 47575 | Transformation of a sum of... |
affinecomb1 47576 | Combination of two real af... |
affinecomb2 47577 | Combination of two real af... |
affineid 47578 | Identity of an affine comb... |
1subrec1sub 47579 | Subtract the reciprocal of... |
resum2sqcl 47580 | The sum of two squares of ... |
resum2sqgt0 47581 | The sum of the square of a... |
resum2sqrp 47582 | The sum of the square of a... |
resum2sqorgt0 47583 | The sum of the square of t... |
reorelicc 47584 | Membership in and outside ... |
rrx2pxel 47585 | The x-coordinate of a poin... |
rrx2pyel 47586 | The y-coordinate of a poin... |
prelrrx2 47587 | An unordered pair of order... |
prelrrx2b 47588 | An unordered pair of order... |
rrx2pnecoorneor 47589 | If two different points ` ... |
rrx2pnedifcoorneor 47590 | If two different points ` ... |
rrx2pnedifcoorneorr 47591 | If two different points ` ... |
rrx2xpref1o 47592 | There is a bijection betwe... |
rrx2xpreen 47593 | The set of points in the t... |
rrx2plord 47594 | The lexicographical orderi... |
rrx2plord1 47595 | The lexicographical orderi... |
rrx2plord2 47596 | The lexicographical orderi... |
rrx2plordisom 47597 | The set of points in the t... |
rrx2plordso 47598 | The lexicographical orderi... |
ehl2eudisval0 47599 | The Euclidean distance of ... |
ehl2eudis0lt 47600 | An upper bound of the Eucl... |
lines 47605 | The lines passing through ... |
line 47606 | The line passing through t... |
rrxlines 47607 | Definition of lines passin... |
rrxline 47608 | The line passing through t... |
rrxlinesc 47609 | Definition of lines passin... |
rrxlinec 47610 | The line passing through t... |
eenglngeehlnmlem1 47611 | Lemma 1 for ~ eenglngeehln... |
eenglngeehlnmlem2 47612 | Lemma 2 for ~ eenglngeehln... |
eenglngeehlnm 47613 | The line definition in the... |
rrx2line 47614 | The line passing through t... |
rrx2vlinest 47615 | The vertical line passing ... |
rrx2linest 47616 | The line passing through t... |
rrx2linesl 47617 | The line passing through t... |
rrx2linest2 47618 | The line passing through t... |
elrrx2linest2 47619 | The line passing through t... |
spheres 47620 | The spheres for given cent... |
sphere 47621 | A sphere with center ` X `... |
rrxsphere 47622 | The sphere with center ` M... |
2sphere 47623 | The sphere with center ` M... |
2sphere0 47624 | The sphere around the orig... |
line2ylem 47625 | Lemma for ~ line2y . This... |
line2 47626 | Example for a line ` G ` p... |
line2xlem 47627 | Lemma for ~ line2x . This... |
line2x 47628 | Example for a horizontal l... |
line2y 47629 | Example for a vertical lin... |
itsclc0lem1 47630 | Lemma for theorems about i... |
itsclc0lem2 47631 | Lemma for theorems about i... |
itsclc0lem3 47632 | Lemma for theorems about i... |
itscnhlc0yqe 47633 | Lemma for ~ itsclc0 . Qua... |
itschlc0yqe 47634 | Lemma for ~ itsclc0 . Qua... |
itsclc0yqe 47635 | Lemma for ~ itsclc0 . Qua... |
itsclc0yqsollem1 47636 | Lemma 1 for ~ itsclc0yqsol... |
itsclc0yqsollem2 47637 | Lemma 2 for ~ itsclc0yqsol... |
itsclc0yqsol 47638 | Lemma for ~ itsclc0 . Sol... |
itscnhlc0xyqsol 47639 | Lemma for ~ itsclc0 . Sol... |
itschlc0xyqsol1 47640 | Lemma for ~ itsclc0 . Sol... |
itschlc0xyqsol 47641 | Lemma for ~ itsclc0 . Sol... |
itsclc0xyqsol 47642 | Lemma for ~ itsclc0 . Sol... |
itsclc0xyqsolr 47643 | Lemma for ~ itsclc0 . Sol... |
itsclc0xyqsolb 47644 | Lemma for ~ itsclc0 . Sol... |
itsclc0 47645 | The intersection points of... |
itsclc0b 47646 | The intersection points of... |
itsclinecirc0 47647 | The intersection points of... |
itsclinecirc0b 47648 | The intersection points of... |
itsclinecirc0in 47649 | The intersection points of... |
itsclquadb 47650 | Quadratic equation for the... |
itsclquadeu 47651 | Quadratic equation for the... |
2itscplem1 47652 | Lemma 1 for ~ 2itscp . (C... |
2itscplem2 47653 | Lemma 2 for ~ 2itscp . (C... |
2itscplem3 47654 | Lemma D for ~ 2itscp . (C... |
2itscp 47655 | A condition for a quadrati... |
itscnhlinecirc02plem1 47656 | Lemma 1 for ~ itscnhlineci... |
itscnhlinecirc02plem2 47657 | Lemma 2 for ~ itscnhlineci... |
itscnhlinecirc02plem3 47658 | Lemma 3 for ~ itscnhlineci... |
itscnhlinecirc02p 47659 | Intersection of a nonhoriz... |
inlinecirc02plem 47660 | Lemma for ~ inlinecirc02p ... |
inlinecirc02p 47661 | Intersection of a line wit... |
inlinecirc02preu 47662 | Intersection of a line wit... |
pm4.71da 47663 | Deduction converting a bic... |
logic1 47664 | Distribution of implicatio... |
logic1a 47665 | Variant of ~ logic1 . (Co... |
logic2 47666 | Variant of ~ logic1 . (Co... |
pm5.32dav 47667 | Distribution of implicatio... |
pm5.32dra 47668 | Reverse distribution of im... |
exp12bd 47669 | The import-export theorem ... |
mpbiran3d 47670 | Equivalence with a conjunc... |
mpbiran4d 47671 | Equivalence with a conjunc... |
dtrucor3 47672 | An example of how ~ ax-5 w... |
ralbidb 47673 | Formula-building rule for ... |
ralbidc 47674 | Formula-building rule for ... |
r19.41dv 47675 | A complex deduction form o... |
rspceb2dv 47676 | Restricted existential spe... |
rmotru 47677 | Two ways of expressing "at... |
reutru 47678 | Two ways of expressing "ex... |
reutruALT 47679 | Alternate proof for ~ reut... |
ssdisjd 47680 | Subset preserves disjointn... |
ssdisjdr 47681 | Subset preserves disjointn... |
disjdifb 47682 | Relative complement is ant... |
predisj 47683 | Preimages of disjoint sets... |
vsn 47684 | The singleton of the unive... |
mosn 47685 | "At most one" element in a... |
mo0 47686 | "At most one" element in a... |
mosssn 47687 | "At most one" element in a... |
mo0sn 47688 | Two ways of expressing "at... |
mosssn2 47689 | Two ways of expressing "at... |
unilbss 47690 | Superclass of the greatest... |
inpw 47691 | Two ways of expressing a c... |
mof0 47692 | There is at most one funct... |
mof02 47693 | A variant of ~ mof0 . (Co... |
mof0ALT 47694 | Alternate proof for ~ mof0... |
eufsnlem 47695 | There is exactly one funct... |
eufsn 47696 | There is exactly one funct... |
eufsn2 47697 | There is exactly one funct... |
mofsn 47698 | There is at most one funct... |
mofsn2 47699 | There is at most one funct... |
mofsssn 47700 | There is at most one funct... |
mofmo 47701 | There is at most one funct... |
mofeu 47702 | The uniqueness of a functi... |
elfvne0 47703 | If a function value has a ... |
fdomne0 47704 | A function with non-empty ... |
f1sn2g 47705 | A function that maps a sin... |
f102g 47706 | A function that maps the e... |
f1mo 47707 | A function that maps a set... |
f002 47708 | A function with an empty c... |
map0cor 47709 | A function exists iff an e... |
fvconstr 47710 | Two ways of expressing ` A... |
fvconstrn0 47711 | Two ways of expressing ` A... |
fvconstr2 47712 | Two ways of expressing ` A... |
fvconst0ci 47713 | A constant function's valu... |
fvconstdomi 47714 | A constant function's valu... |
f1omo 47715 | There is at most one eleme... |
f1omoALT 47716 | There is at most one eleme... |
iccin 47717 | Intersection of two closed... |
iccdisj2 47718 | If the upper bound of one ... |
iccdisj 47719 | If the upper bound of one ... |
mreuniss 47720 | The union of a collection ... |
clduni 47721 | The union of closed sets i... |
opncldeqv 47722 | Conditions on open sets ar... |
opndisj 47723 | Two ways of saying that tw... |
clddisj 47724 | Two ways of saying that tw... |
neircl 47725 | Reverse closure of the nei... |
opnneilem 47726 | Lemma factoring out common... |
opnneir 47727 | If something is true for a... |
opnneirv 47728 | A variant of ~ opnneir wit... |
opnneilv 47729 | The converse of ~ opnneir ... |
opnneil 47730 | A variant of ~ opnneilv . ... |
opnneieqv 47731 | The equivalence between ne... |
opnneieqvv 47732 | The equivalence between ne... |
restcls2lem 47733 | A closed set in a subspace... |
restcls2 47734 | A closed set in a subspace... |
restclsseplem 47735 | Lemma for ~ restclssep . ... |
restclssep 47736 | Two disjoint closed sets i... |
cnneiima 47737 | Given a continuous functio... |
iooii 47738 | Open intervals are open se... |
icccldii 47739 | Closed intervals are close... |
i0oii 47740 | ` ( 0 [,) A ) ` is open in... |
io1ii 47741 | ` ( A (,] 1 ) ` is open in... |
sepnsepolem1 47742 | Lemma for ~ sepnsepo . (C... |
sepnsepolem2 47743 | Open neighborhood and neig... |
sepnsepo 47744 | Open neighborhood and neig... |
sepdisj 47745 | Separated sets are disjoin... |
seposep 47746 | If two sets are separated ... |
sepcsepo 47747 | If two sets are separated ... |
sepfsepc 47748 | If two sets are separated ... |
seppsepf 47749 | If two sets are precisely ... |
seppcld 47750 | If two sets are precisely ... |
isnrm4 47751 | A topological space is nor... |
dfnrm2 47752 | A topological space is nor... |
dfnrm3 47753 | A topological space is nor... |
iscnrm3lem1 47754 | Lemma for ~ iscnrm3 . Sub... |
iscnrm3lem2 47755 | Lemma for ~ iscnrm3 provin... |
iscnrm3lem3 47756 | Lemma for ~ iscnrm3lem4 . ... |
iscnrm3lem4 47757 | Lemma for ~ iscnrm3lem5 an... |
iscnrm3lem5 47758 | Lemma for ~ iscnrm3l . (C... |
iscnrm3lem6 47759 | Lemma for ~ iscnrm3lem7 . ... |
iscnrm3lem7 47760 | Lemma for ~ iscnrm3rlem8 a... |
iscnrm3rlem1 47761 | Lemma for ~ iscnrm3rlem2 .... |
iscnrm3rlem2 47762 | Lemma for ~ iscnrm3rlem3 .... |
iscnrm3rlem3 47763 | Lemma for ~ iscnrm3r . Th... |
iscnrm3rlem4 47764 | Lemma for ~ iscnrm3rlem8 .... |
iscnrm3rlem5 47765 | Lemma for ~ iscnrm3rlem6 .... |
iscnrm3rlem6 47766 | Lemma for ~ iscnrm3rlem7 .... |
iscnrm3rlem7 47767 | Lemma for ~ iscnrm3rlem8 .... |
iscnrm3rlem8 47768 | Lemma for ~ iscnrm3r . Di... |
iscnrm3r 47769 | Lemma for ~ iscnrm3 . If ... |
iscnrm3llem1 47770 | Lemma for ~ iscnrm3l . Cl... |
iscnrm3llem2 47771 | Lemma for ~ iscnrm3l . If... |
iscnrm3l 47772 | Lemma for ~ iscnrm3 . Giv... |
iscnrm3 47773 | A completely normal topolo... |
iscnrm3v 47774 | A topology is completely n... |
iscnrm4 47775 | A completely normal topolo... |
isprsd 47776 | Property of being a preord... |
lubeldm2 47777 | Member of the domain of th... |
glbeldm2 47778 | Member of the domain of th... |
lubeldm2d 47779 | Member of the domain of th... |
glbeldm2d 47780 | Member of the domain of th... |
lubsscl 47781 | If a subset of ` S ` conta... |
glbsscl 47782 | If a subset of ` S ` conta... |
lubprlem 47783 | Lemma for ~ lubprdm and ~ ... |
lubprdm 47784 | The set of two comparable ... |
lubpr 47785 | The LUB of the set of two ... |
glbprlem 47786 | Lemma for ~ glbprdm and ~ ... |
glbprdm 47787 | The set of two comparable ... |
glbpr 47788 | The GLB of the set of two ... |
joindm2 47789 | The join of any two elemen... |
joindm3 47790 | The join of any two elemen... |
meetdm2 47791 | The meet of any two elemen... |
meetdm3 47792 | The meet of any two elemen... |
posjidm 47793 | Poset join is idempotent. ... |
posmidm 47794 | Poset meet is idempotent. ... |
toslat 47795 | A toset is a lattice. (Co... |
isclatd 47796 | The predicate "is a comple... |
intubeu 47797 | Existential uniqueness of ... |
unilbeu 47798 | Existential uniqueness of ... |
ipolublem 47799 | Lemma for ~ ipolubdm and ~... |
ipolubdm 47800 | The domain of the LUB of t... |
ipolub 47801 | The LUB of the inclusion p... |
ipoglblem 47802 | Lemma for ~ ipoglbdm and ~... |
ipoglbdm 47803 | The domain of the GLB of t... |
ipoglb 47804 | The GLB of the inclusion p... |
ipolub0 47805 | The LUB of the empty set i... |
ipolub00 47806 | The LUB of the empty set i... |
ipoglb0 47807 | The GLB of the empty set i... |
mrelatlubALT 47808 | Least upper bounds in a Mo... |
mrelatglbALT 47809 | Greatest lower bounds in a... |
mreclat 47810 | A Moore space is a complet... |
topclat 47811 | A topology is a complete l... |
toplatglb0 47812 | The empty intersection in ... |
toplatlub 47813 | Least upper bounds in a to... |
toplatglb 47814 | Greatest lower bounds in a... |
toplatjoin 47815 | Joins in a topology are re... |
toplatmeet 47816 | Meets in a topology are re... |
topdlat 47817 | A topology is a distributi... |
catprslem 47818 | Lemma for ~ catprs . (Con... |
catprs 47819 | A preorder can be extracte... |
catprs2 47820 | A category equipped with t... |
catprsc 47821 | A construction of the preo... |
catprsc2 47822 | An alternate construction ... |
endmndlem 47823 | A diagonal hom-set in a ca... |
idmon 47824 | An identity arrow, or an i... |
idepi 47825 | An identity arrow, or an i... |
funcf2lem 47826 | A utility theorem for prov... |
isthinc 47829 | The predicate "is a thin c... |
isthinc2 47830 | A thin category is a categ... |
isthinc3 47831 | A thin category is a categ... |
thincc 47832 | A thin category is a categ... |
thinccd 47833 | A thin category is a categ... |
thincssc 47834 | A thin category is a categ... |
isthincd2lem1 47835 | Lemma for ~ isthincd2 and ... |
thincmo2 47836 | Morphisms in the same hom-... |
thincmo 47837 | There is at most one morph... |
thincmoALT 47838 | Alternate proof for ~ thin... |
thincmod 47839 | At most one morphism in ea... |
thincn0eu 47840 | In a thin category, a hom-... |
thincid 47841 | In a thin category, a morp... |
thincmon 47842 | In a thin category, all mo... |
thincepi 47843 | In a thin category, all mo... |
isthincd2lem2 47844 | Lemma for ~ isthincd2 . (... |
isthincd 47845 | The predicate "is a thin c... |
isthincd2 47846 | The predicate " ` C ` is a... |
oppcthin 47847 | The opposite category of a... |
subthinc 47848 | A subcategory of a thin ca... |
functhinclem1 47849 | Lemma for ~ functhinc . G... |
functhinclem2 47850 | Lemma for ~ functhinc . (... |
functhinclem3 47851 | Lemma for ~ functhinc . T... |
functhinclem4 47852 | Lemma for ~ functhinc . O... |
functhinc 47853 | A functor to a thin catego... |
fullthinc 47854 | A functor to a thin catego... |
fullthinc2 47855 | A full functor to a thin c... |
thincfth 47856 | A functor from a thin cate... |
thincciso 47857 | Two thin categories are is... |
0thincg 47858 | Any structure with an empt... |
0thinc 47859 | The empty category (see ~ ... |
indthinc 47860 | An indiscrete category in ... |
indthincALT 47861 | An alternate proof for ~ i... |
prsthinc 47862 | Preordered sets as categor... |
setcthin 47863 | A category of sets all of ... |
setc2othin 47864 | The category ` ( SetCat ``... |
thincsect 47865 | In a thin category, one mo... |
thincsect2 47866 | In a thin category, ` F ` ... |
thincinv 47867 | In a thin category, ` F ` ... |
thinciso 47868 | In a thin category, ` F : ... |
thinccic 47869 | In a thin category, two ob... |
prstcval 47872 | Lemma for ~ prstcnidlem an... |
prstcnidlem 47873 | Lemma for ~ prstcnid and ~... |
prstcnid 47874 | Components other than ` Ho... |
prstcbas 47875 | The base set is unchanged.... |
prstcleval 47876 | Value of the less-than-or-... |
prstclevalOLD 47877 | Obsolete proof of ~ prstcl... |
prstcle 47878 | Value of the less-than-or-... |
prstcocval 47879 | Orthocomplementation is un... |
prstcocvalOLD 47880 | Obsolete proof of ~ prstco... |
prstcoc 47881 | Orthocomplementation is un... |
prstchomval 47882 | Hom-sets of the constructe... |
prstcprs 47883 | The category is a preorder... |
prstcthin 47884 | The preordered set is equi... |
prstchom 47885 | Hom-sets of the constructe... |
prstchom2 47886 | Hom-sets of the constructe... |
prstchom2ALT 47887 | Hom-sets of the constructe... |
postcpos 47888 | The converted category is ... |
postcposALT 47889 | Alternate proof for ~ post... |
postc 47890 | The converted category is ... |
mndtcval 47893 | Value of the category buil... |
mndtcbasval 47894 | The base set of the catego... |
mndtcbas 47895 | The category built from a ... |
mndtcob 47896 | Lemma for ~ mndtchom and ~... |
mndtcbas2 47897 | Two objects in a category ... |
mndtchom 47898 | The only hom-set of the ca... |
mndtcco 47899 | The composition of the cat... |
mndtcco2 47900 | The composition of the cat... |
mndtccatid 47901 | Lemma for ~ mndtccat and ~... |
mndtccat 47902 | The function value is a ca... |
mndtcid 47903 | The identity morphism, or ... |
grptcmon 47904 | All morphisms in a categor... |
grptcepi 47905 | All morphisms in a categor... |
nfintd 47906 | Bound-variable hypothesis ... |
nfiund 47907 | Bound-variable hypothesis ... |
nfiundg 47908 | Bound-variable hypothesis ... |
iunord 47909 | The indexed union of a col... |
iunordi 47910 | The indexed union of a col... |
spd 47911 | Specialization deduction, ... |
spcdvw 47912 | A version of ~ spcdv where... |
tfis2d 47913 | Transfinite Induction Sche... |
bnd2d 47914 | Deduction form of ~ bnd2 .... |
dffun3f 47915 | Alternate definition of fu... |
setrecseq 47918 | Equality theorem for set r... |
nfsetrecs 47919 | Bound-variable hypothesis ... |
setrec1lem1 47920 | Lemma for ~ setrec1 . Thi... |
setrec1lem2 47921 | Lemma for ~ setrec1 . If ... |
setrec1lem3 47922 | Lemma for ~ setrec1 . If ... |
setrec1lem4 47923 | Lemma for ~ setrec1 . If ... |
setrec1 47924 | This is the first of two f... |
setrec2fun 47925 | This is the second of two ... |
setrec2lem1 47926 | Lemma for ~ setrec2 . The... |
setrec2lem2 47927 | Lemma for ~ setrec2 . The... |
setrec2 47928 | This is the second of two ... |
setrec2v 47929 | Version of ~ setrec2 with ... |
setrec2mpt 47930 | Version of ~ setrec2 where... |
setis 47931 | Version of ~ setrec2 expre... |
elsetrecslem 47932 | Lemma for ~ elsetrecs . A... |
elsetrecs 47933 | A set ` A ` is an element ... |
setrecsss 47934 | The ` setrecs ` operator r... |
setrecsres 47935 | A recursively generated cl... |
vsetrec 47936 | Construct ` _V ` using set... |
0setrec 47937 | If a function sends the em... |
onsetreclem1 47938 | Lemma for ~ onsetrec . (C... |
onsetreclem2 47939 | Lemma for ~ onsetrec . (C... |
onsetreclem3 47940 | Lemma for ~ onsetrec . (C... |
onsetrec 47941 | Construct ` On ` using set... |
elpglem1 47944 | Lemma for ~ elpg . (Contr... |
elpglem2 47945 | Lemma for ~ elpg . (Contr... |
elpglem3 47946 | Lemma for ~ elpg . (Contr... |
elpg 47947 | Membership in the class of... |
pgindlem 47948 | Lemma for ~ pgind . (Cont... |
pgindnf 47949 | Version of ~ pgind with ex... |
pgind 47950 | Induction on partizan game... |
sbidd 47951 | An identity theorem for su... |
sbidd-misc 47952 | An identity theorem for su... |
gte-lte 47957 | Simple relationship betwee... |
gt-lt 47958 | Simple relationship betwee... |
gte-lteh 47959 | Relationship between ` <_ ... |
gt-lth 47960 | Relationship between ` < `... |
ex-gt 47961 | Simple example of ` > ` , ... |
ex-gte 47962 | Simple example of ` >_ ` ,... |
sinhval-named 47969 | Value of the named sinh fu... |
coshval-named 47970 | Value of the named cosh fu... |
tanhval-named 47971 | Value of the named tanh fu... |
sinh-conventional 47972 | Conventional definition of... |
sinhpcosh 47973 | Prove that ` ( sinh `` A )... |
secval 47980 | Value of the secant functi... |
cscval 47981 | Value of the cosecant func... |
cotval 47982 | Value of the cotangent fun... |
seccl 47983 | The closure of the secant ... |
csccl 47984 | The closure of the cosecan... |
cotcl 47985 | The closure of the cotange... |
reseccl 47986 | The closure of the secant ... |
recsccl 47987 | The closure of the cosecan... |
recotcl 47988 | The closure of the cotange... |
recsec 47989 | The reciprocal of secant i... |
reccsc 47990 | The reciprocal of cosecant... |
reccot 47991 | The reciprocal of cotangen... |
rectan 47992 | The reciprocal of tangent ... |
sec0 47993 | The value of the secant fu... |
onetansqsecsq 47994 | Prove the tangent squared ... |
cotsqcscsq 47995 | Prove the tangent squared ... |
ifnmfalse 47996 | If A is not a member of B,... |
logb2aval 47997 | Define the value of the ` ... |
comraddi 48004 | Commute RHS addition. See... |
mvlraddi 48005 | Move the right term in a s... |
mvrladdi 48006 | Move the left term in a su... |
assraddsubi 48007 | Associate RHS addition-sub... |
joinlmuladdmuli 48008 | Join AB+CB into (A+C) on L... |
joinlmulsubmuld 48009 | Join AB-CB into (A-C) on L... |
joinlmulsubmuli 48010 | Join AB-CB into (A-C) on L... |
mvlrmuld 48011 | Move the right term in a p... |
mvlrmuli 48012 | Move the right term in a p... |
i2linesi 48013 | Solve for the intersection... |
i2linesd 48014 | Solve for the intersection... |
alimp-surprise 48015 | Demonstrate that when usin... |
alimp-no-surprise 48016 | There is no "surprise" in ... |
empty-surprise 48017 | Demonstrate that when usin... |
empty-surprise2 48018 | "Prove" that false is true... |
eximp-surprise 48019 | Show what implication insi... |
eximp-surprise2 48020 | Show that "there exists" w... |
alsconv 48025 | There is an equivalence be... |
alsi1d 48026 | Deduction rule: Given "al... |
alsi2d 48027 | Deduction rule: Given "al... |
alsc1d 48028 | Deduction rule: Given "al... |
alsc2d 48029 | Deduction rule: Given "al... |
alscn0d 48030 | Deduction rule: Given "al... |
alsi-no-surprise 48031 | Demonstrate that there is ... |
5m4e1 48032 | Prove that 5 - 4 = 1. (Co... |
2p2ne5 48033 | Prove that ` 2 + 2 =/= 5 `... |
resolution 48034 | Resolution rule. This is ... |
testable 48035 | In classical logic all wff... |
aacllem 48036 | Lemma for other theorems a... |
amgmwlem 48037 | Weighted version of ~ amgm... |
amgmlemALT 48038 | Alternate proof of ~ amgml... |
amgmw2d 48039 | Weighted arithmetic-geomet... |
young2d 48040 | Young's inequality for ` n... |
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